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

 Vs not in OR formula butVsin
model, so OR formula controls
confounding:

logit P(X) = a + bE + Σ gi Vi

+ E Σ dj Wj

No interaction:

alldj¼ 0 )ROR¼exp (b)
"
constant

logit P(X)¼aþbEþ~giVi
"
confounding
effects adjusted

Although the coefficients of theVterms do not

appear in the odds ratio formula, these terms

are still part of the fitted model. Thus, the odds

ratio formula not only reflects the interaction

effects in the model but also controls for the

confounding variables in the model.

In contrast, if the model contains no interac-

tion terms, then, equivalently, all thedjcoeffi-

cients are 0; the odds ratio formula thus

reduces to ROR equals to e tob, wherebis

the coefficient of the exposure variable E.

Here, theodds ratio is a fixed constant, so that

its value does not change with different values

of the independent variables. The model in this

case reduces to logit P(X) equalsaplusbtimes

Eplus the sum of the main effect terms involv-

ing theVs and contains no product terms. For

this model, we can say that e tobrepresents an

odds ratio thatadjusts for the potential con-

founding effects of the control variables C 1

throughCpdefined in terms of theVs.

As an example of the use of the odds ratio

formula for theE, V, Wmodel, we return to

the CHD study example we described earlier.

The CHD study model contained eight inde-

pendent variables. The model is restated here

as logit P(X) equalsaplusbtimes CAT plus the

sum of five main effect terms plus the sum of

two interaction terms.

The five main effect terms in this model

account for the potential confounding effects

of the variables AGE through HPT. The two

product terms account for the potential inter-

action effects of CHL and HPT with CAT.

For this example, the odds ratio formula

reduces to the expression ROR equals e to the

quantityb plus the sumd 1 times CHL plus

d 2 times HPT.

EXAMPLE

The model:

logit P (X)¼aþbCAT

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}þg^1 AGEþg^2 CHLþg^3 SMKþg^4 ECGþg^5 HPT

main effects

þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}CATðÞd 1 CHLþd 2 HPT

interaction effects

logit PðXÞ¼aþbCAT

þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g 1 AGEþg 2 CHLþg 3 SMKþg 4 ECGþg 5 HPT

main effects: confounding

þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}CATðÞd 1 CHLþd 2 HPT

product terms: interaction

ROR¼expðÞbþd 1 CHLþd 2 HPT

60 2. Important Special Cases of the Logistic Model