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

Solve for P(D¼1|X)and
P(D¼2|X) in terms of P(D¼0|X).

PðD¼ 1 jXÞ¼PðD¼ 0 jXÞexp½h 1 ðXފ

PðD¼ 2 jXÞ¼PðD¼ 0 jXÞexp½h 2 ðXފ

PðD¼ 0 jXÞþPðD¼ 0 jXÞexp½h 1 ðXފ

þPðD¼ 0 jXÞexp½h 2 ðXފ¼ 1

Factoring out P(D¼0|X):

PðD¼ 0 jXÞ½ 1 þexph 1 ðXÞ

þexph 2 ðXފ¼ 1

With some algebra, we find that

PðD¼ 0 jXÞ

¼

1

1 þexp½h 1 ðXފþexp½h 2 ðXފ

and that

PðD¼ 1 jXÞ

¼

exp½h 1 ðXފ

1 þexp½h 1 ðXފþexp½h 2 ðXފ

and that

PðD¼ 2 jXÞ

¼

exp½h 2 ðXފ

1 þexp½h 1 ðXފþexp½h 2 ðXފ

L , joint probability of observed
data.
The ML method chooses parame-
ter estimates that maximizeL

Rearranging these equations allows us to solve

for the probability that the outcome is in cate-

gory 1, and for the probability that the outcome

is in category 2, in terms of the probability that

the outcome is in category 0.

The probability that the outcome is in cate-

gory 1 is equal to the probability that the out-

come is in category 0 times e to theh 1 (X).

Similarly, the probability that the outcome is

in category 2 is equal to the probability that the

outcome is in category 0 times e to theh 2 (X).

These quantities can be substituted into the

total probability equation and summed to 1.

With some simple algebra, we can see that the

probability that the outcome is in category 0 is

1 divided by the quantity 1 plus e to theh 1 (X)

plus e to theh 2 (X).

Substituting this value into our earlier equa-

tion for the probability that the outcome is in

category 1, we obtain the probability that the

outcome is in category 1 as e to the h 1 (X)

divided by one plus e to theh 1 (X) plus e to the

h 2 (X).

The probability that the outcome is in category

2 can be found in a similar way, as shown on

the left.

Recall that the likelihood function (L) repre-

sents the joint probability of observing the

data that have been collected and that the

method of maximum likelihood (ML) chooses

that estimator of the set of unknown para-

meters that maximizes the likelihood.

Presentation: VI. Likelihood Function for Polytomous Model 451