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