Introduction to Categorical Data Analysis

190 MULTICATEGORY LOGIT MODELS
6.3.1 Adjacent-Categories Logits
One approach forms logits for all pairs of adjacent categories. The adjacent-categories
logits are
� �
πj+1
log , j = 1,...,J − 1
πj
For J = 3, these logits are log(π2/π1) and log(π3/π2).
With a predictor x, the adjacent-categories logit model has form
log
� πj+1
πj
A simpler proportional odds version of the model is
log
� πj+1
πj
�
= αj + βj x, j = 1,...,J − 1 (6.5)
�
= αj + βx, j = 1,...,J − 1 (6.6)
For it, the effects {βj = β} of x on the odds of making the higher instead of the
lower response are identical for each pair of adjacent response categories. Like the
cumulative logit model (6.4) of proportional odds form, this model has a single
parameter rather than J − 1 parameters for the effect of x. This makes it simpler to
summarize an effect.
The adjacent-categories logits, like the baseline-category logits, determine the
logits for all pairs of response categories. For the simpler model (6.6), the coefficient
of x for the logit, log(πa/πb), equals β(a − b). The effect depends on the distance
between categories, so this model recognizes the ordering of the response scale.
6.3.2 Example: Political Ideology Revisited
Let’s return to Table 6.7 and model political ideology using the adjacent-categories
logit model (6.6) of proportional odds form. Let x = 0 for Democrats and x = 1 for
Republicans.
Software reports that the party affiliation effect is ˆβ = 0.435. The estimated odds
that a Republican’s ideology classification is in category j + 1 instead of j are
exp( ˆβ) = 1.54 times the estimated odds for Democrats. This is the estimated odds
ratio for each of the four 2 × 2 tables consisting of a pair of adjacent columns of
Table 6.7. For instance, the estimated odds of “slightly conservative” instead of “moderate”
ideology are 54% higher for Republicans than for Democrats. The estimated
odds ratio for an arbitrary pair of columns a and b equals exp[ ˆβ(a − b)]. The estimated
odds that a Republican’s ideology is “very conservative” (category 5) instead
of “very liberal” (category 1) are exp[0.435(5 − 1)] =(1.54) 4 = 5.7 times those for
Democrats.