Introduction to Categorical Data Analysis

258 MODELS FOR MATCHED PAIRS
from the discrepancy between n13 and n31. For that pair, the standardized residual
equals (44 − 17)/ √ (44 + 17) = 3.5. Consumers of High Point changed to Sanka
more often than the reverse. Otherwise, the symmetry model fits most of the table
fairly well.
The quasi-symmetry model has G 2 = 10.0 and X 2 = 8.5, with df = 6. Permitting
the marginal distributions to differ yields a better fit than the symmetry model
provides. We will next see how to use this information to construct a test of marginal
homogeneity.
8.4.4 Testing Marginal Homogeneity Using Symmetry and Quasi-Symmetry
For the quasi-symmetry model, log(πij /πji) = βi − βj for all i and j, marginal
homogeneity is the special case in which all βi = 0. This special case is the symmetry
model. In other words, for the quasi-symmetry model, marginal homogeneity is
equivalent to symmetry.
To test marginal homogeneity, we can test the null hypothesis that the symmetry
(S) model holds against the alternative hypothesis of quasi symmetry (QS). The
likelihood-ratio test compares the G2 goodness-of-fit statistics,
G 2 (S | QS) = G 2 (S) − G 2 (QS)
For I × I tables, the test has df = I − 1.
For Table 8.5, the 5 × 5 table on choice of coffee brand at two purchases,
G 2 (S) = 22.5 and G 2 (QS) = 10.0. The difference G 2 (S | QS) = 12.5, based on
df = 4, provides evidence of differing marginal distributions (P = 0.014).
Section 8.3.1 described other tests of H0: marginal homogeneity, based on the
ML fit under H0 and using pi+ − p+i, i = 1,...,I. For the coffee data, these gave
similar results as using G 2 (S | QS). Those other tests do not assume that the quasisymmetry
model holds. In practice, however, for nominal classifications the statistic
G 2 (S | QS) usually captures most of the information about marginal heterogeneity
even if the quasi-symmetry model shows lack of fit.
8.4.5 An Ordinal Quasi-Symmetry Model
The symmetry and quasi-symmetry models treat the classifications as nominal. A
special case of quasi-symmetry often is useful when the categories are ordinal. Let
u1 ≤ u2 ≤···≤uI denote ordered scores for both the row and column categories.
The ordinal quasi-symmetry model is
log(πij /πji) = β(uj − ui) (8.11)
This is a special case of the quasi-symmetry model (8.10) in which {βi} have a linear
trend. The symmetry model is the special case β = 0.