The Dependent Groups t-Test
Suppose that the individual observations in the two samples, Treatment and Control, actually represented linked pairs: for example, a brother and a sister, or two vehicles of the same brand and model. In that case, you can calculate a correlation between the two sets of scores.
If the correlation is large enough, then much of the variability in the scores can be attributed to the correlation. In effect, you remove that variability from the standard error of the difference between means and allocate it to the correlation.
The result is that the standard error of the difference between means becomes smaller. Along with it, the critical value gets lower. And that increases the power of the t-test. See Figure 4.
Figure 4 Accounting for the correlation increases the statistical power, much as ANCOVA can increase the power of an ANOVA.
Just as occurred in Figure 3, Figure 4 shows that the standard error of the difference between means has been reduced—this time to 4.24 (see cell F23). Once again, reducing the size of the standard error has the effect of lowering the critical value, this time to 7.34 (see cell F24).
You can see the source of this effect by examining the formula for the standard error of the difference between the means. Here is the formula for the standard error when the means are of independent groups, with the same number of observations per group, or n:
In contrast, here is the formula for the standard error when the means are of groups that comprise paired observations, or dependent groups:
Notice that the second formula above is the same as the first, except that the second formula subtracts a term whose size is a function of the size of the correlation r between the two groups.
Therefore, the larger the correlation, the smaller the standard error. And the smaller the standard error, the smaller the critical value (recall that the critical value is the product of the standard error and the t value associated with the size of alpha).