### 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

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: .

#### Generalized Logit Model

For a vector of explanatory variables , define the linear predictors , and let denote the probability of obtaining the response value i: By the delta method, A 100(1 )% confidence level for is given by where is the estimated expected probability of response i, 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 .