Introduction to Categorical Data Analysis

290 MODELING CORRELATED, CLUSTERED RESPONSES
Consider the insomnia study of Problem 9.3.2 from the previous section. Let
y1 be the initial time to fall asleep, let Y2 be the follow-up time, with explanatory
variable x defining the two treatment groups (1 = active drug, 0 = placebo). We will
now treat Y2 as an ordinal response and y1 as an explanatory variable, using scores
{10, 25, 45, 75}. In the model
logit[P(Y2 ≤ j)]=αj + β1x + β2y1
(9.2)
β1 compares the follow-up distributions for the treatments, controlling for the initial
observation. This models the follow-up response (Y2), conditional on y1, rather than
marginal distributions of (Y1,Y2). It’s the type of model for an ordinal response that
Section 6.2 discussed. Here, the initial response y1 plays the role of an explanatory
variable.
From software for ordinary cumulative logit models, the ML treatment effect
estimate is ˆβ1 = 0.885 (SE = 0.246). This provides strong evidence that follow-up
time to fall asleep is lower for the active drug group. For any given value for the
initial response, the estimated odds of falling asleep by a particular time for the active
treatment are exp(0.885) = 2.4 times those for the placebo group. Exercise 9.12
considers alternative analyses for these data.
9.4.4 Transitional Models Relate to Loglinear Models
Effects in transitional models differ from effects in marginal models, both in magnitude
and in their interpretation. The effect of a predictor xj on Yt is conditional
on yt−1 in a transitional model, but it ignores yt−1 in a marginal model. Effects in
transitional models are often considerably weaker than effects in marginal models,
because conditioning on a previous response attenuates the effect of a predictor.
Transitional models have connections with the loglinear models of Chapter 7,
which described joint distributions. Associations in loglinear models are conditional
on the other response variables. In addition, a joint distribution of (Y1,Y2,...,YT ) can
be factored into the distribution of Y1, the distribution of Y2 given Y1, the distribution
of Y3 given Y1 and Y2, and so forth.
PROBLEMS
9.1 Refer to Table 7.3 on high school students’ use of alcohol, cigarettes, and
marijuana. View the table as matched triplets.
a. Construct the marginal distribution for each substance. Find the sample
proportions of students who used (i) alcohol, (ii) cigarettes, (iii) marijuana.
b. Specify a marginal model that could be fitted as a way of comparing the
margins. Explain how to interpret the parameters in the model. State the
hypothesis, in terms of the model parameters, that corresponds to marginal
homogeneity.