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.
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 backtransforming the corresponding measures for the linear predictor, as shown in the following table:
Link 
Predicted Probability 
100(1–)% Confidence Limits 

LOGIT 


PROBIT 


CLOGLOG 


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: .
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 .
Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.