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

REFERENCES FOR CHOICE OF Vs
AND Ws FROM Cs

 Chap. 6: Modeling Strategy
Guidelines

 Epidemiologic Research,
Chap. 21

Assume:Vs andWs areCs or subset
ofCs

NOTE

Ws ARE SUBSET OF Vs

logitPðXÞ¼aþbEþg 1 V 1 þg 2 V 2

þþgp 1 Vp 1 þd 1 EW 1

þd 2 EW 2 þþdp 2 EWp 2 ,

where
b¼coefficient ofE
gs¼coefficient ofVs
ds¼coefficient ofWs

logit P(X)¼aþbE

þ~

p 1

i¼ 1

giViþE~

p 2

j¼ 1

djWj

It is beyond the scope of this chapter to discuss

the subtleties involved in the particular choice

of theVs andWs from theCs for a given model.

More depth is provided in a separate chapter

(Chap. 6) on modeling strategies and in Chap.

21 ofEpidemiologic Researchby Kleinbaum,

Kupper, and Morgenstern.

In most applications, the Vs will be theCs

themselves or some subset of theCs and the

Ws will also be theCs themselves or some sub-

set thereof. For example, if theCs are AGE,

RACE, and SEX, then theVs may be AGE,

RACE, and SEX, and theWs may be AGE and

SEX, the latter two variables being a subset of

theCs. Here the number of Vvariables, p 1 ,

equals 3, and the number ofWvariables,p 2 ,

equals 2, so thatk, which gives the total num-

ber of variables in the model, isp 1 plusp 2 plus 1

equals 6.

Note, as we describe further in Chap. 6, that

you cannot have aWin the model that is not

also contained in the model as aV; that is,Ws

have to be a subset of theVs. For instance, we

cannot allow a model whoseVs are AGE and

RACE and whose Ws are AGE and SEX

because the SEX variable is not contained in

the model as aVterm.

A logistic model incorporating this special case

containing theE, V, andWvariables defined

above can be written in logit form as shown

here.

Note thatbis the coefficient of the single expo-

sure variableE, thegs are coefficients of poten-

tial confounding variables denoted by theVs,

and theds are coefficients of potential interac-

tion effects involvingEseparately with each of

theWs.

We can factor out theEfrom each of the inter-

action terms, so that the model may be more

simply written as shown here. This is the form

of the model that we will use henceforth in this

presentation.

EXAMPLE

C 1 ¼AGE,C 2 ¼RACE,C 3 ¼SEX

V 1 ¼AGE,V 2 ¼RACE,V 3 ¼SEX

W 1 ¼AGE,W 2 ¼SEX

p 1 ¼3,p 2 ¼2,k¼p 1 þp 2 þ 1 ¼ 6

EXAMPLE

V 1 = AGE, V 2 = RACE

W 1 = AGE, W 2 = SEX

58 2. Important Special Cases of the Logistic Model