6.1 Paired Comparisons
Suppose each of \(n\) units is measured under two conditions, giving \(\underline {X}_{j1}\) and \(\underline {X}_{j2}\) for unit \(j\). The pairing must be respected: treating \(2n\) vectors as two independent samples of size \(n\) throws away the very structure the design was built to exploit.
Form the differences \[\underline {D}_j = \underline {X}_{j1}-\underline {X}_{j2}, \qquad j = 1,\dots ,n,\] and suppose \(\underline {D}_j \sim N_p\left (\underline {\delta },\Sigma _d\right )\) independently. The problem is now a one-sample problem in the differences, and Section 5.4 applies unchanged.
Theorem 6.1 (Paired \(T^{2}\)). With \(\overline {\underline {D}}\) and \(S_d\) the sample mean and covariance of the differences, under \(H_0: \underline {\delta }=\underline {0}\) \[T^{2} = n\,\overline {\underline {D}}'\,S_d^{-1}\,\overline {\underline {D}} \ \sim \ \frac {(n-1)p}{n-p}\,F_{p,\,n-p},\] and \(H_0\) is rejected at level \(\alpha \) when \(T^{2}\) exceeds \(\frac {(n-1)p}{n-p}F_{p,n-p,\alpha }\).
Note 6.2. Everything about the paired analysis follows from the single act of taking differences. The covariance between the two conditions — which is usually substantial, since the same unit is measured twice — disappears into \(\Sigma _d\) and is never estimated separately. This is exactly why pairing is worth doing: a large positive covariance makes \(\Sigma _d\) small, and a small \(\Sigma _d\) makes the test sensitive.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.