A binary response yij = 1 or 0 for observation j on subject i, i = 1,…, n, j = 1,…,

d. Let ȳ.j = ∑

i yij∕n, ȳi. = ∑

j yij∕d, and ȳ = ∑

i

∑

j yij∕nd. Regard

{yi+} as fixed, and suppose each way to allocate the yi+ “successes” to the d observations is equally likely. Show that E(yij) = ȳi.

, var(yij) = ȳi.

(1 − ȳi.

), and cov(yij, yik) = −ȳi.

(1 − ȳi.

)∕(d − 1) for j ≠ k. For large n with independent subjects, explain why (ȳ.1,…, ȳ.d) is approximately multivariate normal with pairwise correlation ???? = −1∕(d − 1). Conclude that Cochran’s Q statistic

(Cochran 1950)

Q = n2(d − 1) ∑d j=1(ȳ.j − ȳ)

2 d

∑n i=1 ȳi.

(1 − ȳi.

)

has an approximate chi-squared distribution with df = (d − 1) for testing homogeneity of the d marginal distributions. Show that Q is unaffected by deleting all observations for which yi1 = ⋯ = yid.