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

eij¼YijPðYij¼ 1 jXÞ;

where

PðYij¼ 1 jXÞ¼m

¼

1

1 þexp ðb 0 þ~

p

h¼ 1

bhXhijþb 0 iÞ

"#

GLM GEE

Model Yi¼miþeij Yij¼mijþeij
R Independent Correlated
G ——

GLMM:Yij¼g^1 ðX;b;b 0 iÞ
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
mij

þeij

User specifies covariance struc-

tures forR, G,or both

GEE:correlation structure speci-
fied

GLMM:covariance structure spe-
cified

Covariance structure contains
parameters for both the variance
and covariance

The residual variation (eij) accounts for the dif-

ference between the observed value ofYand

the mean response, P(Y¼1|X), for a given sub-

ject. The random effect (b 0 i), on the other hand,

allows for randomness in the modeling of

the mean response [i.e., P(Y¼1|X)], which is

modeled using both fixed (bs) and random (bs)

effects.

For GLM and GEE models, the outcomeYis

modeled as the sum of the mean and the resid-

ual variation [Y¼mþeij, where the mean (m)

is fixed] determined by the subject’s pattern of

covariates. For GEE, the residual variation is

modeled with a correlation structure, whereas

for GLM, the residual variation (theRmatrix)

is modeled as independent. Neither GLM nor

GEE models contain aGmatrix, as they do not

contain any random effects (b).

In contrast, for GLMMs, the mean also con-

tains a random component (b 0 i). With GLMM,

the user can specify a covariance structure for

theGmatrix (for the random effects), theR

matrix (for the residual variation), or both.

Even if theGandRmatrices are modeled to

contain zero correlations separately, the com-

bination of both matrices in the model gener-

ally forms a correlated random structure for

the response variable.

Another difference between a GEE model and a

mixed model (e.g., MLM) is that a correlation

structure is specified with a GEE model,

whereas a covariance structure is specified

with a mixed model. A covariance structure con-

tains parameters for both the variance and

covariance, whereas a correlation structure con-

tains parameters for just the correlation. Thus,

with a covariance structure, there are additional

variance parameters and relationships among

those parameters (e.g., variance heterogeneity)

to consider (see Practice Exercises 7–9).

Presentation: IV. The Generalized Linear Mixed Model Approach 583