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

Extending model to G outcomes

Outcome variable has G levels:
(0, 1, 2,...,G1)

ln

PðD¼gjXÞ

PðD¼ 0 jXÞ



¼agþ~

k

i¼ 1

bgiXi;

whereg¼1, 2,...,G 1

Calculation of ORs and CIs as
before

Likelihood ratio test

Wald tests

)

same

procedures

Likelihood ratio test

 2 lnLreducedð 2 lnLfullÞ

w^2

with df¼number of parameters
set to zero underH 0 (¼G1if
k¼1)

Wald test

Z¼

^bg 1

s^bg 1

Nð 0 ; 1 Þ;

whereg¼1, 2,...,G 1

The model also easily extends for outcomes

with more than three levels.

Assume that the outcome hasGlevels (0, 1,

2,...,G1). There are nowG1 possible

comparisons with the reference category.

If the reference category is 0, we can define the

model in terms ofG1 expressions of the

following form: the log odds of the probability

that the outcome is in categorygdivided by the

probability the outcome is in category 0 equals

agplus the summation of thekindependent

variables times theirbgcoefficients.

The odds ratios and corresponding confidence

intervals for theG1 comparisons of cate-

gorygto category 0 are calculated in the man-

ner previously described. There are nowG 1

estimated odds ratios and corresponding con-

fidence intervals, for the effect of each inde-

pendent variable in the model.

The likelihood ratio test and Wald test are also

calculated as before.

For the likelihood ratio test, we test G 1

parameter estimates simultaneously for each

independent variable. Thus, for testing one

independent variable, we haveG1 degrees

of freedom for the chi-square test statistic com-

paring the reduced and full models.

We can also perform a Wald test to examine the

significance of individual betas. We haveG 1

coefficients that can be tested for each inde-

pendent variable. As before, the set of coeffi-

cients must either be retained or dropped.

Presentation: V. Extending the Polytomous Model toGOutcomes 449