Introduction to Categorical Data Analysis

10.3 EXTENSIONS TO MULTINOMIAL RESPONSES 311
Table 10.7. Results of Fitting Cumulative Logit Models
(with Standard Errors in Parentheses) to Table 9.6
Marginal Random Effects
Effect GEE (GLMM) ML
Treatment 0.034 (0.238) 0.058 (0.366)
Occasion 1.038 (0.168) 1.602 (0.283)
Treatment × occasion 0.708 (0.244) 1.081 (0.380)
Now, let yit denote the response for subject i at occasion t. The random-intercept
model
logit[P(Yit ≤ j)]=ui + αj + β1t + β2x + β3(t × x)
takes the linear predictor from the marginal model and adds a random effect ui.
The random effect is assumed to be the same for each cumulative probability. A
subject with a relatively high ui, for example, would have relatively high cumulative
probabilities, and hence a relatively high chance of falling at the low end of the ordinal
scale.
Table 10.7 also shows results of fitting this model. Results are substantively similar
to the marginal model. The response distributions are similar initially for the two
treatment groups, but the interaction suggests that at the follow-up response the active
treatment group tends to fall asleep more quickly. We conclude that the time to fall
asleep decreases more for the active treatment group than for the placebo group.
From Table 10.7, estimates and standard errors are about 50% larger for the GLMM
than for the marginal model. This reflects the relatively large heterogeneity. The
random effects have estimated standard deviation ˆσ = 1.90. This corresponds to a
strong association between the responses at the two occasions.
10.3.2 Bivariate Random Effects and Association Heterogeneity
The examples so far have used univariate random effects, taking the form of random
intercepts. Sometimes it is sensible to have a multivariate random effect, for example
to allow a slope as well as an intercept to be random.
We illustrate using Table 10.8, from three of 41 studies that compared a new
surgery with an older surgery for treating ulcers. The analyses below use data from
all 41 studies, which you can see at the text web site. The response was whether the
surgery resulted in the adverse event of recurrent bleeding (1 = yes, 0 = no).
As usual, to compare two groups on a binary response with data stratified on a third
variable, we can analyze the strength of association in the 2 × 2 tables and investigate
how that association varies (if at all) among the strata. When the strata are themselves
a sample, such as different studies for a meta analysis, or schools, or medical clinics,
a random effects approach is natural. We then use a separate random effect for each