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

 SomeCs matched by design

 RemainingCs not matched

D¼ð 0 ; 1 Þdisease

X 1 ¼E¼ð 0 ; 1 Þexposure

Some Xs: V 1 i dummy variables
(matched strata)

SomeXs: V 2 jvariables (potential
confounders)

SomeXs: product termsEWj
(Note:Ws usuallyV 2 s)

The model:

logit PðXÞ¼aþbE

þ~g 1 iV 1 i

|fflffl{zfflffl}

matching

þ~g 2 jV 2 j

|fflffl{zfflffl}

confounders

þE~dkWk

|ffl{zffl}

interaction

We assume that some of theseCvariables have

been matched in the study design, either using

pair-matching orR-to-1 matching. The remain-

ingCvariables have not been matched, but it is

of interest to control for them, nevertheless.

Given the above context, we now define the

following set of variables to be incorporated

into a logistic model for matched data. We

have a (0, 1) disease variableDand a (0, 1)

exposure variableX 1 equal toE.

We also have a collection ofXs which are dummy

variables to indicate the different matched strata;

these variables are denoted asV 1 variables.

Further, we have a collection ofXs which are

defined from theCs not involved in the match-

ing and represent potential confounders in

addition to the matched variables. These poten-

tial confounders are denoted asV 2 variables.

And finally, we have a collection ofXs which

are product terms of the form E times W,

where theWs denote potential interaction vari-

ables. Note that theWs will usually be defined

in terms of theV 2 variables.

The logistic model for matched analysis is then

given in logit form as shown here. In this

model, theg 1 is are coefficients of the dummy

variables for the matching strata, theg 2 is are

the coefficients of the potential confounders

not involved in the matching, and thedjs are

the coefficients of the interaction variables.

As an example of dummy variables defined for

matched strata, consider a study involving

pair-matching by AGE, RACE, and SEX, con-

taining 100 matched pairs. Then, the above

model requires defining 99 dummy variables

to incorporate the 100 matched pairs.

We can define these dummy variables asV 1 i

equals 1 if an individual falls into the ith

matched pair and 0 otherwise. Thus, it follows

thatV 11 equals 1 if an individual is in the first

matched pair and 0 otherwise,V 12 equals 1 if

an individual is in the second matched pair

and 0 otherwise, and so on up toV1, 99, which

equals 1 if an individual is in the 99th matched

pair and 0 otherwise.

EXAMPLE

Pair-matching by AGE, RACE, SEX

100 matched pairs

99 dummy variables

V 1 i¼^1 ifith matched pair

0 otherwise



i¼ 1 ; 2 ;...; 99

V 11 ¼^1 if first matched pair

0 otherwise



V 12 ¼^1 if second matched pair

0 otherwise



.

.

.

V 1 ; 99 ¼^1 if^99 th matched pair

0 otherwise



Presentation: IV. The Logistic Model for Matched Data 399