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

No interaction model:

logit PðÞ¼X aþb 1 E 1 þb 2 E 2 þ...

þbk 1 Ek 1 þ~

p 1

i¼ 1

giVi

logit PðÞ¼X aþb 1 OCC 1 þb 2 OCC 2

þb 3 OCC 3 þ~

p 1

i¼ 1

giVi

SpecifyE*andE**in terms ofk 1
dummy variables where

E¼(E 1 ,E 2 ,...,Ek 1 )

Generally, defineE*andE**as

E*¼(E 1 *,E 2 *,...,Ek* 1 )

and

E*¼(E 1 **,E 2 **,...,Ek** 1 )

So, for example, with occupational status, we

define three dummy variables OCC 1 , OCC 2 ,

and OCC 3 to reflect four occupational cate-

gories, where OCCiis defined to take on the

value 1 for a person in theith occupational

category and 0 otherwise, foriranging from

1 to 3. Note that for this choice of dummy

variables, the referent group is the fourth occu-

pational category, for which OCC 1 ¼OCC 2 ¼

OCC 3 ¼0.

A no interaction model for a nominal exposure

variable with k categories then takes the

form logit P(X) equalsaplusb 1 timesE 1 plus

b 2 timesE 2 and so on up tobk 1 timesEk 1 plus

the usual set ofVterms, where theEiare the

dummy variables described above.

The corresponding model for four occupational

status categories then becomes logit P(X)

equals a plus b 1 times OCC 1 plus b 2 times

OCC 2 plusb 3 times OCC 3 plus theVterms.

To obtain an odds ratio from the above model,

we need to specify two categoriesE*andE**

of the nominal exposure variable to be com-

pared, and we need to define these categories

in terms of thek1 dummy variables. Note

that we have usedbold letterstoidentify the

two categories of E; this has been done because

theEvariable is a collection of dummy vari-

ables rather than a single variable.

For the occupational status example, suppose

we want an odds ratio comparing occupational

category 3 with occupational category 1. Here,

E*represents category 3 andE**represents cat-

egory 1. In terms of the three dummy variables

for occupational status, then,E*is defined by

OCC 1 *¼0, OCC 2 *¼0, and OCC 3 *¼ 1, whereas

E**is defined by OCC 1 **¼1, OCC 2 **¼0, and

OCC 3 **¼0.

More generally, categoryE*is defined by the

dummy variable valuesE 1 *,E 2 *, and so on up to

Ek* 1 , which are 0s or 1s. Similarly, categoryE 1 **

is defined by the valuesE 1 **,E 2 **, and so on up to

Ek** 1 , which is a different specification of 0s

or 1s.

EXAMPLE

E¼OCC withk¼ 4 )k 1 ¼ 3

OCC 1 , OCC 2 ,

OCC 3

where OCCi¼

 1 if categoryi

0 if otherwise

fori¼1, 2, 3 (referent: category 4)

EXAMPLE

E= occupational status (four

categories)

E*¼category 3 vs. E**¼category 1

E*¼(OCC* 1 ¼0, OCC 2 *¼0, OCC* 3 ¼1)

E**¼(OCC** 1 ¼1, OCC** 2 ¼0,

OCC** 3 ¼0)

Presentation: IV. The Model and Odds Ratio for a Nominal Exposure Variable 83