When comparing groups in your data, you can have either independent or dependent samples. The type of samples in your design impacts sample size requirements, statistical power, the proper analysis, and even your study’s costs. Understanding the implications of each type of sample can help you design a better study.

For example, we often think about increasing sample size to enhance the statistical power of your test. A larger sample size increases your chance of detecting an effect that exists in the population. That’s a great approach! However, strategically using dependent samples can also increase your test’s statistical power without the expense of increasing your sample size.

In this post, I’ll define independent and dependent samples, explain their pros and cons, highlight the appropriate analyses for each type, and illustrate how dependent groups can increase your statistical power.

A quick note about terminology. In a study, you measure an outcome variable for people or objects. I’ll refer to subjects throughout this post to refer to both cases. Additionally, I also use samples and groups synonymously. For example, the term “dependent samples” means the same thing as dependent groups.

## Independent Samples vs. Dependent Samples

Hypothesis tests and statistical modeling that compare groups have assumptions about the nature of those groups. Choosing the correct test or model depends on knowing which type of group you have. Additionally, when designing your study, selecting the best type can help you tailor the design to meet your needs.

### Independent samples

In independent samples, subjects in one group do not provide information about subjects in other groups. Each group contains different subjects and there is no meaningful way to pair them. Independent groups are more common in hypothesis testing.

For example, the following experiments use independent samples:

- A medication trial has a control group and a treatment group that contain different subjects.
- A study assesses the strength of a part made from different alloys. Each alloy sample contains different parts.

Studies that use independent samples estimate between-subject effects. These effects are the differences between groups, such as the mean difference. For example, in the medication study, the effect is the mean difference between the treatment and control groups. The focus is on comparing group properties rather than individuals. The sample size for this type of study is the total number of subjects in all groups.

### Dependent samples

In dependent samples, subjects in one group *do* provide information about subjects in other groups. The groups contain either the same set of subjects or different subjects that the analysts have paired meaningfully.

Groups are frequently dependent because they contain the same subjects—that’s the most common example. However, that’s not always the case. Groups with different subjects can be dependent samples if the subjects in one group provide information about the subjects in the other group. For example, statisticians often consider different samples that include pairs of siblings to be dependent because one sibling can provide information about another sibling for some measurements. Other studies use matched pairs. In these studies, the researchers deliberately pair subjects with very similar characteristics. While matched pairs are different people, the statistical analysis treats them as the same person because they are intentionally very similar.

For example, the following studies use dependent samples:

- A training program assessment takes pretest and posttest scores from the same group of people.
- A paint durability study applies different types of paint to portions of the same wooden boards. All paint types on the same board are considered paired.

Studies that use dependent samples estimate within-subject effects. These effects are the differences between paired subjects, such as the subjects’ mean change. For example, the training program assessment estimates the mean change for subjects from the pretest to the posttest. The emphasis is on the differences between paired subjects. The sample size for this type of study is the number of pairs.

Terms such as paired, repeated measurements, within-subject effects, matched pairs, and pretest/posttest indicate that the groups are dependent.

### Groups in Datasets

Understanding how researchers record the data can also provide hints about the types of groups. For example, the data look similar in the two worksheets below.

However, the Subject ID column in the second dataset unequivocally indicates these are paired data—dependent groups. This column reveals that each row pertains to one subject and that there are multiple observations for each subject. While it’s possible to have dependent samples without an identifier column, analysts typically include them.

For dependent groups, the focus is on the differences between measurements for each subject. Consequently, if you can meaningfully subtract values in a row, that’s a sure sign of dependency. For example, each row represents one individual in the paired dataset, so assessing the difference between values makes sense.

Conversely, for the independent samples dataset, each group contains a different set of individuals that the researchers chose randomly. Each row in this dataset does not pertain to a single subject. Consequently, it does not make sense to subtract the values between pairs of random people.

## Pros and Cons of Independent and Dependent Samples

When thinking about comparing groups, you frequently picture independent groups. For instance, when you imagine comparing a treatment group to a control group, you’re probably assuming these groups contain different subjects. However, by understanding the pros and cons of independent and dependent samples, you can design a study to meet your needs more effectively. The best choice depends on the subject matter and requirements of your experiment. Consider the following while deciding your approach.

### Advantages of Independent Samples

When your study uses independent samples, you test each subject once. When you’re working with human subjects, a single test can be advantageous for several reasons. With a single assessment per person, you don’t need to worry about subjects learning how to perform better, getting bored with multiple tests, and how the passage of time affects each person. By testing subjects once, you can rule out various time and order effects that can influence how scores change.

When you are testing physical items, you only need to test each item once. If the testing damages or alters the items, it’s not possible to test them multiple times.

### Disadvantages of Independent Samples

Because each group contains different subjects, there can be a wide variety of subject specific factors that influence how they respond to the test. While random assignment to groups can reduce systematic differences between groups, these subject specific factors are not controlled.

Differences between participants in the groups can affect the results. Statisticians refer to these differences as participant variables and they include age, gender, and social background, among many other possibilities.

The additional variability that participant variables create reduces statistical power. You generally need larger sample sizes with independent samples.

### Advantages of Dependent Samples

The primary advantage of dependent samples is that you measure the same subjects across different conditions, which allows them to be their own controls. They have the same unique mix of participant variables during all measurements, removing them as sources of variation. Keep this lower variability in mind during my practical demonstration later in this post!

For example, in a pretest/posttest analysis, you will see how each subject reacts to both tests. This method allows the study to focus on the changes within individuals rather than differences between groups of different people.

The net effect is a gain in statistical power. You generally need smaller sample sizes with dependent groups. Additionally, reducing the sample size can decrease a study’s costs, which is particularly helpful when it is difficult or expensive to obtain subjects.

### Disadvantages of Dependent Samples

When working with human subjects, you will need to test them multiple times with dependent samples. During repeated testing, subjects can learn more about the tests and figure out how to improve their scores; they might get bored with being tested multiple times; or their test scores might change as a natural result of time passing. In other words, the multiple testing and the passage of time become factors than can influence the measurement, potentially making it challenging to isolate the treatment’s effect.

For example, if the test scores for the training program increase from the pretest to the posttest, the training program might not cause the change. Instead, participants might be learning how to take the test better!

Researchers can mitigate some of these problems. For example, they can include control groups for comparison and change the order of tests for subsets of subjects. However, in general, designs that use dependent groups make it easier for alternatives to explain the changes.

In some cases, using dependent samples is not possible. For example, with destructive testing of material objects, you can only test them once!

As a researcher, weigh the benefits and drawbacks of both types of samples. Some types of research will lend themselves to one approach or the other.

## Types of Statistical Analyses for Independent and Dependent Groups

After choosing the type of samples and conducting the experiment, you need to use the correct statistical analysis. The table displays pairs of related analyses for independent and dependent samples.

Several notes about the table.

While analyses for dependent groups typically focus on individual changes, McNemar’s test is an exception. That test compares the overall proportions of two dependent groups.

Regression and ANOVA can model both independent and dependent samples. It’s just a matter of specifying the correct model.

**Related posts**: Repeated Measures ANOVA and How to do t-tests

## Example of Dependent Groups and their Extra Statistical Power

I’m closing with an example that illustrates the extra statistical power that dependent samples can provide. Imagine two studies that, by an amazing coincidence, obtain the same measurements exactly. The only difference is that one has independent groups, while the other has dependent groups.

It should go without saying, but I’ll say it anyway—you will never run a 2-sample t-test and a paired t-test on the same dataset in practice. The two designs are entirely incompatible. However, I’m going to do just that to illustrate the difference in power.

For this experiment, we’re assessing a fictional drug that supposedly increases IQ scores. One experiment uses a control group and a treatment group that have different subjects. The other uses the same set of subjects for a pretest and a posttest. You can download the CSV dataset to try it yourself: IndDepSamples.

First, let’s analyze the dataset as a 2-sample t-test.

Drats! The treatment group has a higher mean than the control group, but the results are not statistically significant.

Ok, now let’s use the paired t-test.

Hurray! The posttest scores are higher and the results are significant!

The data are the same for both analyses and the differences between samples are the same (-11.62). The 2-sample t-test uses a sample size of 30 (two groups with 15 per group), while the paired t-test has only 15 subjects, but the researchers test them twice. Why is the paired t-test with the dependent samples statistically significant while the 2-sample t-test with independent samples is not significant?

## Understanding the Different Results

The analyses make different assumptions about the nature of the samples. For the 2-sample t-test, the two groups contain entirely different individuals. While the treatment group has a higher mean IQ score than the control group, we don’t know each subject’s starting score because there was no pretest. Perhaps the treatment group started with higher scores by chance? We don’t know for sure if anyone’s scores increased after taking the drug. This uncertainty reduces the test’s power.

On the other hand, the paired t-test assumes that the pretest and posttest scores are from the same people. From the data, we *know* all 15 participants saw their scores increase from the pretest to the posttest by an average of 11.63 points. That’s a pretty powerful contrast to the independent samples where we don’t know if any IQ scores increased during the study. While we can be reasonably confident that their scores increased, we’re not sure why. It’s possible that their experience taking the pretest helped them do better on the posttest. Tradeoffs! Maybe next time we’ll include a control group and perform repeated measures ANOVA.

For a more statistical explanation, think back to what I said about dependent samples eliminating participant variables as a source of variability. You can see the reduced variability in the statistical output. The 2-sample t-test uses the pooled standard deviation for both groups, which the output indicates is about 19. However, the paired t-test uses the standard deviation of the *differences*, and that is much lower at only 6.81. In t-tests, variability is noise that can obscure the signal. Consequently, higher variability reduces statistical power. For more information on this aspect, read my post about how t-tests work.

If you’re planning your next study, consider whether you should use independent or dependent samples. Throughout this post, you learned that each approach has its own benefits and drawbacks. Determine which one works best for your study.

Read more about the related topic of independent and identically distributed (IID) data.

Ingrid Holzapfel says

Here’s some questions;

Statistics are used especially in psychology, sociology and economics. Why? Consider, in psychology, why it is paired with experimental method

Jim Frost says

Hi Ingrid,

For your question, I’m going to assume you’re referring to inferential statistics because those methods really extend the usefulness of experiments.

Inferential statistics are a set of analyses that allow you to use sample data to draw conclusions about an entire population. These procedures are very important for scientific studies, including in the areas you mention. Imagine a psychology study that is looking at a treatment for a psychological condition. For this study, the scientists will gather a sample of study participants. The scientists don’t want to know whether the treatment works for only this relatively small group of people in the study. That’s not very helpful for everyone else! Instead, these scientists want to understand how the treatment will work in a larger population. They can use inferential statistics to take the results from their sample and generalize them to an entire population. That makes their study much more useful!

By pairing these statistical procedures with experiments, it allows the researchers to draw conclusions about how effective the treatment is for an entire population, not just the small group of subjects in the experiment.

Thanks for writing with the great question!

Hugo Currie says

Statistics are often seen as untrustworthy, and used to prove whatever a person may want to prove. What are some of the common suspicions about the use of statistics?

Jim Frost says

Hi Hugo,

Whenever I read about someone’s statistical analysis, my first concern is about how they collected their data. Did they collect their data from a group of friends who already share their opinion? Or, did they randomly sample people? Data collecting can completely change the results of the analysis.

After data collection, I’d want to understand the specifics about how they analyzed their data. Many analyses can be twisted or misused to give whatever answer the person wants. However, if you collect data properly and use analyses properly, statistics tend to give the correct answers. But, it’s important to understand all the details about how they arrive at their conclusions.

Finally, the best way to protect yourself from someone else misusing statistics is to become knowledgeable in statistics yourself. By understanding statistics, you’ll be able to know what to look for to see through someone else’s statistical trickery!

Thanks for writing and best wishes!

JUDITH LIPOPAI MUSEVE says

Thank you Jim. I have benefited from your article

Antonio Padua Jr says

Thanks for that Jim. I will give the article a read with that in mind. Tony

Jeremy says

Thank you, Jim. This is an excellent refresher on t-tests and introduces new terminology, i.e., dependent and independent samples. It’s always good to look “under the hood” and see how things work.

Tony Padua says

Hi Jim, before I try to wade through your lengthy article, right away I feel this is not the way my study of statistics is going.

My direction, post college statistics, is now data science, machine learning supervised learning alogorithms, as with logistic regression. What little I’ve learned about unsupervised learning is a single project using k-means clustering.

What I know about data sets are pre-processing, data wrangling, modeling and cross validation.

Part of the data wrangling process includes choosing among variables a dependent target variable.

When I saw your title Dependent Samples, I need to understand what is the value here for me.

Let me guess you write for purposes of medical research, not analytics in business.

Thank you for your time.

Tony Padua

Jim Frost says

Hi Tony,

Thanks for your thoughts. Every time I write a blog post, I have no doubt that it will be more helpful for some and less helpful for others. Everyone has their own unique needs. Such is life. I’m sorry this post didn’t help you specifically, but I’m sure it helped others.

I write about all sorts of topics that will be helpful for people learning statistics in a broad range of contexts, including business and machine learning. In this post, some of the content focuses on issues for designing experiments. While that seems to be relevant to scientific fields, many businesses also design experiments. Additionally, I’m sure businesses collect multiple measurements on subjects. This post addresses how that affects the analyses you must use, how to interpret the results, as well as the benefits and risks in terms of explaining the results. This information should be useful in many contexts, such as businesses.

At some point, I plan to write about analyses such as k-means clustering.

Finally, every time I write a post, I include what it’ll be about and the benefits of reading the post right at the beginning. This information allows everyone to decide for themselves if they should read it.

Cheers

Fida Mahy U Deen says

Thank You for such a great effort in the area of statistics.

Well done!

Keep it up!

Best Wishes!

Weldesenbet says

Thank you very much Mr. Jim for your effort of sharing your knowledge about the concepts of statistics area.