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

Choice of W values depends on
investigator

Thisexpressionforthe oddsratio tells usthatwe

obtain a different value for the estimated odds

ratio depending on the values specified for CHL

and HPT. As previously mentioned, this should

make sense conceptually because CHL and HPT

are the only two effect modifiers in the model,

and the value of the odds ratio changes as the

values of the effect modifiers change.

To get a numerical value for the odds ratio, we

consider, for example, the specific values CHL

equal to 220 and HPT equal to 1. Plugging these

into the odds ratio formula, we obtain e to the

0.2028, which equals 1.22.

As a second example, we consider CHL equal to

200 and HPT equal to 0. Here, the odds ratio

becomes e to 1.1506, which equals 3.16.

Thus, we see that depending on the values of

the effect modifiers we will get different values

for the estimated odds ratios. Note that each

estimated odds ratio obtained adjusts for the

confounding effects of all five control variables

because these five variables are contained in

the fitted model asVvariables.

In general, when faced with an odds ratio

expression involving effect modifiers (W), the

choice of values for theWvariables depends

primarily on the interest of the investigator.

Typically, the investigator will choose a range

of values for each interaction variable in

the odds ratio formula; this choice will lead to

a table of estimated odds ratios, such as the

one presented here, for a range of CHL values

and the two values of HPT. From such a table,

together with a table of confidence intervals, the

investigator can interpret the exposure–disease

relationship.

As a second example, we consider a model con-

taining no interaction terms from the same

Evans County data set of 609 white males.

The variables in the model are the exposure

variable CAT, and fiveV variables, namely,

AGE, CHL, SMK, ECG, and HPT. This model

is written in logit form as shown here.

EXAMPLE (continued)

ROR varies with values of CHL and HPT

effect modifiers

 CHL¼220, HPT¼ 1

RORd ¼exp½ 12 : 6894 þ 0 :0692 220ðÞ

 2 :3318 1ðފ

¼expðÞ¼ 0 : 2028 1 : 22

 CHL¼200, HPT¼ 0

RORd ¼exp½ 12 : 6894 þ 0 :0692 200ðÞ

 2 :3318 0ðފ

¼expðÞ¼ 1 : 1506 3 : 16

CHL¼220, HPT¼ 1 )RORd ¼1.22

CHL¼200, HPT¼ 0 )RORd ¼3.16

controls for the confounding effects of

AGE, CHL, SMK, ECG, and HPT

EXAMPLE

TABLE OF POINT ESTIMATESRORd

HPT¼0 HPT¼ 1

CHL¼ 180 0.79 0.08

CHL¼ (^200) 3.16 0.31
CHL¼ (^220) 12.61 1.22
CHL¼ 240 50.33 4.89
EXAMPLE
No interaction model for Evans
County data (n¼609)
logit P(X)¼aþbCAT
þg 1 AGEþg 2 CHL
þg 3 SMKþg 4 ECG
þg 5 HPT
62 2. Important Special Cases of the Logistic Model