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

Chapter 3: Computing OR for dif-
ferent codings

VII. Interval Estimation:
Interaction

No interaction: simple formula

Interaction: complex formula

A detailed discussion of the effect of different

codings of the exposure variable on the compu-

tation of the odds ratio is described in Chap. 3

of this text. It is beyond the scope of this

presentation to describe in detail the effect of

different codings on the corresponding confi-

dence interval for the odds ratio. We do, how-

ever, provide a simple example to illustrate this

situation.

SupposeX 3 is coded as (1, 1) instead of (0, 1),

so that1 denotes unexposed persons and 1

denotes exposed persons. Then, the odds ratio

expression for the effect ofX 3 is given by e to 1

minus1 timesb 3 , which is e to 2 timesb 3. The

corresponding 95% confidence interval for

the odds ratio is then given by exponentiating

the confidence limits for the parameter 2b 3 ,as

shown here; that is, the previous confidence

interval formula is modified by multiplying^b 3

and its standard error by the number 2.

The above confidence interval formulae involv-

ing a single parameter assume that there are no

interaction effects in the model. When there is

interaction, the confidence interval formula

must be modified from what we have given so

far. Because the general confidence interval

formula is quite complex when there is interac-

tion, our discussion of the modifications

required will proceed by example.

Suppose we focus on Model 3, which is again

shown here, and we assume that the variable

X 3 is a (0, 1) exposure variable of interest. Then

the formula for the estimated odds ratio for the

effect ofX 3 controlling for the variablesX 1 and

X 2 is given by the exponential of the quantity^b 3

plus^b 4 timesX 1 plus^b 5 times X 2 , where^b 4 and

^b 5 are the estimated coefficients of the interac-

tion termsX 1 X 3 andX 2 X 3 in the model.

EXAMPLE

X 3 coded as

 1 unexposed

1 exposed



OR¼exp½Š¼ 1 ð 1 Þb 3 e^2 b^3

95 %CI:exp 2 ^b 3  1 : 96  2 sb^ 3



EXAMPLE

Model 3: X 3 ¼(0, 1) exposure

logitP 3 (X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3

þb 4 X 1 X 3 þb 5 X 2 X 3

dOR¼exp^b 3 þ^b 4 X 1 þ^b 5 X 2



142 5. Statistical Inferences Using Maximum Likelihood Techniques