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

IV. Extending the Ordinal
Model

PðDgjXÞ¼
1

1 þexp½ðagþ~

k

i¼ 1

biXiފ

;

where g¼1, 2, 3,...,G 1

Note:P(D0|X)¼ 1

odds¼

PðDgjXÞ

PðD<gjXÞ

¼expðagþ~

k

i¼ 1

biXjÞ

OR¼exp(bi), ifXiis coded (0, 1)

Expanding the model to add more independent

variables is straightforward. The model withk

independent variables is shown on the left.

Theoddsfor the outcome greater than or equal

to levelgis then e to the quantityagplus the

summation theXifor each of thekindependent

variable times its beta.

The odds ratio is calculated in the usual man-

ner as e to thebi,ifXiis coded 0 or 1. As in

standard logistic regression, the use of multi-

ple independent variables allows for the esti-

mation of an odds ratio for one variable

controlling for the effects of the other covari-

ates in the model.

To illustrate, we return to our endometrial

tumor grade example. Suppose we wish to con-

sider the effects of estrogen use as well as

RACE on GRADE. ESTROGEN is coded as 1

for ever user and 0 for never user.

The model now contains two predictor vari-

ables:X 1 ¼RACE andX 2 ¼ESTROGEN.

EXAMPLE

D¼GRADE¼

0 if well differentiated

1 if moderately

differentiated

2 if poorly differentiated

8

>>>

<

>>>

:

X 1 ¼RACE¼

0 if white

1 if black

8

<

:

X 2 ¼ESTROGEN¼

0 if never user

1 if ever user

8

<

:

PðDgjXÞ¼

1

1 þexp½ðagþb 1 X 1 þb 2 X 2 ފ;

whereX 1 ¼RACEð 0 ; 1 Þ

X 2 ¼ESTROGENð 0 ; 1 Þ

g¼ 1 ; 2

476 13. Ordinal Logistic Regression