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

Model for three categories, one
predictor (X 1 = AGEGP):

ln

PðD¼ 1 jX 1 Þ

PðD¼ 0 jX 1 Þ



¼a 1 þb 11 X 1

ln

PðD¼ 2 jX 1 Þ

PðD¼ 0 jX 1 Þ



¼a 2 þb 21 X 1

2 vs. 0

1 vs. 0

a 1 b 11

a 2 b 21

III. Odds Ratio with Three
Categories

^a 1 ^a 2

b^ 11 b^ 21

)

Estimates obtained

as in SLR

Special case for one predictor
whereX 1 ¼1orX 1 ¼ 0

Because our example has three outcome cate-

gories and one predictor (i.e., AGEGP), our

polytomous model requires two regression

expressions. One expression gives the log of

the probability that the outcome is in category

1 divided by the probability that the outcome is

in category 0, which equalsa 1 plusb 11 timesX 1.

We are alsosimultaneouslymodeling the log of

the probability that the outcome is in category

2 divided by the probability that the outcome is

in category 0, which equalsa 2 plusb 21 timesX 1.

Both the alpha and beta terms have a subscript

to indicate which comparison is being made

(i.e., category 1 vs. 0 or category 2 vs. 0).

Once a polytomous logistic regression model

has been fit and the parameters (intercepts

and beta coefficients) have been estimated,

we can then calculate estimates of the disease–

exposure association in a similar manner to the

methods used in standard logistic regression

(SLR).

Consider the special case in which the only

independent variable is the exposure variable

and the exposure is coded 0 and 1. To assess

the effect of the exposure on the outcome, we

compareX 1 ¼1toX 1 ¼0.

Presentation: III. Odds Ratio with Three Categories 437