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

Coding RORd

ðÞAa¼ 1 ;b¼ 0 RORdA¼exp



^bAþ~p^2

j¼ 1

^djAWj



ðÞBa¼ 1 ;b¼ 1 RORdB¼exp



2 ^bBþ~

p 2

j¼ 1

2 ^djBWj



(C)a = 100, b = 0 RORC = exp 100 bC + Σ 100 djCWj
j=1

p 2

same value

although

different

codings

different values

for different

codings

RORA = RORB = RORC bA ≠^ bB ≠^ bC

djA ≠ djB ≠ djC

Thus, depending on the coding scheme forE,

the odds ratio will be calculated differently.

Nevertheless, even though^band the^djwill be

different for different coding schemes, the final

odds ratio value will be the same as long as the

correct formula is used for the corresponding

coding scheme.

As shown here for the three examples above,

which are labeled A, B, and C, the three com-

puted odds ratios will be the same, even

though the estimates^band^djused to compute

these odds ratios will be different for different

codings.

As a numerical example, we consider a model

that contains no interaction terms from a data

set of 609 white males from Evans County,

Georgia. The study is a follow-up study to

determine the development of coronary heart

disease (CHD) over 9 years of follow-up. The

variables in the model are CAT, a dichotomous

exposure variable, and fiveVvariables, namely,

AGE, CHL, SMK, ECG, and HPT.

This model is written inlogit formas logit P(X)

equalsaplusbtimes CAT plus the sum of five

main effect termsg 1 times AGE plusg 2 times

CHL, and so on up throughg 5 times HPT.

We first describe the results from fitting this

model when CAT is coded as a (0, 1) variable.

Then, we contrast these results with other cod-

ings of CAT.

Because this model contains no interaction

terms and CAT is coded as (0, 1), the odds ratio

expression for the CAT, CHD association is

given by e to^b,where^bis the estimated coeffi-

cient of the exposure variable CAT.

EXAMPLE: No Interaction Model

Evans County follow-up study:

n¼609 white males

D¼CHD status

E¼CAT, dichotomous

V 1 ¼AGE,V 2 = CHL,V 3 = SMK,

V 4 ¼ECG,V 5 = HPT

logit PðXÞ¼aþbCATþg 1 AGE

þg 2 CHLþg 3 SMK

þg 4 ECGþg 5 HPT

CAT: (0, 1) vs. other codings

RORd ¼exp^b



78 3. Computing the Odds Ratio in Logistic Regression