Introductory Biostatistics

1. 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.


2. 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 xiފyi

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