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

Equivalent model definition

odds¼

PðDgjX 1 Þ

1 PðDgjX 1 Þ

¼

PðDgjX 1 Þ

PðD<gjX 1 Þ

¼

1

1 þexp½ðagþb 1 X 1 ފ

exp½ðagþb 1 X 1 ފ

1 þexp½ðagþb 1 X 1 ފ

¼expðagþb 1 X 1 Þ

Proportional

odds model:

vs. Standard

logistic

model:

no g subscript

Proportional odds
model:

vs. Polytomous

model:

P(D ³ g|X)P(D = g|X)

Alternate model formulation:

key differences

odds =

where g = 1, 2, 3, ... , G–1

and D* = 1, 2, ... , G

P(D* £ g | X 1 )

P(D* > g | X 1 )

g subscript

b 1 bg 1

=exp(a*g – b∗ 1 X 1 ),

Comparing formulations

b 1 ¼b* 1

butag¼a*g

The model can be defined equivalently in terms

of the odds of an inequality. If we substitute the

formula P(Dg|X 1 ) into the expression for the

odds and then perform some algebra (as shown

on the left), we find that theoddsis equal to

e to the quantityagplusb 1 X 1.

The proportional odds model is written differ-

ently from the standard logistic model. The

model is formulated as the probability of an

inequality, that is, that the outcomeDis greater

than or equal tog.

The model also differs from the polytomous

model in an important way. The beta is not sub-

scripted byg. This is consistent with the propor-

tional odds assumption that only one parameter

is required for each independent variable.

An alternate formulation of the proportional

odds model is to define the model as the odds

ofD* less than or equal toggiven the exposure

is equal to e to the quantitya*gb* 1 X 1 , where

g¼1, 2, 3,...,G1 and whereD*¼1, 2,...,G.

The two key differences with this formulation

are the direction of the inequality (D*g) and

the negative sign before the parameterb* 1 .In

terms of the beta coefficients, these two key

differences “cancel out” so thatb 1 ¼b* 1. Conse-

quently, if the same data are fit for each formu-

lation of the model, the same parameter

estimates of beta would be obtained for each

model. However, the intercepts for the two

formulations differ asag¼ag*.

Presentation: II. Ordinal Logistic Regression: The Proportional Odds Model 469