In paired sample hypothesis testing, a sample from the population is chosen and two measurements for each element in the sample are taken. Each set of measurements is considered a sample, but the samples are not independent of one another. Paired samples are also called matched samples or repeated measures. Examples include:
- Comparing the driving skills of people before and after they take a driver’s training course
- Determining whether the attitudes of husbands and wives differ regarding capital punishment
- Determining the effectiveness of a mosquito repellent by applying the repellent to the right arm of each subject in the sample (but not the left arm) and determining whether the right arms have fewer bites than the left arms.
Here we have two samples for the random variables x and y and test the null hypothesis that the population mean of x and y are equal. This is equivalent to the one-sample t-test on the random variable z = x – y. Specifically, we test the null hypothesis H0: μz = 0, which is equivalent to H0: μx = μy.
The multivariate case is very similar to the univariate case. Now we have two samples for the random vectors X and Y and test the null hypothesis that the population mean vectors of X and Y are equal. This is equivalent to the One Sample Hotelling’s T2 Test on the random vector Z = X – Y. Specifically, we test the null hypothesis H0: μZ = 0, which is equivalent to H0: μX = μY. We illustrate the approach used in the following example.
Example 1: The shoe company from Example 1 of One Sample Hotelling’s T2 Statistic is considering phasing out an existing shoe model (Model 2) by the prototype described in Example 1 of One Sample Hotelling’s T2 Statistic. The company had the same subjects evaluate both Model 1 and Model 2, and looked to see if there was a significant difference between the two models which would help them decide whether to replace Model 2 by Model 1. The sample is shown in Figure 1.
Figure 1 – Data for paired samples Hotelling’s T2 model
Essentially we perform the same analysis as in One Sample Hotelling’s T2 Statistic, with the goal being 0 for the difference between the two evaluations for each criteria. We repeat the analysis in the next few figures.
Figure 2 – Goal for difference between the two models
Figure 3 – Output shows significant difference between the two models
From Figure 3, we see there is a significant difference between the two shoe models. We next determine for which criteria there is a significant difference, using both 95% simultaneous confidence intervals and 95% Bonferroni confidence intervals.
Figure 4 – 95% simultaneous confidence intervals
Figure 5 – 95% Bonferroni confidence intervals