The primary regression coe‰cientb 1 (andb 0 , which is often not needed) can
be estimated iteratively using a computer-packaged program such as SAS.
From the results, we can obtained a point estimate
ORORd¼ebb^^1and its 95% confidence interval
exp½bb^ 1 G 1 :96 SEðbb^ 1 Þ9.1.3 E¤ect of Measurement Scale
It should be noted that the odds ratio, used as a measure of association between
the binary dependent variable and a covariate, depends on the coding scheme
for a binary covariate and for a continuous covariateX, the scale with which to
measureX. For example, if we use the following coding for a factor,
Xi¼ 1 if the subject is not exposed
1 if the subject is exposedthen
lnðodds;nonexposedÞ¼b 0 b 1
lnðodds;exposedÞ¼b 0 þb 1so that
OR¼exp½lnðodds;exposedÞlnðodds;nonexposedÞ¼e^2 b^1and its 95% confidence interval,
exp½ 2 ðbb^ 1 G 1 :96 SEðbb^ 1 ÞÞOf course, the estimate ofb 1 under the new coding scheme is only half of that
under the former scheme; therefore, the (numerical) estimate of the OR remains
unchanged. The following example, however, will show the clear e¤ect of a
measurement scale in the case of a continuous independent variable.
Example 9.2 Refer to the data for patients diagnosed as having cancer of the
prostate in Example 9.1 (Table 9.1) and suppose that we want to investigate the
relationship between nodal involvement found at surgery and the level of acid
phosphatase in blood serum in two di¤erent ways using either (a)X¼acid or
(b)X¼log 10 ðacidÞ.
320 LOGISTIC REGRESSION