The LOGISTIC Procedure

Linear Predictor, Predicted Probability, and Confidence Limits

This section describes how predicted probabilities and confidence limits are calculated by using the maximum likelihood estimates (MLEs) obtained from PROC LOGISTIC. For a specific example, see the section Getting Started: LOGISTIC Procedure. Predicted probabilities and confidence limits can be output to a data set with the OUTPUT statement.

Binary and Cumulative Response Models

For a vector of explanatory variables , the linear predictor


is estimated by


where and are the MLEs of and . The estimated standard error of is , which can be computed as the square root of the quadratic form , where is the estimated covariance matrix of the parameter estimates. The asymptotic confidence interval for is given by


where is the percentile point of a standard normal distribution.

The predicted probability and the confidence limits for are obtained by back-transforming the corresponding measures for the linear predictor, as shown in the following table:


Predicted Probability

100(1–)% Confidence Limits




The CONTRAST statement also enables you to estimate the exponentiated contrast, . The corresponding standard error is , and the confidence limits are computed by exponentiating those for the linear predictor: .

Generalized Logit Model

For a vector of explanatory variables , define the linear predictors , and let denote the probability of obtaining the response value :


By the delta method,


A 100(1)% confidence level for is given by


where is the estimated expected probability of response , and is obtained by evaluating at .

Note that the contrast and exponentiated contrast , their standard errors, and their confidence intervals are computed in the same fashion as for the cumulative response models, replacing with .