2.6 The paired comparison design

Section 1 showed that blocking can remove a large source of variation from a comparison. The simplest blocked design has blocks of two units.

The model for a paired design is \[ y_{ij} = \mu _i + \beta _j + \varepsilon _{ij} , \qquad i = 1, 2 , \quad j = 1, \dots , n , \] where \(\beta _j\) is the effect of the \(j\)th pair (block) and the \(\varepsilon _{ij}\) are independent \(N(0, \sigma ^2)\). The within-pair differences \[ d_j = y_{1j} - y_{2j} = \mu _1 - \mu _2 + \varepsilon _{1j} - \varepsilon _{2j} \] do not contain \(\beta _j\) at all. They are independent \(N(\mu _d, \sigma _d^2)\) with \(\mu _d = \mu _1 - \mu _2\) and \(\sigma _d^2 = 2\sigma ^2\), so the comparison reduces to a one-sample problem. By Theorem 2.1.2, \[ T = \frac {\bar d - \mu _d}{s_d/\sqrt n} \sim t_{n-1} , \] and \(H_0 : \mu _1 = \mu _2\) is rejected if \(|\bar d|/(s_d/\sqrt n) > t_{\alpha /2,\,n-1}\). This is the paired \(t\) test.

The two analyses differ because the cars differ. The car-to-car variation is part of \(s_p^2\) but cancels from every \(d_j\), and the correlation between the two tyres on a car is \(0.989\). In general, if the two observations in a pair have common variance \(\sigma _y^2\) and correlation \(\rho \), then \[ \Var (\bar d) = \frac {\Var (y_1 - y_2)}{n} = \frac {2\sigma _y^2(1 - \rho )}{n} , \] against \(2\sigma _y^2/n\) for two independent samples of size \(n\). Pairing divides the variance by \(1/(1 - \rho )\), here by about \(90\).