Consider the comparison of two success probabilities in

(a) the two binomial situation of Section 4.5 with m = n, and

(b) the matched-pairs situation of Section 4.9. Suppose the matching is completely at random, that is, a random sample of 2n subjects, obtained from a population of size N(2n ≤ N), is divided at random into n pairs, and the two treatments B and Bc are assigned at random within each pair.

(i) The UMP unbiased test for design

(a) (Fisher’s exact test) is always more powerful than the UMP unbiased test for design

(b) (McNemar’s test).

(ii) Let Xi (respectively Yi) be 1 or 0 as the 1st (respectively 2nd) member of the ith pair is a success or failure. Then the correlation coefficient of Xi and Yi can be positive or negative and tends to zero as N → ∞.

[(ii): Assume that the kth member of the population has probability of success P(k)

A under treatment A and P(k)

A˜ under A˜.]