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

Ordinal

Variable Parameter

Intercept a 1 ,a 2 ,...,aG 1
X 1 b 1

Polytomous

Variable Parameter

Intercept a 1 ,a 2 ,...,aG 1
X 1 b 11 ,b 21 ,...,b(G1)1

Odds arenotinvariant

Proportional odds model:Gout-
come levels and one predictor(X)

PðDgjX 1 Þ¼

1

1 þexp½ðagþb 1 X 1 ފ

;

whereg¼1, 2,...,G 1

1 PðDgjX 1 Þ

¼ 1 

1

1 þexp½ðagþb 1 X 1 ފ

¼

exp½ðagþb 1 X 1 ފ

1 þexp½ðagþb 1 X 1 ފ

¼PðD<gjX 1 Þ

This implies that if there areGoutcome cate-

gories, there is only one parameter (b) for each

of the predictors variables (e.g.,b 1 for predictor

X 1 ). However, there is still a separate intercept

term (ag) for each of theG1 comparisons.

This contrasts with polytomous logistic regres-

sion, where there areG1 parameters for each

predictor variable, as well as a separate inter-

cept for each of theG1 comparisons.

The assumption of the invariance of the odds

ratio regardless of cut-point isnotthe same as

assuming that theoddsfor a given exposure

pattern is invariant. Using our previous exam-

ple, for a given exposure levelE(e.g.,E¼0),

the odds comparing categories greater than or

equal to 1 to less than 1 doesnotequal the odds

comparing categories greater than or equal to 4

to less than 4.

We now present the form for the proportional

odds model with an outcome (D) withGlevels

(D¼0, 1, 2,...,G1) and one independent

variable (X 1 ). The probability that the disease

outcome is in a category greater than or equal

tog, given the exposure, is 1 over 1 plus e to the

negative of the quantityagplusb 1 X 1.

The probability that the disease outcome is in a

categorylessthan g is equal to 1 minus the

probability that the disease outcome is greater

than or equal to categoryg.

EXAMPLE

odds(D1) 6 ¼odds(D4)

where, forE¼0,

oddsðD 1 Þ¼

PðD 1 jE¼ 0 Þ

PðD< 1 jE¼ 0 Þ

oddsðD 4 Þ¼

PðD 4 jE¼ 0 Þ

PðD< 4 jE¼ 0 Þ

but

ORðD 1 Þ¼ORðD 4 Þ

468 13. Ordinal Logistic Regression