## Receiver Operating Characteristic Curves

In a sample of *n* individuals, suppose *n*_{1} individuals are
observed to have a certain condition or event. Let
this group be denoted by *C*_{1}, and let the group of the
remaining *n*_{2}=*n*-*n*_{1} individuals who do not have the condition be
denoted by *C*_{2}. Risk
factors are identified for the sample, and a logistic regression
model is fitted to the data. For the *j*th individual,
an estimated probability of the event of interest is
calculated. Note that is computed directly without resorting
to the one-step approximation, as used in the calculation of the
classification table.
Suppose the *n* individuals undergo a test for predicting
the event and the test is based on the estimated probability
of the event. Higher values of this estimated probability are assumed
to be associated with the event. A receiver operating characteristic
(ROC) curve can be
constructed by varying the cutpoint that
determines which estimated event probabilities are considered
to predict the event.
For each cutpoint *z*, the following measures
can be output to a data set using the OUTROC= option:

where *I*(.) is the indicator function.

Note that _POS_(*z*) is the number of correctly predicted event responses,
_NEG_(*z*) is the number of correctly predicted nonevent responses,
_FALPOS_(*z*) is the number of falsely predicted event responses,
_FALNEG_(*z*) is the number of falsely predicted nonevent responses,
_SENSIT_(*z*) is the sensitivity of the test, and
_1MSPEC_(*z*) is one minus the specificity of the test.

A plot of the ROC curve can be constructed by
using the PLOT or GPLOT procedure with the OUTROC= data set and
plotting sensitivity
(_SENSIT_) against 1-specificity (_1MSPEC_).
The area under the ROC curve, as determined by the trapezoidal rule,
is given by the statistic *c* in the "Association of
Predicted Probabilities and Observed Responses" table.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.