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

logit PðXÞ¼aþbEþ~

i

g 1 iV 1 i

þ~

j

g 2 jV 2 jþE~

k

dkWk;

where

V 1 i¼dummy variables for

matching strata

V 2 j¼other covariates (not

matched)

Wk¼effect modifiers defined

from other covariates

Frequency matching

(small no. of strata)

+

Unconditional ML estimation may be

used if ‘‘appropriate’’

ðConditional ML always unbiasedg)

Four types of stratum:

Type 1

E E

D 11

D 00

11

Ppairs

concordant

Type 2

E E

D 10

D 01

11

Qpairs

discordant

Type 3

E E

D 01

D 10

11

Rpairs

discordant

Type 4

E E

D 00

D 11

11

Spairs

concordant

The logistic model for matched follow-up stud-

ies is shown at the left. This model is essentially

the same model as we defined for case-control

studies, except that the matching strata are

now defined by exposed/unexposed matched

sets instead of by case/control matched sets.

The model shown here allows for interaction

between the exposure of interest and the con-

trol variables that are not involved in the

matching.

If frequency matching is used, then the number

of matching strata will typically be small rela-

tive to the total sample size, so it is appropriate

to consider using unconditional ML estimation

for fitting the model. Nevertheless, as when

pooling exchangeable matched sets results

from individual matching, conditional ML esti-

mation will always provide unbiased estimates

(but may yield less precise estimates than

obtained from unconditional ML estimation).

In matched-pair follow-up studies, each of the

matched sets (i.e., strata) can take one of four

types, shown at the left. This is analogous to

the four types of stratum for a matched case-

control study, except here each stratum con-

tains one exposed subject and one unexposed

subject rather than one case and control.

The first of the four types of stratum describes

a “concordant” pair for which both the exposed

and unexposed have the disease. We assume

there arePpairs of this type.

The second type describes a “discordant pair”

in which the exposed subject is diseased and an

unexposed subject is not diseased. We assume

Qpairs of this type.

The third type describes a “discordant pair” in

which the exposed subject is nondiseased and

the unexposed subject is diseased. We assume

Rpairs of this type.

410 11. Analysis of Matched Data Using Logistic Regression