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

For each subject:

logit of baseline risk = (b 0 + b 0 i)

b 0 i = subject-specific intercept

No random effect for RX

+

RX effect is same for each subject

i.e., exp(b 1 )

Mixed logistic model (MLM)

Variable Coefficient

Standard

Error

Wald
p-value
INTERCEPT 0.2285 0.3583 0.5274
RX 0.3445 0.4425 0.4410

Odds ratio and 95% CI:

ORd¼expð 0 : 3445 Þ¼ 1 : 41

95 %CI¼ð 0 : 593 ; 3 : 360 Þ

Model comparison

Model ORd s^b

MLM 1.41 0.4425
GEE 1.35 0.3868
CLR 1.50 0.5271

With this model, each subject has his/her own

baseline risk, the logit of which is the intercept

plus the random effect (b 0 þb 0 i). The sum

(b 0 þb 0 i) is typically called the subject-specific

intercept. The amount of variation in the

baseline risk is determined by the variance

ðÞsb 02 ofb 0 i.

In addition to the intercept, we could have

added another random effect allowing the

treatment (RX) effect to also vary by subject

(see Practice Exercises 4–9). By not adding this

additional random effect, there is an assump-

tion that the odds ratio for the effect of treat-

ment is the same for each subject, exp(b 1 ).

The output obtained from running the MLM

on the heartburn data is presented on the left.

This model was run using SAS’s GLIMMIX

procedure. (See the Computer Appendix for

details and an example of program coding.)

The odds ratio estimate for the effect of treat-

ment for relieving heartburn is exp(0.3445)¼

1.41. The 95% confidence interval is (0.593,

3.360).

The odds ratio estimate using this model is

slightly larger than the estimate obtained

(1.35) using the GEE approach, but somewhat

smaller than the estimate obtained (1.50) using

the conditional logistic regression approach.

The standard error at 0.4425 is also larger

than what was obtained in the GEE model

(0.3868), but smaller than in the conditional

logistic regression (0.5271).

Presentation: IV. The Generalized Linear Mixed Model Approach 581