Introduction to Categorical Data Analysis

8.3 COMPARING MARGINS OF SQUARE CONTINGENCY TABLES 253
Let πij = P(Y1 = i, Y2 = j). Marginal homogeneity is
P(Y1 = i) = P(Y2 = i) for i = 1,...,I
that is, each row marginal probability equals the corresponding column marginal
probability.
8.3.1 Marginal Homogeneity and Nominal Classifications
One way to test H0: marginal homogeneity compares ML fitted values {ˆμij } that
satisfy marginal homogeneity to {nij } using G2 or X2 statistics. The df = I − 1.
The ML fit of marginal homogeneity is obtained iteratively.
Another way generalizes the McNemar test. It tests H0: marginal homogeneity by
exploiting the large-sample normality of marginal proportions. Let di = pi+ − p+i
compare the marginal proportions in column i and row i. Let d be a vector of the first
I − 1 differences. It is redundant to include dI , since � di = 0. Under H0, E(d) = 0
and the estimated covariance matrix of d is ˆV0/n, where ˆV0 has elements
ˆvij 0 =−(pij + pji) for i �= j
ˆvii0 = pi+ + p+i − 2pii
Now, d has a large-sample multivariate normal distribution. The quadratic form
W0 = nd ′ ˆV −1
0 d (8.6)
is a score test statistic. It is asymptotically chi-squared with df = I − 1. For I = 2,
W0 simplifies to the McNemar statistic, the square of equation (8.1).
8.3.2 Example: Coffee Brand Market Share
A survey recorded the brand choice for a sample of buyers of instant coffee. At a later
coffee purchase by these subjects, the brand choice was again recorded. Table 8.5
shows results for five brands of decaffinated coffee. The cell counts on the “main
diagonal” (the cells for which the row variable outcome is the same as the column
variable outcome) are relatively large. This indicates that most buyers did not change
their brand choice.
The table also shows the ML fitted values that satisfy marginal homogeneity.
Comparing these to the observed cell counts gives G 2 = 12.6 and X 2 = 12.4(df = 4).
The P -values are less than 0.015 for testing H0: marginal homogeneity. (Table A.11
in the Appendix shows how software can obtain the ML fit and test statistics.) The
statistic (8.6) using differences in sample marginal proportions gives similar results,
equaling 12.3 with df = 4.