hypothesis example marketing

How to write a hypothesis for marketing experimentation

Creating your strongest marketing hypothesis

The potential for your marketing improvement depends on the strength of your testing hypotheses.

But where are you getting your test ideas from? Have you been scouring competitor sites, or perhaps pulling from previous designs on your site? The web is full of ideas and you’re full of ideas – there is no shortage of inspiration, that’s for sure.

Coming up with something you  want  to test isn’t hard to do. Coming up with something you  should  test can be hard to do.

Hard – yes. Impossible? No. Which is good news, because if you can’t create hypotheses for things that should be tested, your test results won’t mean mean much, and you probably shouldn’t be spending your time testing.

Taking the time to write your hypotheses correctly will help you structure your ideas, get better results, and avoid wasting traffic on poor test designs.

With this post, we’re getting advanced with marketing hypotheses, showing you how to write and structure your hypotheses to gain both business results and marketing insights!

By the time you finish reading, you’ll be able to:

• Distinguish a solid hypothesis from a time-waster, and
• Structure your solid hypothesis to get results  and  insights

To make this whole experience a bit more tangible, let’s track a sample idea from…well…idea to hypothesis.

Let’s say you identified a call-to-action (CTA)* while browsing the web, and you were inspired to test something similar on your own lead generation landing page. You think it might work for your users! Your idea is:

“My page needs a new CTA.”

*A call-to-action is the point where you, as a marketer, ask your prospect to do something on your page. It often includes a button or link to an action like “Buy”, “Sign up”, or “Request a quote”.

The basics: The correct marketing hypothesis format

A well-structured hypothesis provides insights whether it is proved, disproved, or results are inconclusive.

You should never phrase a marketing hypothesis as a question. It should be written as a statement that can be rejected or confirmed.

Further, it should be a statement geared toward revealing insights – with this in mind, it helps to imagine each statement followed by a  reason :

• Changing _______ into ______ will increase [conversion goal], because:
• Changing _______ into ______ will decrease [conversion goal], because:
• Changing _______ into ______ will not affect [conversion goal], because:

Each of the above sentences ends with ‘because’ to set the expectation that there will be an explanation behind the results of whatever you’re testing.

It’s important to remember to plan ahead when you create a test, and think about explaining why the test turned out the way it did when the results come in.

Level up: Moving from a good to great hypothesis

Understanding what makes an idea worth testing is necessary for your optimization team.

If your tests are based on random ideas you googled or were suggested by a consultant, your testing process still has its training wheels on. Great hypotheses aren’t random. They’re based on rationale and aim for learning.

Hypotheses should be based on themes and analysis that show potential conversion barriers.

At Conversion, we call this investigation phase the “Explore Phase” where we use frameworks like the LIFT Model to understand the prospect’s unique perspective. (You can read more on the the full optimization process here).

A well-founded marketing hypothesis should also provide you with new, testable clues about your users regardless of whether or not the test wins, loses or yields inconclusive results.

These new insights should inform future testing: a solid hypothesis can help you quickly separate worthwhile ideas from the rest when planning follow-up tests.

“Ultimately, what matters most is that you have a hypothesis going into each experiment and you design each experiment to address that hypothesis.” – Nick So, VP of Delivery

Here’s a quick tip :

If you’re about to run a test that isn’t going to tell you anything new about your users and their motivations, it’s probably not worth investing your time in.

Let’s take this opportunity to refer back to your original idea:

Ok, but  what now ? To get actionable insights from ‘a new CTA’, you need to know why it behaved the way it did. You need to ask the right question.

To test the waters, maybe you changed the copy of the CTA button on your lead generation form from “Submit” to “Send demo request”. If this change leads to an increase in conversions, it could mean that your users require more clarity about what their information is being used for.

That’s a potential insight.

Based on this insight, you could follow up with another test that adds copy around the CTA about next steps: what the user should anticipate after they have submitted their information.

For example, will they be speaking to a specialist via email? Will something be waiting for them the next time they visit your site? You can test providing more information, and see if your users are interested in knowing it!

That’s the cool thing about a good hypothesis: the results of the test, while important (of course) aren’t the only component driving your future test ideas. The insights gleaned lead to further hypotheses and insights in a virtuous cycle.

It’s based on a science

The term “hypothesis” probably isn’t foreign to you. In fact, it may bring up memories of grade-school science class; it’s a critical part of the  scientific method .

The scientific method in testing follows a systematic routine that sets ideation up to predict the results of experiments via:

• Collecting data and information through observation
• Creating tentative descriptions of what is being observed
• Forming  hypotheses  that predict different outcomes based on these observations
• Testing your  hypotheses
• Analyzing the data, drawing conclusions and insights from the results

Don’t worry! Hypothesizing may seem ‘sciency’, but it doesn’t have to be complicated in practice.

Hypothesizing simply helps ensure the results from your tests are quantifiable, and is necessary if you want to understand how the results reflect the change made in your test.

A strong marketing hypothesis allows testers to use a structured approach in order to discover what works, why it works, how it works, where it works, and who it works on.

“My page needs a new CTA.” Is this idea in its current state clear enough to help you understand what works? Maybe. Why it works? No. Where it works? Maybe. Who it works on? No.

Your idea needs refining.

Let’s pull back and take a broader look at the lead generation landing page we want to test.

Imagine the situation: you’ve been diligent in your data collection and you notice several recurrences of Clarity pain points – meaning that there are many unclear instances throughout the page’s messaging.

Rather than focusing on the CTA right off the bat, it may be more beneficial to deal with the bigger clarity issue.

Now you’re starting to think about solving your prospects conversion barriers rather than just testing random ideas!

If you believe the overall page is unclear, your overarching theme of inquiry might be positioned as:

• “Improving the clarity of the page will reduce confusion and improve [conversion goal].”

By testing a hypothesis that supports this clarity theme, you can gain confidence in the validity of it as an actionable marketing insight over time.

If the test results are negative : It may not be worth investigating this motivational barrier any further on this page. In this case, you could return to the data and look at the other motivational barriers that might be affecting user behavior.

If the test results are positive : You might want to continue to refine the clarity of the page’s message with further testing.

Typically, a test will start with a broad idea — you identify the changes to make, predict how those changes will impact your conversion goal, and write it out as a broad theme as shown above. Then, repeated tests aimed at that theme will confirm or undermine the strength of the underlying insight.

Building marketing hypotheses to create insights

You believe you’ve identified an overall problem on your landing page (there’s a problem with clarity). Now you want to understand how individual elements contribute to the problem, and the effect these individual elements have on your users.

It’s game time  – now you can start designing a hypothesis that will generate insights.

You believe your users need more clarity. You’re ready to dig deeper to find out if that’s true!

If a specific question needs answering, you should structure your test to make a single change. This isolation might ask: “What element are users most sensitive to when it comes to the lack of clarity?” and “What changes do I believe will support increasing clarity?”

At this point, you’ll want to boil down your overarching theme…

• Improving the clarity of the page will reduce confusion and improve [conversion goal].

…into a quantifiable hypothesis that isolates key sections:

• Changing the wording of this CTA to set expectations for users (from “submit” to “send demo request”) will reduce confusion about the next steps in the funnel and improve order completions.

Does this answer what works? Yes: changing the wording on your CTA.

Does this answer why it works? Yes: reducing confusion about the next steps in the funnel.

Does this answer where it works? Yes: on this page, before the user enters this theoretical funnel.

Does this answer who it works on? No, this question demands another isolation. You might structure your hypothesis more like this:

• Changing the wording of the CTA to set expectations for users (from “submit” to “send demo request”) will reduce confusion  for visitors coming from my email campaign  about the next steps in the funnel and improve order completions.

Now we’ve got a clear hypothesis. And one worth testing!

What makes a great hypothesis?

1. It’s testable.

2. It addresses conversion barriers.

3. It aims at gaining marketing insights.

Let’s compare:

The original idea : “My page needs a new CTA.”

Following the hypothesis structure : “A new CTA on my page will increase [conversion goal]”

The first test implied a problem with clarity, provides a potential theme : “Improving the clarity of the page will reduce confusion and improve [conversion goal].”

The potential clarity theme leads to a new hypothesis : “Changing the wording of the CTA to set expectations for users (from “submit” to “send demo request”) will reduce confusion about the next steps in the funnel and improve order completions.”

Final refined hypothesis : “Changing the wording of the CTA to set expectations for users (from “submit” to “send demo request”) will reduce confusion for visitors coming from my email campaign about the next steps in the funnel and improve order completions.”

Which test would you rather your team invest in?

Before you start your next test, take the time to do a proper analysis of the page you want to focus on. Do preliminary testing to define bigger issues, and use that information to refine and pinpoint your marketing hypothesis to give you forward-looking insights.

Doing this will help you avoid time-wasting tests, and enable you to start getting some insights for your team to keep testing!

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Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on August 15, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism. Run a free check.

Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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• Free Resources

A/B Testing in Digital Marketing: Example of four-step hypothesis framework

by Daniel Burstein , Senior Director, Content & Marketing, MarketingSherpa and MECLABS Institute

This article was originally published in the MarketingSherpa email newsletter .

If you are a marketing expert — whether in a brand’s marketing department or at an advertising agency — you may feel the need to be absolutely sure in an unsure world.

What should the headline be? What images should we use? Is this strategy correct? Will customers value this promo?

This is the stuff you’re paid to know. So you may feel like you must boldly proclaim your confident opinion.

But you can’t predict the future with 100% accuracy. You can’t know with absolute certainty how humans will behave. And let’s face it, even as marketing experts we’re occasionally wrong.

It’s not bad, it’s healthy. And the most effective way to overcome that doubt is by testing our marketing creative to see what really works.

Developing a hypothesis

After we published Value Sequencing: A step-by-step examination of a landing page that generated 638% more conversions , a MarketingSherpa reader emailed us and asked …

Great stuff Daniel. Much appreciated. I can see you addressing all the issues there.

I thought I saw one more opportunity to expand on what you made. Would you consider adding the IF, BY, WILL, BECAUSE to the control/treatment sections so we can see what psychology you were addressing so we know how to create the hypothesis to learn from what the customer is currently doing and why and then form a test to address that? The video today on customer theory was great (Editor’s Note: Part of the MarketingExperiments YouTube Live series ) . I think there is a way to incorporate that customer theory thinking into this article to take it even further.

Developing a hypothesis is an essential part of marketing experimentation. Qualitative-based research should inform hypotheses that you test with real-world behavior.

The hypotheses help you discover how accurate those insights from qualitative research are. If you engage in hypothesis-driven testing, then you ensure your tests are strategic (not just based on a random idea) and built in a way that enables you to learn more and more about the customer with each test.

And that methodology will ultimately lead to greater and greater lifts over time, instead of a scattershot approach where sometimes you get a lift and sometimes you don’t, but you never really know why.

Here is a handy tool to help you in developing hypotheses — the MECLABS Four-Step Hypothesis Framework.

As the reader suggests, I will use the landing page test referenced in the previous article as an example. ( Please note: While the experiment in that article was created with a hypothesis-driven approach, this specific four-step framework is fairly new and was not in common use by the MECLABS team at that time, so I have created this specific example after the test was developed based on what I see in the test).

Here is what the hypothesis would look like for that test, and then we’ll break down each part individually:

If we emphasize the process-level value by adding headlines, images and body copy, we will generate more leads because the value of a longer landing page in reducing the anxiety of calling a TeleAgent outweighs the additional friction of a longer page.

IF: Summary description

The hypothesis begins with an overall statement about what you are trying to do in the experiment. In this case, the experiment is trying to emphasize the process-level value proposition (one of the four essential levels of value proposition ) of having a phone call with a TeleAgent.

The control landing page was emphasizing the primary value proposition of the brand itself.

The treatment landing page is essentially trying to answer this value proposition question: If I am your ideal customer, why should I call a TeleAgent rather than take any other action to learn more about my Medicare options?

The control landing page was asking a much bigger question that customers weren’t ready to say “yes” to yet, and it was overlooking the anxiety inherent in getting on a phone call with someone who might try to sell you something: If I am your ideal customer, why should I buy from your company instead of any other company.

This step answers WHAT you are trying to do.

BY: Remove, add, change

The next step answers HOW you are going to do it.

As Flint McGlaughlin, CEO and Managing Director of MECLABS Institute teaches, there are only three ways to improve performance: removing, adding or changing .

In this case, the team focused mostly on adding — adding headlines, images and body copy that highlighted the TeleAgents as trusted advisors.

“Adding” can be counterintuitive for many marketers. The team’s original landing page was short. Conventional wisdom says customers won’t read long landing pages. When I’m presenting to a group of marketers, I’ll put a short and long landing page on a slide and ask which page they think achieved better results.

Invariably I will hear, “Oh, the shorter page. I would never read something that long.”

That first-person statement is a mistake. Your marketing creative should not be based on “I” — the marketer. It should be based on “they” — the customer.

Most importantly, you need to focus on the customer at a specific point in time — when he or she is in the mindspace of considering to take an action like purchase a product or in need of more information before they decide to download a whitepaper. And sometimes in these situations, longer landing pages perform better.

In the case of this landing page, even the customer may not necessarily favor a long landing page all the time. But in the real-world situation when they are considering whether to call a TeleAgent or not, the added value helps more customers decide to take the action.

WILL: Improve performance

This is your KPI (key performance indicator). This step answers another HOW question: How do you know your hypothesis has been supported or refuted?

You can choose secondary metrics to monitor during your test as well. This might help you interpret the customer behavior observed in the test.

But ultimately, the hypothesis should rest on a single metric.

For this test, the goal was to generate more leads. And the treatment did — 638% more leads.

BECAUSE: Customer insight

This last step answers a WHY question — why did the customers act this way?

This helps you determine what you can learn about customers based on the actions observed in the experiment.

This is ultimately why you test. To learn about the customer and continually refine your company’s customer theory .

In this case, the team theorized that the value of a longer landing page in reducing the anxiety of calling a TeleAgent outweighs the additional friction of a longer landing page.

And the test results support that hypothesis.

Related Resources

The Hypothesis and the Modern-Day Marketer

Boost your Conversion Rate with a MECLABS Quick Win Intensive

Designing Hypotheses that Win: A four-step framework for gaining customer wisdom and generating marketing results

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“A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty until found effective.”

– Edward Teller, Nuclear Physicist

During my first brainstorming meeting on my first project at McKinsey, this very serious partner, who had a PhD in Physics, looked at me and said, “So, Joe, what are your main hypotheses.” I looked back at him, perplexed, and said, “Ummm, my what?” I was used to people simply asking, “what are your best ideas, opinions, thoughts, etc.” Over time, I began to understand the importance of hypotheses and how it plays an important role in McKinsey’s problem solving of separating ideas and opinions from facts.

What is a Hypothesis?

“Hypothesis” is probably one of the top 5 words used by McKinsey consultants. And, being hypothesis-driven was required to have any success at McKinsey. A hypothesis is an idea or theory, often based on limited data, which is typically the beginning of a thread of further investigation to prove, disprove or improve the hypothesis through facts and empirical data.

The first step in being hypothesis-driven is to focus on the highest potential ideas and theories of how to solve a problem or realize an opportunity.

Let’s go over an example of being hypothesis-driven.

Let’s say you own a website, and you brainstorm ten ideas to improve web traffic, but you don’t have the budget to execute all ten ideas. The first step in being hypothesis-driven is to prioritize the ten ideas based on how much impact you hypothesize they will create.

The second step in being hypothesis-driven is to apply the scientific method to your hypotheses by creating the fact base to prove or disprove your hypothesis, which then allows you to turn your hypothesis into fact and knowledge. Running with our example, you could prove or disprove your hypothesis on the ideas you think will drive the most impact by executing:

1. An analysis of previous research and the performance of the different ideas 2. A survey where customers rank order the ideas 3. An actual test of the ten ideas to create a fact base on click-through rates and cost

While there are many other ways to validate the hypothesis on your prioritization , I find most people do not take this critical step in validating a hypothesis. Instead, they apply bad logic to many important decisions. An idea pops into their head, and then somehow it just becomes a fact.

One of my favorite lousy logic moments was a CEO who stated,

“I’ve never heard our customers talk about price, so the price doesn’t matter with our products, and I’ve decided we’re going to raise prices.”

Luckily, his management team was able to do a survey to dig deeper into the hypothesis that customers weren’t price-sensitive. Well, of course, they were and through the survey, they built a fantastic fact base that proved and disproved many other important hypotheses.

Why is being hypothesis-driven so important?

Imagine if medicine never actually used the scientific method. We would probably still be living in a world of lobotomies and bleeding people. Many organizations are still stuck in the dark ages, having built a house of cards on opinions disguised as facts, because they don’t prove or disprove their hypotheses. Decisions made on top of decisions, made on top of opinions, steer organizations clear of reality and the facts necessary to objectively evolve their strategic understanding and knowledge. I’ve seen too many leadership teams led solely by gut and opinion. The problem with intuition and gut is if you don’t ever prove or disprove if your gut is right or wrong, you’re never going to improve your intuition. There is a reason why being hypothesis-driven is the cornerstone of problem solving at McKinsey and every other top strategy consulting firm.

How do you become hypothesis-driven?

Most people are idea-driven, and constantly have hypotheses on how the world works and what they or their organization should do to improve. Though, there is often a fatal flaw in that many people turn their hypotheses into false facts, without actually finding or creating the facts to prove or disprove their hypotheses. These people aren’t hypothesis-driven; they are gut-driven.

The conversation typically goes something like “doing this discount promotion will increase our profits” or “our customers need to have this feature” or “morale is in the toilet because we don’t pay well, so we need to increase pay.” These should all be hypotheses that need the appropriate fact base, but instead, they become false facts, often leading to unintended results and consequences. In each of these cases, to become hypothesis-driven necessitates a different framing.

• Instead of “doing this discount promotion will increase our profits,” a hypothesis-driven approach is to ask “what are the best marketing ideas to increase our profits?” and then conduct a marketing experiment to see which ideas increase profits the most.

• Instead of “our customers need to have this feature,” ask the question, “what features would our customers value most?” And, then conduct a simple survey having customers rank order the features based on value to them.

• Instead of “morale is in the toilet because we don’t pay well, so we need to increase pay,” conduct a survey asking, “what is the level of morale?” what are potential issues affecting morale?” and what are the best ideas to improve morale?”

Beyond, watching out for just following your gut, here are some of the other best practices in being hypothesis-driven:

Listen to Your Intuition

Your mind has taken the collision of your experiences and everything you’ve learned over the years to create your intuition, which are those ideas that pop into your head and those hunches that come from your gut. Your intuition is your wellspring of hypotheses. So listen to your intuition, build hypotheses from it, and then prove or disprove those hypotheses, which will, in turn, improve your intuition. Intuition without feedback will over time typically evolve into poor intuition, which leads to poor judgment, thinking, and decisions.

Constantly Be Curious

I’m always curious about cause and effect. At Sports Authority, I had a hypothesis that customers that received service and assistance as they shopped, were worth more than customers who didn’t receive assistance from an associate. We figured out how to prove or disprove this hypothesis by tying surveys to transactional data of customers, and we found the hypothesis was true, which led us to a broad initiative around improving service. The key is you have to be always curious about what you think does or will drive value, create hypotheses and then prove or disprove those hypotheses.

Validate Hypotheses

You need to validate and prove or disprove hypotheses. Don’t just chalk up an idea as fact. In most cases, you’re going to have to create a fact base utilizing logic, observation, testing (see the section on Experimentation ), surveys, and analysis.

Be a Learning Organization

The foundation of learning organizations is the testing of and learning from hypotheses. I remember my first strategy internship at Mercer Management Consulting when I spent a good part of the summer combing through the results, findings, and insights of thousands of experiments that a banking client had conducted. It was fascinating to see the vastness and depth of their collective knowledge base. And, in today’s world of knowledge portals, it is so easy to disseminate, learn from, and build upon the knowledge created by companies.

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• 9.4 Full Hypothesis Test Examples
• Introduction
• 1.1 Definitions of Statistics, Probability, and Key Terms
• 1.2 Data, Sampling, and Variation in Data and Sampling
• 1.3 Levels of Measurement
• 1.4 Experimental Design and Ethics
• Chapter Review
• 2.1 Display Data
• 2.2 Measures of the Location of the Data
• 2.3 Measures of the Center of the Data
• 2.4 Sigma Notation and Calculating the Arithmetic Mean
• 2.5 Geometric Mean
• 2.6 Skewness and the Mean, Median, and Mode
• 2.7 Measures of the Spread of the Data
• Formula Review
• Bringing It Together: Homework
• 3.1 Terminology
• 3.2 Independent and Mutually Exclusive Events
• 3.3 Two Basic Rules of Probability
• 3.4 Contingency Tables and Probability Trees
• 3.5 Venn Diagrams
• Bringing It Together: Practice
• 4.1 Hypergeometric Distribution
• 4.2 Binomial Distribution
• 4.3 Geometric Distribution
• 4.4 Poisson Distribution
• 5.1 Properties of Continuous Probability Density Functions
• 5.2 The Uniform Distribution
• 5.3 The Exponential Distribution
• 6.1 The Standard Normal Distribution
• 6.2 Using the Normal Distribution
• 6.3 Estimating the Binomial with the Normal Distribution
• 7.1 The Central Limit Theorem for Sample Means
• 7.2 Using the Central Limit Theorem
• 7.3 The Central Limit Theorem for Proportions
• 7.4 Finite Population Correction Factor
• 8.1 A Confidence Interval for a Population Standard Deviation, Known or Large Sample Size
• 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case
• 8.3 A Confidence Interval for A Population Proportion
• 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
• 9.1 Null and Alternative Hypotheses
• 9.2 Outcomes and the Type I and Type II Errors
• 9.3 Distribution Needed for Hypothesis Testing
• 10.1 Comparing Two Independent Population Means
• 10.2 Cohen's Standards for Small, Medium, and Large Effect Sizes
• 10.3 Test for Differences in Means: Assuming Equal Population Variances
• 10.4 Comparing Two Independent Population Proportions
• 10.5 Two Population Means with Known Standard Deviations
• 10.6 Matched or Paired Samples
• 11.1 Facts About the Chi-Square Distribution
• 11.2 Test of a Single Variance
• 11.3 Goodness-of-Fit Test
• 11.4 Test of Independence
• 11.5 Test for Homogeneity
• 11.6 Comparison of the Chi-Square Tests
• 12.1 Test of Two Variances
• 12.2 One-Way ANOVA
• 12.3 The F Distribution and the F-Ratio
• 12.4 Facts About the F Distribution
• 13.1 The Correlation Coefficient r
• 13.2 Testing the Significance of the Correlation Coefficient
• 13.3 Linear Equations
• 13.4 The Regression Equation
• 13.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation
• 13.6 Predicting with a Regression Equation
• 13.7 How to Use Microsoft Excel® for Regression Analysis
• A | Statistical Tables
• B | Mathematical Phrases, Symbols, and Formulas

Tests on Means

Example 9.8.

Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds . His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims . For the 15 swims, Jeffrey's mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α = 0.05.

Set up the Hypothesis Test:

Since the problem is about a mean, this is a test of a single population mean .

Set the null and alternative hypothesis:

In this case there is an implied challenge or claim. This is that the goggles will reduce the swimming time. The effect of this is to set the hypothesis as a one-tailed test. The claim will always be in the alternative hypothesis because the burden of proof always lies with the alternative. Remember that the status quo must be defeated with a high degree of confidence, in this case 95 % confidence. The null and alternative hypotheses are thus:

H 0 : μ ≥ 16.43   H a : μ < 16.43

For Jeffrey to swim faster, his time will be less than 16.43 seconds. The "<" tells you this is left-tailed.

Determine the distribution needed:

Random variable: X ¯ X ¯ = the mean time to swim the 25-yard freestyle.

Distribution for the test statistic:

The sample size is less than 30 and we do not know the population standard deviation so this is a t-test. and the proper formula is: t c = X ¯ - μ 0 σ / n t c = X ¯ - μ 0 σ / n

μ 0 = 16.43 comes from H 0 and not the data. X ¯ X ¯ = 16. s = 0.8, and n = 15.

Our step 2, setting the level of significance, has already been determined by the problem, .05 for a 95 % significance level. It is worth thinking about the meaning of this choice. The Type I error is to conclude that Jeffrey swims the 25-yard freestyle, on average, in less than 16.43 seconds when, in fact, he actually swims the 25-yard freestyle, on average, in 16.43 seconds. (Reject the null hypothesis when the null hypothesis is true.) For this case the only concern with a Type I error would seem to be that Jeffery’s dad may fail to bet on his son’s victory because he does not have appropriate confidence in the effect of the goggles.

To find the critical value we need to select the appropriate test statistic. We have concluded that this is a t-test on the basis of the sample size and that we are interested in a population mean. We can now draw the graph of the t-distribution and mark the critical value. For this problem the degrees of freedom are n-1, or 14. Looking up 14 degrees of freedom at the 0.05 column of the t-table we find 1.761. This is the critical value and we can put this on our graph.

Step 3 is the calculation of the test statistic using the formula we have selected. We find that the calculated test statistic is 2.08, meaning that the sample mean is 2.08 standard deviations away from the hypothesized mean of 16.43.

Step 4 has us compare the test statistic and the critical value and mark these on the graph. We see that the test statistic is in the tail and thus we move to step 4 and reach a conclusion. The probability that an average time of 16 minutes could come from a distribution with a population mean of 16.43 minutes is too unlikely for us to accept the null hypothesis. We cannot accept the null.

Step 5 has us state our conclusions first formally and then less formally. A formal conclusion would be stated as: “With a 95% level of significance we cannot accept the null hypothesis that the swimming time with goggles comes from a distribution with a population mean time of 16.43 minutes.” Less formally, “With 95% significance we believe that the goggles improves swimming speed”

If we wished to use the p-value system of reaching a conclusion we would calculate the statistic and take the additional step to find the probability of being 2.08 standard deviations from the mean on a t-distribution. This value is .0187. Comparing this to the α-level of .05 we see that we cannot accept the null. The p-value has been put on the graph as the shaded area beyond -2.08 and it shows that it is smaller than the hatched area which is the alpha level of 0.05. Both methods reach the same conclusion that we cannot accept the null hypothesis.

The mean throwing distance of a football for Marco, a high school freshman quarterback, is 40 yards, with a standard deviation of two yards. The team coach tells Marco to adjust his grip to get more distance. The coach records the distances for 20 throws. For the 20 throws, Marco’s mean distance was 45 yards. The coach thought the different grip helped Marco throw farther than 40 yards. Conduct a hypothesis test using a preset α = 0.05. Assume the throw distances for footballs are normal.

First, determine what type of test this is, set up the hypothesis test, find the p -value, sketch the graph, and state your conclusion.

Example 9.9

Jane has just begun her new job as on the sales force of a very competitive company. In a sample of 16 sales calls it was found that she closed the contract for an average value of 108 dollars with a standard deviation of 12 dollars. Test at 5% significance that the population mean is at least 100 dollars against the alternative that it is less than 100 dollars. Company policy requires that new members of the sales force must exceed an average of $100 per contract during the trial employment period. Can we conclude that Jane has met this requirement at the significance level of 95%?

• H 0 : µ ≤ 100 H a : µ > 100 The null and alternative hypothesis are for the parameter µ because the number of dollars of the contracts is a continuous random variable. Also, this is a one-tailed test because the company has only an interested if the number of dollars per contact is below a particular number not "too high" a number. This can be thought of as making a claim that the requirement is being met and thus the claim is in the alternative hypothesis.
• Test statistic: t c = x ¯ − µ 0 s n = 108 − 100 ( 12 16 ) = 2.67 t c = x ¯ − µ 0 s n = 108 − 100 ( 12 16 ) = 2.67
• Critical value: t a = 1.753 t a = 1.753 with n-1 degrees of freedom= 15

The test statistic is a Student's t because the sample size is below 30; therefore, we cannot use the normal distribution. Comparing the calculated value of the test statistic and the critical value of t t ( t a ) ( t a ) at a 5% significance level, we see that the calculated value is in the tail of the distribution. Thus, we conclude that 108 dollars per contract is significantly larger than the hypothesized value of 100 and thus we cannot accept the null hypothesis. There is evidence that supports Jane's performance meets company standards.

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, state your conclusion, and identify the Type I errors.

Example 9.10

A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses salad dressings is working properly when 8 ounces are dispensed. Suppose that the average amount dispensed in a particular sample of 35 bottles is 7.91 ounces with a variance of 0.03 ounces squared, s 2 s 2 . Is there evidence that the machine should be stopped and production wait for repairs? The lost production from a shutdown is potentially so great that management feels that the level of significance in the analysis should be 99%.

Again we will follow the steps in our analysis of this problem.

STEP 1 : Set the Null and Alternative Hypothesis. The random variable is the quantity of fluid placed in the bottles. This is a continuous random variable and the parameter we are interested in is the mean. Our hypothesis therefore is about the mean. In this case we are concerned that the machine is not filling properly. From what we are told it does not matter if the machine is over-filling or under-filling, both seem to be an equally bad error. This tells us that this is a two-tailed test: if the machine is malfunctioning it will be shutdown regardless if it is from over-filling or under-filling. The null and alternative hypotheses are thus:

STEP 2 : Decide the level of significance and draw the graph showing the critical value.

This problem has already set the level of significance at 99%. The decision seems an appropriate one and shows the thought process when setting the significance level. Management wants to be very certain, as certain as probability will allow, that they are not shutting down a machine that is not in need of repair. To draw the distribution and the critical value, we need to know which distribution to use. Because this is a continuous random variable and we are interested in the mean, and the sample size is greater than 30, the appropriate distribution is the normal distribution and the relevant critical value is 2.575 from the normal table or the t-table at 0.005 column and infinite degrees of freedom. We draw the graph and mark these points.

STEP 3 : Calculate sample parameters and the test statistic. The sample parameters are provided, the sample mean is 7.91 and the sample variance is .03 and the sample size is 35. We need to note that the sample variance was provided not the sample standard deviation, which is what we need for the formula. Remembering that the standard deviation is simply the square root of the variance, we therefore know the sample standard deviation, s, is 0.173. With this information we calculate the test statistic as -3.07, and mark it on the graph.

STEP 4 : Compare test statistic and the critical values Now we compare the test statistic and the critical value by placing the test statistic on the graph. We see that the test statistic is in the tail, decidedly greater than the critical value of 2.575. We note that even the very small difference between the hypothesized value and the sample value is still a large number of standard deviations. The sample mean is only 0.08 ounces different from the required level of 8 ounces, but it is 3 plus standard deviations away and thus we cannot accept the null hypothesis.

STEP 5 : Reach a Conclusion

Three standard deviations of a test statistic will guarantee that the test will fail. The probability that anything is within three standard deviations is almost zero. Actually it is 0.0026 on the normal distribution, which is certainly almost zero in a practical sense. Our formal conclusion would be “ At a 99% level of significance we cannot accept the hypothesis that the sample mean came from a distribution with a mean of 8 ounces” Or less formally, and getting to the point, “At a 99% level of significance we conclude that the machine is under filling the bottles and is in need of repair”.

Hypothesis Test for Proportions

Just as there were confidence intervals for proportions, or more formally, the population parameter p of the binomial distribution, there is the ability to test hypotheses concerning p .

The population parameter for the binomial is p . The estimated value (point estimate) for p is p′ where p′ = x/n , x is the number of successes in the sample and n is the sample size.

When you perform a hypothesis test of a population proportion p , you take a simple random sample from the population. The conditions for a binomial distribution must be met, which are: there are a certain number n of independent trials meaning random sampling, the outcomes of any trial are binary, success or failure, and each trial has the same probability of a success p . The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np′ and nq′ must both be greater than five ( np′ > 5 and nq′ > 5). In this case the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = np μ = np and σ = npq σ = npq . Remember that q = 1 – p q = 1 – p . There is no distribution that can correct for this small sample bias and thus if these conditions are not met we simply cannot test the hypothesis with the data available at that time. We met this condition when we first were estimating confidence intervals for p .

Again, we begin with the standardizing formula modified because this is the distribution of a binomial.

Substituting p 0 p 0 , the hypothesized value of p , we have:

This is the test statistic for testing hypothesized values of p , where the null and alternative hypotheses take one of the following forms:

The decision rule stated above applies here also: if the calculated value of Z c shows that the sample proportion is "too many" standard deviations from the hypothesized proportion, the null hypothesis cannot be accepted. The decision as to what is "too many" is pre-determined by the analyst depending on the level of significance required in the test.

Example 9.11

The mortgage department of a large bank is interested in the nature of loans of first-time borrowers. This information will be used to tailor their marketing strategy. They believe that 50% of first-time borrowers take out smaller loans than other borrowers. They perform a hypothesis test to determine if the percentage is the same or different from 50% . They sample 100 first-time borrowers and find 53 of these loans are smaller that the other borrowers. For the hypothesis test, they choose a 5% level of significance.

STEP 1 : Set the null and alternative hypothesis.

H 0 : p = 0.50   H a : p ≠ 0.50

The words "is the same or different from" tell you this is a two-tailed test. The Type I and Type II errors are as follows: The Type I error is to conclude that the proportion of borrowers is different from 50% when, in fact, the proportion is actually 50%. (Reject the null hypothesis when the null hypothesis is true). The Type II error is there is not enough evidence to conclude that the proportion of first time borrowers differs from 50% when, in fact, the proportion does differ from 50%. (You fail to reject the null hypothesis when the null hypothesis is false.)

STEP 2 : Decide the level of significance and draw the graph showing the critical value

The level of significance has been set by the problem at the 95% level. Because this is two-tailed test one-half of the alpha value will be in the upper tail and one-half in the lower tail as shown on the graph. The critical value for the normal distribution at the 95% level of confidence is 1.96. This can easily be found on the student’s t-table at the very bottom at infinite degrees of freedom remembering that at infinity the t-distribution is the normal distribution. Of course the value can also be found on the normal table but you have go looking for one-half of 95 (0.475) inside the body of the table and then read out to the sides and top for the number of standard deviations.

STEP 3 : Calculate the sample parameters and critical value of the test statistic.

The test statistic is a normal distribution, Z, for testing proportions and is:

For this case, the sample of 100 found 53 first-time borrowers were different from other borrowers. The sample proportion, p′ = 53/100= 0.53 The test question, therefore, is : “Is 0.53 significantly different from .50?” Putting these values into the formula for the test statistic we find that 0.53 is only 0.60 standard deviations away from .50. This is barely off of the mean of the standard normal distribution of zero. There is virtually no difference from the sample proportion and the hypothesized proportion in terms of standard deviations.

STEP 4 : Compare the test statistic and the critical value.

The calculated value is well within the critical values of ± 1.96 standard deviations and thus we cannot reject the null hypothesis. To reject the null hypothesis we need significant evident of difference between the hypothesized value and the sample value. In this case the sample value is very nearly the same as the hypothesized value measured in terms of standard deviations.

STEP 5 : Reach a conclusion

The formal conclusion would be “At a 95% level of significance we cannot reject the null hypothesis that 50% of first-time borrowers have the same size loans as other borrowers”. Less formally we would say that “There is no evidence that one-half of first-time borrowers are significantly different in loan size from other borrowers”. Notice the length to which the conclusion goes to include all of the conditions that are attached to the conclusion. Statisticians for all the criticism they receive, are careful to be very specific even when this seems trivial. Statisticians cannot say more than they know and the data constrain the conclusion to be within the metes and bounds of the data.

Try It 9.11

A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.

Example 9.12

Suppose a consumer group suspects that the proportion of households that have three or more cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey 150 households with the result that 43 of the households have three or more cell phones.

Here is an abbreviate version of the system to solve hypothesis tests applied to a test on a proportions.

Example 9.13

The National Institute of Standards and Technology provides exact data on conductivity properties of materials. Following are conductivity measurements for 11 randomly selected pieces of a particular type of glass.

1.11; 1.07; 1.11; 1.07; 1.12; 1.08; .98; .98 1.02; .95; .95 Is there convincing evidence that the average conductivity of this type of glass is greater than one? Use a significance level of 0.05.

Let’s follow a four-step process to answer this statistical question.

• H 0 : μ ≤ 1
• H a : μ > 1
• Plan : We are testing a sample mean without a known population standard deviation with less than 30 observations. Therefore, we need to use a Student's-t distribution. Assume the underlying population is normal.
• Do the calculations and draw the graph .
• State the Conclusions : We cannot accept the null hypothesis. It is reasonable to state that the data supports the claim that the average conductivity level is greater than one.

Example 9.14

In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users (the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type I error.

• H 0 : p ≤ 0.00034
• H a : p > 0.00034

If we commit a Type I error, we are essentially accepting a false claim. Since the claim describes cancer-causing environments, we want to minimize the chances of incorrectly identifying causes of cancer.

• We will be testing a sample proportion with x = 172 and n = 420,019. The sample is sufficiently large because we have np' = 420,019(0.00034) = 142.8, nq' = 420,019(0.99966) = 419,876.2, two independent outcomes, and a fixed probability of success p' = 0.00034. Thus we will be able to generalize our results to the population.

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From Hypothesis to Results: Mastering the Art of Marketing Experiments

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Suppose you’re trying to convince your friend to watch your favorite movie. You could either tell them about the intriguing plot or show them the exciting trailer.

To find out which approach works best, you try both methods with different friends and see which one gets more people to watch the movie.

Marketing experiments work in much the same way, allowing businesses to test different marketing strategies, gather feedback from their target audience, and make data-driven decisions that lead to improved outcomes and growth.

By testing different approaches and measuring their outcomes, companies can identify what works best for their unique target audience and adapt their marketing strategies accordingly. This leads to more efficient use of marketing resources and results in higher conversion rates, increased customer satisfaction, and, ultimately, business growth.

Marketing experiments are the backbone of building an organization’s culture of learning and curiosity, encouraging employees to think outside the box and challenge the status quo.

In this article, we will delve into the fundamentals of marketing experiments, discussing their key elements and various types. By the end, you’ll be in a position to start running these tests and securing better marketing campaigns with explosive results.

Why Digital Marketing Experiments Matter

One of the most effective ways to drive growth and optimize marketing strategies is through digital marketing experiments. These experiments provide invaluable insights into customer preferences, behaviors, and the overall effectiveness of marketing efforts, making them an essential component of any digital marketing strategy.

Digital marketing experiments matter for several reasons:

• Customer-centric approach: By conducting experiments, businesses can gain a deeper understanding of their target audience’s preferences and behaviors. This enables them to tailor their marketing efforts to better align with customer needs, resulting in more effective and engaging campaigns.
• Data-driven decision-making: Marketing experiments provide quantitative data on the performance of different marketing strategies and tactics. This empowers businesses to make informed decisions based on actual results rather than relying on intuition or guesswork. Ultimately, this data-driven approach leads to more efficient allocation of resources and improved marketing outcomes.
• Agility and adaptability: Businesses must be agile and adaptable to keep up with emerging trends and technologies. Digital marketing experiments allow businesses to test new ideas, platforms, and strategies in a controlled environment, helping them stay ahead of the curve and quickly respond to changing market conditions.
• Continuous improvement: Digital marketing experiments facilitate an iterative process of testing, learning, and refining marketing strategies. This ongoing cycle of improvement enables businesses to optimize their marketing efforts, drive better results, and maintain a competitive edge in the digital marketplace.
• ROI and profitability: By identifying which marketing tactics are most effective, businesses can allocate their marketing budget more efficiently and maximize their return on investment. This increased profitability can be reinvested into the business, fueling further growth and success.

Developing a culture of experimentation allows businesses to continuously improve their marketing strategies, maximize their ROI, and avoid being left behind by the competition.

The Fundamentals of Digital Marketing Experiments

Marketing experiments are structured tests that compare different marketing strategies, tactics, or assets to determine which one performs better in achieving specific objectives.

These experiments use a scientific approach, which involves formulating hypotheses, controlling variables, gathering data, and analyzing the results to make informed decisions.

Marketing experiments provide valuable insights into customer preferences and behaviors, enabling businesses to optimize their marketing efforts and maximize returns on investment (ROI).

There are several types of marketing experiments that businesses can use, depending on their objectives and available resources.

The most common types include:

A/B testing

A/B testing, also known as split testing, is a simple yet powerful technique that compares two variations of a single variable to determine which one performs better.

In an A/B test, the target audience is randomly divided into two groups: one group is exposed to version A (the control). In contrast, the other group is exposed to version B (the treatment). The performance of both versions is then measured and compared to identify the one that yields better results.

A/B testing can be applied to various marketing elements, such as headlines, calls-to-action, email subject lines, landing page designs, and ad copy. The primary advantage of A/B testing is its simplicity, making it easy for businesses to implement and analyze.

Multivariate testing

Multivariate testing is a more advanced technique that allows businesses to test multiple variables simultaneously.

In a multivariate test, several elements of a marketing asset are modified and combined to create different versions. These versions are then shown to different segments of the target audience, and their performance is measured and compared to determine the most effective combination of variables.

Multivariate testing is beneficial when optimizing complex marketing assets, such as websites or email templates, with multiple elements that may interact with one another. However, this method requires a larger sample size and more advanced analytical tools compared to A/B testing.

Pre-post analysis

Pre-post analysis involves comparing the performance of a marketing strategy before and after implementing a change.

This type of experiment is often used when it is not feasible to conduct an A/B or multivariate test, such as when the change affects the entire customer base or when there are external factors that cannot be controlled.

While pre-post analysis can provide useful insights, it is less reliable than A/B or multivariate testing because it does not account for potential confounding factors. To obtain accurate results from a pre-post analysis, businesses must carefully control for external influences and ensure that the observed changes are indeed due to the implemented modifications.

How To Start Growth Marketing Experiments

To conduct effective marketing experiments, businesses must pay attention to the following key elements:

Clear objectives

Having clear objectives is crucial for a successful marketing experiment. Before starting an experiment, businesses must identify the specific goals they want to achieve, such as increasing conversions, boosting engagement, or improving click-through rates. Clear objectives help guide the experimental design and ensure the results are relevant and actionable.

Hypothesis-driven approach

A marketing experiment should be based on a well-formulated hypothesis that predicts the expected outcome. A reasonable hypothesis is specific, testable, and grounded in existing knowledge or data. It serves as the foundation for experimental design and helps businesses focus on the most relevant variables and outcomes.

Proper experimental design

A marketing experiment requires a well-designed test that controls for potential confounding factors and ensures the reliability and validity of the results. This includes the random assignment of participants, controlling for external influences, and selecting appropriate variables to test. Proper experimental design increases the likelihood that observed differences are due to the tested variables and not other factors.

Adequate sample size

A successful marketing experiment requires an adequate sample size to ensure the results are statistically significant and generalizable to the broader target audience. The required sample size depends on the type of experiment, the expected effect size, and the desired level of confidence. In general, larger sample sizes provide more reliable and accurate results but may also require more resources to conduct the experiment.

Data-driven analysis

Marketing experiments rely on a data-driven analysis of the results. This involves using statistical techniques to determine whether the observed differences between the tested variations are significant and meaningful. Data-driven analysis helps businesses make informed decisions based on empirical evidence rather than intuition or gut feelings.

By understanding the fundamentals of marketing experiments and following best practices, businesses can gain valuable insights into customer preferences and behaviors, ultimately leading to improved outcomes and growth.

Setting up Your First Marketing Experiment

Embarking on your first marketing experiment can be both exciting and challenging. Following a systematic approach, you can set yourself up for success and gain valuable insights to improve your marketing efforts.

Here’s a step-by-step guide to help you set up your first marketing experiment.

Identifying your marketing objectives

Before diving into your experiment, it’s essential to establish clear marketing objectives. These objectives will guide your entire experiment, from hypothesis formulation to data analysis.

Consider what you want to achieve with your marketing efforts, such as increasing website conversions, improving open email rates, or boosting social media engagement.

Make sure your objectives are specific, measurable, achievable, relevant, and time-bound (SMART) to ensure that they are actionable and provide meaningful insights.

Formulating a hypothesis

With your marketing objectives in mind, the next step is formulating a hypothesis for your experiment. A hypothesis is a testable prediction that outlines the expected outcome of your experiment. It should be based on existing knowledge, data, or observations and provide a clear direction for your experimental design.

For example, suppose your objective is to increase email open rates. In that case, your hypothesis might be, “Adding the recipient’s first name to the email subject line will increase the open rate by 10%.” This hypothesis is specific, testable, and clearly linked to your marketing objective.

Designing the experiment

Once you have a hypothesis in place, you can move on to designing your experiment. This involves several key decisions:

Choosing the right testing method:

Select the most appropriate testing method for your experiment based on your objectives, hypothesis, and available resources.

As discussed earlier, common testing methods include A/B, multivariate, and pre-post analyses. Choose the method that best aligns with your goals and allows you to effectively test your hypothesis.

Selecting the variables to test:

Identify the specific variables you will test in your experiment. These should be directly related to your hypothesis and marketing objectives. In the email open rate example, the variable to test would be the subject line, specifically the presence or absence of the recipient’s first name.

When selecting variables, consider their potential impact on your marketing objectives and prioritize those with the greatest potential for improvement. Also, ensure that the variables are easily measurable and can be manipulated in your experiment.

Identifying the target audience:

Determine the target audience for your experiment, considering factors such as demographics, interests, and behaviors. Your target audience should be representative of the larger population you aim to reach with your marketing efforts.

When segmenting your audience for the experiment, ensure that the groups are as similar as possible to minimize potential confounding factors.

In A/B or multivariate testing, this can be achieved through random assignment, which helps control for external influences and ensures a fair comparison between the tested variations.

Executing the experiment

With your experiment designed, it’s time to put it into action.

This involves several key considerations:

Timing and duration:

Choose the right timing and duration for your experiment based on factors such as the marketing channel, target audience, and the nature of the tested variables.

The duration of the experiment should be long enough to gather a sufficient amount of data for meaningful analysis but not so long that it negatively affects your marketing efforts or causes fatigue among your target audience.

In general, aim for a duration that allows you to reach a predetermined sample size or achieve statistical significance. This may vary depending on the specific experiment and the desired level of confidence.

Monitoring the experiment:

During the experiment, monitor its progress and performance regularly to ensure that everything is running smoothly and according to plan. This includes checking for technical issues, tracking key metrics, and watching for any unexpected patterns or trends.

If any issues arise during the experiment, address them promptly to prevent potential biases or inaccuracies in the results. Additionally, avoid making changes to the experimental design or variables during the experiment, as this can compromise the integrity of the results.

Analyzing the results

Once your experiment has concluded, it’s time to analyze the data and draw conclusions.

This involves two key aspects:

Statistical significance:

Statistical significance is a measure of the likelihood that the observed differences between the tested variations are due to the variables being tested rather than random chance. To determine statistical significance, you will need to perform a statistical test, such as a t-test or chi-squared test, depending on the nature of your data.

Generally, a result is considered statistically significant if the probability of the observed difference occurring by chance (the p-value) is less than a predetermined threshold, often set at 0.05 or 5%. This means there is a 95% confidence level that the observed difference is due to the tested variables and not random chance.

Practical significance:

While statistical significance is crucial, it’s also essential to consider the practical significance of your results. This refers to the real-world impact of the observed differences on your marketing objectives and business goals.

To assess practical significance, consider the effect size of the observed difference (e.g., the percentage increase in email open rates) and the potential return on investment (ROI) of implementing the winning variation. This will help you determine whether the experiment results are worth acting upon and inform your marketing decisions moving forward.

A systematic approach to designing growth marketing experiments helps you to design, execute, and analyze your experiment effectively, ultimately leading to better marketing outcomes and business growth.

Examples of Successful Marketing Experiments

In this section, we will explore three fictional case studies of successful marketing experiments that led to improved marketing outcomes. These examples will demonstrate the practical application of marketing experiments across different channels and provide valuable lessons that can be applied to your own marketing efforts.

Example 1: Redesigning a website for increased conversions

AcmeWidgets, an online store selling innovative widgets, noticed that its website conversion rate had plateaued.

They conducted a marketing experiment to test whether a redesigned landing page could improve conversions. They hypothesized that a more visually appealing and user-friendly design would increase conversion rates by 15%.

AcmeWidgets used A/B testing to compare their existing landing page (the control) with a new, redesigned version (the treatment). They randomly assigned website visitors to one of the two landing pages. They tracked conversions over a period of four weeks.

At the end of the experiment, AcmeWidgets found that the redesigned landing page had a conversion rate 18% higher than the control. The results were statistically significant, and the company decided to implement the new design across its entire website.

As a result, AcmeWidgets experienced a substantial increase in sales and revenue.

Example 2: Optimizing email marketing campaigns

EcoTravel, a sustainable travel agency, wanted to improve the open rates of their monthly newsletter. They hypothesized that adding a sense of urgency to the subject line would increase open rates by 10%.

To test this hypothesis, EcoTravel used A/B testing to compare two different subject lines for their newsletter:

• “Discover the world’s most beautiful eco-friendly destinations” (control)
• “Last chance to book: Explore the world’s most beautiful eco-friendly destinations” (treatment)

EcoTravel sent the newsletter to a random sample of their subscribers. Half received the control subject line, and the other half received the treatment. They then tracked the open rates for both groups over one week.

The results of the experiment showed that the treatment subject line, which included a sense of urgency, led to a 12% increase in open rates compared to the control.

Based on these findings, EcoTravel incorporated a sense of urgency in their future email subject lines to boost newsletter engagement.

Example 3: Improving social media ad performance

FitFuel, a meal delivery service for fitness enthusiasts, was looking to improve its Facebook ad campaign’s click-through rate (CTR). They hypothesized that using an image of a satisfied customer enjoying a FitFuel meal would increase CTR by 8% compared to their current ad featuring a meal image alone.

FitFuel conducted an A/B test on their Facebook ad campaign, comparing the performance of the control ad (meal image only) with the treatment ad (customer enjoying a meal). They targeted a similar audience with both ad variations and measured the CTR over two weeks. The experiment revealed that the treatment ad, featuring the customer enjoying a meal, led to a 10% increase in CTR compared to the control ad. FitFuel decided to update its

Facebook ad campaign with the new image, resulting in a more cost-effective campaign and higher return on investment.

Lessons learned from these examples

These fictional examples of successful marketing experiments highlight several key takeaways:

• Clearly defined objectives and hypotheses: In each example, the companies had specific marketing objectives and well-formulated hypotheses, which helped guide their experiments and ensure relevant and actionable results.
• Proper experimental design: Each company used the appropriate testing method for their experiment and carefully controlled variables, ensuring accurate and reliable results.
• Data-driven decision-making: The companies analyzed the data from their experiments to make informed decisions about implementing changes to their marketing strategies, ultimately leading to improved outcomes.
• Continuous improvement: These examples demonstrate that marketing experiments can improve marketing efforts continuously. By regularly conducting experiments and applying the lessons learned, businesses can optimize their marketing strategies and stay ahead of the competition.
• Relevance across channels: Marketing experiments can be applied across various marketing channels, such as website design, email campaigns, and social media advertising. Regardless of the channel, the principles of marketing experimentation remain the same, making them a valuable tool for marketers in diverse industries.

By learning from these fictional examples and applying the principles of marketing experimentation to your own marketing efforts, you can unlock valuable insights, optimize your marketing strategies, and achieve better results for your business.

Common Pitfalls of Marketing Experiments and How to Avoid Them

Conducting marketing experiments can be a powerful way to optimize your marketing strategies and drive better results.

However, it’s important to be aware of common pitfalls that can undermine the effectiveness of your experiments. In this section, we will discuss some of these pitfalls and provide tips on how to avoid them.

Insufficient sample size

An insufficient sample size can lead to unreliable results and limit the generalizability of your findings. When your sample size is too small, you run the risk of not detecting meaningful differences between the tested variations or incorrectly attributing the observed differences to random chance.

To avoid this pitfall, calculate the required sample size for your experiment based on factors such as the expected effect size, the desired level of confidence, and the type of statistical test you will use.

In general, larger sample sizes provide more reliable and accurate results but may require more resources to conduct the experiment. Consider adjusting your experimental design or testing methods to accommodate a larger sample size if necessary.

Lack of clear objectives

Your marketing experiment may not provide meaningful or actionable insights without clear objectives. Unclear objectives can lead to poorly designed experiments, irrelevant variables, or difficulty interpreting the results.

To prevent this issue, establish specific, measurable, achievable, relevant, and time-bound (SMART) objectives before starting your experiment. These objectives should guide your entire experiment, from hypothesis formulation to data analysis, and ensure that your findings are relevant and useful for your marketing efforts.

Confirmation bias

Confirmation bias occurs when you interpret the results of your experiment in a way that supports your pre-existing beliefs or expectations. This can lead to inaccurate conclusions and suboptimal marketing decisions.

To minimize confirmation bias, approach your experiments with an open mind and be willing to accept results that challenge your assumptions.

Additionally, involve multiple team members in the data analysis process to ensure diverse perspectives and reduce the risk of individual biases influencing the interpretation of the results.

Overlooking external factors

External factors, such as changes in market conditions, seasonal fluctuations, or competitor actions, can influence the results of your marketing experiment and potentially confound your findings. Ignoring these factors may lead to inaccurate conclusions about the effectiveness of your marketing strategies.

To account for external factors, carefully control for potential confounding variables during the experimental design process. This might involve using random assignment, testing during stable periods, or controlling for known external influences.

Consider running follow-up experiments or analyzing historical data to confirm your findings and rule out the impact of external factors.

Tips for avoiding these pitfalls

By being aware of these common pitfalls and following best practices, you can ensure the success of your marketing experiments and obtain valuable insights for your marketing efforts. Here are some tips to help you avoid these pitfalls:

• Plan your experiment carefully: Invest time in the planning stage to establish clear objectives, calculate an adequate sample size, and design a robust experiment that controls for potential confounding factors.
• Use a hypothesis-driven approach: Formulate a specific, testable hypothesis based on existing knowledge or data to guide your experiment and focus on the most relevant variables and outcomes.
• Monitor your experiment closely: Regularly check the progress of your experiment, address any issues that arise, and ensure that your experiment is running smoothly and according to plan.
• Analyze your data objectively: Use statistical techniques to determine the significance of your results and consider the practical implications of your findings before making marketing decisions.
• Learn from your experiments: Apply the lessons learned from your experiments to continuously improve your marketing strategies and stay ahead of the competition.

By avoiding these common pitfalls and following best practices, you can increase the effectiveness of your marketing experiments, gain valuable insights into customer preferences and behaviors, and ultimately drive better results for your business.

Building a Culture of Experimentation

To truly reap the benefits of marketing experiments, it’s essential to build a culture of experimentation within your organization. This means fostering an environment where curiosity, learning, data-driven decision-making, and collaboration are valued and encouraged.

Encouraging curiosity and learning within your organization

Cultivating curiosity and learning starts with leadership. Encourage your team to ask questions, explore new ideas, and embrace a growth mindset.

Promote ongoing learning by providing resources, such as training programs, workshops, or access to industry events, that help your team stay up-to-date with the latest marketing trends and techniques.

Create a safe environment where employees feel comfortable sharing their ideas and taking calculated risks. Emphasize the importance of learning from both successes and failures and treat every experiment as an opportunity to grow and improve.

Adopting a data-driven mindset

A data-driven mindset is crucial for successful marketing experimentation. Encourage your team to make decisions based on data rather than relying on intuition or guesswork. This means analyzing the results of your experiments objectively, using statistical techniques to determine the significance of your findings, and considering the practical implications of your results before making marketing decisions.

To foster a data-driven culture, invest in the necessary tools and technologies to collect, analyze, and visualize data effectively. Train your team on how to use these tools and interpret the data to make informed marketing decisions.

Regularly review your data-driven efforts and adjust your strategies as needed to continuously improve and optimize your marketing efforts.

Integrating experimentation into your marketing strategy

Establish a systematic approach to conducting marketing experiments to fully integrate experimentation into your marketing strategy. This might involve setting up a dedicated team or working group responsible for planning, executing, and analyzing experiments or incorporating experimentation as a standard part of your marketing processes.

Create a roadmap for your marketing experiments that outlines each project’s objectives, hypotheses, and experimental designs. Monitor the progress of your experiments and adjust your roadmap as needed based on the results and lessons learned.

Ensure that your marketing team has the necessary resources, such as time, budget, and tools, to conduct experiments effectively. Set clear expectations for the role of experimentation in your marketing efforts and emphasize its importance in driving better results and continuous improvement.

Collaborating across teams for a holistic approach

Marketing experiments often involve multiple teams within an organization, such as design, product, sales, and customer support. Encourage cross-functional collaboration to ensure a holistic approach to experimentation and leverage each team’s unique insights and expertise.

Establish clear communication channels and processes for sharing information and results from your experiments. This might involve regular meetings, shared documentation, or internal presentations to keep all stakeholders informed and engaged.

Collaboration also extends beyond your organization. Connect with other marketing professionals, industry experts, and thought leaders to learn from their experiences, share your own insights, and stay informed about the latest trends and best practices in marketing experimentation.

By building a culture of experimentation within your organization, you can unlock valuable insights, optimize your marketing strategies, and drive better results for your business.

Encourage curiosity and learning, adopt a data-driven mindset, integrate experimentation into your marketing strategy, and collaborate across teams to create a strong foundation for marketing success.

If you’re new to marketing experiments, don’t be intimidated—start small and gradually expand your efforts as your confidence grows. By embracing a curious and data-driven mindset, even small-scale experiments can lead to meaningful insights and improvements.

As you gain experience, you can tackle more complex experiments and further refine your marketing strategies.

Remember, continuous learning and improvement is the key to success in marketing experimentation. By regularly conducting experiments, analyzing the results, and applying the lessons learned, you can stay ahead of the competition and drive better results for your business.

So, take the plunge and start experimenting today—your marketing efforts will be all the better.

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A Beginner’s Guide to Hypothesis Testing in Business

• 30 Mar 2021

Becoming a more data-driven decision-maker can bring several benefits to your organization, enabling you to identify new opportunities to pursue and threats to abate. Rather than allowing subjective thinking to guide your business strategy, backing your decisions with data can empower your company to become more innovative and, ultimately, profitable.

If you’re new to data-driven decision-making, you might be wondering how data translates into business strategy. The answer lies in generating a hypothesis and verifying or rejecting it based on what various forms of data tell you.

Below is a look at hypothesis testing and the role it plays in helping businesses become more data-driven.

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What Is Hypothesis Testing?

To understand what hypothesis testing is, it’s important first to understand what a hypothesis is.

A hypothesis or hypothesis statement seeks to explain why something has happened, or what might happen, under certain conditions. It can also be used to understand how different variables relate to each other. Hypotheses are often written as if-then statements; for example, “If this happens, then this will happen.”

Hypothesis testing , then, is a statistical means of testing an assumption stated in a hypothesis. While the specific methodology leveraged depends on the nature of the hypothesis and data available, hypothesis testing typically uses sample data to extrapolate insights about a larger population.

Hypothesis Testing in Business

When it comes to data-driven decision-making, there’s a certain amount of risk that can mislead a professional. This could be due to flawed thinking or observations, incomplete or inaccurate data , or the presence of unknown variables. The danger in this is that, if major strategic decisions are made based on flawed insights, it can lead to wasted resources, missed opportunities, and catastrophic outcomes.

The real value of hypothesis testing in business is that it allows professionals to test their theories and assumptions before putting them into action. This essentially allows an organization to verify its analysis is correct before committing resources to implement a broader strategy.

As one example, consider a company that wishes to launch a new marketing campaign to revitalize sales during a slow period. Doing so could be an incredibly expensive endeavor, depending on the campaign’s size and complexity. The company, therefore, may wish to test the campaign on a smaller scale to understand how it will perform.

In this example, the hypothesis that’s being tested would fall along the lines of: “If the company launches a new marketing campaign, then it will translate into an increase in sales.” It may even be possible to quantify how much of a lift in sales the company expects to see from the effort. Pending the results of the pilot campaign, the business would then know whether it makes sense to roll it out more broadly.

Related: 9 Fundamental Data Science Skills for Business Professionals

Key Considerations for Hypothesis Testing

1. alternative hypothesis and null hypothesis.

In hypothesis testing, the hypothesis that’s being tested is known as the alternative hypothesis . Often, it’s expressed as a correlation or statistical relationship between variables. The null hypothesis , on the other hand, is a statement that’s meant to show there’s no statistical relationship between variables being tested. It’s typically the exact opposite of whatever is stated in the alternative hypothesis.

For example, consider a company’s leadership team who historically and reliably sees $12 million in monthly revenue. They want to understand if reducing the price of their services will attract more customers and, in turn, increase revenue.

In this case, the alternative hypothesis may take the form of a statement such as: “If we reduce the price of our flagship service by five percent, then we’ll see an increase in sales and realize revenues greater than $12 million in the next month.”

The null hypothesis, on the other hand, would indicate that revenues wouldn’t increase from the base of $12 million, or might even decrease.

2. Significance Level and P-Value

Statistically speaking, if you were to run the same scenario 100 times, you’d likely receive somewhat different results each time. If you were to plot these results in a distribution plot, you’d see the most likely outcome is at the tallest point in the graph, with less likely outcomes falling to the right and left of that point.

With this in mind, imagine you’ve completed your hypothesis test and have your results, which indicate there may be a correlation between the variables you were testing. To understand your results' significance, you’ll need to identify a p-value for the test, which helps note how confident you are in the test results.

In statistics, the p-value depicts the probability that, assuming the null hypothesis is correct, you might still observe results that are at least as extreme as the results of your hypothesis test. The smaller the p-value, the more likely the alternative hypothesis is correct, and the greater the significance of your results.

3. One-Sided vs. Two-Sided Testing

When it’s time to test your hypothesis, it’s important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests , or one-tailed and two-tailed tests, respectively.

Typically, you’d leverage a one-sided test when you have a strong conviction about the direction of change you expect to see due to your hypothesis test. You’d leverage a two-sided test when you’re less confident in the direction of change.

4. Sampling

To perform hypothesis testing in the first place, you need to collect a sample of data to be analyzed. Depending on the question you’re seeking to answer or investigate, you might collect samples through surveys, observational studies, or experiments.

A survey involves asking a series of questions to a random population sample and recording self-reported responses.

Observational studies involve a researcher observing a sample population and collecting data as it occurs naturally, without intervention.

Finally, an experiment involves dividing a sample into multiple groups, one of which acts as the control group. For each non-control group, the variable being studied is manipulated to determine how the data collected differs from that of the control group.

Learning How to Perform Hypothesis Testing

Hypothesis testing is a complex process involving different moving pieces that can allow an organization to effectively leverage its data and inform strategic decisions.

If you’re interested in better understanding hypothesis testing and the role it can play within your organization, one option is to complete a course that focuses on the process. Doing so can lay the statistical and analytical foundation you need to succeed.

Are you interested in improving your data literacy? Download our Beginner’s Guide to Data & Analytics to learn how you can leverage the power of data for professional and organizational success.

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What Are Examples of a Hypothesis?

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A hypothesis is an explanation for a set of observations. Here are examples of a scientific hypothesis.

Although you could state a scientific hypothesis in various ways, most hypotheses are either "If, then" statements or forms of the null hypothesis . The null hypothesis is sometimes called the "no difference" hypothesis. The null hypothesis is good for experimentation because it's simple to disprove. If you disprove a null hypothesis, that is evidence for a relationship between the variables you are examining.

Examples of Null Hypotheses

• Hyperactivity is unrelated to eating sugar.
• All daisies have the same number of petals.
• The number of pets in a household is unrelated to the number of people living in it.
• A person's preference for a shirt is unrelated to its color.

Examples of If, Then Hypotheses

• If you get at least 6 hours of sleep, you will do better on tests than if you get less sleep.
• If you drop a ball, it will fall toward the ground.
• If you drink coffee before going to bed, then it will take longer to fall asleep.
• If you cover a wound with a bandage, then it will heal with less scarring.

Improving a Hypothesis to Make It Testable

You may wish to revise your first hypothesis in order to make it easier to design an experiment to test. For example, let's say you have a bad breakout the morning after eating a lot of greasy food. You may wonder if there is a correlation between eating greasy food and getting pimples. You propose the hypothesis:

Eating greasy food causes pimples.

Next, you need to design an experiment to test this hypothesis. Let's say you decide to eat greasy food every day for a week and record the effect on your face. Then, as a control, you'll avoid greasy food for the next week and see what happens. Now, this is not a good experiment because it does not take into account other factors such as hormone levels, stress, sun exposure, exercise, or any number of other variables that might conceivably affect your skin.

The problem is that you cannot assign cause to your effect . If you eat french fries for a week and suffer a breakout, can you definitely say it was the grease in the food that caused it? Maybe it was the salt. Maybe it was the potato. Maybe it was unrelated to diet. You can't prove your hypothesis. It's much easier to disprove a hypothesis.

So, let's restate the hypothesis to make it easier to evaluate the data:

Getting pimples is unaffected by eating greasy food.

So, if you eat fatty food every day for a week and suffer breakouts and then don't break out the week that you avoid greasy food, you can be pretty sure something is up. Can you disprove the hypothesis? Probably not, since it is so hard to assign cause and effect. However, you can make a strong case that there is some relationship between diet and acne.

If your skin stays clear for the entire test, you may decide to accept your hypothesis . Again, you didn't prove or disprove anything, which is fine

• What Is a Hypothesis? (Science)
• Null Hypothesis Definition and Examples
• What Are the Elements of a Good Hypothesis?
• Understanding Simple vs Controlled Experiments
• What Is a Testable Hypothesis?
• What 'Fail to Reject' Means in a Hypothesis Test
• Null Hypothesis Examples
• Scientific Method Vocabulary Terms
• How To Design a Science Fair Experiment
• Scientific Hypothesis Examples
• Six Steps of the Scientific Method
• An Example of a Hypothesis Test
• Definition of a Hypothesis
• Scientific Method Flow Chart
• How to Conduct a Hypothesis Test

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Expert Advice on Developing a Hypothesis for Marketing Experimentation 

• Conversion Rate Optimization

Simbar Dube

Every marketing experimentation process has to have a solid hypothesis. 

That’s a must – unless you want to be roaming in the dark and heading towards a dead-end in your experimentation program.

Hypothesizing is the second phase of our SHIP optimization process here at Invesp.

It comes after we have completed the research phase. 

This is an indication that we don’t just pull a hypothesis out of thin air. We always make sure that it is based on research data. 

But having a research-backed hypothesis doesn’t mean that the hypothesis will always be correct. In fact, tons of hypotheses bear inconclusive results or get disproved. 

The main idea of having a hypothesis in marketing experimentation is to help you gain insights – regardless of the testing outcome. 

By the time you finish reading this article, you’ll know: 

• The essential tips on what to do when crafting a hypothesis for marketing experiments
• How a marketing experiment hypothesis works 

How experts develop a solid hypothesis

The basics: marketing experimentation hypothesis.

A hypothesis is a research-based statement that aims to explain an observed trend and create a solution that will improve the result. This statement is an educated, testable prediction about what will happen.

It has to be stated in declarative form and not as a question.

“ If we add magnification info, product video and making virtual mirror buttons, will that improve engagement? ” is not declarative, but “ Improving the experience of product pages by adding magnification info, product video and making virtual mirror buttons will increase engagement ” is.

Here’s a quick example of how a hypothesis should be phrased: 

• Replacing ___ with __ will increase [conversion goal] by [%], because:
• Removing ___ and __ will decrease [conversion goal] by [%], because:
• Changing ___ into __ will not affect [conversion goal], because:
• Improving  ___ by  ___will increase [conversion goal], because: 

As you can see from the above sentences, a good hypothesis is written in clear and simple language. Reading your hypothesis should tell your team members exactly what you thought was going to happen in an experiment.

Another important element of a good hypothesis is that it defines the variables in easy-to-measure terms, like who the participants are, what changes during the testing, and what the effect of the changes will be: 

Example : Let’s say this is our hypothesis: 

Displaying full look items on every “continue shopping & view your bag” pop-up and highlighting the value of having a full look will improve the visibility of a full look, encourage visitors to add multiple items from the same look and that will increase the average order value, quantity with cross-selling by 3% .

Who are the participants : 

Visitors. 

What changes during the testing : 

Displaying full look items on every “continue shopping & view your bag” pop-up and highlighting the value of having a full look…

What the effect of the changes will be:  

Will improve the visibility of a full look, encourage visitors to add multiple items from the same look and that will increase the average order value, quantity with cross-selling by 3% .

Don’t bite off more than you can chew! Answering some scientific questions can involve more than one experiment, each with its own hypothesis. so, you have to make sure your hypothesis is a specific statement relating to a single experiment.

How a Marketing Experimentation Hypothesis Works

Assuming that you have done conversion research and you have identified a list of issues ( UX or conversion-related problems) and potential revenue opportunities on the site. The next thing you’d want to do is to prioritize the issues and determine which issues will most impact the bottom line.

Having ranked the issues you need to test them to determine which solution works best. At this point, you don’t have a clear solution for the problems identified. So, to get better results and avoid wasting traffic on poor test designs, you need to make sure that your testing plan is guided. 

This is where a hypothesis comes into play. 

For each and every problem you’re aiming to address, you need to craft a hypothesis for it – unless the problem is a technical issue that can be solved right away without the need to hypothesize or test. 

One important thing you should note about an experimentation hypothesis is that it can be implemented in different ways.  

This means that one hypothesis can have four or five different tests as illustrated in the image above. Khalid Saleh , the Invesp CEO, explains: 

“There are several ways that can be used to support one single hypothesis. Each and every way is a possible test scenario. And that means you also have to prioritize the test design you want to start with. Ultimately the name of the game is you want to find the idea that has the biggest possible impact on the bottom line with the least amount of effort. We use almost 18 different metrics to score all of those.”

In one of the recent tests we launched after watching video recordings, viewing heatmaps, and conducting expert reviews, we noticed that:  

• Visitors were scrolling to the bottom of the page to fill out a calculator so as to get a free diet plan. 
• Brand is missing 
• Too many free diet plans – and this made it hard for visitors to choose and understand.  
• No value proposition on the page
• The copy didn’t mention the benefits of the paid program
• There was no clear CTA for the next action

To help you understand, let’s have a look at how the original page looked like before we worked on it: 

So our aim was to make the shopping experience seamless for visitors, make the page more appealing and not confusing. In order to do that, here is how we phrased the hypothesis for the page above: 

Improving the experience of optin landing pages by making the free offer accessible above the fold and highlighting the next action with a clear CTA and will increase the engagement on the offer and increase the conversion rate by 1%.

For this particular hypothesis, we had two design variations aligned to it:

The two above designs are different, but they are aligned to one hypothesis. This goes on to show how one hypothesis can be implemented in different ways. Looking at the two variations above – which one do you think won?

Yes, you’re right, V2 was the winner. 

Considering that there are many ways you can implement one hypothesis, so when you launch a test and it fails, it doesn’t necessarily mean that the hypothesis was wrong. Khalid adds:

“A single failure of a test doesn’t mean that the hypothesis is incorrect. Nine times out of ten it’s because of the way you’ve implemented the hypothesis. Look at the way you’ve coded and look at the copy you’ve used – you are more likely going to find something wrong with it. Always be open.” 

So there are three things you should keep in mind when it comes to marketing experimentation hypotheses: 

• It takes a while for this hypothesis to really fully test it.
• A single failure doesn’t necessarily mean that the hypothesis is incorrect.
• Whether a hypothesis is proved or disproved, you can still learn something about your users.

I know it’s never easy to develop a hypothesis that informs future testing – I mean it takes a lot of intense research behind the scenes, and tons of ideas to begin with. So, I reached out to six CRO experts for tips and advice to help you understand more about developing a solid hypothesis and what to include in it. 

Maurice   says that a solid hypothesis should have not more than one goal: 

Maurice Beerthuyzen – CRO/CXO Lead at ClickValue “Creating a hypothesis doesn’t begin at the hypothesis itself. It starts with research. What do you notice in your data, customer surveys, and other sources? Do you understand what happens on your website? When you notice an opportunity it is tempting to base one single A/B test on one hypothesis. Create hypothesis A and run a single test, and then move forward to the next test. With another hypothesis. But it is very rare that you solve your problem with only one hypothesis. Often a test provides several other questions. Questions which you can solve with running other tests. But based on that same hypothesis! We should not come up with a new hypothesis for every test. Another mistake that often happens is that we fill the hypothesis with multiple goals. Then we expect that the hypothesis will work on conversion rate, average order value, and/or Click Through Ratio. Of course, this is possible, but when you run your test, your hypothesis can only have one goal at once. And what if you have two goals? Just split the hypothesis then create a secondary hypothesis for your second goal. Every test has one primary goal. What if you find a winner on your secondary hypothesis? Rerun the test with the second hypothesis as the primary one.”

Jon believes that a strong hypothesis is built upon three pillars:

Jon MacDonald – President and Founder of The Good Respond to an established challenge – The challenge must have a strong background based on data, and the background should state an established challenge that the test is looking to address. Example: “Sign up form lacks proof of value, incorrectly assuming if users are on the page, they already want the product.” Propose a specific solution – What is the one, the single thing that is believed will address the stated challenge? Example: “Adding an image of the dashboard as a background to the signup form…”. State the assumed impact – The assumed impact should reference one specific, measurable optimization goal that was established prior to forming a hypothesis. Example: “…will increase signups.” So, if your hypothesis doesn’t have a specific, measurable goal like “will increase signups,” you’re not really stating a test hypothesis!”

Matt uses his own hypothesis builder to collate important data points into a single hypothesis. 

Matt Beischel – Founder of Corvus CRO Like Jon, Matt also breaks down his hypothesis writing process into three sections. Unlike Jon, Matt sections are: Comprehension Response Outcome I set it up so that the names neatly match the “CRO.” It’s a sort of “mad-libs” style fill-in-the-blank where each input is an important piece of information for building out a robust hypothesis. I consider these the minimum required data points for a good hypothesis; if you can’t completely fill out the form, then you don’t have a good hypothesis. Here’s a breakdown of each data point: Comprehension – Identifying something that can be improved upon Problem: “What is a problem we have?” Observation Method: “How did we identify the problem?” Response – Change that can cause improvement Variation: “What change do we think could solve the problem?” Location: “Where should the change occur?” Scope: “What are the conditions for the change?” Audience: “Who should the change affect?” Outcome – Measurable result of the change that determines the success Behavior Change : “What change in behavior are we trying to affect?” Primary KPI: “What is the important metric that determines business impact?” Secondary KPIs: “Other metrics that will help reinforce/refute the Primary KPI” Something else to consider is that I have a “user first” approach to formulating hypotheses. My process above is always considered within the context of how it would first benefit the user. Now, I do feel that a successful experiment should satisfy the needs of BOTH users and businesses, but always be in favor of the user. Notice that “Behavior Change” is the first thing listed in Outcome, not primary business KPI. Sure, at the end of the day you are working for the business’s best interests (both strategically and financially), but placing the user first will better inform your decision making and prioritization; there’s a reason that things like personas, user stories, surveys, session replays, reviews, etc. exist after all. A business-first ideology is how you end up with dark patterns and damaging brand credibility.”

One of the many mistakes that CROs make when writing a hypothesis is that they are focused on wins and not on insights. Shiva advises against this mindset:

Shiva Manjunath – Marketing Manager and CRO at Gartner “Test to learn, not test to win. It’s a very simple reframe of hypotheses but can have a magnitude of difference. Here’s an example: Test to Win Hypothesis: If I put a product video in the middle of the product page, I will improve add to cart rates and improve CVR. Test to Learn Hypothesis: If I put a product video on the product page, there will be high engagement with the video and it will positively influence traffic What you’re doing is framing your hypothesis, and test, in a particular way to learn as much as you can. That is where you gain marketing insights. The more you run ‘marketing insight’ tests, the more you will win. Why? The more you compound marketing insight learnings, your win velocity will start to increase as a proxy of the learnings you’ve achieved. Then, you’ll have a higher chance of winning in your tests – and the more you’ll be able to drive business results.”

Lorenzo  says it’s okay to focus on achieving a certain result as long as you are also getting an answer to: “Why is this event happening or not happening?”

Lorenzo Carreri – CRO Consultant “When I come up with a hypothesis for a new or iterative experiment, I always try to find an answer to a question. It could be something related to a problem people have or an opportunity to achieve a result or a way to learn something. The main question I want to answer is “Why is this event happening or not happening?” The question is driven by data, both qualitative and quantitative. The structure I use for stating my hypothesis is: From [data source], I noticed [this problem/opportunity] among [this audience of users] on [this page or multiple pages]. So I believe that by [offering this experiment solution], [this KPI] will [increase/decrease/stay the same].

Jakub Linowski says that hypotheses are meant to hold researchers accountable:

Jakub Linowski – Chief Editor of GoodUI “They do this by making your change and prediction more explicit. A typical hypothesis may be expressed as: If we change (X), then it will have some measurable effect (A). Unfortunately, this oversimplified format can also become a heavy burden to your experiment design with its extreme reductionism. However you decide to format your hypotheses, here are three suggestions for more flexibility to avoid limiting yourself. One Or More Changes To break out of the first limitation, we have to admit that our experiments may contain a single or multiple changes. Whereas the classic hypothesis encourages a single change or isolated variable, it’s not the only way we can run experiments. In the real world, it’s quite normal to see multiple design changes inside a single variation. One valid reason for doing this is when wishing to optimize a section of a website while aiming for a greater effect. As more positive changes compound together, there are times when teams decide to run bigger experiments. An experiment design (along with your hypotheses) therefore should allow for both single or multiple changes. One Or More Metrics A second limitation of many hypotheses is that they often ask us to only make a single prediction at a time. There are times when we might like to make multiple guesses or predictions to a set of metrics. A simple example of this might be a trade-off experiment with a guess of increased sales but decreased trial signups. Being able to express single or multiple metrics in our experimental designs should therefore be possible. Estimates, Directional Predictions, Or Unknowns Finally, traditional hypotheses also tend to force very simple directional predictions by asking us to guess whether something will increase or decrease. In reality, however, the fidelity of predictions can be higher or lower. On one hand, I’ve seen and made experiment estimations that contain specific numbers from prior data (ex: increase sales by 14%). While at other times it should also be acceptable to admit the unknown and leave the prediction blank. One example of this is when we are testing a completely novel idea without any prior data in a highly exploratory type of experiment. In such cases, it might be dishonest to make any sort of predictions and we should allow ourselves to express the unknown comfortably.”

Conclusion 

So there you have it! Before you jump on launching a test, start by making sure that your hypothesis is solid and backed by research. Ask yourself the questions below when crafting a hypothesis for marketing experimentation:

• Is the hypothesis backed by research?
• Can the hypothesis be tested?
• Does the hypothesis provide insights?
• Does the hypothesis set the expectation that there will be an explanation behind the results of whatever you’re testing?

Don’t worry! Hypothesizing may seem like a very complicated process, but it’s not complicated in practice especially when you have done proper research.

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11 A/B Testing Examples From Real Businesses

Published: April 21, 2023

Whether you're looking to increase revenue, sign-ups, social shares, or engagement, A/B testing and optimization can help you get there.

But for many marketers out there, the tough part about A/B testing is often finding the right test to drive the biggest impact — especially when you're just getting started. So, what's the recipe for high-impact success?

Truthfully, there is no one-size-fits-all recipe. What works for one business won't work for another — and finding the right metrics and timing to test can be a tough problem to solve. That’s why you need inspiration from A/B testing examples.

In this post, let's review how a hypothesis will get you started with your testing, and check out excellent examples from real businesses using A/B testing. While the same tests may not get you the same results, they can help you run creative tests of your own.

A/B Testing Hypothesis Examples

A hypothesis can make or break your experiment, especially when it comes to A/B testing. When creating your hypothesis, you want to make sure that it’s:

• Focused on one specific problem you want to solve or understand
• Able to be proven or disproven
• Focused on making an impact (bringing higher conversion rates, lower bounce rate, etc.)

When creating a hypothesis, following the "If, then" structure can be helpful, where if you changed a specific variable, then a particular result would happen.

Here are some examples of what that would look like in an A/B testing hypothesis:

• Shortening contact submission forms to only contain required fields would increase the number of sign-ups.
• Changing the call-to-action text from "Download now" to "Download this free guide" would increase the number of downloads.
• Reducing the frequency of mobile app notifications from five times per day to two times per day will increase mobile app retention rates.
• Using featured images that are more contextually related to our blog posts will contribute to a lower bounce rate.
• Greeting customers by name in emails will increase the total number of clicks.

Let’s go over some real-life examples of A/B testing to prepare you for your own.

A/B Testing Examples

Website a/b testing examples, 1. hubspot academy's homepage hero image.

Most websites have a homepage hero image that inspires users to engage and spend more time on the site. This A/B testing example shows how hero image changes can impact user behavior and conversions.

Based on previous data, HubSpot Academy found that out of more than 55,000 page views, only .9% of those users were watching the video on the homepage. Of those viewers, almost 50% watched the full video.

Chat transcripts also highlighted the need for clearer messaging for this useful and free resource.

That's why the HubSpot team decided to test how clear value propositions could improve user engagement and delight.

A/B Test Method

HubSpot used three variants for this test, using HubSpot Academy conversion rate (CVR) as the primary metric. Secondary metrics included CTA clicks and engagement.

Variant A was the control.

For variant B, the team added more vibrant images and colorful text and shapes. It also included an animated "typing" headline.

Variant C also added color and movement, as well as animated images on the right-hand side of the page.

As a result, HubSpot found that variant B outperformed the control by 6%. In contrast, variant C underperformed the control by 1%. From those numbers, HubSpot was able to project that using variant B would lead to about 375 more sign ups each month.

2. FSAstore.com’s Site Navigation

Every marketer will have to focus on conversion at some point. But building a website that converts is tough.

FSAstore.com is an ecommerce company supplying home goods for Americans with a flexible spending account.

This useful site could help the 35 million+ customers that have an FSA. But the website funnel was overwhelming. It had too many options, especially on category pages. The team felt that customers weren't making purchases because of that issue.

To figure out how to appeal to its customers, this company tested a simplified version of its website. The current site included an information-packed subheader in the site navigation.

To test the hypothesis, this A/B testing example compared the current site to an update without the subheader.

This update showed a clear boost in conversions and FSAstore.com saw a 53.8% increase in revenue per visitor.

3. Expoze’s Web Page Background

The visuals on your web page are important because they help users decide whether they want to spend more time on your site.

In this A/B testing example, Expoze.io decided to test the background on its homepage.

The website home page was difficult for some users to read because of low contrast. The team also needed to figure out how to improve page navigation while still representing the brand.

First, the team did some research and created several different designs. The goals of the redesign were to improve the visuals and increase attention to specific sections of the home page, like the video thumbnail.

They used AI-generated eye tracking as they designed to find the best designs before A/B testing. Then they ran an A/B heatmap test to see whether the new or current design got the most attention from visitors.

The new design showed a big increase in attention, with version B bringing over 40% more attention to the desired sections of the home page.

This design change also brought a 25% increase in CTA clicks. The team believes this is due to the added contrast on the page bringing more attention to the CTA button, which was not changed.

4. Thrive Themes’ Sales Page Optimization

Many landing pages showcase testimonials. That's valuable content and it can boost conversion.

That's why Thrive Themes decided to test a new feature on its landing pages — customer testimonials .

In the control, Thrive Themes had been using a banner that highlighted product features, but not how customers felt about the product.

The team decided to test whether adding testimonials to a sales landing page could improve conversion rates.

In this A/B test example, the team ran a 6-week test with the control against an updated landing page with testimonials.

This change netted a 13% increase in sales. The control page had a 2.2% conversion rate, but the new variant showed a 2.75% conversion rate.

Email A/B Testing Examples

5. hubspot's email subscriber experience.

Getting users to engage with email isn't an easy task. That's why HubSpot decided to A/B test how alignment impacts CTA clicks.

HubSpot decided to change text alignment in the weekly emails for subscribers to improve the user experience. Ideally, this improved experience would result in a higher click rate.

For the control, HubSpot sent centered email text to users.

For variant B, HubSpot sent emails with left-justified text.

HubSpot found that emails with left-aligned text got fewer clicks than the control. And of the total left-justified emails sent, less than 25% got more clicks than the control.

6. Neurogan’s Deal Promotion

Making the most of email promotion is important for any company, especially those in competitive industries.

This example uses the power of current customers for increasing email engagement.

Neurogan wasn't always offering the right content to its audience and it was having a hard time competing with a flood of other new brands.

An email agency audited this brand's email marketing, then focused efforts on segmentation. This A/B testing example starts with creating product-specific offers. Then, this team used testing to figure out which deals were best for each audience.

These changes brought higher revenue for promotions and higher click rates. It also led to a new workflow with a 37% average open rate and a click rate of 3.85%.

For more on how to run A/B testing for your campaigns, check out this free A/B testing kit .

Social Media A/B Testing Examples

7. vestiaire’s tiktok awareness campaign.

A/B testing examples like the one below can help you think creatively about what to test and when. This is extra helpful if your business is working with influencers and doesn't want to impact their process while working toward business goals.

Fashion brand Vestaire wanted help growing the brand on TikTok. It was also hoping to increase awareness with Gen Z audiences for its new direct shopping feature.

Vestaire's influencer marketing agency asked eight influencers to create content with specific CTAs to meet the brand's goals. Each influencer had extensive creative freedom and created a range of different social media posts.

Then, the agency used A/B testing to choose the best-performing content and promoted this content with paid advertising .

This testing example generated over 4,000 installs. It also decreased the cost per install by 50% compared to the brand's existing presence on Instagram and YouTube.

8. Underoutfit’s Promotion of User-Generated Content on Facebook

Paid advertising is getting more expensive, and clickthrough rates decreased through the end of 2022 .

To make the most of social ad spend, marketers are using A/B testing to improve ad performance. This approach helps them test creative content before launching paid ad campaigns, like in the examples below.

Underoutfit wanted to increase brand awareness on Facebook.

To meet this goal, it decided to try adding branded user-generated content. This brand worked with an agency and several creators to create branded content to drive conversion.

Then, Underoutfit ran split testing between product ads and the same ads combined with the new branded content ads. Both groups in the split test contained key marketing messages and clear CTA copy.

The brand and agency also worked with Meta Creative Shop to make sure the videos met best practice standards.

The test showed impressive results for the branded content variant, including a 47% higher clickthrough rate and 28% higher return on ad spend.

9. Databricks’ Ad Performance on LinkedIn

Pivoting to a new strategy quickly can be difficult for organizations. This A/B testing example shows how you can use split testing to figure out the best new approach to a problem.

Databricks , a cloud software tool, needed to raise awareness for an event that was shifting from in-person to online .

To connect with a large group of new people in a personalized way, the team decided to create a LinkedIn Message Ads campaign. To make sure the messages were effective, it used A/B testing to tweak the subject line and message copy.

The third variant of the copy featured a hyperlink in the first sentence of the invitation. Compared to the other two variants, this version got nearly twice as many clicks and conversions.

Mobile A/B Testing Example

7. hubspot's mobile calls-to-action.

On this blog, you'll notice anchor text in the introduction, a graphic CTA at the bottom, and a slide-in CTA when you scroll through the post. Once you click on one of these offers, you'll land on a content offer page.

While many users access these offers from a desktop or laptop computer, many others plan to download these offers to mobile devices.

But on mobile, users weren't finding the CTA buttons as quickly as they could on a computer. That's why HubSpot tested mobile design changes to improve the user experience.

Previous A/B tests revealed that HubSpot's mobile audience was 27% less likely to click through to download an offer. Also, less than 75% of mobile users were scrolling down far enough to see the CTA button.

So, HubSpot decided to test different versions of the offer page CTA, using conversion rate (CVR) as the primary metric. For secondary metrics, the team measured CTA clicks for each CTA, as well as engagement.

HubSpot used four variants for this test.

For variant A, the control, the traditional placement of CTAs remained unchanged.

For variant B, the team redesigned the hero image and added a sticky CTA bar.

For variant C, the redesigned hero was the only change.

For variant D, the team redesigned the hero image and repositioned the slider.

All variants outperformed the control for the primary metric, CVR. Variant C saw a 10% increase, variant B saw a 9% increase, and variant D saw an 8% increase.

From those numbers, HubSpot was able to project that using variant C on mobile would lead to about 1,400 more content leads and almost 5,700 more form submissions each month.

11. Hospitality.net’s Mobile Booking

Businesses need to keep up with quick shifts in mobile devices to create a consistently strong customer experience.

A/B testing examples like the one below can help your business streamline this process.

Hospitality.net offered both simplified and dynamic mobile booking experiences. The simplified experience showed a limited number of available dates and the design is for smaller screens. The dynamic experience is for the larger mobile device screens. It shows a wider range of dates and prices.

But the brand wasn’t sure which mobile optimization strategy would be better for conversion.

This brand believed that customers would prefer the dynamic experience and that it would get more conversions. But it chose to test these ideas with a simple A/B test. Over 34 days, it sent half of the mobile visitors to the simplified mobile experience, and half to the dynamic experience, with over 100,000 visitors total.

This A/B testing example showed a 33% improvement in conversion. It also helped confirm the brand's educated guesses about mobile booking preferences.

A/B Testing Takeaways for Marketers

A lot of different factors can go into A/B testing, depending on your business needs. But there are a few key things to keep in mind:

• Every A/B test should start with a hypothesis focused on one specific problem that you can test.
• Make sure you’re testing a control variable (your original version) and a treatment variable (a new version that you think will perform better).
• You can test various things, like landing pages, CTAs, emails, or mobile app designs.
• The best way to understand if your results mean something is to figure out the statistical significance of your test.
• There are a variety of goals to focus on for A/B testing (increased site traffic, lower bounce rates, etc.), but you should be able to test, support, prove, and disprove your hypothesis.
• When testing, make sure you’re splitting your sample groups equally and randomly, so your data is viable and not due to chance.
• Take action based on the results you observe.

Start Your Next A/B Test Today

You can see amazing results from the A/B testing examples above. These businesses were able to take action on goals because they started testing. If you want to get great results, you've got to get started, too.

Editor's note: This post was originally published in October 2014 and has been updated for comprehensiveness.

Don't forget to share this post!

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A/B Testing: Example of a good hypothesis

Want to know the secret to always running successful tests?

The answer is to formulate a hypothesis .

Now when I say it’s always successful, I’m not talking about always increasing your Key Performance Indicator (KPI). You can “lose” a test, but still be successful.

That sounds like an oxymoron, but it’s not. If you set up your test strategically, even if the test decreases your KPI, you gain a learning , which is a success! And, if you win, you simultaneously achieve a lift and a learning. Double win!

The way you ensure you have a strategic test that will produce a learning is by centering it around a strong hypothesis.

So, what is a hypothesis?

By definition, a hypothesis is a proposed statement made on the basis of limited evidence that can be proved or disproved and is used as a starting point for further investigation.

Let’s break that down:

It is a proposed statement.

• A hypothesis is not fact, and should not be argued as right or wrong until it is tested and proven one way or the other.

It is made on the basis of limited (but hopefully some ) evidence.

• Your hypothesis should be informed by as much knowledge as you have. This should include data that you have gathered, any research you have done, and the analysis of the current problems you have performed.

It can be proved or disproved.

• A hypothesis pretty much says, “I think by making this change , it will cause this effect .” So, based on your results, you should be able to say “this is true” or “this is false.”

It is used as a starting point for further investigation.

• The key word here is starting point . Your hypothesis should be formed and agreed upon before you make any wireframes or designs as it is what guides the design of your test. It helps you focus on what elements to change, how to change them, and which to leave alone.

How do I write a hypothesis?

The structure of your basic hypothesis follows a CHANGE: EFFECT framework.

While this is a truly scientific and testable template, it is very open-ended. Even though this hypothesis, “Changing an English headline into a Spanish headline will increase clickthrough rate,” is perfectly valid and testable, if your visitors are English-speaking, it probably doesn’t make much sense.

So now the question is …

How do I write a GOOD hypothesis?

To quote my boss Tony Doty , “This isn’t Mad Libs.”

We can’t just start plugging in nouns and verbs and conclude that we have a good hypothesis. Your hypothesis needs to be backed by a strategy. And, your strategy needs to be rooted in a solution to a problem .

So, a more complete version of the above template would be something like this:

In order to have a good hypothesis, you don’t necessarily have to follow this exact sentence structure, as long as it is centered around three main things:

Presumed problem

Proposed solution

Anticipated result

After you’ve completed your analysis and research, identify the problem that you will address. While we need to be very clear about what we think the problem is, you should leave it out of the hypothesis since it is harder to prove or disprove. You may want to come up with both a problem statement and a hypothesis .

For example:

Problem Statement: “The lead generation form is too long, causing unnecessary friction .”

Hypothesis: “By changing the amount of form fields from 20 to 10, we will increase number of leads.”

When you are thinking about the solution you want to implement, you need to think about the psychology of the customer. What psychological impact is your proposed problem causing in the mind of the customer?

For example, if your proposed problem is “There is a lack of clarity in the sign-up process,” the psychological impact may be that the user is confused.

Now think about what solution is going to address the problem in the customer’s mind. If they are confused, we need to explain something better, or provide them with more information. For this example, we will say our proposed solution is to “Add a progress bar to the sign-up process.”  This leads straight into the anticipated result.

If we reduce the confusion in the visitor’s mind (psychological impact) by adding the progress bar, what do we foresee to be the result? We are anticipating that it would be more people completing the sign-up process. Your proposed solution and your KPI need to be directly correlated.

Note: Some people will include the psychological impact in their hypothesis. This isn’t necessarily wrong, but we do have to be careful with assumptions. If we say that the effect will be “Reduced confusion and therefore increase in conversion rate,” we are assuming the reduced confusion is what made the impact. While this may be correct, it is not measureable and it is hard to prove or disprove.

To summarize, your hypothesis should follow a structure of: “If I change this, it will have this effect,” but should always be informed by an analysis of the problems and rooted in the solution you deemed appropriate.

Related Resources:

A/B Testing 101: How to get real results from optimization

The True Value of Data

15 Years of Marketing Research in 11 Minutes

Marketing Analytics: 6 simple steps for interpreting your data

Website A/B Testing: 4 tips to beat an unbeatable landing page

Online Cart: 6 ideas to test and optimize your checkout process

B2B Gamification: Autodesk’s two approaches to in-trial marketing [Video]

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The 21 Psychological Elements that Power Effective Web Design (Part 3)

The 21 Psychological Elements that Power Effective Web Design (Part 2)

The 21 Psychological Elements that Power Effective Web Design (Part 1)

Thanks for the article. I’ve been trying to wrap my head around this type of testing because I’d like to use it to see the effectiveness on some ads. This article really helped. Thanks Again!

Hey Lauren, I am just getting to the point that I have something to perform A-B testing on. This post led me to this site which will and already has become a help in what to test and how to test .

Again, thanks for getting me here .

Good article. I have been researching different approaches to writing testing hypotheses and this has been a help. The only thing I would add is that it can be useful to capture the insight/justification within the hypothesis statement. IF i do this, THEN I expect this result BECAUSE I have this insight.

@Kaya Great!

Good article – but technically you can never prove an hypothesis, according to the principle of falsification (Popper), only fail to disprove the null hypothesis.

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How to Write a Great Hypothesis

Hypothesis Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis, operational definitions, types of hypotheses, hypotheses examples.

• Collecting Data

Frequently Asked Questions

A hypothesis is a tentative statement about the relationship between two or more  variables. It is a specific, testable prediction about what you expect to happen in a study.

One hypothesis example would be a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. It is only at this point that researchers begin to develop a testable hypothesis. Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore a number of factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk wisdom that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis.   In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in a number of different ways. One of the basic principles of any type of scientific research is that the results must be replicable.   By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. How would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

In order to measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming other people. In this situation, the researcher might utilize a simulated task to measure aggressiveness.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative sample of the population and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• Complex hypothesis: "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "Children who receive a new reading intervention will have scores different than students who do not receive the intervention."
• "There will be no difference in scores on a memory recall task between children and adults."

Examples of an alternative hypothesis:

• "Children who receive a new reading intervention will perform better than students who did not receive the intervention."
• "Adults will perform better on a memory task than children." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when it would be impossible or difficult to  conduct an experiment . These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a correlational study can then be used to look at how the variables are related. This type of research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

A Word From Verywell

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

• "Staying up late will lead to worse test performance the next day."
• "People who consume one apple each day will visit the doctor fewer times each year."
• "Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

• The research question
• The independent variable (IV)
• The dependent variable (DV)
• The proposed relationship between the IV and DV

Castillo M. The scientific method: a need for something better? . AJNR Am J Neuroradiol. 2013;34(9):1669-71. doi:10.3174/ajnr.A3401

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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The ‘Hypothesis-Driven’ Marketing Strategy

The best way to have a good idea is to have a lot of ideas. Continually test and refine a network of hypotheses to keep marketing relevant.

There were times when a good idea now, would still be a good idea next year. You could confidently plan ahead marketing activities for months, and marketing plans lasted an entire year. There was a “BIG idea” approach to marketing strategy, developed by top marketers with almost supernatural powers. It was based on the opinion of people who knew it all.

Now, fast forward 20 years later, it seems even the present is going… ehm, fast forward. Big marketing plans become outdated before execution starts . In 2016, most static marketing documents become useless pretty fast.

The traditional BIG-idea-marketing-plan becomes less relevant every month from its creation onward. You’d wish it was just ‘useless’, but in can be even worse. When you’re chained to activities you agreed upon 10 months ago that now even counteract your cause, you’d prefer not having any marketing plan over a binding outdated plan.

Wouldn’t it be great if your marketing strategy would actually become móre valuable month-over-month? That’s the whole point of investing right?

[bctt tweet="Make your marketing plan more valuable month-over-month. Make it hypothesis-driven!" username="Boardview_io"]

To keep your marketing strategy aligned and relevant throughout the year, you have to take a radically different approach. Let me introduce you to the network of hypotheses that is your new agile marketing plan .

The traditional versus the ‘hypothesis-driven’ marketing plan

When we were younger, there was this thing called the ‘market research’ phase. Surely, there are some situations left when extensive research yields useful information, but in today’s complex and volatile environments there is often no time for elaborate studies.

There is no time for defining marketing challenges properly and there is no time for listing a generous collection of solutions to be carefully considered by a panel of ‘yesterday’s experts’ that gather once a month to drink wine, eat French cheese and do a bit of marketing. And who then bet the entire budget on a single horse.

When environments are so volatile that new challenges pop up and new information becomes available before you can make sense of yesterday’s best practices: go hard and go early. Leave the competition to figure out how you pulled ahead, in a 6-month long investigation.

Nowadays research is not a phase in marketing. Today, marketing = research. It is about continuously adjusting activities to find the perfect product-market fit and associated proposition. Select the most promising hypothetical goal or solution early on and attempt to refute or confirm it. Ship early, get your products out the door and see what sticks.

Or as leading strategy professor Liedtka puts it:

“In an environment of ever-increasing information availability and decreasing time to think, the ability to develop good hypotheses and test them efficiently is critical.”

Create a network of hypotheses while keeping an overview

Moving with the times and being agile is not a game of “who can plow through chaos the fastest”. It is not about doing more phone calls or patching up a host of confusing problems quicker than the competition. Agile marketing, when you do it right, involves a lot of strategizing.

In essence, the agile marketing plan is a network made up of hypotheses about which goals to pursue to take the company from the present to the desired future. What we call a ‘goal tree’ is a visualization of which commercial results are expected to be required for the achievement of higher goals.

[bctt tweet="An agile marketing plan is a continuously updating network of strategic hypotheses." username="Boardview_io"]

Any result achieved for any of the sub goals should have an influence on their respective parent goal. Think of it as a provisional, dynamic but well aligned value chain of goals that ends with marketing success.

The agile marketing strategy does not include descriptions of tasks to be performed. Those will be decided upon along the way, based on the latest input.

Continuously update the network after hypothesis confirmation or refute

When you’re done outlining the first version of your network of hypotheses in a goal tree: go live! Your hypothesis driven marketing strategy that will undergo a theoretically infinite number of iterations along the way based on new insights acquired.

Now is the time consumers can judge your performance. Continuously monitor how your customers like what you did. Your hypotheses about what goes on in the complex and fragile mind of the consumer may well be refuted, but when you’re working with an agile marketing plan, that’s nothing to worry about.

You use information the consumer has given you to your own advantage and formulate another, more suitable, hypothesis to be tested in the next sprint. You iterate to improve goals and their alignment in the ‘goal tree’ to get ever closer to your big goal. In a hypothesis driven strategy you replace the big plan with the big goal.

However, you may never reach your big goal. Just as you are about to reach it, consumers are likely to change their behavior. But you will be light on your feet, conscious of where they went and ready to chase them down much quicker than the competition.

You can never really win at marketing, there is no referee blowing a whistle to say the game has finished, but with an agile marketing plan, my hypothesis is that you will be ahead of every other marketing organization out there. Which makes you a winner in my book.

Ready to give it a go? Try Boardview for free .

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• Product Management

How to Generate and Validate Product Hypotheses

What is a product hypothesis.

A hypothesis is a testable statement that predicts the relationship between two or more variables. In product development, we generate hypotheses to validate assumptions about customer behavior, market needs, or the potential impact of product changes. These experimental efforts help us refine the user experience and get closer to finding a product-market fit.

Product hypotheses are a key element of data-driven product development and decision-making. Testing them enables us to solve problems more efficiently and remove our own biases from the solutions we put forward.

Here’s an example: ‘If we improve the page load speed on our website (variable 1), then we will increase the number of signups by 15% (variable 2).’ So if we improve the page load speed, and the number of signups increases, then our hypothesis has been proven. If the number did not increase significantly (or not at all), then our hypothesis has been disproven.

In general, product managers are constantly creating and testing hypotheses. But in the context of new product development , hypothesis generation/testing occurs during the validation stage, right after idea screening .

Now before we go any further, let’s get one thing straight: What’s the difference between an idea and a hypothesis?

Idea vs hypothesis

Innovation expert Michael Schrage makes this distinction between hypotheses and ideas – unlike an idea, a hypothesis comes with built-in accountability. “But what’s the accountability for a good idea?” Schrage asks. “The fact that a lot of people think it’s a good idea? That’s a popularity contest.” So, not only should a hypothesis be tested, but by its very nature, it can be tested.

At Railsware, we’ve built our product development services on the careful selection, prioritization, and validation of ideas. Here’s how we distinguish between ideas and hypotheses:

Idea: A creative suggestion about how we might exploit a gap in the market, add value to an existing product, or bring attention to our product. Crucially, an idea is just a thought. It can form the basis of a hypothesis but it is not necessarily expected to be proven or disproven.

• We should get an interview with the CEO of our company published on TechCrunch.
• Why don’t we redesign our website?
• The Coupler.io team should create video tutorials on how to export data from different apps, and publish them on YouTube.
• Why not add a new ‘email templates’ feature to our Mailtrap product?

Hypothesis: A way of framing an idea or assumption so that it is testable, specific, and aligns with our wider product/team/organizational goals.

Examples: 

• If we add a new ‘email templates’ feature to Mailtrap, we’ll see an increase in active usage of our email-sending API.
• Creating relevant video tutorials and uploading them to YouTube will lead to an increase in Coupler.io signups.
• If we publish an interview with our CEO on TechCrunch, 500 people will visit our website and 10 of them will install our product.

Now, it’s worth mentioning that not all hypotheses require testing . Sometimes, the process of creating hypotheses is just an exercise in critical thinking. And the simple act of analyzing your statement tells whether you should run an experiment or not. Remember: testing isn’t mandatory, but your hypotheses should always be inherently testable.

Let’s consider the TechCrunch article example again. In that hypothesis, we expect 500 readers to visit our product website, and a 2% conversion rate of those unique visitors to product users i.e. 10 people. But is that marginal increase worth all the effort? Conducting an interview with our CEO, creating the content, and collaborating with the TechCrunch content team – all of these tasks take time (and money) to execute. And by formulating that hypothesis, we can clearly see that in this case, the drawbacks (efforts) outweigh the benefits. So, no need to test it.

In a similar vein, a hypothesis statement can be a tool to prioritize your activities based on impact. We typically use the following criteria:

• The quality of impact
• The size of the impact
• The probability of impact

This lets us organize our efforts according to their potential outcomes – not the coolness of the idea, its popularity among the team, etc.

Now that we’ve established what a product hypothesis is, let’s discuss how to create one.

Start with a problem statement

Before you jump into product hypothesis generation, we highly recommend formulating a problem statement. This is a short, concise description of the issue you are trying to solve. It helps teams stay on track as they formalize the hypothesis and design the product experiments. It can also be shared with stakeholders to ensure that everyone is on the same page.

The statement can be worded however you like, as long as it’s actionable, specific, and based on data-driven insights or research. It should clearly outline the problem or opportunity you want to address.

Here’s an example: Our bounce rate is high (more than 90%) and we are struggling to convert website visitors into actual users. How might we improve site performance to boost our conversion rate?

How to generate product hypotheses

Now let’s explore some common, everyday scenarios that lead to product hypothesis generation. For our teams here at Railsware, it’s when:

• There’s a problem with an unclear root cause e.g. a sudden drop in one part of the onboarding funnel. We identify these issues by checking our product metrics or reviewing customer complaints.
• We are running ideation sessions on how to reach our goals (increase MRR, increase the number of users invited to an account, etc.)
• We are exploring growth opportunities e.g. changing a pricing plan, making product improvements , breaking into a new market.
• We receive customer feedback. For example, some users have complained about difficulties setting up a workspace within the product. So, we build a hypothesis on how to help them with the setup.

BRIDGES framework for ideation

When we are tackling a complex problem or looking for ways to grow the product, our teams use BRIDGeS – a robust decision-making and ideation framework. BRIDGeS makes our product discovery sessions more efficient. It lets us dive deep into the context of our problem so that we can develop targeted solutions worthy of testing.

Between 2-8 stakeholders take part in a BRIDGeS session. The ideation sessions are usually led by a product manager and can include other subject matter experts such as developers, designers, data analysts, or marketing specialists. You can use a virtual whiteboard such as Figjam or Miro (see our Figma template ) to record each colored note.

In the first half of a BRIDGeS session, participants examine the Benefits, Risks, Issues, and Goals of their subject in the ‘Problem Space.’ A subject is anything that is being described or dealt with; for instance, Coupler.io’s growth opportunities. Benefits are the value that a future solution can bring, Risks are potential issues they might face, Issues are their existing problems, and Goals are what the subject hopes to gain from the future solution. Each descriptor should have a designated color.

After we have broken down the problem using each of these descriptors, we move into the Solution Space. This is where we develop solution variations based on all of the benefits/risks/issues identified in the Problem Space (see the Uber case study for an in-depth example).

In the Solution Space, we start prioritizing those solutions and deciding which ones are worthy of further exploration outside of the framework – via product hypothesis formulation and testing, for example. At the very least, after the session, we will have a list of epics and nested tasks ready to add to our product roadmap.

How to write a product hypothesis statement

Across organizations, product hypothesis statements might vary in their subject, tone, and precise wording. But some elements never change. As we mentioned earlier, a hypothesis statement must always have two or more variables and a connecting factor.

1. Identify variables

Since these components form the bulk of a hypothesis statement, let’s start with a brief definition.

First of all, variables in a hypothesis statement can be split into two camps: dependent and independent. Without getting too theoretical, we can describe the independent variable as the cause, and the dependent variable as the effect . So in the Mailtrap example we mentioned earlier, the ‘add email templates feature’ is the cause i.e. the element we want to manipulate. Meanwhile, ‘increased usage of email sending API’ is the effect i.e the element we will observe.

Independent variables can be any change you plan to make to your product. For example, tweaking some landing page copy, adding a chatbot to the homepage, or enhancing the search bar filter functionality.

Dependent variables are usually metrics. Here are a few that we often test in product development:

• Number of sign-ups
• Number of purchases
• Activation rate (activation signals differ from product to product)
• Number of specific plans purchased
• Feature usage (API activation, for example)
• Number of active users

Bear in mind that your concept or desired change can be measured with different metrics. Make sure that your variables are well-defined, and be deliberate in how you measure your concepts so that there’s no room for misinterpretation or ambiguity.

For example, in the hypothesis ‘Users drop off because they find it hard to set up a project’ variables are poorly defined. Phrases like ‘drop off’ and ‘hard to set up’ are too vague. A much better way of saying it would be: If project automation rules are pre-defined (email sequence to responsible, scheduled tickets creation), we’ll see a decrease in churn. In this example, it’s clear which dependent variable has been chosen and why.

And remember, when product managers focus on delighting users and building something of value, it’s easier to market and monetize it. That’s why at Railsware, our product hypotheses often focus on how to increase the usage of a feature or product. If users love our product(s) and know how to leverage its benefits, we can spend less time worrying about how to improve conversion rates or actively grow our revenue, and more time enhancing the user experience and nurturing our audience.

2. Make the connection

The relationship between variables should be clear and logical. If it’s not, then it doesn’t matter how well-chosen your variables are – your test results won’t be reliable.

To demonstrate this point, let’s explore a previous example again: page load speed and signups.

Through prior research, you might already know that conversion rates are 3x higher for sites that load in 1 second compared to sites that take 5 seconds to load. Since there appears to be a strong connection between load speed and signups in general, you might want to see if this is also true for your product.

Here are some common pitfalls to avoid when defining the relationship between two or more variables:

Relationship is weak. Let’s say you hypothesize that an increase in website traffic will lead to an increase in sign-ups. This is a weak connection since website visitors aren’t necessarily motivated to use your product; there are more steps involved. A better example is ‘If we change the CTA on the pricing page, then the number of signups will increase.’ This connection is much stronger and more direct.

Relationship is far-fetched. This often happens when one of the variables is founded on a vanity metric. For example, increasing the number of social media subscribers will lead to an increase in sign-ups. However, there’s no particular reason why a social media follower would be interested in using your product. Oftentimes, it’s simply your social media content that appeals to them (and your audience isn’t interested in a product).

Variables are co-dependent. Variables should always be isolated from one another. Let’s say we removed the option “Register with Google” from our app. In this case, we can expect fewer users with Google workspace accounts to register. Obviously, it’s because there’s a direct dependency between variables (no registration with Google→no users with Google workspace accounts).

3. Set validation criteria

First, build some confirmation criteria into your statement . Think in terms of percentages (e.g. increase/decrease by 5%) and choose a relevant product metric to track e.g. activation rate if your hypothesis relates to onboarding. Consider that you don’t always have to hit the bullseye for your hypothesis to be considered valid. Perhaps a 3% increase is just as acceptable as a 5% one. And it still proves that a connection between your variables exists.

Secondly, you should also make sure that your hypothesis statement is realistic . Let’s say you have a hypothesis that ‘If we show users a banner with our new feature, then feature usage will increase by 10%.’ A few questions to ask yourself are: Is 10% a reasonable increase, based on your current feature usage data? Do you have the resources to create the tests (experimenting with multiple variations, distributing on different channels: in-app, emails, blog posts)?

Null hypothesis and alternative hypothesis

In statistical research, there are two ways of stating a hypothesis: null or alternative. But this scientific method has its place in hypothesis-driven development too…

Alternative hypothesis: A statement that you intend to prove as being true by running an experiment and analyzing the results. Hint: it’s the same as the other hypothesis examples we’ve described so far.

Example: If we change the landing page copy, then the number of signups will increase.

Null hypothesis: A statement you want to disprove by running an experiment and analyzing the results. It predicts that your new feature or change to the user experience will not have the desired effect.

Example: The number of signups will not increase if we make a change to the landing page copy.

What’s the point? Well, let’s consider the phrase ‘innocent until proven guilty’ as a version of a null hypothesis. We don’t assume that there is any relationship between the ‘defendant’ and the ‘crime’ until we have proof. So, we run a test, gather data, and analyze our findings — which gives us enough proof to reject the null hypothesis and validate the alternative. All of this helps us to have more confidence in our results.

Now that you have generated your hypotheses, and created statements, it’s time to prepare your list for testing.

Prioritizing hypotheses for testing

Not all hypotheses are created equal. Some will be essential to your immediate goal of growing the product e.g. adding a new data destination for Coupler.io. Others will be based on nice-to-haves or small fixes e.g. updating graphics on the website homepage.

Prioritization helps us focus on the most impactful solutions as we are building a product roadmap or narrowing down the backlog . To determine which hypotheses are the most critical, we use the MoSCoW framework. It allows us to assign a level of urgency and importance to each product hypothesis so we can filter the best 3-5 for testing.

MoSCoW is an acronym for Must-have, Should-have, Could-have, and Won’t-have. Here’s a breakdown:

• Must-have – hypotheses that must be tested, because they are strongly linked to our immediate project goals.
• Should-have – hypotheses that are closely related to our immediate project goals, but aren’t the top priority.
• Could-have – hypotheses of nice-to-haves that can wait until later for testing. 
• Won’t-have – low-priority hypotheses that we may or may not test later on when we have more time.

How to test product hypotheses

Once you have selected a hypothesis, it’s time to test it. This will involve running one or more product experiments in order to check the validity of your claim.

The tricky part is deciding what type of experiment to run, and how many. Ultimately, this all depends on the subject of your hypothesis – whether it’s a simple copy change or a whole new feature. For instance, it’s not necessary to create a clickable prototype for a landing page redesign. In that case, a user-wide update would do.

On that note, here are some of the approaches we take to hypothesis testing at Railsware:

A/B testing

A/B or split testing involves creating two or more different versions of a webpage/feature/functionality and collecting information about how users respond to them.

Let’s say you wanted to validate a hypothesis about the placement of a search bar on your application homepage. You could design an A/B test that shows two different versions of that search bar’s placement to your users (who have been split equally into two camps: a control group and a variant group). Then, you would choose the best option based on user data. A/B tests are suitable for testing responses to user experience changes, especially if you have more than one solution to test.

Prototyping

When it comes to testing a new product design, prototyping is the method of choice for many Lean startups and organizations. It’s a cost-effective way of collecting feedback from users, fast, and it’s possible to create prototypes of individual features too. You may take this approach to hypothesis testing if you are working on rolling out a significant new change e.g adding a brand-new feature, redesigning some aspect of the user flow, etc. To control costs at this point in the new product development process , choose the right tools — think Figma for clickable walkthroughs or no-code platforms like Bubble.

Deliveroo feature prototype example

Let’s look at how feature prototyping worked for the food delivery app, Deliveroo, when their product team wanted to ‘explore personalized recommendations, better filtering and improved search’ in 2018. To begin, they created a prototype of the customer discovery feature using web design application, Framer.

One of the most important aspects of this feature prototype was that it contained live data — real restaurants, real locations. For test users, this made the hypothetical feature feel more authentic. They were seeing listings and recommendations for real restaurants in their area, which helped immerse them in the user experience, and generate more honest and specific feedback. Deliveroo was then able to implement this feedback in subsequent iterations.

Asking your users

Interviewing customers is an excellent way to validate product hypotheses. It’s a form of qualitative testing that, in our experience, produces better insights than user surveys or general user research. Sessions are typically run by product managers and involve interviewing one customer at a time. They can be conducted in person or online (through a virtual call center , for instance) and last anywhere between 30 minutes to 1 hour.

Although CustDev interviews may require more effort to execute than other tests (the process of finding participants, devising questions, organizing interviews, and honing interview skills can be time-consuming), it’s still a highly rewarding approach. You can quickly validate assumptions by asking customers about their pain points, concerns, habits, processes they follow, and analyzing how your solution fits into all of that.

Wizard of Oz

The Wizard of Oz approach is suitable for gauging user interest in new features or functionalities. It’s done by creating a prototype of a fake or future feature and monitoring how your customers or test users interact with it.

For example, you might have a hypothesis that your number of active users will increase by 15% if you introduce a new feature. So, you design a new bare-bones page or simple button that invites users to access it. But when they click on the button, a pop-up appears with a message such as ‘coming soon.’

By measuring the frequency of those clicks, you could learn a lot about the demand for this new feature/functionality. However, while these tests can deliver fast results, they carry the risk of backfiring. Some customers may find fake features misleading, making them less likely to engage with your product in the future.

User-wide updates

One of the speediest ways to test your hypothesis is by rolling out an update for all users. It can take less time and effort to set up than other tests (depending on how big of an update it is). But due to the risk involved, you should stick to only performing these kinds of tests on small-scale hypotheses. Our teams only take this approach when we are almost certain that our hypothesis is valid.

For example, we once had an assumption that the name of one of Mailtrap ’s entities was the root cause of a low activation rate. Being an active Mailtrap customer meant that you were regularly sending test emails to a place called ‘Demo Inbox.’ We hypothesized that the name was confusing (the word ‘demo’ implied it was not the main inbox) and this was preventing new users from engaging with their accounts. So, we updated the page, changed the name to ‘My Inbox’ and added some ‘to-do’ steps for new users. We saw an increase in our activation rate almost immediately, validating our hypothesis.

Feature flags

Creating feature flags involves only releasing a new feature to a particular subset or small percentage of users. These features come with a built-in kill switch; a piece of code that can be executed or skipped, depending on who’s interacting with your product.

Since you are only showing this new feature to a selected group, feature flags are an especially low-risk method of testing your product hypothesis (compared to Wizard of Oz, for example, where you have much less control). However, they are also a little bit more complex to execute than the others — you will need to have an actual coded product for starters, as well as some technical knowledge, in order to add the modifiers ( only when… ) to your new coded feature.

Let’s revisit the landing page copy example again, this time in the context of testing.

So, for the hypothesis ‘If we change the landing page copy, then the number of signups will increase,’ there are several options for experimentation. We could share the copy with a small sample of our users, or even release a user-wide update. But A/B testing is probably the best fit for this task. Depending on our budget and goal, we could test several different pieces of copy, such as:

• The current landing page copy
• Copy that we paid a marketing agency 10 grand for
• Generic copy we wrote ourselves, or removing most of the original copy – just to see how making even a small change might affect our numbers.

Remember, every hypothesis test must have a reasonable endpoint. The exact length of the test will depend on the type of feature/functionality you are testing, the size of your user base, and how much data you need to gather. Just make sure that the experiment running time matches the hypothesis scope. For instance, there is no need to spend 8 weeks experimenting with a piece of landing page copy. That timeline is more appropriate for say, a Wizard of Oz feature.

Recording hypotheses statements and test results

Finally, it’s time to talk about where you will write down and keep track of your hypotheses. Creating a single source of truth will enable you to track all aspects of hypothesis generation and testing with ease.

At Railsware, our product managers create a document for each individual hypothesis, using tools such as Coda or Google Sheets. In that document, we record the hypothesis statement, as well as our plans, process, results, screenshots, product metrics, and assumptions.

We share this document with our team and stakeholders, to ensure transparency and invite feedback. It’s also a resource we can refer back to when we are discussing a new hypothesis — a place where we can quickly access information relating to a previous test.

Understanding test results and taking action

The other half of validating product hypotheses involves evaluating data and drawing reasonable conclusions based on what you find. We do so by analyzing our chosen product metric(s) and deciding whether there is enough data available to make a solid decision. If not, we may extend the test’s duration or run another one. Otherwise, we move forward. An experimental feature becomes a real feature, a chatbot gets implemented on the customer support page, and so on.

Something to keep in mind: the integrity of your data is tied to how well the test was executed, so here are a few points to consider when you are testing and analyzing results:

Gather and analyze data carefully. Ensure that your data is clean and up-to-date when running quantitative tests and tracking responses via analytics dashboards. If you are doing customer interviews, make sure to record the meetings (with consent) so that your notes will be as accurate as possible.

Conduct the right amount of product experiments. It can take more than one test to determine whether your hypothesis is valid or invalid. However, don’t waste too much time experimenting in the hopes of getting the result you want. Know when to accept the evidence and move on.

Choose the right audience segment. Don’t cast your net too wide. Be specific about who you want to collect data from prior to running the test. Otherwise, your test results will be misleading and you won’t learn anything new.

Watch out for bias. Avoid confirmation bias at all costs. Don’t make the mistake of including irrelevant data just because it bolsters your results. For example, if you are gathering data about how users are interacting with your product Monday-Friday, don’t include weekend data just because doing so would alter the data and ‘validate’ your hypothesis.

• Not all failed hypotheses should be treated as losses. Even if you didn’t get the outcome you were hoping for, you may still have improved your product. Let’s say you implemented SSO authentication for premium users, but unfortunately, your free users didn’t end up switching to premium plans. In this case, you still added value to the product by streamlining the login process for paying users.
• Yes, taking a hypothesis-driven approach to product development is important. But remember, you don’t have to test everything . Use common sense first. For example, if your website copy is confusing and doesn’t portray the value of the product, then you should still strive to replace it with better copy – regardless of how this affects your numbers in the short term.

Wrapping Up

The process of generating and validating product hypotheses is actually pretty straightforward once you’ve got the hang of it. All you need is a valid question or problem, a testable statement, and a method of validation. Sure, hypothesis-driven development requires more of a time commitment than just ‘giving it a go.’ But ultimately, it will help you tune the product to the wants and needs of your customers.

If you share our data-driven approach to product development and engineering, check out our services page to learn more about how we work with our clients!

Marketing Research Hypothesis Examples

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Marketing Research Design & Analysis 2019

5 hypothesis testing.

This chapter is primarily based on Field, A., Miles J., & Field, Z. (2012): Discovering Statistics Using R. Sage Publications, chapters 5, 9, 15, 18 .

You can download the corresponding R-Code here

5.1 Introduction

We test hypotheses because we are confined to taking samples – we rarely work with the entire population. In the previous chapter, we introduced the standard error (i.e., the standard deviation of a large number of hypothetical samples) as an estimate of how well a particular sample represents the population. We also saw how we can construct confidence intervals around the sample mean \(\bar x\) by computing \(SE_{\bar x}\) as an estimate of \(\sigma_{\bar x}\) using \(s\) as an estimate of \(\sigma\) and calculating the 95% CI as \(\bar x \pm 1.96 * SE_{\bar x}\) . Although we do not know the true population mean ( \(\mu\) ), we might have an hypothesis about it and this would tell us how the corresponding sampling distribution looks like. Based on the sampling distribution of the hypothesized population mean, we could then determine the probability of a given sample assuming that the hypothesis is true .

Let us again begin by assuming we know the entire population using the example of music listening times among students from the previous example. As a reminder, the following plot shows the distribution of music listening times in the population of WU students.

In this example, the population mean ( \(\mu\) ) is equal to 19.98, and the population standard deviation \(\sigma\) is equal to 14.15.

5.1.1 The null hypothesis

Let us assume that we were planning to take a random sample of 50 students from this population and our hypothesis was that the mean listening time is equal to some specific value \(\mu_0\) , say \(10\) . This would be our null hypothesis . The null hypothesis refers to the statement that is being tested and is usually a statement of the status quo, one of no difference or no effect. In our example, the null hypothesis would state that there is no difference between the true population mean \(\mu\) and the hypothesized value \(\mu_0\) (in our example \(10\) ), which can be expressed as follows:

\[ H_0: \mu = \mu_0 \] When conducting research, we are usually interested in providing evidence against the null hypothesis. If we then observe sufficient evidence against it and our estimate is said to be significant. If the null hypothesis is rejected, this is taken as support for the alternative hypothesis . The alternative hypothesis assumes that some difference exists, which can be expressed as follows:

\[ H_1: \mu \neq \mu_0 \] Accepting the alternative hypothesis in turn will often lead to changes in opinions or actions. Note that while the null hypothesis may be rejected, it can never be accepted based on a single test. If we fail to reject the null hypothesis, it means that we simply haven’t collected enough evidence against the null hypothesis to disprove it. In classical hypothesis testing, there is no way to determine whether the null hypothesis is true. Hypothesis testing provides a means to quantify to what extent the data from our sample is in line with the null hypothesis.

In order to quantify the concept of “sufficient evidence” we look at the theoretical distribution of the sample means given our null hypothesis and the sample standard error. Using the available information we can infer the sampling distribution for our null hypothesis. Recall that the standard deviation of the sampling distribution (i.e., the standard error of the mean) is given by \(\sigma_{\bar x}={\sigma \over \sqrt{n}}\) , and thus can be computed as follows:

Since we know from the central limit theorem that the sampling distribution is normal for large enough samples, we can now visualize the expected sampling distribution if our null hypothesis was in fact true (i.e., if the was no difference between the true population mean and the hypothesized mean of 10).

We also know that 95% of the probability is within 1.96 standard deviations from the mean. Values higher than that are rather unlikely, if our hypothesis about the population mean was indeed true. This is shown by the shaded area, also known as the “rejection region”. To test our hypothesis that the population mean is equal to \(10\) , let us take a random sample from the population.

The mean listening time in the sample (black line) \(\bar x\) is 18.59. We can already see from the graphic above that such a value is rather unlikely under the hypothesis that the population mean is \(10\) . Intuitively, such a result would therefore provide evidence against our null hypothesis. But how could we quantify specifically how unlikely it is to obtain such a value and decide whether or not to reject the null hypothesis? Significance tests can be used to provide answers to these questions.

5.1.2 Statistical inference on a sample

5.1.2.1 test statistic, 5.1.2.1.1 z-scores.

Let’s go back to the sampling distribution above. We know that 95% of all values will fall within 1.96 standard deviations from the mean. So if we could express the distance between our sample mean and the null hypothesis in terms of standard deviations, we could make statements about the probability of getting a sample mean of the observed magnitude (or more extreme values). Essentially, we would like to know how many standard deviations ( \(\sigma_{\bar x}\) ) our sample mean ( \(\bar x\) ) is away from the population mean if the null hypothesis was true ( \(\mu_0\) ). This can be formally expressed as follows:

\[ \bar x- \mu_0 = z \sigma_{\bar x} \]

In this equation, z will tell us how many standard deviations the sample mean \(\bar x\) is away from the null hypothesis \(\mu_0\) . Solving for z gives us:

\[ z = {\bar x- \mu_0 \over \sigma_{\bar x}}={\bar x- \mu_0 \over \sigma / \sqrt{n}} \]

This standardized value (or “z-score”) is also referred to as a test statistic . Let’s compute the test statistic for our example above:

To make a decision on whether the difference can be deemed statistically significant, we now need to compare this calculated test statistic to a meaningful threshold. In order to do so, we need to decide on a significance level \(\alpha\) , which expresses the probability of finding an effect that does not actually exist (i.e., Type I Error). You can find a detailed discussion of this point at the end of this chapter. For now, we will adopt the widely accepted significance level of 5% and set \(\alpha\) to 0.05. The critical value for the normal distribution and \(\alpha\) = 0.05 can be computed using the qnorm() function as follows:

We use 0.975 and not 0.95 since we are running a two-sided test and need to account for the rejection region at the other end of the distribution. Recall that for the normal distribution, 95% of the total probability falls within 1.96 standard deviations of the mean, so that higher (absolute) values provide evidence against the null hypothesis. Generally, we speak of a statistically significant effect if the (absolute) calculated test statistic is larger than the (absolute) critical value. We can easily check if this is the case in our example:

Since the absolute value of the calculated test statistic is larger than the critical value, we would reject \(H_0\) and conclude that the true population mean \(\mu\) is significantly different from the hypothesized value \(\mu_0 = 10\) .

5.1.2.1.2 t-statistic

You may have noticed that the formula for the z-score above assumes that we know the true population standard deviation ( \(\sigma\) ) when computing the standard deviation of the sampling distribution ( \(\sigma_{\bar x}\) ) in the denominator. However, the population standard deviation is usually not known in the real world and therefore represents another unknown population parameter which we have to estimate from the sample. We saw in the previous chapter that we usually use \(s\) as an estimate of \(\sigma\) and \(SE_{\bar x}\) as and estimate of \(\sigma_{\bar x}\) . Intuitively, we should be more conservative regarding the critical value that we used above to assess whether we have a significant effect to reflect this uncertainty about the true population standard deviation. That is, the threshold for a “significant” effect should be higher to safeguard against falsely claiming a significant effect when there is none. If we replace \(\sigma_{\bar x}\) by it’s estimate \(SE_{\bar x}\) in the formula for the z-score, we get a new test statistic (i.e, the t-statistic ) with its own distribution (the t-distribution ):

\[ t = {\bar x- \mu_0 \over SE_{\bar x}}={\bar x- \mu_0 \over s / \sqrt{n}} \]

Here, \(\bar X\) denotes the sample mean and \(s\) the sample standard deviation. The t-distribution has more probability in its “tails”, i.e. farther away from the mean. This reflects the higher uncertainty introduced by replacing the population standard deviation by its sample estimate. Intuitively, this is particularly relevant for small samples, since the uncertainty about the true population parameters decreases with increasing sample size. This is reflected by the fact that the exact shape of the t-distribution depends on the degrees of freedom , which is the sample size minus one (i.e., \(n-1\) ). To see this, the following graph shows the t-distribution with different degrees of freedom for a two-tailed test and \(\alpha = 0.05\) . The grey curve shows the normal distribution.

Notice that as \(n\) gets larger, the t-distribution gets closer and closer to the normal distribution, reflecting the fact that the uncertainty introduced by \(s\) is reduced. To summarize, we now have an estimate for the standard deviation of the distribution of the sample mean (i.e., \(SE_{\bar x}\) ) and an appropriate distribution that takes into account the necessary uncertainty (i.e., the t-distribution). Let us now compute the t-statistic according to the formula above:

Notice that the value of the t-statistic is higher compared to the z-score (4.29). This can be attributed to the fact that by using the \(s\) as and estimate of \(\sigma\) , we underestimate the true population standard deviation. Hence, the critical value would need to be larger to adjust for this. This is what the t-distribution does. Let us compute the critical value from the t-distribution with n - 1 degrees of freedom.

Again, we use 0.975 and not 0.95 since we are running a two-sided test and need to account for the rejection region at the other end of the distribution. Notice that the new critical value based on the t-distributionis larger, to reflect the uncertainty when estimating \(\sigma\) from \(s\) . Now we can see that the calculated test statistic is still larger than the critical value.

The following graphics shows that the calculated test statistic (red line) falls into the rejection region so that in our example, we would reject the null hypothesis that the true population mean is equal to \(10\) .

Decision: Reject \(H_0\) , given that the calculated test statistic is larger than critical value.

Something to keep in mind here is the fact the test statistic is a function of the sample size. This, as \(n\) gets large, the test statistic gets larger as well and we are more likely to find a significant effect. This reflects the decrease in uncertainty about the true population mean as our sample size increases.

5.1.2.2 P-values

In the previous section, we computed the test statistic, which tells us how close our sample is to the null hypothesis. The p-value corresponds to the probability that the test statistic would take a value as extreme or more extreme than the one that we actually observed, assuming that the null hypothesis is true . It is important to note that this is a conditional probability : we compute the probability of observing a sample mean (or a more extreme value) conditional on the assumption that the null hypothesis is true. The pnorm() function can be used to compute this probability. It is the cumulative probability distribution function of the `normal distribution. Cumulative probability means that the function returns the probability that the test statistic will take a value less than or equal to the calculated test statistic given the degrees of freedom. However, we are interested in obtaining the probability of observing a test statistic larger than or equal to the calculated test statistic under the null hypothesis (i.e., the p-value). Thus, we need to subtract the cumulative probability from 1. In addition, since we are running a two-sided test, we need to multiply the probability by 2 to account for the rejection region at the other side of the distribution.

This value corresponds to the probability of observing a mean equal to or larger than the one we obtained from our sample, if the null hypothesis was true. As you can see, this probability is very low. A small p-value signals that it is unlikely to observe the calculated test statistic under the null hypothesis. To decide whether or not to reject the null hypothesis, we would now compare this value to the level of significance ( \(\alpha\) ) that we chose for our test. For this example, we adopt the widely accepted significance level of 5%, so any test results with a p-value < 0.05 would be deemed statistically significant. Note that the p-value is directly related to the value of the test statistic. The relationship is such that the higher (lower) the value of the test statistic, the lower (higher) the p-value.

Decision: Reject \(H_0\) , given that the p-value is smaller than 0.05.

5.1.2.3 Confidence interval

For a given statistic calculated for a sample of observations (e.g., listening times), a 95% confidence interval can be constructed such that in 95% of samples, the true value of the true population mean will fall within its limits. If the parameter value specified in the null hypothesis (here \(10\) ) does not lie within the bounds, we reject \(H_0\) . Building on what we learned about confidence intervals in the previous chapter, the 95% confidence interval based on the t-distribution can be computed as follows:

\[ CI_{lower} = {\bar x} - t_{1-{\alpha \over 2}} * SE_{\bar x} \\ CI_{upper} = {\bar x} + t_{1-{\alpha \over 2}} * SE_{\bar x} \]

It is easy to compute this interval manually:

The interpretation of this interval is as follows: if we would (hypothetically) take 100 samples and calculated the mean and confidence interval for each of them, then the true population mean would be included in 95% of these intervals. The CI is informative when reporting the result of your test, since it provides an estimate of the uncertainty associated with the test result. From the test statistic or the p-value alone, it is not easy to judge in which range the true population parameter is located. The CI provides an estimate of this range.

Decision: Reject \(H_0\) , given that the parameter value from the null hypothesis ( \(10\) ) is not included in the interval.

To summarize, you can see that we arrive at the same conclusion (i.e., reject \(H_0\) ), irrespective if we use the test statistic, the p-value, or the confidence interval. However, keep in mind that rejecting the null hypothesis does not prove the alternative hypothesis (we can merely provide support for it). Rather, think of the p-value as the chance of obtaining the data we’ve collected assuming that the null hypothesis is true. You should report the confidence interval to provide an estimate of the uncertainty associated with your test results.

5.1.3 Choosing the right test

The test statistic, as we have seen, measures how close the sample is to the null hypothesis and often follows a well-known distribution (e.g., normal, t, or chi-square). To select the correct test, various factors need to be taken into consideration. Some examples are:

• On what scale are your variables measured (categorical vs. continuous)?
• Do you want to test for relationships or differences?
• If you test for differences, how many groups would you like to test?
• For parametric tests, are the assumptions fulfilled?

The previous discussion used a one sample t-test as an example, which requires that variable is measured on an interval or ratio scale. If you are confronted with other settings, the following flow chart provides a rough guideline on selecting the correct test:

Flowchart for selecting an appropriate test (source: McElreath, R. (2016): Statistical Rethinking, p. 2)

For a detailed overview over the different type of tests, please also refer to this overview by the UCLA.

5.1.3.1 Parametric vs. non-parametric tests

A basic distinction can be made between parametric and non-parametric tests. Parametric tests require that variables are measured on an interval or ratio scale and that the sampling distribution follows a known distribution. Non-Parametric tests on the other hand do not require the sampling distribution to be normally distributed (a.k.a. “assumption free tests”). These tests may be used when the variable of interest is measured on an ordinal scale or when the parametric assumptions do not hold. They often rely on ranking the data instead of analyzing the actual scores. By ranking the data, information on the magnitude of differences is lost. Thus, parametric tests are more powerful if the sampling distribution is normally distributed. In this chapter, we will first focus on parametric tests and cover non-parametric tests later.

5.1.3.2 One-tailed vs. two-tailed test

For some tests you may choose between a one-tailed test versus a two-tailed test . The choice depends on the hypothesis you specified, i.e., whether you specified a directional or a non-directional hypotheses. In the example above, we used a non-directional hypothesis . That is, we stated that the mean is different from the comparison value \(\mu_0\) , but we did not state the direction of the effect. A directional hypothesis states the direction of the effect. For example, we might test whether the population mean is smaller than a comparison value:

\[ H_0: \mu \ge \mu_0 \\ H_1: \mu < \mu_0 \]

Similarly, we could test whether the population mean is larger than a comparison value:

\[ H_0: \mu \le \mu_0 \\ H_1: \mu > \mu_0 \]

Connected to the decision of how to phrase the hypotheses (directional vs. non-directional) is the choice of a one-tailed test versus a two-tailed test . Let’s first think about the meaning of a one-tailed test. Using a significance level of 0.05, a one-tailed test means that 5% of the total area under the probability distribution of our test statistic is located in one tail. Thus, under a one-tailed test, we test for the possibility of the relationship in one direction only, disregarding the possibility of a relationship in the other direction. In our example, a one-tailed test could test either if the mean listening time is significantly larger or smaller compared to the control condition, but not both. Depending on the direction, the mean listening time is significantly larger (smaller) if the test statistic is located in the top (bottom) 5% of its probability distribution.

The following graph shows the critical values that our test statistic would need to surpass so that the difference between the population mean and the comparison value would be deemed statistically significant.

It can be seen that under a one-sided test, the rejection region is at one end of the distribution or the other. In a two-sided test, the rejection region is split between the two tails. As a consequence, the critical value of the test statistic is smaller using a one-tailed test, meaning that it has more power to detect an effect. Having said that, in most applications, we would like to be able catch effects in both directions, simply because we can often not rule out that an effect might exist that is not in the hypothesized direction. For example, if we would conduct a one-tailed test for a mean larger than some specified value but the mean turns out to be substantially smaller, then testing a one-directional hypothesis ($H_0: _0 $) would not allow us to conclude that there is a significant effect because there is not rejection at this end of the distribution.

5.1.4 Summary

As we have seen, the process of hypothesis testing consists of various steps:

• Formulate null and alternative hypotheses
• Select an appropriate test
• Choose the level of significance ( \(\alpha\) )
• Descriptive statistics and data visualization
• Conduct significance test
• Report results and draw a marketing conclusion

In the following, we will go through the individual steps using examples for different tests.

5.2 One sample t-test

The example we used in the introduction was an example of the one sample t-test and we computed all statistics by hand to explain the underlying intuition. When you conduct hypothesis tests using R, you do not need to calculate these statistics by hand, since there are build-in routines to conduct the steps for you. Let us use the same example again to see how you would conduct hypothesis tests in R.

1. Formulate null and alternative hypotheses

The null hypothesis states that there is no difference between the true population mean \(\mu\) and the hypothesized value (i.e., \(10\) ), while the alternative hypothesis states the opposite:

\[ H_0: \mu = 10 \\ H_1: \mu \neq 10 \]

2. Select an appropriate test

Because we would like to test if the mean of a variable is different from a specified threshold, the one-sample t-test is appropriate. The assumptions of the test are 1) that the variable is measured using an interval or ratio scale, and 2) that the sampling distribution is normal. Both assumptions are met since 1) listening time is a ratio scale, and 2) we deem the sample size (n = 50) large enough to assume a normal sampling distribution according to the central limit theorem.

3. Choose the level of significance

We choose the conventional 5% significance level.

4. Descriptive statistics and data visualization

Provide descriptive statistics using the stat.desc() function:

From this, we can already see that the mean is different from the hypothesized value. The question however remains, whether this difference is significantly different, given the sample size and the variability in the data. Since we only have one continuous variable, we can visualize the distribution in a histogram.

5. Conduct significance test

In the beginning of the chapter, we saw, how you could conduct significance test by hand. However, R has built-in routines that you can use to conduct the analyses. The t.test() function can be used to conduct the test. To test if the listening time among WU students was 10, you can use the following code:

Note that if you would have stated a directional hypothesis (i.e., the mean is either greater or smaller than 10 hours), you could easily amend the code to conduct a one sided test by changing the argument alternative from 'two.sided' to either 'less' or 'greater' .

6. Report results and draw a marketing conclusion

Note that the results are the same as above, when we computed the test by hand. You could summarize the results as follows:

On average, the listening times in our sample were different form 10 hours per month (Mean = 18.99 hours, SE = 1.78). This difference was significant t(49) = 5.058, p < .05 (95% CI = [15.42; 22.56]). Based on this evidence, we can conclude that the mean in our sample is significantly lower compared to the hypothesized population mean of \(10\) hours, providing evidence against the null hypothesis.

Note that in the reporting above, the number 49 in parenthesis refers to the degrees of freedom that are available from the output.

5.3 Comparing two means

In the one-sample test above, we tested the hypothesis that the population mean has some specific value \(\mu_0\) using data from only one sample. In marketing (as in many other disciplines), you will often be confronted with a situation where you wish to compare the means of two groups. For example, you may conduct an experiment and randomly split your sample into two groups, one of which receives a treatment (experimental group) while the other doesn’t (control group). In this case, the units (e.g., participants, products) in each group are different (‘between-subjects design’) and the samples are said to be independent. Hence, we would use a independent-means t-test . If you run an experiment with two experimental conditions and the same units (e.g., participants, products) were observed in both experimental conditions, the sample is said to be dependent in the sense that you have the same units in each group (‘within-subjects design’). In this case, we would need to conduct an dependent-means t-test . Both tests are described in the following sections, beginning with the independent-means t-test.

5.3.1 Independent-means t-test

Using an independent-means t-test, we can compare the means of two possibly different populations. It is, for example, quite common for online companies to test new service features by running an experiment and randomly splitting their website visitors into two groups: one is exposed to the website with the new feature (experimental group) and the other group is not exposed to the new feature (control group). This is a typical A/B-Test scenario.

As an example, imagine that a music streaming service would like to introduce a new playlist feature that let’s their users access playlists created by other users. The goal is to analyse how the new service feature impacts the listening time of users. The service randomly splits a representative subset of their users into two groups and collects data about their listening times over one month. Let’s create a data set to simulate such a scenario.

This data set contains two variables: the variable hours indicates the music listening times (in hours) and the variable group indicates from which group the observation comes, where ‘A’ refers to the control group (with the standard service) and ‘B’ refers to the experimental group (with the new playlist feature). Let’s first look at the descriptive statistics by group using the describeBy function:

From this, we can already see that there is a difference in means between groups A and B. We can also see that the number of observations is different, as is the standard deviation. The question that we would like to answer is whether there is a significant difference in mean listening times between the groups. Remember that different users are contained in each group (‘between-subjects design’) and that the observations in one group are independent of the observations in the other group. Before we will see how you can easily conduct an independent-means t-test, let’s go over some theory first.

5.3.1.1 Theory

As a starting point, let us label the unknown population mean of group A (control group) in our experiment \(\mu_1\) , and that of group B (experimental group) \(\mu_2\) . In this setting, the null hypothesis would state that the mean in group A is equal to the mean in group B:

\[ H_0: \mu_1=\mu_2 \]

This is equivalent to stating that the difference between the two groups ( \(\delta\) ) is zero:

\[ H_0: \mu_1 - \mu_2=0=\delta \]

That is, \(\delta\) is the new unknown population parameter, so that the null and alternative hypothesis become:

\[ H_0: \delta = 0 \\ H_1: \delta \ne 0 \]

Remember that we usually don’t have access to the entire population so that we can not observe \(\delta\) and have to estimate is from a sample statistic, which we define as \(d = \bar x_1-\bar x_2\) , i.e., the difference between the sample means from group a ( \(\bar x_1\) ) and group b ( \(\bar x_2\) ). But can we really estimate \(d\) from \(\delta\) ? Remember from the previous chapter, that we could estimate \(\mu\) from \(\bar x\) , because if we (hypothetically) take a larger number of samples, the distribution of the means of these samples (the sampling distribution) will be normally distributed and its mean will be (in the limit) equal to the population mean. It turns out that we can use the same underlying logic here. The above samples were drawn from two different populations with \(\mu_1\) and \(\mu_2\) . Let us compute the difference in means between these two populations:

This means that the true difference between the mean listening times of groups a and b is -7.42. Let us now repeat the exercise from the previous chapter: let us repeatedly draw a large number of \(20,000\) random samples of 100 users from each of these populations, compute the difference (i.e., \(d\) , our estimate of \(\delta\) ), store the difference for each draw and create a histogram of \(d\) .

This gives us the sampling distribution of the mean differences between the samples. You will notice that this distribution follows a normal distribution and is centered around the true difference between the populations. This means that, on average, the difference between two sample means \(d\) is a good estimate of \(\delta\) . In our example, the difference between \(\bar x_1\) and \(\bar x_2\) is:

Now that we have \(d\) as an estimate of \(\delta\) , how can we find out if the observed difference is significantly different from the null hypothesis (i.e., \(\delta = 0\) )?

Recall from the previous section, that the standard deviation of the sampling distribution \(\sigma_{\bar x}\) (i.e., the standard error) gives us indication about the precision of our estimate. Further recall that the standard error can be calculated as \(\sigma_{\bar x}={\sigma \over \sqrt{n}}\) . So how can we calculate the standard error of the difference between two population means? According to the variance sum law , to find the variance of the sampling distribution of differences, we merely need to add together the variances of the sampling distributions of the two populations that we are comparing. To find the standard error, we only need to take the square root of the variance (because the standard error is the standard deviation of the sampling distribution and the standard deviation is the square root of the variance), so that we get:

\[ \sigma_{\bar x_1-\bar x_2} = \sqrt{{\sigma_1^2 \over n_1}+{\sigma_2^2 \over n_2}} \]

But recall that we don’t actually know the true population standard deviation, so we use \(SE_{\bar x_1-\bar x_2}\) as an estimate of \(\sigma_{\bar x_1-\bar x_2}\) :

\[ SE_{\bar x_1-\bar x_2} = \sqrt{{s_1^2 \over n_1}+{s_2^2 \over n_2}} \]

Hence, for our example, we can calculate the standard error as follows:

Recall from above that we can calculate the t-statistic as:

\[ t= {\bar x - \mu_0 \over {s \over \sqrt{n}}} \]

Exchanging \(\bar x\) for \(d\) , we get

\[ t= {(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2) \over {\sqrt{{s_1^2 \over n_1}+{s_2^2 \over n_2}}}} \]

Note that according to our hypothesis \(\mu_1-\mu_2=0\) , so that we can calculate the t-statistic as:

Following the example of our one sample t-test above, we would now need to compare this calculated test statistic to a critical value in order to assess if \(d\) is sufficiently far away from the null hypothesis to be statistically significant. To do this, we would need to know the exact t-distribution, which depends on the degrees of freedom. The problem is that deriving the degrees of freedom in this case is not that obvious. If we were willing to assume that \(\sigma_1=\sigma_2\) , the correct t-distribution has \(n_1 -1 + n_2-1\) degrees of freedom (i.e., the sum of the degrees of freedom of the two samples). However, because in real life we don not know if \(\sigma_1=\sigma_2\) , we need to account for this additional uncertainty. We will not go into detail here, but R automatically uses a sophisticated approach to correct the degrees of freedom called the Welch’s correction, as we will see in the subsequent application.

5.3.1.2 Application

The section above explained the theory behind the independent-means t-test and showed how to compute the statistics manually. Obviously you don’t have to compute these statistics by hand in this section shows you how to conduct an independent-means t-test in R using the example from above.

We wish to analyze whether there is a significant difference in music listening times between groups A and B. So our null hypothesis is that the means from the two populations are the same (i.e., there is no difference), while the alternative hypothesis states the opposite:

\[ H_0: \mu_1=\mu_2\\ H_1: \mu_1 \ne \mu_2 \]

Since we have a ratio scaled variable (i.e., listening times) and two independent groups, where the mean of one sample is independent of the group of the second sample (i.e., the groups contain different units), the independent-means t-test is appropriate.

We can compute the descriptive statistics for each group separately, using the describeBy() function:

This already shows us that the mean between groups A and B are different. We can visualize the data using a plot of means, boxplot, and a histogram.

To conduct the independent means t-test, we can use the t.test() function:

The results showed that listening times were higher in the experimental group B (Mean = 28.50, SE = 1.7) compared to the control group (Mean = 18.11, SE = 1.22). This means that the listening times were 10.39 hours higher on average in the experimental group (B), compared to the control group (A). An independent-means t-test showed that this difference is significant t(195.73) = -4.9646, p < .05 (95% CI = [-14.514246,-6.261264]).

5.3.2 Dependent-means t-test

While the independent-means t-test is used when different units (e.g., participants, products) were assigned to the different condition, the dependent-means t-test is used when there are two experimental conditions and the same units (e.g., participants, products) were observed in both experimental conditions.

Imagine, for example, a slightly different experimental setup for the above experiment. Imagine that we do not assign different users to the groups, but that a sample of 100 users gets to use the music streaming service with the new feature for one month and we compare the music listening times of these users during the month of the experiment with the listening time in the previous month. Let us generate data for this example:

Note that the data set has almost the same structure as before only that we know have two variables representing the listening times of each user in the month before the experiment and during the month of the experiment when the new feature was tested.

5.3.2.1 Theory

In this case, we want to test the hypothesis that there is no difference in mean the mean listening times between the two months. This can be expressed as follows:

\[ H_0: \mu_D = 0 \\ \] Note that the hypothesis only refers to one population, since both observations come from the same units (i.e., users). To use consistent notation, we replace \(\mu_D\) with \(\delta\) and get:

\[ H_0: \delta = 0 \\ H_1: \delta \neq 0 \]

where \(\delta\) denotes the difference between the observed listening times from the two consecutive months of the same users . As is the previous example, since we do not observe the entire population, we estimate \(\delta\) based on the sample using \(d\) , which is the difference in mean listening time between the two months for our sample. Note that we assume that everything else (e.g., number of new releases) remained constant over the two month to keep it simple. We can show as above that the sampling distribution follows a normal distribution with a mean that is (in the limit) the same as the population mean. This means, again, that the difference in sample means is a good estimate for the difference in population means. Let’s compute a new variable \(d\) , which is the difference between two month.

Note that we now have a new variable, which is the difference in listening times (in hours) between the two months. The mean of this difference is:

Again, we use \(SE_{\bar x}\) as an estimate of \(\sigma_{\bar x}\) :

\[ SE_{\bar d}={s \over \sqrt{n}} \] Hence, we can compute the standard error as:

The test statistic is therefore:

\[ t = {\bar d- \mu_0 \over SE_{\bar d}} \] on 99 (i.e., n-1) degrees of freedom. Now we can compute the t-statistic as follows:

Note that in the case of the dependent-means t-test, we only base our hypothesis on one population and hence there is only one population variance. This is because in the dependent sample test, the observations come from the same observational units (i.e., users). Hence, there is no unsystematic variation due to potential differences between users that were assigned to the experimental groups. This means that the influence of unobserved factors (unsystematic variation) relative to the variation due to the experimental manipulation (systematic variation) is not as strong in the dependent-means test compared to the independent-means test and we don’t need to correct for differences in the population variances.

5.3.2.2 Application

Again, we don’t have to compute all this by hand since the t.test(...) function can be used to do it for us. Now we have to use the argument paired=TRUE to let R know that we are working with dependent observations.

We would like to the test if there is a difference in music listening times between the two consecutive months, so our null hypothesis is that there is no difference, while the alternative hypothesis states the opposite:

\[ H_0: \mu_D = 0 \\ H_0: \mu_D \ne 0 \]

Since we have a ratio scaled variable (i.e., listening times) and two observations of the same group of users (i.e., the groups contain the same units), the dependent-means t-test is appropriate.

We can compute the descriptive statistics for each month separately, using the describe() function:

This already shows us that the mean between the two months are different. We can visiualize the data using a plot of means, boxplot, and a histogram.

To plot the data, we need to do some restructuring first, since the variables are now stored in two different columns (“hours_a” and “hours_b”). This is also known as the “wide” format. To plot the data we need all observations to be stored in one variable. This is also known as the “long” format. We can use the melt(...) function from the reshape2 package to “melt” the two variable into one column to plot the data.

Now we are ready to plot the data:

To conduct the independent means t-test, we can use the t.test() function with the argument paired = TRUE :

On average, the same users used the service more when it included the new feature (M = 25.96, SE = 1.68) compared to the service without the feature (M = 20.99, SE = 1.34). This difference was significant t(99) = 2.3781, p < .05 (95% CI = [0.82, 9.12]).

5.3.3 Further considerations

5.3.3.1 type i and type ii errors.

When choosing the level of significance ( \(\alpha\) ). It is important to note that the choice of the significance level affects the type 1 and type 2 error:

• Type I error: When we believe there is a genuine effect in our population, when in fact there isn’t. Probability of type I error ( \(\alpha\) ) = level of significance.
• Type II error: When we believe that there is no effect in the population, when in fact there is.

This following table shows the possible outcomes of a test (retain vs. reject \(H_0\) ), depending on whether \(H_0\) is true or false in reality.

5.3.3.2 Significance level, sample size, power, and effect size

When you plan to conduct an experiment, there are some factors that are under direct control of the researcher:

• Significance level ( \(\alpha\) ) : The probability of finding an effect that does not genuinely exist.
• Sample size (n) : The number of observations in each group of the experimental design.

Unlike α and n, which are specified by the researcher, the magnitude of β depends on the actual value of the population parameter. In addition, β is influenced by the effect size (e.g., Cohen’s d), which can be used to determine a standardized measure of the magnitude of an observed effect. The following parameters are affected more indirectly:

• Power (1-β) : The probability of finding an effect that does genuinely exists.
• Effect size (d) : Standardized measure of the effect size under the alternate hypothesis.

Although β is unknown, it is related to α. For example, if we would like to be absolutely sure that we do not falsely identify an effect which does not exist (i.e., make a type I error), this means that the probability of identifying an effect that does exist (i.e., 1-β) decreases and vice versa. Thus, an extremely low value of α (e.g., α = 0.0001) will result in intolerably high β errors. A common approach is to set α=0.05 and 1-β=0.80.

Unlike the t-value of our test, the effect size (d) is unaffected by the sample size and can be categorized as follows (see Cohen, J. 1988):

• 0.2 (small effect)
• 0.5 (medium effect)
• 0.8 (large effect)

In order to test more subtle effects (smaller effect sizes), you need a larger sample size compared to the test of more obvious effects. In this paper , you can find a list of examples for different effect sizes and the number of observations you need to reliably find an effect of that magnitude. Although the exact effect size is unknown before the experiment, you might be able to make a guess about the effect size (e.g., based on previous studies).

If you wish to obtain a standardized measure of the effect, you may compute the effect size (Cohen’s d) using the cohensD() function from the lsr package. Using the examples from the independent-means t-test above, we would use:

According to the thresholds defined above, this effect would be judged to be a small-medium effect.

For the dependent-means t-test, we would use:

According to the thresholds defined above, this effect would also be judged to be a small-medium effect.

When constructing an experimental design, your goal should be to maximize the power of the test while maintaining an acceptable significance level and keeping the sample as small as possible. To achieve this goal, you may use the pwr package, which let’s you compute n , d , alpha , and power . You only need to specify three of the four input variables to get the fourth.

For example, what sample size do we need (per group) to identify an effect with d = 0.6, α = 0.05, and power = 0.8:

Or we could ask, what is the power of our test with 51 observations in each group, d = 0.6, and α = 0.05:

5.3.3.3 P-values, stopping rules and p-hacking

From my experience, students tend to place a lot of weight on p-values when interpreting their research findings. It is therefore important to note some points that hopefully help to put the meaning of a “significant” vs. “insignificant” test result into perspective.

Significant result

• Even if the probability of the effect being a chance result is small (e.g., less than .05) it doesn’t necessarily mean that the effect is important.
• Very small and unimportant effects can turn out to be statistically significant if the sample size is large enough.

Insignificant result

• If the probability of the effect occurring by chance is large (greater than .05), the alternative hypothesis is rejected. However, this does not mean that the null hypothesis is true.
• Although an effect might not be large enough to be anything other than a chance finding, it doesn’t mean that the effect is zero.
• In fact, two random samples will always have slightly different means that would deemed to be statistically significant if the samples were large enough.

Thus, you should not base your research conclusion on p-values alone!

It is also crucial to determine the sample size before you run the experiment or before you start your analysis. Why? Consider the following example:

• You run an experiment
• After each respondent you analyze the data and look at the mean difference between the two groups with a t-test
• You stop when you have a significant effect

This is called p-hacking and should be avoided at all costs. Assuming that both groups come from the same population (i.e., there is no difference in the means): What is the likelihood that the result will be significant at some point? In other words, what is the likelihood that you will draw the wrong conclusion from your data that there is an effect, while there is none? This is shown in the following graph using simulated data - the color red indicates significant test results that arise although there is no effect (i.e., false positives).

Figure 5.1: p-hacking (red indicates false positives)

5.4 Comparing several means

This chapter is primarily based on Field, A., Miles J., & Field, Z. (2012): Discovering Statistics Using R. Sage Publications, chapters 10 & 12 .

5.4.1 Introduction

In the previous section we learned how to compare means using a t-test. The t-test has some limitations since it only lets you compare 2 means and you can only use it with one independent variable. However, often we would like to compare means from 3 or more groups. In addition, there may be instances in which you manipulate more than one independent variable. For these applications, ANOVA (ANalysis Of VAriance) can be used. Hence, to conduct ANOVA you need:

• A metric dependent variable (i.e., measured using an interval or ratio scale)
• One or more non-metric (categorical) independent variables (also called factors)

A treatment is a particular combination of factor levels, or categories. One-way ANOVA is used when there is only one categorical variable (factor). In this case, a treatment is the same as a factor level. N-way ANOVA is used with two or more factors. Note that we are only going to talk about a single independent variable in the context of ANOVA. If you have multiple independent variables please refere to the chapter on Regression .

Let’s use an example to see how ANOVA works. Similar to the previous example it is also imaginable that the music streaming service experiments with a recommendation system for user created playlists. We now have three groups, the control group “A” with the current system, treatment group “B” who have access to playlists created by other users but are not shown recommendations and treatment group “C” who are shown recommendations for user created playlists. As always, we load and inspect the data first:

The null hypothesis, typically, is that all means are equal (non-directional hypothesis). Hence, in our case:

\[H_0: \mu_1 = \mu_2 = \mu_3\]

The alternative hypothesis is simply that the means are not all equal, i.e.,

\[H_1: \textrm{Means are not all equal}\]

If you wanted to put this in mathematical notation, you could also write:

\[H_1: \exists {i,j}: {\mu_i \ne \mu_j} \]

To get a first impression if there are any differences in listening times across the experimental groups, we use the describeBy(...) function from the psych package:

In addition, you should visualize the data using appropriate plots:

Figure 5.2: Plot of means

Note that ANOVA is an omnibus test, which means that we test for an overall difference between groups. Hence, the test will only tell you if the group means are different, but it won’t tell you exactly which groups are different from another.

So why don’t we then just conduct a series of t-tests for all combinations of groups (i.e., A vs. B, A vs. C, B vs. C)? The reason is that if we assume each test to be independent, then there is a 5% probability of falsely rejecting the null hypothesis (Type I error) for each test. In our case:

• A vs. B (α = 0.05)
• A vs. C (α = 0.05)
• B vs. C (α = 0.05)

This means that the overall probability of making a Type I error is 1-(0.95 3 ) = 0.143, since the probability of no Type I error is 0.95 for each of the three tests. Consequently, the Type I error probability would be 14.3%, which is above the conventional standard of 5%. This is also known as the family-wise or experiment-wise error.

5.4.2 Decomposing variance

The basic concept underlying ANOVA is the decomposition of the variance in the data. There are three variance components which we need to consider:

• We calculate how much variability there is between scores: Total sum of squares (SS T )
• We then calculate how much of this variability can be explained by the model we fit to the data (i.e., how much variability is due to the experimental manipulation): Model sum of squares (SS M )
• … and how much cannot be explained (i.e., how much variability is due to individual differences in performance): Residual sum of squares (SS R )

The following figure shows the different variance components using a generalized data matrix:

Decomposing variance

The total variation is determined by the variation between the categories (due to our experimental manipulation) and the within-category variation that is due to extraneous factors (e.g., promotion of artists on a social network):

\[SS_T= SS_M+SS_R\]

To get a better feeling how this relates to our data set, we can look at the data in a slightly different way. Specifically, we can use the dcast(...) function from the reshape2 package to convert the data to wide format:

In this example, X 1 from the generalized data matrix above would refer to the factor level “A”, X 2 to the level “B”, and X 3 to the level “C”. Y 11 refers to the first data point in the first row (i.e., “13”), Y 12 to the second data point in the first row (i.e., “21”), etc.. The grand mean ( \(\overline{Y}\) ) and the category means ( \(\overline{Y}_c\) ) can be easily computed:

To see how each variance component can be derived, let’s look at the data again. The following graph shows the individual observations by experimental group:

Figure 5.3: Sum of Squares

5.4.2.1 Total sum of squares

To compute the total variation in the data, we consider the difference between each observation and the grand mean. The grand mean is the mean over all observations in the data set. The vertical lines in the following plot measure how far each observation is away from the grand mean:

Figure 5.4: Total Sum of Squares

The formal representation of the total sum of squares (SS T ) is:

\[ SS_T= \sum_{i=1}^{N} (Y_i-\bar{Y})^2 \]

This means that we need to subtract the grand mean from each individual data point, square the difference, and sum up over all the squared differences. Thus, in our example, the total sum of squares can be calculated as:

\[ \begin{align} SS_T =&(13−24.67)^2 + (14−24.67)^2 + … + (2−24.67)^2\\ &+(21−24.67)^2 + (18-24.67)^2 + … + (17−24.67)^2\\ &+(30−24.67)^2 + (37−24.67)^2 + … + (28−24.67)^2\\ &=30855.64 \end{align} \]

You could also compute this in R using:

For the subsequent analyses, it is important to understand the concept behind the degrees of freedom . Remember that in order to estimate a population value from a sample, we need to hold something in the population constant. In ANOVA, the df are generally one less than the number of values used to calculate the SS. For example, when we estimate the population mean from a sample, we assume that the sample mean is equal to the population mean. Then, in order to estimate the population mean from the sample, all but one scores are free to vary and the remaining score needs to be the value that keeps the population mean constant. In our example, we used all 300 observations to calculate the sum of square, so the total degrees of freedom (df T ) are:

\[\begin{equation} \begin{split} df_T = N-1=300-1=299 \end{split} \tag{5.1} \end{equation}\]

5.4.2.2 Model sum of squares

Now we know that there are 26646.33 units of total variation in our data. Next, we compute how much of the total variation can be explained by the differences between groups (i.e., our experimental manipulation). To compute the explained variation in the data, we consider the difference between the values predicted by our model for each observation (i.e., the group mean) and the grand mean. The group mean refers to the mean value within the experimental group. The vertical lines in the following plot measure how far the predicted value for each observation (i.e., the group mean) is away from the grand mean:

Figure 5.5: Model Sum of Squares

The formal representation of the model sum of squares (SS M ) is:

\[ SS_M= \sum_{j=1}^{c} n_j(\bar{Y}_j-\bar{Y})^2 \]

where c denotes the number of categories (experimental groups). This means that we need to subtract the grand mean from each group mean, square the difference, and sum up over all the squared differences. Thus, in our example, the model sum of squares can be calculated as:

\[ \begin{align} SS_M &= 100*(15.47−24.67)^2 + 100*(24.88−24.67)^2 + 100*(33.66−24.67)^2 \\ &= 21321.21 \end{align} \]

You could also compute this manually in R using:

In this case, we used the three group means to calculate the sum of squares, so the model degrees of freedom (df M ) are:

\[ df_M= c-1=3-1=2 \]

5.4.2.3 Residual sum of squares

Lastly, we calculate the amount of variation that cannot be explained by our model. In ANOVA, this is the sum of squared distances between what the model predicts for each data point (i.e., the group means) and the observed values. In other words, this refers to the amount of variation that is caused by extraneous factors, such as differences between product characteristics of the products in the different experimental groups. The vertical lines in the following plot measure how far each observation is away from the group mean:

Figure 5.6: Residual Sum of Squares

The formal representation of the residual sum of squares (SS R ) is:

\[ SS_R= \sum_{j=1}^{c} \sum_{i=1}^{n} ({Y}_{ij}-\bar{Y}_{j})^2 \]

This means that we need to subtract the group mean from each individual observation, square the difference, and sum up over all the squared differences. Thus, in our example, the model sum of squares can be calculated as:

\[ \begin{align} SS_R =& (13−14.34)^2 + (14−14.34)^2 + … + (2−14.34)^2 \\ +&(21−24.7)^2 + (18−24.7)^2 + … + (17−24.7)^2 \\ +& (30−34.99)^2 + (37−34.99)^2 + … + (28−34.99)^2 \\ =& 9534.43 \end{align} \]

In this case, we used the 10 values for each of the SS for each group, so the residual degrees of freedom (df R ) are:

\[ \begin{align} df_R=& (n_1-1)+(n_2-1)+(n_3-1) \\ =&(100-1)+(100-1)+(100-1)=297 \end{align} \]

5.4.2.4 Effect strength

Once you have computed the different sum of squares, you can investigate the effect strength. \(\eta^2\) is a measure of the variation in Y that is explained by X:

\[ \eta^2= \frac{SS_M}{SS_T}=\frac{21321.21}{30855.64}=0.69 \]

To compute this in R:

The statistic can only take values between 0 and 1. It is equal to 0 when all the category means are equal, indicating that X has no effect on Y. In contrast, it has a value of 1 when there is no variability within each category of X but there is some variability between categories.

5.4.2.5 Test of significance

How can we determine whether the effect of X on Y is significant?

• First, we calculate the fit of the most basic model (i.e., the grand mean)
• Then, we calculate the fit of the “best” model (i.e., the group means)
• A good model should fit the data significantly better than the basic model
• The F-statistic or F-ratio compares the amount of systematic variance in the data to the amount of unsystematic variance

The F-statistic uses the ratio of mean square related to X (explained variation) and the mean square related to the error (unexplained variation):

\(\frac{SS_M}{SS_R}\)

However, since these are summed values, their magnitude is influenced by the number of scores that were summed. For example, to calculate SS M we only used the sum of 3 values (the group means), while we used 30 and 27 values to calculate SS T and SS R , respectively. Thus, we calculate the average sum of squares (“mean square”) to compare the average amount of systematic vs. unsystematic variation by dividing the SS values by the degrees of freedom associated with the respective statistic.

Mean square due to X:

\[ MS_M= \frac{SS_M}{df_M}=\frac{SS_M}{c-1}=\frac{21321.21}{(3-1)} \]

Mean square due to error:

\[ MS_R= \frac{SS_R}{df_R}=\frac{SS_R}{N-c}=\frac{9534.43}{(300-3)} \]

Now, we compare the amount of variability explained by the model (experiment), to the error in the model (variation due to extraneous variables). If the model explains more variability than it can’t explain, then the experimental manipulation has had a significant effect on the outcome (DV). The F-radio can be derived as follows:

\[ F= \frac{MS_M}{MS_R}=\frac{\frac{SS_M}{c-1}}{\frac{SS_R}{N-c}}=\frac{\frac{21321.21}{(3-1)}}{\frac{9534.43}{(300-3)}}=332.08 \]

You can easily compute this in R:

This statistic follows the F distribution with (m = c – 1) and (n = N – c) degrees of freedom. This means that, like the \(\chi^2\) distribution, the shape of the F-distribution depends on the degrees of freedom. In this case, the shape depends on the degrees of freedom associated with the numerator and denominator used to compute the F-ratio. The following figure shows the shape of the F-distribution for different degrees of freedom:

The F distribution

The outcome of the test is one of the following:

• If the null hypothesis of equal category means is not rejected, then the independent variable does not have a significant effect on the dependent variable
• If the null hypothesis is rejected, then the effect of the independent variable is significant

For 2 and 297 degrees of freedom, the critical value of F is 3.026 for α=0.05. As usual, you can either look up these values in a table or use the appropriate function in R:

The output tells us that the calculated test statistic exceeds the critical value. We can also show the test result visually:

Visual depiction of the test result

Thus, we conclude that because F CAL = 332.08 > F CR = 3.03, H 0 is rejected!

Interpretation: one or more of the differences between means are statistically significant.

Reporting: There was a significant effect of promotion on sales levels, F(2,297) = 332.08, p < 0.05, \(\eta^2\) = 0.69.

Remember: This doesn’t tell us where the differences between groups lie. To find out which group means exactly differ, we need to use post-hoc procedures (see below).

You don’t have to compute these statistics manually! Luckily, there is a function for ANOVA in R, which does the above calculations for you as we will see in the next section.

5.4.3 One-way ANOVA

5.4.3.1 basic anova.

As already indicated, one-way ANOVA is used when there is only one categorical variable (factor). Before conducting ANOVA, you need to check if the assumptions of the test are fulfilled. The assumptions of ANOVA are discussed in the following sections.

Independence of observations

The observations in the groups should be independent. Because we randomly assigned the listeners to the experimental conditions, this assumption can be assumed to be met.

Distributional assumptions

ANOVA is relatively immune to violations to the normality assumption when sample sizes are large due to the Central Limit Theorem. However, if your sample is small (i.e., n < 30 per group) you may nevertheless want to check the normality of your data, e.g., by using the Shapiro-Wilk test or QQ-Plot. In our example, we have 100 observations in each group which is plenty but let’s create another example with only 10 observations in each group. In the latter case we cannot rely on the Central Limit Theorem and we should test the normality of our data. This can be done using the Shapiro-Wilk Test, which has the Null Hypothesis that the data is normally distributed. Hence, an insignificant test results means that the data can be assumed to be approximately normally distributed:

Since the test result is insignificant for all groups, we can conclude that the data approximately follow a normal distribution.

We could also test the distributional assumptions visually using a Q-Q plot (i.e., quantile-quantile plot). This plot can be used to assess if a set of data plausibly came from some theoretical distribution such as the Normal distribution. Since this is just a visual check, it is somewhat subjective. But it may help us to judge if our assumption is plausible, and if not, which data points contribute to the violation. A Q-Q plot is a scatterplot created by plotting two sets of quantiles against one another. If both sets of quantiles came from the same distribution, we should see the points forming a line that’s roughly straight. In other words, Q-Q plots take your sample data, sort it in ascending order, and then plot them versus quantiles calculated from a theoretical distribution. Quantiles are often referred to as “percentiles” and refer to the points in your data below which a certain proportion of your data fall. Recall, for example, the standard Normal distribution with a mean of 0 and a standard deviation of 1. Since the 50th percentile (or 0.5 quantile) is 0, half the data lie below 0. The 95th percentile (or 0.95 quantile), is about 1.64, which means that 95 percent of the data lie below 1.64. The 97.5th quantile is about 1.96, which means that 97.5% of the data lie below 1.96. In the Q-Q plot, the number of quantiles is selected to match the size of your sample data.

To create the Q-Q plot for the normal distribution, you may use the qqnorm() function, which takes the data to be tested as an argument. Using the qqline() function subsequently on the data creates the line on which the data points should fall based on the theoretical quantiles. If the individual data points deviate a lot from this line, it means that the data is not likely to follow a normal distribution.

Figure 5.7: Q-Q plot 1

Figure 5.8: Q-Q plot 2

Figure 5.9: Q-Q plot 3

The Q-Q plots suggest an approximately Normal distribution. If the assumption had been violated, you might consider transforming your data or resort to a non-parametric test.

Homogeneity of variance

Let’s return to our original dataset with 100 observations in each group for the rest of the analysis.

You can test the homogeneity of variances in R using Levene’s test:

The null hypothesis of the test is that the group variances are equal. Thus, if the test result is significant it means that the variances are not equal. If we cannot reject the null hypothesis (i.e., the group variances are not significantly different), we can proceed with the ANOVA as follows:

You can see that the p-value is smaller than 0.05. This means that, if there really was no difference between the population means (i.e., the Null hypothesis was true), the probability of the observed differences (or larger differences) is less than 5%.

To compute η 2 from the output, we can extract the relevant sum of squares as follows

You can see that the results match the results from our manual computation above ( \(\eta^2 =\) 0.69).

The aov() function also automatically generates some plots that you can use to judge if the model assumptions are met. We will inspect two of the plots here.

We will use the first plot to inspect if the residual variances are equal across the experimental groups:

Generally, the residual variance (i.e., the range of values on the y-axis) should be the same for different levels of our independent variable. The plot shows, that there are some slight differences. Notably, the range of residuals is higher in group “B” than in group “C”. However, the differences are not that large and since the Levene’s test could not reject the Null of equal variances, we conclude that the variances are similar enough in this case.

The second plot can be used to test the assumption that the residuals are approximately normally distributed. We use a Q-Q plot to test this assumption:

The plot suggests that, the residuals are approximately normally distributed. We could also test this by extracting the residuals from the anova output using the resid() function and using the Shapiro-Wilk test:

Confirming the impression from the Q-Q plot, we cannot reject the Null that the residuals are approximately normally distributed.

Note that if Levene’s test would have been significant (i.e., variances are not equal), we would have needed to either resort to non-parametric tests (see below), or compute the Welch’s F-ratio instead:

You can see that the results are fairly similar, since the variances turned out to be fairly equal across groups.

5.4.3.2 Post-hoc tests

Provided that significant differences were detected by the overall ANOVA you can find out which group means are different using post hoc procedures. Post hoc procedures are designed to conduct pairwise comparisons of all different combinations of the treatment groups by correcting the level of significance for each test such that the overall Type I error rate (α) across all comparisons remains at 0.05.

In other words, we rejected H 0 : μ 1 = μ 2 = μ 3 , and now we would like to test:

\[H_0: \mu_1 = \mu_2\]

\[H_0: \mu_1 = \mu_3\]

\[H_0: \mu_2 = \mu_3\]

There are several post hoc procedures available to choose from. In this tutorial, we will cover Bonferroni and Tukey’s HSD (“honest significant differences”). Both tests control for family-wise error. Bonferroni tends to have more power when the number of comparisons is small, whereas Tukey’ HSDs is better when testing large numbers of means.

5.4.3.2.1 Bonferroni

One of the most popular (and easiest) methods to correct for the family-wise error rate is to conduct the individual t-tests and divide α by the number of comparisons („k“):

\[ p_{CR}= \frac{\alpha}{k} \]

In our example with three groups:

\[p_{CR}= \frac{0.05}{3}=0.017\]

Thus, the “corrected” critical p-value is now 0.017 instead of 0.05 (i.e., the critical t value is higher). You can implement the Bonferroni procedure in R using:

In the output, you will get the corrected p-values for the individual tests. In our example, we can reject H 0 of equal means for all three tests, since p < 0.05 for all combinations of groups.

Note the difference between the results from the post-hoc test compared to individual t-tests. For example, when we test the “B” vs. “C” groups, the result from a t-test would be:

Usually the p-value is lower in the t-test, reflecting the fact that the family-wise error is not corrected (i.e., the test is less conservative). In this case the p-value is extremely small in both cases and thus indistinguishable.

5.4.3.2.2 Tukey’s HSD

Tukey’s HSD also compares all possible pairs of means (two-by-two combinations; i.e., like a t-test, except that it corrects for family-wise error rate).

Test statistic:

\[\begin{equation} \begin{split} HSD= q\sqrt{\frac{MS_R}{n_c}} \end{split} \tag{5.2} \end{equation}\]

• q = value from studentized range table (see e.g., here )
• MS R = Mean Square Error from ANOVA
• n c = number of observations per group
• Decision: Reject H 0 if

\[|\bar{Y}_i-\bar{Y}_j | > HSD\]

The value from the studentized range table can be obtained using the qtukey() function.

\[HSD= 3.33\sqrt{\frac{33.99}{100}}=1.94\]

Since all mean differences between groups are larger than 1.906, we can reject the null hypothesis for all individual tests, confirming the results from the Bonferroni test. To compute Tukey’s HSD, we can use the appropriate function from the multcomp package.

We may also plot the result for the mean differences incl. their confidence intervals:

Figure 5.10: Tukey’s HSD

You can see that the CIs do not cross zero, which means that the true difference between group means is unlikely zero.

Reporting of post hoc results:

The post hoc tests based on Bonferroni and Tukey’s HSD revealed that people listened to music significantly more when:

• they had access to user created playlists vs. those who did not,
• they got recommendations vs. those who did not. This is true for both the control group “A” as well as treatment “B”.

The following video summarizes how to conduct a one-way ANOVA in R

5.5 Non-parametric tests

Non-Parametric tests do not require the sampling distribution to be normally distributed (a.k.a. “assumption free tests”). These tests may be used when the variable of interest is measured on an ordinal scale or when the parametric assumptions do not hold. They often rely on ranking the data instead of analyzing the actual scores. By ranking the data, information on the magnitude of differences is lost. Thus, parametric tests are more powerful if the sampling distribution is normally distributed.

When should you use non-parametric tests?

• When your DV is measured on an ordinal scale
• When your data is better represented by the median (e.g., there are outliers that you can’t remove)
• When the assumptions of parametric tests are not met (e.g., normally distributed sampling distribution)
• You have a very small sample size (i.e., the central limit theorem does not apply)

5.5.1 Mann-Whitney U Test (a.k.a. Wilcoxon rank-sum test)

The Mann-Whitney U test is a non-parametric test of differences between groups, similar to the two sample t-test. In contrast to the two sample t-test it only requires ordinally scaled data and relies on weaker assumptions. Thus it is often useful if the assumptions of the t-test are violated, especially if the data is not on a ratio scale. The following assumptions must be fulfilled for the test to be applicable:

• The dependent variable is at least ordinally scaled (i.e. a ranking between values can be established)
• The independent variable has only two levels
• A between-subjects design is used (i.e., the subjects are not matched across conditions)

Intuitively, the test compares the frequency of low and high ranks between groups. Under the null hypothesis, the amount of high and low ranks should be roughly equal in the two groups. This is achieved through comparing the expected sum of ranks to the actual sum of ranks.

As an example, we will be using data obtained from a field experiment with random assignment. In a music download store, new releases were randomly assigned to an experimental group and sold at a reduced price (i.e., 7.95€), or a control group and sold at the standard price (9.95€). A representative sample of 102 new releases were sampled and these albums were randomly assigned to the experimental groups (i.e., 51 albums per group). The sales were tracked over one day.

Let’s load and investigate the data first:

Inspect descriptives (overall and by group).

Create boxplot and plot of means.

Figure 5.11: Boxplot

Let’s assume that one of the parametric assumptions has been violated and we needed to conduct a non-parametric test. Then, the Mann-Whitney U test is implemented in R using the function wilcox.test() . Using the ranking data as an independent variable and the listening time as a dependent variable, the test could be executed as follows:

The p-value is smaller than 0.05, which leads us to reject the null hypothesis, i.e. the test yields evidence that the new service feature leads to higher music listening times.

5.5.2 Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric test used to analyze the difference between paired observations, analogously to the paired t-test. It can be used when measurements come from the same observational units but the distributional assumptions of the paired t-test do not hold, because it does not require any assumptions about the distribution of the measurements. Since we subtract two values, however, the test requires that the dependent variable is at least interval scaled, meaning that intervals have the same meaning for different points on our measurement scale.

Under the null hypothesis \(H_0\) , the differences of the measurements should follow a symmetric distribution around 0, meaning that, on average, there is no difference between the two matched samples. \(H_1\) states that the distributions mean is non-zero.

As an example, let’s consider a slightly different experimental setup for the music download store. Imagine that new releases were either sold at a reduced price (i.e., 7.95€), or at the standard price (9.95€). Every time a customer came to the store, the prices were randomly determined for every new release. This means that the same 51 albums were either sold at the standard price or at the reduced price and this price was determined randomly. The sales were then recorded over one day. Note the difference to the previous case, where we randomly split the sample and assigned 50% of products to each condition. Now, we randomly vary prices for all albums between high and low prices.

Again, let’s assume that one of the prarametric assumptions has been violated and we needed to conduct a non-parametric test. Then the Wilcoxon signed-rank test can be performed with the same command as the Mann-Whitney U test, provided that the argument paired is set to TRUE .

Using the 95% confidence level, the result would suggest a significant effect of price on sales (i.e., p < 0.05).

5.5.3 Kruskal-Wallis test

• When the dependent variable is measured at an ordinal scale and we want to compare more than 2 means
• When the assumptions of independent ANOVA are not met (e.g., assumptions regarding the sampling distribution in small samples)

The Kruskal–Wallis test is the non-parametric counterpart of the one-way independent ANOVA. It is designed to test for significant differences in population medians when you have more than two samples (otherwise you would use the Mann-Whitney U-test). The theory is very similar to that of the Mann–Whitney U-test since it is also based on ranked data. The Kruskal-Wallis test is carried out using the kruskal.test() function. Using the same data as before, we type:

The test-statistic follows a chi-square distribution and since the test is significant (p < 0.05), we can conclude that there are significant differences in population medians. Provided that the overall effect is significant, you may perform a post hoc test to find out which groups are different. To get a first impression, we can plot the data using a boxplot:

Figure 5.12: Boxplot

To test for differences between groups, we can, for example, apply post hoc tests according to Nemenyi for pairwise multiple comparisons of the ranked data using the appropriate function from the PMCMR package.

The results reveal that there is a significant difference between the “low” and “high” promotion groups. Note that the results are different compared to the results from the parametric test above. This difference occurs because non-parametric tests have more power to detect differences between groups since we lose information by ranking the data. Thus, you should rely on parametric tests if the assumptions are met.

5.6 Categorical data

In some instances, you will be confronted with differences between proportions, rather than differences between means. For example, you may conduct an A/B-Test and wish to compare the conversion rates between two advertising campaigns. In this case, your data is binary (0 = no conversion, 1 = conversion) and the sampling distribution for such data is binomial. While binomial probabilities are difficult to calculate, we can use a Normal approximation to the binomial when n is large (>100) and the true likelihood of a 1 is not too close to 0 or 1.

Let’s use an example: assume a call center where service agents call potential customers to sell a product. We consider two call center agents:

• Service agent 1 talks to 300 customers and gets 200 of them to buy (conversion rate=2/3)
• Service agent 2 talks to 300 customers and gets 100 of them to buy (conversion rate=1/3)

As always, we load the data first:

Next, we create a table to check the relative frequencies:

We could also plot the data to visualize the frequencies using ggplot:

Figure 5.13: proportion of conversions per agent (stacked bar chart)

… or using the mosaicplot() function:

Figure 5.14: proportion of conversions per agent (mosaic plot)

5.6.1 Confidence intervals for proportions

Recall that we can use confidence intervals to determine the range of values that the true population parameter will take with a certain level of confidence based on the sample. Similar to the confidence interval for means, we can compute a confidence interval for proportions. The (1- \(\alpha\) )% confidence interval for proportions is approximately

\[ CI = p\pm z_{1-\frac{\alpha}{2}}*\sqrt{\frac{p*(1-p)}{N}} \]

where \(\sqrt{p(1-p)}\) is the equivalent to the standard deviation in the formula for the confidence interval for means. Based on the equation, it is easy to compute the confidence intervals for the conversion rates of the call center agents:

Similar to testing for differences in means, we could also ask: Is agent 1 twice as likely as agent 2 to convert a customer? Or, to state it formally:

\[H_0: \pi_1=\pi_2 \\ H_1: \pi_1\ne \pi_2\]

where \(\pi\) denotes the population parameter associated with the proportion in the respective population. One approach to test this is based on confidence intervals to estimate the difference between two populations. We can compute an approximate confidence interval for the difference between the proportion of successes in group 1 and group 2, as:

\[ CI = p_1-p_2\pm z_{1-\frac{\alpha}{2}}*\sqrt{\frac{p_1*(1-p_1)}{n_1}+\frac{p_2*(1-p_2)}{n_2}} \]

If the confidence interval includes zero, then the data does not suggest a difference between the groups. Let’s compute the confidence interval for differences in the proportions by hand first:

Now we can see that the 95% confidence interval estimate of the difference between the proportion of conversions for agent 1 and the proportion of conversions for agent 2 is between 26% and 41%. This interval tells us the range of plausible values for the difference between the two population proportions. According to this interval, zero is not a plausible value for the difference (i.e., interval does not cross zero), so we reject the null hypothesis that the population proportions are the same.

Instead of computing the intervals by hand, we could also use the prop.test() function:

Note that the prop.test() function uses a slightly different (more accurate) way to compute the confidence interval (Wilson’s score method is used). It is particularly a better approximation for smaller N. That’s why the confidence interval in the output slightly deviates from the manual computation above, which uses the Wald interval.

You can also see that the output from the prop.test() includes the results from a χ 2 test for the equality of proportions (which will be discussed below) and the associated p-value. Since the p-value is less than 0.05, we reject the null hypothesis of equal probability. Thus, the reporting would be:

The test showed that the conversion rate for agent 1 was higher by 33%. This difference is significant χ (1) = 70, p < .05 (95% CI = [0.25,0.41]).

5.6.2 Chi-square test

In the previous section, we saw how we can compute the confidence interval for the difference between proportions to decide on whether or not to reject the null hypothesis. Whenever you would like to investigate the relationship between two categorical variables, the \(\chi^2\) test may be used to test whether the variables are independent of each other. It achieves this by comparing the expected number of observations in a group to the actual values. Let’s continue with the example from the previous section. Under the null hypothesis, the two variables agent and conversion in our contingency table are independent (i.e., there is no relationship). This means that the frequency in each field will be roughly proportional to the probability of an observation being in that category, calculated under the assumption that they are independent. The difference between that expected quantity and the actual quantity can be used to construct the test statistic. The test statistic is computed as follows:

\[ \chi^2=\sum_{i=1}^{J}\frac{(f_o-f_e)^2}{f_e} \]

where \(J\) is the number of cells in the contingency table, \(f_o\) are the observed cell frequencies and \(f_e\) are the expected cell frequencies. The larger the differences, the larger the test statistic and the smaller the p-value.

The observed cell frequencies can easily be seen from the contingency table:

The expected cell frequencies can be calculated as follows:

\[ f_e=\frac{(n_r*n_c)}{n} \]

where \(n_r\) are the total observed frequencies per row, \(n_c\) are the total observed frequencies per column, and \(n\) is the total number of observations. Thus, the expected cell frequencies under the assumption of independence can be calculated as:

To sum up, these are the expected cell frequencies

… and these are the observed cell frequencies

To obtain the test statistic, we simply plug the values into the formula:

The test statistic is \(\chi^2\) distributed. The chi-square distribution is a non-symmetric distribution. Actually, there are many different chi-square distributions, one for each degree of freedom as show in the following figure.

Figure 5.15: The chi-square distribution

You can see that as the degrees of freedom increase, the chi-square curve approaches a normal distribution. To find the critical value, we need to specify the corresponding degrees of freedom, given by:

\[ df=(r-1)*(c-1) \]

where \(r\) is the number of rows and \(c\) is the number of columns in the contingency table. Recall that degrees of freedom are generally the number of values that can vary freely when calculating a statistic. In a 2 by 2 table as in our case, we have 2 variables (or two samples) with 2 levels and in each one we have 1 that vary freely. Hence, in our example the degrees of freedom can be calculated as:

Now, we can derive the critical value given the degrees of freedom and the level of confidence using the qchisq() function and test if the calculated test statistic is larger than the critical value:

Figure 5.16: Visual depiction of the test result

We could also compute the p-value using the pchisq() function, which tells us the probability of the observed cell frequencies if the null hypothesis was true (i.e., there was no association):

The test statistic can also be calculated in R directly on the contingency table with the function chisq.test() .

Since the p-value is smaller than 0.05 (i.e., the calculated test statistic is larger than the critical value), we reject H 0 that the two variables are independent.

Note that the test statistic is sensitive to the sample size. To see this, let’s assume that we have a sample of 100 observations instead of 1000 observations:

You can see that even though the proportions haven’t changed, the test is insignificant now. The following equation lets you compute a measure of the effect size, which is insensitive to sample size:

\[ \phi=\sqrt{\frac{\chi^2}{n}} \]

The following guidelines are used to determine the magnitude of the effect size (Cohen, 1988):

• 0.1 (small effect)
• 0.3 (medium effect)
• 0.5 (large effect)

In our example, we can compute the effect sizes for the large and small samples as follows:

You can see that the statistic is insensitive to the sample size.

Note that the Φ coefficient is appropriate for two dichotomous variables (resulting from a 2 x 2 table as above). If any your nominal variables has more than two categories, Cramér’s V should be used instead:

\[ V=\sqrt{\frac{\chi^2}{n*df_{min}}} \]

where \(df_{min}\) refers to the degrees of freedom associated with the variable that has fewer categories (e.g., if we have two nominal variables with 3 and 4 categories, \(df_{min}\) would be 3 - 1 = 2). The degrees of freedom need to be taken into account when judging the magnitude of the effect sizes (see e.g., here ).

Note that the correct = FALSE argument above ensures that the test statistic is computed in the same way as we have done by hand above. By default, chisq.test() applies a correction to prevent overestimation of statistical significance for small data (called the Yates’ correction). The correction is implemented by subtracting the value 0.5 from the computed difference between the observed and expected cell counts in the numerator of the test statistic. This means that the calculated test statistic will be smaller (i.e., more conservative). Although the adjustment may go too far in some instances, you should generally rely on the adjusted results, which can be computed as follows:

As you can see, the results don’t change much in our example, since the differences between the observed and expected cell frequencies are fairly large relative to the correction.

Caution is warranted when the cell counts in the contingency table are small. The usual rule of thumb is that all cell counts should be at least 5 (this may be a little too stringent though). When some cell counts are too small, you can use Fisher’s exact test using the fisher.test() function.

The Fisher test, while more conservative, also shows a significant difference between the proportions (p < 0.05). This is not surprising since the cell counts in our example are fairly large.

5.6.3 Sample size

To calculate the required sample size when comparing proportions, the power.prop.test() function can be used. For example, we could ask how large our sample needs to be if we would like to compare two groups with conversion rates of 2% and 2.5%, respectively using the conventional settings for \(\alpha\) and \(\beta\) :

The output tells us that we need 13809 observations per group to detect a difference of the desired size.

Marketing in healthcare: Improving the consumer experience

Healthcare consumers have never been more empowered than they are today. The COVID-19 pandemic forced healthcare providers to adapt quickly to continue delivering patient care, including by pivoting to digital care. Seemingly overnight, telehealth went from an industry sidenote to the primary means of seeing noncritical patients. 1 For more, see Oleg Bestsennyy, Greg Gilbert, Alex Harris, and Jennifer Rost, “ Telehealth: A quarter-trillion-dollar post-COVID-19 reality? ,” McKinsey, July 9, 2021. Healthcare consumers also saw global retailers, grocery store chains, and other disrupters demon­strate how convenient it can be to order your health products online for delivery or curbside pickup.

Consumers increasingly expect transparent, predictable, and mobile-friendly experiences, but most healthcare organizations have failed to keep up. Although healthcare providers have more options for how to spend limited marketing budgets, many have chosen to stick with the familiar traditional- and digital-marketing channels—such as billboards and ads on television, on the radio, and in magazines—and with a one-size-fits-all digital presence.

Today, health systems have the same aspirations as companies in any industry: to engender long-term relationships with their consumers. 2 For more on how leaders are thinking about brand loyalty, see Lidiya Chapple, Oren Eizenman, and Jamie Wilkie, “ Winning in loyalty ,” McKinsey, August 2, 2022. Consumer goods, retail, and e-commerce leaders set the original bar and continue to push it higher today. 3 For more, see “ E-commerce: At the center of profitable growth in consumer goods ,” McKinsey, July 5, 2022. A 2021 McKinsey survey of more than 3,000 US healthcare consumers found that satisfied patients are 28 percent less likely to switch providers. 4 McKinsey 2021 CX Provider Journey Pulse, a 15-minute patient experience survey conducted across 3,311 respondents in three US metro areas (Dallas–Fort Worth, Detroit, and Tampa Bay).

Healthcare providers that are ready to rethink their marketing approach to grow and maintain continuity of care—or work to gain back what’s been lost in recent years—have their work cut out for them. In this article, we describe three priorities for progress: align the C-suite, build capabilities methodically through carefully chosen use cases and quick wins, and measure what works. In our experience, healthcare providers that implemented changes focused on improving the consumer experience—including through marketing efforts—saw their revenue increase by up to 20 percent over five years, while costs to serve decreased by up to 30 percent.

Providers that can bring it all together can achieve lofty marketing ambitions. One regional healthcare provider that stood up an agile marketing team as part of its digital transformation prioritized increasing the share of new patients who schedule appointments digitally instead of over the phone or in person. The provider was able to build a backlog of more than 300 test ideas representing a wide variety of goals and priorities, from improving online scheduling to optimizing content on specific pages. The team prioritized half of those tests to launch in a period of only 12 months, yielding an impressive test success rate of about 50 percent. When successful tests were scaled, the healthcare provider tripled the number of new patients from digital channels, compared with before the transformation.

Align the C-suite

The chief marketing officer (CMO) is a well-established member of healthcare providers’ C-suite. However, the CMO requires active involve­ment from the leaders of other functions to deliver modern marketing efforts. These leaders include, most crucially, the chief technology officer (CTO) or chief digital officer (CDO), chief information officer (CIO), and chief financial officer (CFO).

Only when the C-suite is aligned on the importance of marketing to drive growth and continuity of care (and what it takes to do so) will the CMO have the permission and support required to effect change beyond their immediate influence. This moves the CMO role away from basic brand and communication activities that characterize marketing at most healthcare organizations today. These traditional- and digital-marketing activities depend on mass, undefined channels—such as television, newspapers, broadcast emails, and a generic website—that reach all consumers with the same messaging. This style of marketing also often fails to include a compelling call to action, such as “schedule your health visit today,” that encourages healthcare consumers to reach back out to the organization to make an appointment or speak to a specialist about their needs.

To make the leap to more sophisticated marketing capabilities, CMOs could start by defining priority use cases by both financial and nonfinancial returns on investment—from costs to improved health outcomes. The CMO could then work with the CTO or CDO to validate the technical feasibility, with the CIO to assess operational challenges and risks, and with the CFO to calculate the financial ROI. The true power of this cross-functional collaboration can be seen in organizations that, for example, have worked in concert to design and build a single source of truth for consumer data, which is critical to enabling use cases related to advanced marketing capabilities, such as personalization.

Many organizations outside of healthcare—and increasingly within healthcare—have adopted agile marketing to drive high-velocity testing across digital- and traditional-marketing channels. 5 For more, see David Edelman, Jason Heller, and Steven Spittaels, “ Agile marketing: A step-by-step guide ,” McKinsey, November 9, 2016. Speed to market requires quick handoffs, cross-functional collaboration, and enterprise-wide transparency. For example, a healthcare provider could focus on improving its new-patient appointment cancellation rate by A/B testing appointment reminders to determine what works best in terms of number of touchpoints, messaging, and cadence.

The test brief—a standard feature of agile marketing typically reviewed by leads from marketing, finance, and technology—can help align the CMO, CFO, and CTO and enable them to compile a more persuasive case for the CEO and other company leaders. A test brief is a document with codified creative, technical, and measurement details for an agile marketing test. It provides the organization with the details it needs to quantify the number of days (or sometimes hours) necessary to launch a test, as well as to estimate the effort required to design, create, build, and deploy it.

Build sophisticated capabilities by choosing use cases wisely

Today, most healthcare providers have critical capability gaps that stand in the way of mounting an end-to-end, personalized consumer journey:

• A disjointed consumer experience and lack of personalization . Multichannel consumer touchpoints can lead to fragmented, impersonal experiences because of the lack of integration between consumer data and omnichannel engagement platforms.
• Siloed systems . Silos result in a limited ability to track current and potential consumers across channels and devices, as well as no organization-wide access to consumer data tracking tools.
• A lack of consumer-centric data. A lack of data leads to channels without access to a real-time, 360-degree view of consumer care needs and to clinical data that is not augmented with nonclinical data.

The theme of these challenges is fragmented information. Marketers increasingly require hands-on, real-time access to technology. Next-level healthcare marketing is built on a robust, integrated technology stack—and reaching full maturity in marketing-technology capabilities is not a short-term undertaking. No marketing function operates with an unlimited budget, meaning the marketing team will need to prioritize use cases that can build the department’s muscle and momentum.

In our experience, patient scheduling tends to be a priority growth lever for healthcare providers and systems. This makes sense considering that being able to easily schedule an appointment is a critical step in the consumer healthcare journey. Use cases that improve scheduling would thus rise to the top of the priority list and the budgeting conversation. For other providers, use cases could include better patient communication management to facilitate follow-up care.

Reliable collaboration between the marketing and technology teams is paramount to achieving seamless healthcare consumer journeys.

Given that these marketing touchpoints span the entire healthcare consumer experience, from learning about a provider to scheduling a visit and receiving care and follow-up care, reliable collab­oration between the marketing and technology teams is paramount to achieving seamless healthcare consumer journeys (exhibit). And to deliver it, an integrated technology stack (across both digital and traditional channels) is essential.

Measure what works

Marketers use the term “attribution” to describe the process of measuring the effects of marketing efforts and the rate at which they convert consumers to achieve desired consumer outcomes. Consumer outcomes are not the only measure of ROI; the ulti­mate goal is an improved patient experience and potentially improved health outcomes. But attri­bution can serve as a crucial indicator. A simple example would be analyzing the click-through rates of marketing emails to determine what messaging is most effective. Attribution analysis is a critical component of measuring the financial ROI of spending on digital marketing in healthcare—and most providers and systems today are early in the process of building out their marketing-attribution capabilities. As a result, the CFO and CMO are often misaligned on what works, which can greatly affect marketing’s budget allocation.

Common marketing-attribution methods and their application to healthcare

Three attribution tools in particular can help healthcare providers effectively mea­sure what works.

Marketing mix modeling. Regression analysis can help marketers understand the specific effects of every interaction with healthcare consumers. It uses his­torical data to estimate the effects of a particular marketing tactic—such as a radio ad or a personalized email—on patient appointment volume. This method can be used to measure tactics in both traditional marketing channels (such as TV and radio) and digital channels (such as paid search and paid social).

This method does have a few limitations, however, including limited measurement granularity, given that it mostly evaluates media effectiveness at a high level for annual or monthly budget allocation. How­ever, A/B testing can be instrumental in improving revenue forecasting with better estimations of revenue and ROI.

Anonymized data. Anonymized data processing can encrypt personal iden­tifying information (PII) or protected health information (PHI) to protect healthcare consumer information while still enabling marketers to send personalized messages, run longitudinal analyses, and port data securely across technology platforms. For example, digital healthcare platforms can encrypt the individual data fields in a healthcare consumer’s record while still sharing journey-specific marketing communications.

Encryption is vital to healthcare providers’ ability to communicate with healthcare con­sumers and their families based on their own treatment journey and needs. It also transforms the experience of consumers by enabling them to share their data privately and receive offline support outside of the clinical setting. There are limits, of course, to the personalization that is available, and it requires rigorous testing to be compliant with industry regu­lations. Encryption at scale is also a massive undertaking that requires a robust core data infrastructure and operations that enable speed, accuracy, and security. Few providers today encrypt PII as part of processing anonymized consu­mer data, and the few that are doing so are not currently using it to drive personalized consumer experiences.

A/B testing. The A/B (or incrementality) testing method is one of the most popular and well-known marketing tactics. By measuring the difference made by discrete details, such as the performance of an email when the audience is addressed as “healthcare professionals” versus “doctors,” A/B testing can offer marketers an abun­dance of information about consumer conversion and sales—and which conver­sions would not have happened without the marketing campaign.

The A/B testing methodology is widely adopted because its results are accurate and easy to understand, and they don’t require complex analytics capabilities. However, many healthcare providers don’t employ the methodology well or at all. Those that can do it well can see outsize bottom-line impact. One healthcare provider, for example, embarked on an end-to-end consumer transformation journey and deployed more than 100 agile digital A/B tests to optimize patient experience across its locations. In one such initiative, the team improved its conversion rate by 0.15 percentage points—or $2.4 million in scaled revenue impact—by personalizing and targeting communications to different healthcare consumer segments.

Successful attribution is founded on the cross-departmental collaboration modeled and facilitated by the C-suite. Specifically, attribution requires marketing and IT to partner closely to, for example, provide marketing the ability to connect how individual digital channels (such as paid media or the provider’s website) are driving new patient appointments booked via phone calls or online.

Industry regulations require that healthcare data be anonymized, but that does not preclude it from being measured. (For more on three of the most ubiquitous marketing-attribution methods that meet the industry’s criteria for healthcare consumer data, see sidebar, “Common marketing-attribution methods and their application to healthcare.”)

It’s not surprising that healthcare providers increas­ingly see marketing as a growth driver. Modern tech­nology, approaches, and collaborative structures can help CMOs pivot the perception and performance of the marketing function away from being a cost center and toward measurable bottom-line gains while improving the healthcare consumer experience.

Adam Broitman is a partner in McKinsey’s New York office, where Michelle Jimenez is a senior expert. Julie Lowrie is a partner in the Atlanta office.

The authors wish to thank Jessica Buchter, Jenny Cordina, and Eli Stein for their contributions to this article.

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