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

Covariance Þ unique correlation

but correlation Þ unique covariance

Forith subject:

 Rmatrix dimensions depend
on number of observations for
subjecti(ni)

 Gmatrix dimensions depend on
number of random effects (q)

Heartburn data:

R¼ 2 2 matrix

G¼ 1  1matrix(onlyone

random effect)

CLR vs. MLM: subject-specific
effects

CLR:logitmi¼b 0 þb 1 RXþgi,

wheregiis afixedeffect

MLM:logitmi¼b 0 þb 1 RXþb 0 i,

whereb 0 iis arandomeffect

Fixed effectgi: impacts modeling ofm

Random effect b 0 i:usedto
characterize the variance

If a covariance structure is specified, then the

correlation structure can be ascertained. The

reverse is not true, however, since a correlation

matrix does not in itself determine a unique

covariance matrix.

For a given subject, the dimensions of theR

matrix depend on how many observations (ni)

the subject contributes, whereas the dimen-

sions of theGmatrix depend on the number

of random effects (e.g., q) included in the

model. For the heartburn data example, in

which there are two observations per subject

(ni¼2), theRmatrix is a 22 matrix modeled

with zero correlation. The dimensions ofGare

1 1(q¼1) since there is only one random

effect (b 0 i), so there is no covariance structure

to consider forGin this model. Nevertheless,

the combination of theGandRmatrix in this

model provides a way to account for correla-

tion between responses from the same subject.

We can compare the modeling of subject-

specific fixed effects with subject-specific ran-

dom effects by examining the conditional logis-

tic model (CLR) and the mixed logistic model

(MLM) in terms of theith subject’s response.

Using the heartburn data, these models can be

expressed as shown on the left.

The fixed effect,gi, impacts the modeling of the

mean response. The random effect,b 0 i,isa

random variable with an expected value of

zero. Therefore,b 0 idoes not directly contribute

to the modeling of the mean response; rather, it

is used to characterize the variance of the mean

response.

584 16. Other Approaches for Analysis of Correlated Data