What is the Dependent Sample T-Test?
The dependent sample t-test is a member of the t-test family. All tests from the t-test family compare one or more mean scores with each other. The t-test family is based on the t-distribution, sometimes also called Student’s t. Student is the pseudonym used by W. S. Gosset in 1908 to publish the t-distribution based on his empirical findings on the height and the length of the left middle finger of criminals in a local prison.
Within the t-test family the dependent sample T-Test compares the mean scores of one group in different measurements. It is also called the paired t-test, because measurements from one group must be paired with measurements from the other group. The dependent sample t-test is used when the observations or cases in one sample are linked with the cases in the other sample. This is typically the case when repeated measures are taken, or when analyzing similar units or comparable specimen.
Making repeated measurements or pairing observations is very common when conducting experiments or making observations with time lags. Pairing the measured data points is typically done in order to exclude any cofounding or hidden factors (cf. partial correlation). It is also often used to account for individual differences in the baselines, for example pre-existing conditions in clinical research. Consider the example of a drug trial where the participants have individual differences that might have an impact on the outcome of the trial. The typical drug trial splits all participants into a control and the treatment group. The dependent sample t-test can correct for the individual differences or baselines by pairing comparable participants from the treatment and control group. Typical grouping variables are easily obtainable statistics such as age, weight, height, blood pressure. Thus the dependent-sample t-test analyzes the effect of the drug while excluding the influence of different baseline levels of health when the trial began.
Pairing data points and conducting the dependent sample t-test is a common approach to establish causality in a chain of effects. However, the dependent sample t-test only signifies the difference between two mean scores and a direction of change—it does not automatically give a directionality of cause and effect.
Since the pairing is explicitly defined and thus new information added to the data, paired data can always be analyzed with the independent sample t-test as well, but not vice versa. A typical guideline to determine whether the dependent sample t-test is the right test is to answer the following three questions:
- Is there a direct relationship between each pair of observations (e.g., before vs. after scores on the same subject)?
- Are the observations of the data points definitely not random (e.g., they must not be randomly selected specimen of the same population)?
- Do both samples have to have the same number of data points?
If the answer is yes to all three of these questions the dependent sample t-test is the right test, otherwise use the independent sample t-test. In statistical terms the dependent samples t-test requires that the within-group variation, which is a source of measurement errors, can be identified and excluded from the analysis.
The Dependent Sample T-Test in SPSS
Our research question for the dependent sample t-test is as follows:
Do students aptitude test1 scores differ from their aptitude test2 scores?
The dependent samples t-test is found in Analyze/Compare Means/Paired Samples T Test…
We need to specify the paired variable in the dialog box for the dependent samples t-test. We need to inform SPSS what is the before and after measurement. SPSS automatically assumes that the second dimension of the pairing is the case number, i.e. that case number 1 is a pair of measurements between variable 1 and 2.
Although we could specify multiple dependent samples t-test that are executed at the same time, our example only looks at the first and the second aptitude test. Thus we drag & drop ‘Aptitude Test 1‘ into the cell of pair 1 and variable 1, and ‘Aptitude Test 2′ into the cell pair 1 and variable 2. The Options… button allows to define the width of the control interval and how missing values are managed. We leave all settings as they are.
The Output of the Dependent Sample T-Test
The output of the dependent samples t-test consists of only three tables. The first table shows the descriptive statistics of the before and after variable. Here we see that on average the aptitude test score decreased from 29.44 to 24.67, not accounting for individual differences in the baseline.
The second table in the output of the dependent samples t-test shows the correlation analysis between the paired variables. This result is not part of any of the other t-tests in the t-test family. The purpose of the correlation analysis is to show whether the use of dependent samples can increase the reliability of the analysis compared to the independent samples t-test. The higher the correlation coefficient the stronger the strength of association between both variable and thus the higher the impact of pairing the data compared to conducting an unpaired t-test. In our example the Pearson’s bivariate correlation analysis finds a medium negative correlation that is significant with p < 0.001. We can therefore assume that pairing our data has a positive impact on the power of t-test.
The third table contains the actual dependent sample t-statistics. The table includes the mean of the differences Before-After, the standard deviation of that difference, the standard error, the t-value, the degrees of freedom, the p-value and the confidence interval for the difference of the mean scores. Unlike the independent samples t-test it does not include the Levene Test for homoscedasticity.
In our example the dependent samples t-test shows that aptitude scores decreased on average by 4.766 with a standard deviation of 14.939. This results in a t-value of t = 3.300 with 106 degrees of freedom. The t-test is highly significant with p = 0.001. The 95% confidence interval for the average difference of the mean is [1.903, 7.630].
An example of a possible write-up would read as follows:
The dependent samples t-test showed an average reduction in achieved aptitude scores by 4.766 scores in our sample of 107 students. The dependent sample t-test was used to account for individual differences in the aptitude of the students. The observed decrease is highly significant (p = 0.001).Therefore, we can reject the null hypothesis that there is no difference in means and can assume with 99.9% confidence that the observed reduction in aptitude score can also be found in the general population. With a 5% error rate we can assume that the difference in aptitude scores will be between 1.903 and 7.630.
Syntax
T-TEST PAIRS=Apt1 WITH Apt2 (PAIRED)
/CRITERIA=CI(.9500)
/MISSING=ANALYSIS.
