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

VI. Likelihood Function
for Polytomous Model

(Section may be omitted.)

Outcome with three levels

Consider probabilities of three out-
comes:

PðD¼ 0 Þ;PðD¼ 1 Þ;PðD¼ 2 Þ

Logistic regression: dichotomous
outcome:

PðD¼ 1 jXÞ¼

1

1 þexpaþ~

k

i¼ 1

biXi



PðD¼ 0 jXÞ¼ 1 PðD¼ 1 jXÞ

Polytomous regression: three-level
outcome:

PðD¼ 0 jXÞþPðD¼ 1 jXÞ

þPðD¼ 2 jXÞ¼ 1

h 1 ðXÞ¼a 1 þ~

k

i¼ 1

b 1 iXi

h 2 ðXÞ¼a 2 þ~

k

i¼ 1

b 2 iXi

PðD¼ 1 jXÞ

PðD¼ 0 jXÞ

¼exp½h 1 ðXފ

PðD¼ 2 jXÞ

PðD¼ 0 jXÞ

¼exp½h 2 ðXފ

We now present the likelihood function for

polytomous logistic regression. This section

may be omitted without loss of continuity.

We will write the function for an outcome vari-

able with three categories. Once the likelihood

is defined for three outcome categories, it can

easily be extended toGoutcome categories.

We begin by examining the individual prob-

abilities for the three outcomes discussed in

our earlier example, that is, the probabilities

of the tumor being classified as Adenocarci-

noma (D¼0), Adenosquamous (D¼1), or

Other (D¼2).

Recall that in logistic regression with a dichot-

omous outcome variable, we were able to write

an expression for the probability that the out-

come variable was in category 1, as shown on

the left, and for the probability the outcome

was in category 0, which is 1 minus the first

probability.

Similar expressions can be written for a three-

level outcome. As noted earlier, the sum of the

probabilities for the three outcomes must be

equal to 1, the total probability.

To simplify notation, we can leth 1 (X) be equal

toa 1 plus the summation of thekindependent

variables times theirb 1 coefficients andh 2 (X)

be equal toa 2 plus the summation of thek

independent variables times their b 2 coeffi-

cients.

The probability for the outcome being in cate-

gory 1 divided by the probability for the out-

come being in category 0 is modeled as e to the

h 1 (X) and the ratio of probabilities for category

2 and category 0 is modeled as e to theh 2 (X).

450 12. Polytomous Logistic Regression