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

IV. The Generalized Linear
Mixed Model
Approach

Mixed models:

 Random effects

 Fixed effects

 Cluster effect is random
variable

Mixed logistic model (MLM):

Special case of GLMM

Combines GEE and CLR features

GEE

User specifies

g(m) and Ci

GLMM: subject-specific effects random

Subject-specific

effects

CLR

The generalized linear mixed model (GLMM)

provides another approach that can be used for

correlated dichotomous outcomes. GLMM is a

generalization of the linear mixed model.

Mixed models refer to themixingof random

and fixed effects. With this approach, the clus-

ter variable is considered a random effect. This

means that the cluster effect is a random vari-

able following a specified distribution (typi-

cally a normal distribution).

A special case of the GLMM is the mixed logis-

tic model (MLM). This type of model combines

some of the features of the GEE approach and

some of the features of the conditional logistic

regression approach. As with the GEE

approach, the user specifies the logit link func-

tion and a structure (Ci) for modeling response

correlation. As with the conditional logistic

regression approach, subject-specific effects

are directly included in the model. However,

here these subject-specific effects are treated

as random rather than fixed effects. The

model is commonly stated in terms of theith

subject’s mean response (mi).

We again use the heartburn data to illustrate

the model (shown on the left) and state it in

terms of theith subject’s mean response, which

in this case is theith subject’s probability of

heartburn relief. The coefficientb 1 is called a

fixed effect, whereasb 0 i is called a random

effect. The random effect (b 0 i) in this model is

assumed to follow a normal distribution with

mean 0 and variancesb 02. Subject-specific ran-

dom effects are designed to account for the

subject-to-subject variation, which may be

due to unexplained genetic or environmental

factors that are otherwise unaccounted for in

the model. More generally, random effects are

often used when levels of a variable are selected

at random from a large population of possible

levels.

EXAMPLE

Heartburn Relief Study

logit mi = b 0 + b 1 RXi + b 0 i

b 1 = fixed effect

b 0 i = random effect,

where b 0 i is a random variable

~ N(0, sb 02 )

580 16. Other Approaches for Analysis of Correlated Data