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

Empirical estimator generally
recommended.

Reason: robust to

misspecification of

correlation structure

Preferable to specify working corre-
lation structure close to actual one:

 More efficient estimate ofb

 More reliable estimate of varð^bÞ
if number of clusters is small

X. Statistical Tests

In SLR, three tests of significance
of^bhs:

 Likelihood ratio test

 Score test

 Wald test

In GEE models, two tests of^bh:

 Score test

 Wald test

likelihood ratio test

For the estimation of the variance of ^b in

the GEE model, the empirical estimator is

generally recommended over the model-based

estimator since it is more robust to misspecifi-

cation of the correlation structure. This may

seem to imply that if the empirical estimator

is used, it does not matter which correlation

structure is specified. However, choosing a

working correlation that is closer to the actual

one is preferable since there is a gain in effi-

ciency. Additionally, since consistency is an

asymptotic property, if the number of clusters

is small, then even the empirical variance esti-

mate may be unreliable (e.g., may yield incor-

rect confidence intervals) if the correlation

structure is misspecified.

Thelikelihood ratio test, theWald test, and the

Score testcan each be used to test the statistical

significance of regression parameters in a stan-

dard logistic regression (SLR). The formu-

lation of the likelihood ratio statistic relies on

the likelihood function. The formulation of the

Score statistic relies on the score function, (i.e.,

the partial derivatives of the log likelihood).

(Score functions are described in Sect.XI.)

The formulation of the Wald test statistic relies

on the parameter estimate and its variance

estimate.

For GEE models, the likelihood ratio test cannot

be used since a likelihood is never formulated.

However, there is a generalization of the Score

test designed for GEE models. The test statistic

for this Score test is based on the generalized

estimating “score-like” equations that are solved

to produce parameter estimates for the GEE

model. (These “score-like” equations are

described in Sect.XI.) The Wald test can also

be used for GEE models since parameter

estimates for GEE models are asymptotically

normal.

Presentation: X. Statistical Tests 519