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

Illustration

0

0

0

0

0

1

123

3

3

34

4

4

4

2

2

2

1

1

1

0

But, cannot allow

4 1 23

234

ForGcategories)G1 ways to
dichotomize outcome:

D1vs.D<1;

D2vs.D<2,...,

DG1 vs.D<G 1

oddsðDgÞ¼

PðDgÞ

PðD<gÞ

;

whereg¼1, 2, 3,...,G 1

Proportional odds assumption

Same odds ratio regardless of
where categories are dichotomized

To illustrate the proportional odds model,

assume we have an outcome variable with five

categories and consider the four possible ways

to divide the five categories into two collapsed

categories preserving the natural order.

We could compare category 0 to categories 1

through 4, or categories 0 and 1 to categories

2 through 4, or categories 0 through 2 to cate-

gories 3 and 4, or, finally, categories 0 through

3 to category 4. However, we could not com-

bine categories 0 and 4 for comparison with

categories 1, 2, and 3, since that would disrupt

the natural ordering from 0 through 4.

More generally, if an ordinal outcome variable

DhasGcategories (D¼0, 1, 2,...,G1), then

there areG1 ways to dichotomize the out-

come: (D1 vs.D<1;D2 vs.D<2,...,

DG1 vs.D<G1). With this categoriza-

tion ofD, the odds thatDgis equal to the

probability ofDgdivided by the probability

ofD<g, where (g¼1, 2, 3,...,G1).

The proportional odds model makes an impor-

tant assumption. Under this model, the odds

ratio assessing the effect of an exposure vari-

able for any of these comparisons will be the

same regardless of where the cut-point is

made. Suppose we have an outcome with five

levels and one dichotomous exposure (E¼1,

E ¼0). Then, under the proportional odds

assumption, the odds ratio that compares cate-

gories greater than or equal to 1 to less than 1 is

the same as the odds ratio that compares cate-

gories greater than or equal to 4 to less than 4.

In other words, the odds ratio isinvariantto

where the outcome categories are dichotomized.

EXAMPLE

OR (D1)¼OR (D4)

Comparing two exposure groups

e:g:;E¼ 1 vs:E¼ 0 ;

where

ORðD 1 Þ¼

odds½ðD 1 ÞjE¼ 1 Š

odds½ðD 1 ÞjE¼ 0 Š

ORðD 4 Þ¼

odds½ðD 4 ÞjE¼ 1 Š

odds½ðD 4 ÞjE¼ 0 Š

Presentation: II. Ordinal Logistic Regression: The Proportional Odds Model 467