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

D. The likelihood ratio test is used to test

hypotheses about the significance of the

predictor variable(s).

i. With three levels of the outcome variable,

there are two comparisons and two

estimated coefficients for each predictor

ii. The null hypothesis is that each of the 2 beta

coefficients (for a given predictor) is equal

to zero

iii. The test compares the log likelihood of the

full model with the predictor to that of

the reduced model without the predictor.

The test is distributed approximately chi-

square, with 2 df for each predictor tested

E. The Wald test is used to test the significance of

the predictor at a single outcome level. The

procedure is analogous to standard logistic

regression.

V. Extending the polytomous model toGoutcomes

andkpredictors(pages 444–449)

A. The model easily extends to includek

independent variables.

B. The general form of the model forGoutcome

levels is

ln

PðD¼gjXÞ

PðD¼ 0 jXÞ



¼agþ~

k

i¼ 1

bgiXi;

whereg¼ 1 ; 2 ;...;G 1 :

C. The calculation of the odds ratio, confidence

intervals, and hypothesis testing using the

likelihood ratio and Wald tests remains the

same.

D. Interaction terms can be added and tested in a

manner analogous to standard logistic

regression.

VI. Likelihood function for polytomous model

(pages 450–452)

A. For an outcome variable withGcategories, the

likelihood function is

Yn

j¼ 1

GY 1

g¼ 0

PðD¼gjXÞyig; where

yjg¼

1 if thejth subject hasD¼g

0 if otherwise



wherenis the total number of subjects and

g¼0, 1,...,G1.

456 12. Polytomous Logistic Regression