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

III. Odds Ratio for
Arbitrary Coding ofE

Model:

dichotomous, ordinal or interval

logit PðÞ¼X aþbEþ~

p 1

i¼ 1

giVi

þE~

p 2

j¼ 1

djWj

Fitting this no interaction model to the data,

we obtain the estimates listed here.

For this fitted model, then, the odds ratio is

given by e to the power 0.5978, which equals

1.82. Notice that, as should be expected, this

odds ratio is a fixed number as there are no

interaction terms in the model.

Now, if we consider the same data set and the

same model, except that the coding of CAT is

(1, 1) instead of (0, 1), the coefficientb^of CAT

becomes 0.2989, which is one-half of 0.5978.

Thus, for this coding scheme, the odds ratio

is computed as e to 2 times the corresponding

^bof 0.2989, which is the same as e to 0.5978,

or 1.82. We see that, regardless of the coding

scheme used, the final odds ratio result is the

same, as long as the correct odds ratio formula

is used. In contrast, it would be incorrect to use

the (1, 1) coding scheme and then compute

the odds ratio as e to 0.2989.

We now consider the odds ratio formula for

any single exposure variableE, whetherdicho-

tomous, ordinal,orinterval, controlling for a

collection ofCvariables in the context of an

E, V, Wmodel shown again here. That is, we

allow the variableEto be defined arbitrarily of

interest.

EXAMPLE (continued)

(0, 1) coding for CAT

Variable Coefficient

Intercept ^a¼ 6 : 7747

CAT ^b¼ 0 : 5978

AGE ^g 1 ¼ 0 : 0322

CHL ^g 2 ¼ 0 : 0088

SMK ^g 3 ¼ 0 : 8348

ECG ^g 4 ¼ 0 : 3695

HPT ^g 5 ¼ 0 : 4392

RORd ¼expð 0 : 5978 Þ¼ 1 : 82

No interaction model: ROR fixed

ðÞ 1 ; 1 coding for CAT:

^b¼ 0 : 2989 ¼^0 :^5978

2



RORd ¼exp 2 ^b



¼expðÞ 2  0 : 2989

¼expðÞ 0 : 5978

¼ 1 : 82

sameROR as for (0, 1) codingd

Note.RORd 6 ¼expð 0 : 2989 Þ¼ 1 :" 35

incorrect value

Presentation: III. Odds Ratio for Arbitrary Coding ofE 79