## Paired t-tests

The paired t-test is used to compare the values of means from two related samples, for example in a 'before and after' scenario.

The difference between the means of the samples is unlikely to be equal to zero (due to sampling variation) and the hypothesis test is designed to answer the question "Is the observed difference sufficiently large enough to indicate that the alternative hypothesis is true?". The answer comes in the form of a probability - the p-value.

Worked Example
Consider the following study in which standing and supine systolic blood pressures were compared. This study was performed on twelve subjects. Their blood pressures were measured in both positions. It is therefore, a paired samples design.

Subject Standing Supine Difference
1 132 136 4
2 146 145 1
3 135 140 5
4 141 147 6
5 139 142 3
6 162 160 -2
7 128 137 9
8 137 136 -1
9 145 149 4
10 151 158 7
11 131 120 -11
12 143 150 7
Mean 140.83 143.33 2.50
SD 9.49 10.83 5.50

The statistical analysis of paired data is performed on the differences between the pairs, and for this data the mean difference (Supine - Standing) between the blood pressures is 2.50 mmHg. The standard deviation (SD) of the difference is 5.50 mmHg.

Suggested null and alternative hypotheses could be:

H0: There is no difference between the mean blood pressures in the two populations

H1: There is a difference between the mean blood pressures in the two populations

or equivalently

H0: On average there is no difference between the blood pressures in the two populations

H1: On average there is a difference between the blood pressures in the two populations

The computer output from performing a paired samples t-test on the standing and supine blood pressure data gives a p-value of 0.144. Thus the probability of getting a difference of 2.50 mmHg between the mean blood pressures (given that position does not affect blood pressure) is 0.144 or 14.4% or about 1 in 7. This is not sufficiently low to conclude that position does affect mean blood pressure. Therefore, we fail to reject the null hypothesis with this data, and conclude that there is insufficient evidence to suggest a difference between blood pressures, on average, in the two positions.

## The paired t-test

The output for the standing and supine blood pressure example is shown below, and gives the sample means and the mean difference, their standard deviations and the standard errors. The 95% confidence interval for the mean difference is also shown as well as the t-test of the null hypothesis that the "mean difference = 0" versus (vs) the alternative hypothesis that the mean difference is "not = 0". The p-value equals 0.144.

In SPSS
Enter the data from the first sample into one column and the corresponding data from the second sample in a second column, then select
Analyze > Compare Means > Paired-Samples T Test...
The output is as follows:   in Minitab

Enter the data from the first sample into one column and the corresponding data from the second sample in a second column, then select
Stat > Basic Statistics > Paired t...

Paired T-Test and Confidence Interval

Paired T for Supine - Standing

N Mean StDev SE Mean
Supine 12 143.33 10.83 3.13
Standing 12 140.83 9.49 2.74
Difference 12 2.50 5.50 1.59

Assumptions underlying the paired sample t-test
Both the paired and independent sample t-tests make assumptions about the data, although both tests are fairly robust against departures from these assumptions.

In the paired samples t-test it is assumed that the differences, calculated for each pair, have an approximately normal distribution. Techniques are available to test this assumption. An alternative procedure that makes no assumptions about the distribution of the data is the Wilcoxon Test, but this test is less powerful than the paired sample t-test.