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

We now consider how to use the information

provided to obtain an estimated odds ratio

for the fitted model. Because this model con-

tains the product terms CC equal to CAT

CHL, and CH equal to CATHPT, the esti-

mated odds ratio for the effect of CAT must

consider the coefficients of these terms as

well as the coefficient of CAT.

The formula for this estimated odds ratio is

given by the exponential of the quantityb^plus

^d 1 times CHL plus^d 2 times HPT, whereb^equals

12.6894 is the coefficient of CAT,^d 1 equals

0.0692 is the coefficient of the interaction term

CC, and^d 2 equals2.3318 is the coefficient of

the interaction term CH.

Plugging the estimated coefficients into the

odds ratio formula yields the expression:eto

the quantity12.6894 plus 0.0692 times CHL

plus2.3318 times HPT.

To obtain a numerical value from this expres-

sion, it is necessary to specify a value for CHL

and a value for HPT. Different values for CHL

and HPT will, therefore, yield different odds

ratio values, as should be expected because

the model contains interaction terms.

The table shown here illustrates different odds

ratio estimates that can result from specifying

different values of the effect modifiers. In this

table, the values of CHL are 200, 220, and 240;

the values of HPT are 0 and 1, where 1 denotes

a person who has hypertension. The cells

within the table give the estimated odds ratios

computed from the above expression for the

odds ratio for different combinations of CHL

and HPT.

For example, when CHL equals 200 and HPT

equals 0, the estimated odds ratio is given by

3.16; when CHL equals 220 and HPT equals 1,

the estimated odds ratio is 1.22. Note that

each of the estimated odds ratios in this table

describes the association between CAT and

CHD adjusted for the five covariables AGE,

CHL, ECG, SMK, and HPT because each of

the covariables is contained in the model asV

variables.

EXAMPLE (continued)

OR considers coefficients of CAT, CC,d
and CH

ORd¼expð^bþ^d 1 CHLþ^d 2 HPTÞ
where

b^¼ 12 : 6894

^d 1 ¼ 0 : 0692

^d 2 ¼ 2 : 3318

dOR¼exp½ 12 : 6894 þ 0 : 0692 CHL

þð 2 : 3318 ÞHPTŠ

Must specify:

CHL and HPT

effect modifiers

Note.OR different for different valuesd
specified for CHL and HPT

HPT
01
200 3.16 0.31
CHL 220 12.61 1.22

240 50.33 4.89

CHL¼200,HPT¼0:ORd¼3.16

CHL¼220,HPT¼1:ORd¼1.22

OR adjusts for AGE, CHL, ECG,d
SMK, and HPT (theVvariables)

Presentation: V. Overview on Statistical Inferences for Logistic Regression 119