Let yit = 1 or 0 for observation t on

Let y_it = 1 or 0 for observation t on subject i, i = 1,…, n, t = 1, …, T. Let y_.t = ∑_i y_it/n, y_i. = ∑_t y_it/T, and y_.. = ∑_i ∑_t y_it/nT.

a. Regard {y_i+} as fixed. Suppose that each way to allocate the y_i+ “successes” to y_i+ of the observations is equally likely. Show that E(Y_it) = y_i, var(Y_it) = y_i.(1 – y_i), and cov(Y_it, Y_ik) = – y_i.(1 – y_i.)/(T – 1) for t ≠ k. [The covariance is the same for any pair of cells in the same row, and var(∑_t Y_it) = 0 since y_i+ is fixed.]

b. Refer to part (a). For large n with independent subjects, explain why (Y_.1,…, Y_.T) is approximately multivariate normal with pairwise correlation ρ = –1/(T – 1). Conclude that Cochran’s Q statistic (Cochran 1950)

is approximately chi-squared with df = (T – 1). [One way notes that if (X₁,…, X_T) is multivariate normal with common mean and common variance σ² and common correlation ρ for pairs (X_t, X_k), then ∑(X_t – X̅)²/σ²(1 – ρ) is chi-squared with df = (T – 1). See Bhapkar and Somes (1977) for slightly weaker conditions for a chi-squared limiting for Q than those in part (a).]

c. Show that Q is unaffected by deleting cases in which y_i1 = … = y_iT.