Chapter 6: Analysing the Data Part III: Common Statistical Tests

# Example of paired sample t-test

Let us consider a simple example of what is often termed "pre/post" data or "pretest Ð posttest" data. Suppose you wish to test the effect of Prozac on the well-being of depressed individuals, using a standardised "well-being scale" that sums Likert-type items to obtain a score that could range from 0 to 20. Higher scores indicate greater well-being (that is, Prozac is having a positive effect). While there are flaws in this design (e.g., lack of a control group) it will serve as an example of how to analyse such data.

The value that we are interested in is the change score, and we obtain it by taking the difference between time 2 and time one. The following snapshot of an SPSS data window provides the data that we can work with.

Figure 6.9 Data for the paired sample t test.

Notice that we have subtracted the first score away from the second to get a difference score or change score. Person #3's well-being score decreased by one point at the post-test. Person #5 increased their well-being score from 4 point to 10 points. The mean of the Pretest data is 3.33 and the mean of the Post test data is 7.0. The question is, "Is this a significant increase?"

### Step 1: Stating the hypotheses:

Ho: md = 0

H1: md 0

The alternative is two-tailed and alpha = .05

### Step 2: Check assumptions

The assumptions underlying the repeated samples t-test are similar to the one-sample t-test but refer to the set of difference scores.

1. The observations are independent of each other

2. The dependent variable is measured on an interval scale

3. The differences are normally distributed in the population.

The measures are approximately interval scale numbers (self-report scores) and we assume that each person's score has not been influenced by other people's scores. The numbers look to have no major extremes or unusual distribution.

### Step 3: Calculate test statistic

The output is given in Output 6.5. Detailed comments on the output follow.

Paired Samples Statistics. Here, the variables being compared are identified, the Mean, N, Standard Deviation, and Standard Error of the Mean for each variable is given.

Paired Samples Correlations. Here the correlation between each of the pairs of variables is given. Because this is a repeated measures analysis, the same people are measured twice. You would expect a high degree of correlation between the two sets of scores. A person who was fairly low well-being score before the treatment should still have a fairly low well-being score relative to the others after the treatment, even if everyone improved (e.g., cases #2). Similarly, someone who had a high well-being score beforehand will probably have one of the highest well-being score afterward (e.g., case #2), even if every one improves by a certain amount. Here the correlation between the two sets of scores is quite minimal for real data (you can tell they were manufactured). There is no consistent pattern of change (see cases #4 and 8). If there is little correlation between the two sets of scores, you might as well be using an Independent Groups T-test.

Paired Samples Test. Table 1. Here the descriptive statistics for the difference between each pair of variables is given. The mean difference of Ð3.67 is what is actually being tested against zero. Is this a large number or a small number? Is this difference a real one or one that we could reasonably expect due to chance alone?

Notice the 95% Confidence Interval values are also given here. We do not place much emphasis on these in this unit. This information says that the true population mean lies between Ð6.357 and -.9763 with a 95% probability. Notice the hypothesised mean of zero does not fall within this range.

Paired Samples Test. Table 2.

The answer! Here we can determine that the chance of this number occurring by chance alone (given the null hypothesis) is about .014 (or 1.4%).

### Step 4: Evaluate the result

The result is significant t(9) = -3.143, p = .012. We reject the null hypothesis in favour of the alternative. Only once or twice out of every 100 times we repeated this experiment (and the null hypothesis was true) would we get a t-statistic of this size. We therefore conclude that it is more likely to have been due to some systematic, deliberate cause. If all other confounds are eliminated, this systematic cause must have been the Prozac drug.
2 = (-3.143) 2/(-3.143) + 9 = 0.523. So, 52.3% of the variability in well-being scores can be explained by the use of the drug or not.

### Step 5: Interpret the result

A significant increase in well-being occurred (t(9) = -3.14, p = .012,
2 = .52) in the Prozac treated group compared to the control group.

We have found strong evidence that Prozac enhances well-being in depressed individuals. However, the major lack of controls in this study would suggest we keep quiet about it until we can repeat the finding with more stringent safeguards against confounds!