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

aü

bi?

X 1 ;X 2

fixed

;...;Xi;

varies

...;Xk

fixed

A second interpretation is thatagives thelogof

thebackground,orbaseline, odds.

The first interpretation fora, which considers

it as thelog oddsfor a person with 0 values for

allXs, has a serious limitation: There may not

be any person in the population of interest with

zero values on all theXs.

For example, no subject could have zero values

for naturally occurring variables, like age or

weight. Thus, it would not make sense to talk

of a person with zero values for allXs.

The second interpretation forais more appeal-

ing: to describe it as thelogof thebackground,

orbaseline, odds.

By background odds, we mean the odds that

would result for a logistic model without any

Xs at all.

The form of such a model is 1 over 1 plus e to

minusa. We might be interested in this model

to obtain a baseline risk or odds estimate that

ignores all possible predictor variables. Such

an estimate can serve as a starting point for

comparing other estimates of risk or odds

when one or moreXs are considered.

Because we have given an interpretation toa,

can we also give an interpretation tobi? Yes, we

can, in terms of eitheroddsorodds ratios.We

will turn to odds ratios shortly.

With regard to the odds, we need to consider

what happens to the logit when only one of the

Xs varies while keeping the others fixed.

For example, if ourXs are CAT, AGE, and ECG,

we might ask what happens to the logit when

CAT changes from 0 to 1, given an AGE of 40

and an ECG of 0.

To answer this question, we write the model in

logit formasaþb 1 CATþb 2 AGEþb 3 ECG.

EXAMPLE

CAT changes from 0 to 1;

AGE|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼ 40 ;ECG¼ 0

fixed

logit PðXÞ¼aþb 1 CATþb 2 AGE

þb 3 ECG

EXAMPLE (continued)

(2) a¼log of background odds

LIMITATION OF (1)

AllXi¼0 for any individual?

AGE 6 ¼ 0

WEIGHT 6 ¼ 0

DEFINITION OF (2)

background odds: ignores allXs

model:PðÞ¼X^1

1 þea

20 1. Introduction to Logistic Regression