Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

IX. Empirical and Model-
Based Variance
Estimators

GEE estimates have desirable
asymptotic
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properties.

K!1(i.e.,K“large”),

whereK¼# clusters

“Large” is subjective

Two statistical properties of GEE
estimates (if model correct):

1. Consistent

^b!b asK!1

2. Asymptotically normal

^bnormal asK!1

Asymptotic normal property allows:
 Confidence intervals

 Statistical tests

In the next section, we describe two variance

estimators that can be obtained for the fitted

regression coefficients – empirical and model-

based estimators. In addition, we discuss the

effect of misspecification of the correlation

structure on those estimators.

Maximum likelihood estimates in GLM are

appealing because they have desirable asymp-

totic statistical properties. Parameter esti-

mates derived from GEE share some of these

properties. By asymptotic, we mean “as the

number of clusters approaches infinity”. This

is a theoretical concept since the datasets that

we are considering have a finite sample size.

Rather, we can think of these properties as

holding for large samples. Nevertheless, the

determination of what constitutes a “large”

sample is somewhat subjective.

If a GEE model is correctly specified, then the

resultant regression parameter estimates have

two important statistical properties: (1) the

estimates areconsistentand (2) the distribu-

tion of the estimates is asymptotically normal.

A consistent estimator is a parameter estimate

that approaches the true parameter value in

probability. In other words, as the number of

clusters becomes sufficiently large, the differ-

ence between the parameter estimate and the

true parameter approaches zero. Consistency

is an important statistical property since it

implies that the method will asymptotically

arrive at the correct answer. The asymptotic

normal property is also important since know-

ledge of the distribution of the parameter

estimates allows us to construct confidence

intervals and perform statistical tests.

516 14. Logistic Regression for Correlated Data: GEE