Introduction to Categorical Data Analysis

192 MULTICATEGORY LOGIT MODELS
Table 6.11. Outcomes for Pregnant Mice in Developmental
Toxicity Study a
Concentration
Response
(mg/kg per day) Non-live Malformation Normal
0 (controls) 15 1 281
62.5 17 0 225
125 22 7 283
250 38 59 202
500 144 132 9
a Based on results in C. J. Price et al., Fund. Appl. Toxicol., 8: 115–126,
1987. I thank Dr. Louise Ryan for showing me these data.
For instance, given that a fetus was live, for every 100-unit increase in concentration
level, the estimated odds that it was malformed rather than normal changes by a
multiplicative factor of exp(100 × 0.0174) = 5.7.
When models for different continuation-ratio logits have separate parameters, as
in this example, separate fitting of ordinary binary logistic regression models for
different logits gives the same results as simultaneous fitting. The sum of the separate
deviance statistics is an overall goodness-of-fit statistic pertaining to the simultaneous
fitting. For Table 6.11, the deviance G 2 values are 5.8 for the first logit and 6.1 for
the second, each based on df = 3. We summarize the fit by their sum, G 2 = 11.8,
based on df = 6 (P = 0.07).
6.3.5 Overdispersion in Clustered Data
The above analysis treats pregnancy outcomes for different fetuses as independent
observations. In fact, each pregnant mouse had a litter of fetuses, and statistical
dependence may exist among different fetuses from the same litter. The model
also treats fetuses from different litters at a given concentration level as having
the same response probabilities. Heterogeneity of various types among the litters
(for instance, due to different physical conditions of different pregnant mice) would
usually cause these probabilities to vary somewhat among litters. Either statistical
dependence or heterogeneous probabilities violates the binomial assumption. They
typically cause overdispersion – greater variation than the binomial model implies
(recall Section 3.3.3).
For example, consider mice having litters of size 10 at a fixed, low concentration
level. Suppose the average probability of fetus death is low, but some mice have
genetic defects that cause fetuses in their litter to have a high probability of death.
Then, the number of fetuses that die in a litter may vary among pregnant mice to
a greater degree than if the counts were based on identical probabilities. We might
observe death counts (out of 10 fetuses in a litter) such as 1, 0, 1, 10, 0, 2, 10, 1, 0,
10; this is more variability than we expect with binomial variates.