significance; the null hypothesis to be considered is
H 0 :b 1 ¼ 0The reason for interest in testing whether or notb 1 ¼0isthatb 1 ¼0 implies
that there is no relation between the binary dependent variable and the co-
variateX under investigation. Since the likelihood function is rather simple,
one can easily derive, say, the score test for the null hypothesis above; however,
nothing would be gained by going through this exercise. We can simply apply a
chi-square test (if the covariate is binary or categorical) orttest or Wilcoxon
test (if the covariate under investigation is on a continuous scale). Of course,
the application of the logistic model is still desirable, at least in the case of a
continuous covariate, because it would provide a measure of association.
9.1.5 Use of the Logistic Model for Di¤erent Designs
Data for risk determination may come from di¤erent sources, with the two
fundamental designs being retrospective and prospective. Prospective studies
enroll group or groups of subjects and follow them over certain periods of
time—examples include occupational mortality studies and clinical trials—and
observe the occurrence of a certain event of interest such as a disease or death.
Retrospective studies gather past data from selected cases and controls to
determine di¤erences, if any, in the exposure to a suspected risk factor. They
are commonly referred to as case–control studies. It can be seen that the logis-
tic model fits in very well with the prospective or follow-up studies and has
been used successfully to model the ‘‘risk’’ of developing a condition—say, a
disease—over a specified time period as a function of a certain risk factor. In
such applications, after a logistic model has been fitted, one can estimate the
individual riskspðxÞ’s—given the covariate valuex—as well as any risks ratio
or relative risk,
RR¼
pðxiÞ
pðxjÞAs for case–control studies, it can be shown, using the Bayes’ theorem, that if
we have for the population
PrðY¼ 1 ;givenxÞ¼1
1 þexp½ðb 0 þb 1 xÞthen
PrðY¼ 1 ;given the samples and givenxÞ¼1
1 þexp½ðb 0 þb 1 xÞ322 LOGISTIC REGRESSION