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

P(D¼g)¼[P(Dg)]
[P(Dgþ1) ]

Forg¼ 2
P(D¼2)¼P(D2)P(D3)

Use relationship to obtain proba-
bility that individual is in given out-
come category.

Lis product of individual contribu-
tions.

Yn

j¼ 1

GY 1

g¼ 0

PðD¼gjXÞyjg;

where

yjg¼^1 if thejth subject hasD¼g
0 if otherwise



VI. Ordinal vs. Multiple
Standard Logistic
Regressions

Proportional odds model: order of
outcome considered.

Alternative: several logistic regres-
sion models

Original variable: 0, 1, 2, 3
Recoded:
 1vs.< 1,  2vs.< 2 , and
3vs.< 3

In the proportional odds model, we model the

probability ofDg. To obtain an expression

for the probability ofD¼g, we can use the

relationship that the probability (D¼g)is

equal to the probability ofDgminus the

probability ofD(gþ1). For example, the

probability thatDequals 2 is equal to the prob-

ability that D is greater than or equal to

2 minus the probability thatDis greater than

or equal to 3. In this way we can use the model

to obtain an expression for the probability that

an individual is in a specific outcome category

for a given pattern of covariates (X).

The likelihood (L) is then calculated in the same

manner discussed previously in the section

on polytomous regression – that is, by taking

the product of the individual contributions.

The proportional odds model takes into

account the effect of an exposure on an ordered

outcome and yields one odds ratio summariz-

ing that effect across outcome levels. An alter-

native approach is to conduct a series of

logistic regressions with different dichoto-

mized outcome variables. A separate odds

ratio for the effect of the exposure can be

obtained for each of the logistic models.

For example, in a four-level outcome variable,

coded as 0, 1, 2, and 3, we can define three new

outcomes: greater than or equal to 1 vs. less

than 1, greater than or equal to 2 vs. less than

2, and greater than or equal to 3 vs. less than 3.

Presentation: VI. Ordinal vs. Multiple Standard Logistic Regressions 479