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

Subjects:j¼1, 2, 3,...,n

yj 0 ¼

1 if outcome¼ 0

0 otherwise



yj 1 ¼

1 if outcome¼ 1

0 otherwise



yj 2 ¼

1 if outcome¼ 2

0 otherwise



PðD¼ 0 jXÞyj^0 PðD¼ 1 jXÞyj^1

PðD¼ 2 jXÞyj^2

yj 0 þyj 1 þyj 2 ¼ 1

since each subject has one outcome

Yn

j¼ 1

PðD¼ 0 jXÞyj^0 PðD¼ 1 jXÞyj^1 PðD¼ 2 jXÞyj^2

Likelihood for G outcome cate-
gories:

Yn

j¼ 1

GY 1

g¼ 0

PðD¼gjXÞyjg;

where

yjg¼

1 if thejth subject hasD¼g

ðg¼ 0 ; 1 ;...;G 1 Þ

0 if otherwise

8

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>:

Estimated as and bs are those
which maximizeL

Assume that there arensubjects in the dataset,

numbered fromj¼1ton. If the outcome for

subjectjis in category 0, then we let an indica-

tor variable,yj 0 , be equal to 1, otherwiseyj 0 is

equal to 0. We similarly create indicator vari-

ablesyj 1 andyj 2 to indicate whether the sub-

ject’s outcome is in category 1 or category 2.

The contribution of each subject to the likeli-

hood is the probability that the outcome is in

category 0, raised to theyj 0 power, times the

probability that the outcome is in category 1,

raised to theyj 1 , times the probability that the

outcome is in category 2, raised to theyj 2.

Note that each individual subject contributes

to only one of the category probabilities, since

only one of the indicator variables will be non-

zero.

The joint probability for the likelihood is

the product of all the individual subject

probabilities, assuming subject outcomes are

independent.

The likelihood can be generalized to includeG

outcome categories by taking the product of

each individual’s contribution across the G

outcome categories.

The unknown parameters that will be esti-

mated by maximizing the likelihood are the

alphas and betas in the probability that the

disease outcome is in category g, where g

equals 0, 1,...,G1.

452 12. Polytomous Logistic Regression