- The range of the logistic function is between 0 and 1; that makes it suit-
able for use as a probability model, representing individual risk. - The logistic curve has an increasing S-shape with a threshold (Figure 9.1);
that makes it suitable for use as a biological model, representing risk due
to exposure.
Under the simple logistic regression model, the likelihood function is given
by
L¼
Yn
i¼ 1
PrðYi¼yiÞ
¼
Yn
i¼ 1
½expðb 0 þb 1 xiÞyi
1 þexpðb 0 þb 1 xiÞ
yi¼ 0 ; 1
from which we can obtain maximum likelihood estimates of the parametersb 0
and b 1. As mentioned previously, the logistic model has been used both
extensively and successfully to describe the probability of developingðY¼ 1 Þ
some disease over a specified time period as a function of a risk factorX.
9.1.2 Measure of Association
Regression analysis serves two major purposes: (1) control or intervention, and
(2) prediction. In many studies, such as the one in Example 9.1, one important
objective is measuring the strength of a statistical relationship between the
binary dependent variable and each independent variable or covariate mea-
Figure 9.1 General form of a logistic curve.
318 LOGISTIC REGRESSION