Introduction to Categorical Data Analysis

284 MODELING CORRELATED, CLUSTERED RESPONSES
Table 9.5. Estimates and Standard Errors (in
Parentheses) for Logistic Models Fitted to Teratology
Data of Table 9.4
Type of Logistic Model Fitting
Parameter Binomial ML GEE
Intercept 1.14 (0.13) 1.21 (0.27)
Group 2 −3.32 (0.33) −3.37 (0.43)
Group 3 −4.48 (0.73) −4.58 (0.62)
Group 4 −4.13 (0.48) −4.25 (0.60)
Overdispersion None ˆρ = 0.19
Note: Binomial ML assumes no overdispersion; GEE has exchangeable
working correlation. The intercept term gives result for group 1
(placebo) alone.
Suppose an application has positive within-cluster correlation, as often happens
in practice and as seems to be the case here. Then, standard errors for betweencluster
effects (such as comparisons of separate treatment groups) and standard errors
of estimated means within clusters tend to be larger than when the observations
are independent. We see this in Table 9.5 except for group 3 and its comparison
with the placebo group. With positive within-cluster correlation, standard errors for
within-cluster effects, such as a slope for a trend in the repeated measurements on a
subject, tend to be smaller than when the observations are independent.
9.2.5 Limitations of GEE Compared with ML
Because the GEE method specifies the marginal distributions and the correlation
structure but not the complete multivariate distribution, there is no likelihood function.
In this sense, the GEE method is a multivariate type of quasi-likelihood method. So,
its estimates are not ML estimates.
For clustered data, the GEE method is much simpler computationally than ML
and much more readily available in software. However, it has limitations. Because
it does not have a likelihood function, likelihood-ratio methods are not available for
checking fit, comparing models, and conducting inference about parameters. Instead
inference uses statistics, such as Wald statistics, based on the approximate normality
of the estimators together with their estimated covariance matrix. Such inference is
reliable mainly for very large samples. Otherwise, the empirically based standard
errors tend to underestimate the true ones.
Some software attempts to improve on Wald-type inference by making tests available
that mimic the way score tests are constructed, if one had a likelihood function.
These generalized score tests also incorporate empirical information in forming
standard error estimates, and they are preferable to Wald tests.