the association between the response and the set of independent variables
considered together; for example, withR^2 ¼1, we can predict the response
perfectly.
For logistic regression analyses, after fitting a logistic regression model, each
subject’s fitted response probability,^ppi, can be calculated. Using these proba-
bilitiesas values of a separator, we can construct a nonparametric ROC curve-
tracing sensitivities against the estimated false positivities for various cut
points. Such an ROC curve not only makes it easy to determine an optimal cut
point [the point on the curve nearest the top left corner (0,1) which corresponds
to 1.0 sensitivity and 1.0 specificity] but also shows the overall performance of
the fitted logistic regression model; the better the performance, the farther away
the curve is from the diagonal. The areaCunder this ROC curve can be used
as a measure of goodness of fit. The measureCrepresents theseparation power
of the logistic model under consideration; for example, withC¼1, the fitted
response probabilities for subjects withy¼1 and the fitted response proba-
bilities for subjects withy¼0 areseparatedcompletely.
Example 9.9 Refer to the data set on prostate cancer of Example 9.1 (Table
9.1) with all five covariates and fitted results shown in Example 9.4. Using
the estimated regression parameters obtained from Example 9.4, we have
C¼ 0 :845.
Note:The area under the ROC curve, measureC, is provided by SAS’s
PROC LOGISTIC.
Since the measure of goodness of fitChas a meaningful interpretation and
increases when we add an explanatory variable to the model, it can be used as a
criterion in performing stepwise logistic regression instead of thepvalue, which
is easily influenced by the sample size. For example, in the forward selection
procedure, we proceed as follows:
Figure 9.3 Nonparametric ROC curve.338 LOGISTIC REGRESSION