## Classification Table

For binary response data, the response is either an *event* or
a *nonevent*. In PROC LOGISTIC, the response with
Ordered Value 1 is regarded as the *event*, and the response
with Ordered Value 2 is the *nonevent*. PROC LOGISTIC models
the probability of the *event*. From the fitted model,
a predicted *event* probability can be computed for each
observation.
The method to compute a reduced-bias estimate of the predicted
probability is given in the "Predicted Probability of an Event
for Classification" section, which follows.
If
the predicted *event* probability exceeds some cutpoint value ,the observation is predicted to be an *event* observation;
otherwise, it
is predicted as a *nonevent*. A **2×2** frequency table can be
obtained by cross-classifying the observed and predicted
responses.
The CTABLE option produces this table, and the PPROB= option selects
one or more cutpoints. Each cutpoint generates a classification table.
If the PEVENT= option is also specified, a classification table is
produced for each combination of PEVENT= and PPROB= values.
The accuracy of the classification is measured by its
*sensitivity* (the ability to predict an *event* correctly) and
specificity (the ability to predict a *nonevent* correctly).
*Sensitivity* is the proportion of *event*
responses that were
predicted to be *events*. *Specificity* is the
proportion of *nonevent* responses
that were predicted to be *nonevents*.
PROC LOGISTIC also computes three other conditional probabilities:
*false positive rate*, *false negative rate*, and
*rate of correct classification*.
The *false positive rate* is
the proportion
of predicted *event* responses that were observed as *nonevents*.
The *false negative rate* is
the proportion of predicted *nonevent* responses that were observed as
*events*. Given prior probabilities
specified with the PEVENT= option, these conditional
probabilities can be computed as posterior probabilities using
Bayes' theorem.

When you classify a set of binary data, if the same observations used
to fit the model are also used to estimate the classification error,
the resulting error-count estimate is biased. One way of reducing the bias is
to remove the binary observation to be classified
from the data, reestimate the parameters of the model, and
then classify the observation based on the new parameter estimates.
However, it would
be costly to fit the model leaving out each observation one at a time. The
LOGISTIC procedure provides a less expensive one-step approximation
to the preceding parameter estimates. Let **b** be the MLE of the
parameter vector based on all observations.
Let **b**_{j} denote the MLE
computed without the
*j*th observation. The one-step estimate of
**b**_{j} is given by

where

*y*_{j}
- is 1 for an event response and 0 otherwise
*w*_{j}
- is the WEIGHT value
- is the predicted event probability
based on
**b**
*h*_{jj}
- is the
hat diagonal element
with
*n*_{j}=1 and *r*_{j}=*y*_{j}
- is the estimated covariance matrix
of
**b**

Suppose *n*_{1}
of *n* individuals experience an event, for example, a disease.
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
disease be denoted by *C*_{2}.
The *j*th individual is classified as giving
a positive response if the
predicted probability of disease () is large.
The probability is the reduced-bias
estimate based on a one-step approximation given in the
previous section.
For a given cutpoint *z*,
the *j*th individual is predicted to give a positive response
if .Let *B* denote the event that a subject has the disease and
denote the event of not having the disease.
Let *A* denote the event that the subject responds
positively, and let denote the event of responding
negatively.
Results of the classification
are represented by two
conditional probabilities, and , where
is the sensitivity, and is one minus
the specificity.

These probabilities are given by

where *I*(·) is the indicator function.
Bayes' theorem
is used to compute the error rates of the
classification. For
a given prior probability
**Pr(***B*) of the disease, the
false positive rate *P*_{F+} and the false negative rate *P*_{F-}
are given by Fleiss (1981, pp. 4 -5) as follows:

The prior probability **Pr(***B*) can be specified by the PEVENT= option.
If the PEVENT= option is not specified, the sample proportion of
diseased individuals is used; that is, **Pr(***B*) = *n*_{1}/*n*. In such a
case, the false positive
rate and the false negative rate reduce to

Note that for a stratified sampling situation in which *n*_{1} and
*n*_{2} are chosen a priori, *n*_{1}/*n* is not a desirable estimate
of **Pr(***B*). For such situations, the PEVENT= option should be
specified.

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