Paired Samples t-test

What is the t-test for paired samples?

The dependent samples T-test (or paired samples t-test) is a statistical test that determines whether there is a difference between two dependent groups or samples.

The dependent samples T-test, or also known as the dependent t-test, tests whether the mean values of two dependent groups differ significantly from each other.

Why do you need the dependent t-Test?

You need the paired t-test whenever you survey the same group or sample at two points in time. For example, you might be interested in whether a rehabilitation program has a positive effect on physical fitness. Since you can't ask all the people who go to rehab, you use a random sample. You can then use the paired t-test to infer the population from the sample.

What are dependent or paired samples?

In dependent samples, these measured values are available in pairs. The pairs result from repeated measurements, parallelization or matching. This can be the case, for example, in longitudinal studies with several measurement points (time series analyses) or in intervention studies with experimental designs (before-after measurement).

An example of dependent sampling is when the weight of a group of people is measured at two points in time. A person can then be uniquely assigned a weight at the first and second measurement time points and the difference in the measured values can be calculated in each case. If more than two measurement times are available, ANOVA with repeated measures is used.

What is the advantage of a dependent t-test over an independent t-test?

The question of whether to use a dependent t-test or an independent t-test is, of course, already determined as part of the study design, and it is not possible to arbitrarily use either one test or the other. Therefore, the question is rather which type of study makes more sense:

• Conducting a study with one group of participants who are measured twice.
• To conduct a study with two separate groups of participants, each measured once.

The major advantage of a repeated-measures design that then uses the paired t-test is that individual differences between participants can be eliminated. This means that the probability of detecting a (statistically significant) difference, if one exists, is higher with the paired t-test than with the independent t-test.

Example of a paired sample t-test

The t-test for dependent samples has numerous applications, here are three examples.

Research Question and hypotheses

In order to be able to calculate a t-test for dependent samples, you first need to define a question and the hypotheses

Research Question

In a t-test for dependent samples, the general question is: Is there a statistically significant difference between the mean value of two dependent groups?

The questions for the above examples arise as follows:

• Does the new drug help to increase memory performance?
• Does the newly introduced lubricant have an influence on downtimes?
• Has the health consciousness of the German population changed between 2010 and 2015?

Hypotheses

Now the hypothesis can be derived from the question. In the hypothesis, a preliminary, i.e. unsubstantiated, assumption is made which is to be tested. In the case of a t test for dependent samples, the hypotheses are:

• Null hypothesis H₀: The mean value of the two dependent groups is equal.
• Alternative hypothesis H₁: The mean values of the two dependent groups are different.

Assumptions paired t-Test

Of course, the assumptions must be checked before calculating the paired t-test. If the prerequisites 2 and 3 (listed below) are not fulfilled, the Wilcoxon test must be used. The Wilcoxon test is the non-parametric counterpart of the paired t-test.

How does a paired t-test work?

In the paired t-test, the difference is calculated from each paired case. The mean value is then calculated from these differences. Depending on how large the mean value is and how large the standard error of the mean value is, a statement is then made as to how likely it is that this result has arisen by chance.

Calculate t-test for paired samples

For the calculation of the t-test for dependent samples, the difference of each pair from the two groups is first formed. From the resulting differences, the mean value x̄_diff is then calculated.

The calculation of the test statistics t is now equal to the t test for one sample. If there is no difference between the two groups, the mean value of the difference x̄_diff is zero. So the question is, is there a difference between x̄_diff and zero. The test statistic t for the t-test for dependent samples is then calculated as

where is the standard error of the mean value

• = Difference between the groups
• = Mean value of the difference between the two groups
• = Sample size
• = Standard deviation
• = Estimated standard error of mean

Effect size dependent t-test

The indication of the effect size is very important for empirical studies. To make a statement about the effect size in a t-test for dependent samples, the following equation can be used

In general, it can be said about the effect size:

• Effect size r: 0,2 small Effect
• Effect size r: 0,5 medium Effect
• Effect size r: 0,8 large Effect

Calculate with DATAtab

In the paired t-test example it is examined whether the summer holidays have an effect on the physical fitness of statistics students.

So the question is: Do the summer holidays have an effect on the physical fitness of statistics students? For this purpose, a fitness test is carried out once before and once after the holidays for 10 statistics students (2 measurement points).

Since two test results always come from one student, there is a dependency between the two samples. Therefore the paired t-test is used.

After copying the upper table into the dependent t-test calculator you can calculate the t-test. The results are as follows:

Statistics

Correlation

Paired t-test

95% confidence interval of the difference

Interpret t-test for dependent samples

If the calculated p-value is smaller than the specified significance level (usually 5%), the null hypothesis is rejected, otherwise it is not rejected. For the upper example, you can report the results as follows:

The score of the variable before the vacations had lower values (M = 46.4, SD = 11.452) than the score of the variable after the vacations (M = 49.7, SD = 14.095). A t-test for dependent samples showed that this difference was not statistically significant, t(9) = -1.392 p = .197, 95% confidence interval [-8.664, 2.064].

This results in a p-value of 0.197 which is above the defined significance level of 0.05. The t-test result is therefore not significant and the null hypothesis is not rejected.