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

Two odds ratios:

OR 1 (category 1 vs. category 0)
(Adenosquamous vs.
Adenocarcinoma)

OR 2 (category 2 vs. category 0)
(Other vs. Adenocarcinoma)

OR 1 ¼

½PðD¼ 1 jX¼ 1 Þ=PðD¼ 0 jX¼ 1 ފ

½PðD¼ 1 jX¼ 0 Þ=PðD¼ 0 jX¼ 0 ފ

OR 2 ¼

½PðD¼ 2 jX¼ 1 Þ=PðD¼ 0 jX¼ 1 ފ

½PðD¼ 2 jX¼ 0 Þ=PðD¼ 0 jX¼ 0 ފ

Adenosquamous vs. Adenocarci-
noma:

OR 1 ¼

exp½a 1 þb 11 ð 1 ފ

exp½a 1 þb 11 ð 0 ފ

¼eb^11

Other vs. Adenocarcinoma:

OR 2 ¼

exp½a 2 þb 21 ð 1 ފ

exp½a 2 þb 21 ð 0 ފ

¼eb^21

They are different!

OR 1 =eb^11 OR 2 =eb^21

We need to calculate two odds ratios, one that

compares category 1 (Adenosquamous) to

category 0 (Adenocarcinoma) and one that

compares category 2 (Other) to category 0

(Adenocarcinoma).

Recall that we are actually calculating a ratio of

two “odds-like” expressions. However, we con-

tinue the conventional use of the term odds

ratio for our discussion.

Each odds ratio is calculated in a manner sim-

ilar to that used in standard logistic regression.

The two OR formulas are shown on the left.

Using our previously defined probabilities of

the log odds, we substitute the two values of

X 1 for the exposure (i.e., 0 and 1) into those

expressions. After dividing, we see that the

odds ratio for the first comparison (Adenos-

quamous vs. Adenocarcinoma) is e to theb 11.

The odds ratio for the second comparison

(Other vs. Adenocarcinoma) is e to theb 21.

We obtain two different odds ratio expressions,

one utilizingb 11 and the other utilizingb 21.

Thus, quantifying the association between the

exposure and outcome depends on which

levels of the outcome are being compared.

438 12. Polytomous Logistic Regression