Dependent samples t-test is used when you have a within-subjects design with two conditions. Data collected in each condition is related to the other condition, i.e. not independent, because the participants in each condition are the same. The procedures for the dependent samples t-test are very similar to those used for the independent samples t-test. Groups of data are still being compared and a mean difference is calculated for the conditions; however, the data for both conditions come from the same set of participants.
Step 1: State the hypotheses
When setting up the two-tailed hypotheses for dependent samples t-test, population mean difference is used and are represented by µD . The population mean difference is calculated by subtracting the scores in one condition from the scores in the second condition for each participant (i.e. getting a difference score for each participant) and then creating a mean of those difference scores. Oftentimes, this is set up where you subtract the data in the control condition from the data in the experimental condition. A sample mean difference is calculated in the video at the end of the chapter.
The null hypothesis predicts there is no difference between the conditions, i.e. the mean difference will be equal to zero. This indicates that the data from one condition will be equal to the data from another condition; thus, when you subtract one from the other, you get zero. The alternative hypothesis predicts there is a difference between conditions, i.e. the mean difference will not be equal to zero.
H0: µD = 0
H1: µD ≠ 0
If this were a one-tailed test and the researcher expected the experimental condition to have larger values than the control condition, the alternative hypothesis would have a > sign, instead of not equal. This would mean that when you subtract the control condition data point from the experimental condition data point, you should consistently get a positive value for each participant. The null hypothesis would have a < sign to cover all other possible values.
H0: µD < 0
H1: µD > 0
For a one-tailed test where the researcher expected the experimental condition to have smaller values than the control condition, the the alternative hypothesis would have a < sign and the null hypothesis would have a > sign
H0: µD > 0
H1: µD < 0
Step 2: Determine the critical region
Using a t-test table (also available on page 539 of the Morling book) and the degrees of freedom (df) for the sample, a critical value can be determined for the analysis. This is the same procedure that we used for independent t-tests.
For dependent t-test, df = n-1, where n equals the number of participants in the study
Step 3: Calculate your statistics
First, the sample variance (s2) needs to be calculated. You will need to calculate SS for the data as well. A formula for calculating SS for the difference scores is below where D represents the difference in scores between the two conditions. SS is then used to calculate the sample variance.
NOTE: You may also see formulas for dependent samples t-test that say nD, instead of just n. This is to make the distinction that you should be using an n equal to the number of difference scores you have, not the number of individual data points. The number of difference scores you have should also equal the number of participants you have. In this example, you have 3 difference scores and 3 participants; however, you have 6 data points because each participant contributes 2. The n value (or nD value) you should use would be 3, not 6.
Next, the estimated standard error needs to be calculated.
Finally, t can be calculated. Just like with the independent samples t-test, the second part of the numerator (µD) should be equal to zero and can be ignored.
The same concepts for how mean difference, variance, and sample size affect the calculated t value also apply. Mean difference is in the numerator of the equation. When you increase the numerator of a fraction, the overall value of the fraction increases. For example, compare these two simple fractions 1/3 and 2/3. The denominator of the fractions is the same: 3. The numerators are different. As you increase the numerator of the fraction, the overall value increases too, i.e. 2/3 is greater than 1/3. Increasing the mean difference follows the same rule. If you are increasing the mean difference in the numerator, then the overall value of the fraction (i.e. the calculated t value) also increases.
Variance and sample size affect the denominator of the equation. When you increase variance, you decrease the chance of finding a significant t value; when you decrease variance, you increase the chance of finding a significant t value. Increasing sample size is another way to change your chance of getting a significant t. As you know, as you increase your sample size, you also decrease variability. Thus, increasing sample size has the opposite effect of increasing variability. As you increase sample size, you increase your chance of finding a significant t value.
Step 4: State a decision
Compare the calculated t to the critical value for t to determine whether to reject or fail to reject the null hypothesis. If the calculated t falls in your critical region (in the tails of the distribution beyond your critical value), then you reject the null hypothesis. If your calculated t does not fall within your critical region, you fail to reject the null hypothesis.
If you decide to reject the null hypothesis, an effect size should be calculated. Two different methods can be used for effect size and each should give you approximately the same answer:Cohen’s d and r2.
Cohen’s d provides a standardized measure of the mean difference, similar to the way a z-score can be used as a standardized number for a sample mean.
With Cohen’s d, you use the following to interpret the calculation.
| Effect Size | |
| 0.20 | Small |
| 0.50 | Medium |
| 0.80 | Large |
Alternatively, you can use r2, which measures the percentage of variance in the population accounted for by the sample statistics.
With r2, you use the following to interpret the calculation.
| r2 | Effect Size |
| 0.001 | Small |
| 0.09 | Medium |
| 0.25 | Large |
Next, you should write an interpretation of the results in APA style. Your interpretation should minimally include:
- the independent and dependent variables (i.e. what was being tested in the study)
- a narrative explanation of the statistics, and
- the statistical copy presented in APA style: t(df) = calculated t, p < or > alpha, effect size