Introductory Biostatistics

PrðY¼ 1 ;X¼xÞ

PrðY¼ 0 ;X¼xÞ

¼

PrðY¼ 1 Þ

PrðY¼ 0 Þ

PrðX¼x;Y¼ 1 Þ

PrðX¼x;Y¼ 0 Þ

On the right-hand side, the ratio of prior probabilities is a constant with respect
tox, and with our assumption thatXhas a normal distribution, the likelihood
ratio is the ratio of two normal densities. Let

mi¼EðX;Y¼i fori¼ 0 ; 1 Þ

si^2 ¼VarðX;Y¼i fori¼ 0 ; 1 Þ

denote the means and variances of the subjects with events (e.g., cases,Y¼1)
and the subjects without events (e.g., controls,Y¼0), respectively, we can
write

logit¼ln

px

1 
px

¼constantþ

m 1

s 12

m 0

s 02



xþ

1

2

1

s 02

1

s 12



x^2

This result indicates that ifs 12 ands 02 are not equal, we should have a quadratic
model; the model is linear if and only ifs^20 ¼s 12. We often drop the quadratic
term, but the robustness has not been investigated fully.
Let us assume that

s 02 ¼s^21 ¼s^2

so that we have the linear model

logit¼ln

px

1 
px

¼constantþ

m 1 
m 0

s^2

x

the very same linear logistic model as Section 9.1. It can be seen that with this
approach,

b 1 ¼

m 1 
m 0

s^2

which can easily be estimated using sample means of Chapter 2 and pooled
sample variance as used in thettests of Chapter 7. That is,

bb^ 1 ¼x^1 
x^0

sp^2

and it has been shown that this estimate works quite well even if the distribu-
tion ofXis not normal.

340 LOGISTIC REGRESSION