Testing Marginal Homogeneity in a Square Table.

For an I × I table, the hypothesis of marginal homogeneity is H0 : pi· = p·i , i = 1,...,I. A natural statistic for testing this hypothesis is the vector d = (n1· − n·1,...,nI· − n·I )
Clearly, E(di) = pi· − p·i .

Use the results of Exercise 1.6.5 to show that Var(di) = N

(pi· + p·i − 2pii) − (pi· − p·i)

2!

and Cov(dh, di) = N [(phi + pih)+(ph· − p·h)(pi· − p·i)] .

Use the previous exercise to find the large sample distribution of

d. Show that Pr(J

d = 0) = 1. Show that the asymptotic covariance matrix of d has rank I − 1 and thus is not invertible. It is well known that if Y ∼ N(0, V )

with V an s × s nonsingular matrix, then Y

V −1Y ∼ χ2(s). Use this fact along with the asymptotic distribution of d to obtain a test of the hypothesis of marginal homogeneity. Apply the test of marginal homogeneity to the data of Exercise 2.6.10.