This section describes how predicted probabilities and confidence limits are calculated by using the pseudo-estimates (MLEs) obtained from PROC SURVEYLOGISTIC. For a specific example, see the section Getting Started: SURVEYLOGISTIC Procedure. Predicted probabilities and confidence limits can be output to a data set with the OUTPUT statement.
Let is the
th percentile point of a standard normal distribution or a t distribution according to the DF=
specification:
For a row 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
The predicted value and the confidence limits for Pr
are obtained by back-transforming the corresponding measures for the linear predictor.
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For a vector of explanatory variables , 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
.