It turns out that the paired-samples t-test is a special case of the 1-sample t-test. What makes it special is that the sample consists of difference scores instead of a regular variable. Difference scores are calculated from the difference between 2 scores from paired variables. Paired variables come in 3 varieties:
1. Longitudinal Variables (Same person, same variable measured twice): Suppose that you have a sample of people about to enter psychotherapy. You measure depression in each person before therapy begins. After therapy is over, you measure depression in each person again. You'd like to test whether the therapy reduced depression. To do this, you will compute the difference scores for each person by subtracting each person's pre-therapy depression score from the post-therapy depression score.

Person	Pre-Therapy	Post-Therapy	Difference (D)
1	50	51	1
2	54	50	-4
3	75	61	-14
4	71	59	-12
5	65	55	-10

2. Within-Person Contrast (Same person, different variables): Suppose that you have sample of people who rate how much they like math and how much they like literature. You want to know if people tend to like one academic domain better than the other. In order for this to work, both variables must be measured on the same scale. Let's say they rate both domains on a scale of 1 to 10. Then you subtract each person's math score from the corresponding literature score. Alternatively, you could subtract the literature score from the math score. In this case, it does not really matter which score is subtracted from the other. But, in order to interpret the results, you have to remember which score was subtracted from the other.

3. Dyadic Data (Different people, same variable): Suppose that you wish to know if supervisors perceive the work environment to be more positive than supervisees perceive it to be. You find a sample of supervisors and 1 supervisee working under each supervisor and you ask each person to rate the positivity of the work environment on a scale of 0 to 100. Subtract each supervisee's score from the corresponding supervisor's score to compute the difference score.

There are many different kinds of dyads. For some kinds, it is clear which member of the dyad goes in which column. These are called distinguishable dyads. Here are some examples:

For other kinds of dyads, it is arbitrary (chosen without a clear rationale) which person goes in which column. These are called indistinguishable dyads. Here are some examples:

Indistinguishable dyads are especially useful for experimental designs in which one member of the dyad is randomly assigned to one condition and the other member is assigned to the other condition.

Difference scores are symbolized with D instead of X. So if you were to conduct a 1-sample t-test with difference scores, the formula would be:

Note that the formula is exactly the same except the X's have been replaced with D's. So what is the big deal? Well, with a 1-sample t-test, you don't usually know the population mean. When working with difference scores, however, you know exactly what the null hypothesis says the population mean is. Why?

Suppose you have a sample of students who took a spelling test before and after participating in a program designed to improve their spelling. The null hypothesis is that the program had no effect. Therefore the mean spelling ability before (μ₁) is the same as the mean spelling ability after the program (μ₂). Thus if μ₁ = μ₂,then μ₁ - μ₂ = 0. Why? Because any number minus itself is 0 (e.g., 202 - 202 = 0). Even though we have no idea what the μ₁ and μ₂ are, we know that the difference between them (μ_D) is 0.

Knowing that the population mean of the difference scores is 0, we can simplify the paired-samples t-test formula to:

This means that the observed t is equal to the sample mean of the difference scores divided by the estimated standard error of the difference scores.

What is special about this formula is that it contains no population parameters, only sample statistics.

If you are computing a paired-samples t-test by hand or by calculator, you compute the difference scores and proceed exactly as you would with a 1-sample t-test. The sample size is not the number of scores from the original variables but the number of difference scores. Thus, the degrees of freedom is the number of paired scores minus 1. Using the 1-sample t-test spreadsheet from the last lab might make things a little easier. Remember that the population mean is 0 for paired t-tests.

In SPSS, you proceed a little differently than you would with a 1-sample t-test. I will use an example of 38 aggressive children who participated in an anger-management and social skills training program. Each child's aggression was assessed before and after treatment. You wish to know if the treatment was effective in lowering aggression. For the sake of simplicity, let's make this a 2-tailed test because SPSS doesn't have a friendly way of doing 1-tailed paired-samples t-tests. Thus, the null hypothesis is that the mean aggression scores are the same before and after treatment.

Click Analyze-->Compare Means-->Paired Samples T Test

Click both variables in the pair like so they are both highlighted like this:

In this output, you can see in the lower right corner that the p-value is .041. This means that if the null hypothesis were true, the probability of getting an observed t of 2.12 or higher is only .041. This is lower than α of .05, so you would reject the null hypothesis. In ordinary language, your conclusion is that compared to before treatment, the children's aggression was significantly lower after treatment.

Download the worksheet here to answer the questions.
Email it to your GA when you are finished.