Introduction to Categorical Data Analysis

8.4 SYMMETRY AND QUASI-SYMMETRY MODELS FOR SQUARE TABLES 257
8.4.1 Symmetry as a Logistic Model
The symmetry condition has the simple logistic form
log(πij /πji) = 0 for all i and j
The ML fit of the symmetry model has expected frequency estimates
ˆμij = (nij + nji)/2
The fit satisfies ˆμij =ˆμji. It has ˆμii = nii, a perfect fit on the main diagonal. The
residual df for chi-squared goodness-of-fit tests equal I(I − 1)/2.
The standardized residuals for the symmetry model equal
rij = (nij − nji)/ � nij + nji
(8.9)
The two residuals for each pair of categories are redundant, since rij =−rji. The
sum of squared standardized residuals, one for each pair of categories, equals X 2 for
testing the model fit.
8.4.2 Quasi-Symmetry
The symmetry model is so simple that it rarely fits well. For instance, when
the marginal distributions differ substantially, the model fits poorly. One can
accommodate marginal heterogeneity by the quasi-symmetry model,
log(πij /πji) = βi − βj for all i and j (8.10)
One parameter is redundant, and we set βI = 0. The symmetry model is the special
case of equation (8.10) in which all βi = 0. Roughly speaking, the higher the value
of ˆβi, relatively more observations fall in row i compared to column i.
Fitting the quasi-symmetry model requires iterative methods. To use software, treat
each separate pair of cell counts (nij ,nji) as an independent binomial variate, ignoring
the main-diagonal counts. Set up I artificial explanatory variables, corresponding to
the coefficients of the {βi} parameters. For the logit log(πij /πji) for a given pair of
categories, the variable for βi is 1, the variable for βj is −1, and the variables for the
other parameters equal 0. (Table A.13 in the Appendix illustrates this with SAS code.)
One explanatory variable is redundant, corresponding to the redundant parameter. The
fitted marginal totals equal the observed totals. Its residual df = (I − 1)(I − 2)/2.
8.4.3 Example: Coffee Brand Market Share Revisited
Table 8.5 in Section 8.3.2 summarized coffee purchases at two times. The symmetry
model has G 2 = 22.5 and X 2 = 20.4, with df = 10. The lack of fit results primarily