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
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.
the predicted event probability exceeds some cutpoint value ,the observation is predicted to be an event observation;
is predicted as a nonevent. A 2×2 frequency table can be
obtained by cross-classifying the observed and predicted
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
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
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 bj denote the MLE
computed without the
jth observation. The one-step estimate of
bj is given by
of n individuals experience an event, for example, a disease.
Let this group be denoted by C1, and let the group
of the remaining n2=n-n1 individuals who do not have the
disease be denoted by C2.
The jth 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
For a given cutpoint z,
the jth individual is predicted to give a positive response
- is 1 for an event response and 0 otherwise
- is the WEIGHT value
- is the predicted event probability
based on b
- is the
hat diagonal element
with nj=1 and rj=yj
- is the estimated covariance matrix
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
Results of the classification
are represented by two
conditional probabilities, and , where
is the sensitivity, and is one minus
These probabilities are given by
where I(·) is the indicator function.
is used to compute the error rates of the
a given prior probability
Pr(B) of the disease, the
false positive rate PF+ and the false negative rate PF-
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) = n1/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 n1 and
n2 are chosen a priori, n1/n is not a desirable estimate
of Pr(B). For such situations, the PEVENT= option should be
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.