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

V. The Model and Odds
Ratio for Several
Exposure Variables
(No Interaction Case)

q variables:E 1 ,E 2 ,...,Eq

(dichotomous, ordinal, or interval)

No interaction model:

logit PðÞ¼X aþb 1 E 1 þb 2 E 2

þ...þbqEqþ~

p 1

i¼ 1

giVi

 q 6 ¼k1 in general

E*vs.E**

E¼(E 1 ,E 2 ,...,Eq)

E¼(E 1 ,E 2 ,...,Eq)

General formula:E 1 ,E 2 ,...,E 8
(no interaction)

RORE*vs:E**¼exp

h

E* 1 E** 1



b 1

þ E* 2 E** 2



b 2 þ

þ E*qE**q



bq

i

We now consider the odds ratio formula when

there are several different exposure variables in

the model, rather than a single exposure vari-

able with several categories. The formula for

this situation is actually no different than for a

single nominal variable. The different exposure

variables may be denoted byE 1 ,E 2 , and so on

up throughEq. However, rather than being

dummy variables, theseEs can be any kind of

variable – dichotomous, ordinal, or interval.

For example,E 1 may be a (0, 1) variable for

smoking (SMK),E 2 may be an ordinal variable

for physical activity level (PAL), andE 3 may be

the interval variable systolic blood pressure

(SBP).

A no interaction model with several exposure

variables then takes the form logit P(X) equalsa

plusb 1 timesE 1 plusb 2 timesE 2 , and so on up

tobqtimesEqplus the usual set ofVterms. This

model form is the same as that for a single

nominal exposure variable, although this time

there areqEs of any type, whereas previously

we hadk1 dummy variables to indicatek

exposure categories. The corresponding model

involving the three exposure variables SMK,

PAL, and SBP is shown here.

As before, the general odds ratio formula for

several variables requires specifying the values

of the exposure variables for two different per-

sons or groups to be compared – denoted by the

boldE*andE**. CategoryE*is specified by the

variable valuesE 1 *,E 2 *, and so on up toEq*, and

categoryE**is specified by a different collec-

tion of valuesE 1 **,E 2 **, and so on up toEq**.

The general odds ratio formula for comparing

E* vs. E** is given by the formula ROR

equals e to the quantity (E 1

*

E 1 *) times

b 1 plus (E*E**) timesb 2 , and so on up to

(Eq*Eq**) timesbq.

EXAMPLE

logit PðÞ¼X aþb 1 SMKþb 2 PAL

þb 3 SBPþ~

p 1

i¼ 1

giVi

EXAMPLE

E 1 ¼SMK (0,1)

E 2 ¼PAL (ordinal)

E 3 ¼SBP (interval)

Presentation: V. The Model and Odds Ratio for Several Exposure Variables 85