49. Comparison of two designs. Under the assumptions made at the beginning of Section 12, one has the following comparison of the methods of complete randomization and matched pairs. The unit effects and experimental effects 0; and V; are independently normally distributed with variances al, a2 and means E(O;) = Ii and E(V;) = C or 'II as V; corresponds to a control or treatment. With complete randomization, the observations are X; = 0; + V; (i = 1, ... , n) for the controls and Y; = u,,+i + v,,+i (i = 1, . . . , n) for the treated cases, with E(X;) = Ii + t E(Y;) = Ii + 'II. For the matched pairs, if the matching is assumed to be perfect, the X's are as before, but Y; = 0; + V,r+i. UMP unbiased tests are given by (27) for complete randomization and by (59) for matched pairs. The distribution of the test statistic under an alternative l1 = 'II - C is the noncentral t-distribution with noncentrality parameter V;l1/ V2( a2 + an and 2n - 2 degrees of freedom in the first case, and with noncentrality parameter V;l1/ fi a and n - 1 degrees of freedom in the second. Thus the method of matched pairs has the disadvantage of a smaller number of degrees of freedom and the advantage of a larger noncentrality parameter. For a = .05 and l1 = 4, compare the power of the two methods as a function of n when al = 1, a = 2 and when al = 2, a = 1