Introduction to Categorical Data Analysis

256 MODELS FOR MATCHED PAIRS
Table 8.6. Behaviors on Recycling and Driving Less to Help Environment, with Fit
of Ordinal Quasi-Symmetry Model
Drive Less
Recycle Always Often Sometimes Never
Always 12 (12) 43 (43.1) 163 (165.6) 233 (232.8)
Often 4 (3.9) 21 (21) 99 (98.0) 185 (184.5)
Sometimes 4 (1.4) 8 (9.0) 77 (77) 230 (227.3)
Never 0 (0.2) 1 (1.5) 18 (20.7) 132 (132)
and the denominator equals
1(4 + 8 + 18) + 2(4 + 1) + 3(0) = 40
Thus, ˆβ = log(1767/40) = 3.79. The estimated odds ratio is exp( ˆβ) = 1767/40 =
44.2. For each subject the estimated odds of response “always” (instead of the other
three categories) on recycling are 44.2 times the estimated odds of that response for
driving less. This very large estimated odds ratio indicates a substantial effect.
For Table 8.6, ˆβ = 3.79 has SE = 0.180. For H0: β = 0, z = 3.79/0.180 = 21.0
provides extremely strong evidence against the null hypothesis of marginal homogeneity.
Strong evidence also results from the comparison of mean scores. For the scores
{1, 2, 3, 4}, the mean for driving less is [20 + 2(73) + 3(357) + 4(780)]/1230 =
3.54, and the mean for recycling is [451 + 2(309) + 3(319) + 4(151)]/1230 = 2.14.
The test statistic is z = ( ¯x −¯y)/SE = (2.14 − 3.54)/0.0508 =−27.6. The sample
marginal means also indicate that responses tended to be considerably more toward
the low end of the response scale (i.e., more frequent) on recycling than on driving
less.
8.4 SYMMETRY AND QUASI-SYMMETRY MODELS FOR
SQUARE TABLES ∗
The probabilities in a square table satisfy symmetry if
πij = πji
(8.8)
for all pairs of cells. Cell probabilities on one side of the main diagonal are a mirror
image of those on the other side. When symmetry holds, necessarily marginal homogeneity
also holds. When I>2, though, marginal homogeneity can occur without
symmetry. This section shows how to compare marginal distributions using logistic
models for pairs of cells in square tables.