Linear Predictor, Predicted Probability, and Confidence Limits

The LOGISTIC Procedure

Linear Predictor, Predicted Probability, and Confidence Limits

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

For a vector of explanatory variables x, the linear predictor

$\eta_i= g({Pr}(Y\leq i|{x})) = \alpha_i+{\beta}'x 1 \leq i \leq k$

is estimated by

$\hat{\eta}_i=\hat{\alpha}_i+\hat{{\beta}}'x$

where $\hat{\alpha}_i$ and $\hat{{\beta}}$ are the MLEs of $\alpha_i$ and ${\beta}$ . The estimated standard error of ${\eta}_i$ is $\hat{\sigma}({\hat{\eta}}_i)$ ,which can be computed as the square root of the quadratic form $(1, x^'){\hat{V}_{b}}(1, x^')^'$ where $\hat{V}_{b}$ is the estimated covariance matrix of the parameter estimates. The asymptotic $100(1-\alpha)\%$ confidence interval for ${\eta}_i$ is given by

$\hat{\eta}_i+- z_{\alpha/2}\hat{\sigma}({\hat{\eta}}_i)$

where $z_{\alpha/2}$ is the $100(1-\alpha/2)$ percentile point of a standard normal distribution.

The predicted value and the $100(1-\alpha)\%$ confidence limits for Pr $(Y{\leq}i|{x})$ are obtained by back-transforming the corresponding measures for the linear predictor.

Link	Predicted Probability	100(1-0.5 $\alpha$ )% Confidence Limits
LOGIT	$1/(1+e^{-\hat{\eta}_i})$	$1/(1+e^{-\hat{\eta}_i +- z_{\alpha/2}\hat{\sigma}({\hat{\eta}}_i)})$
PROBIT	$\Phi(\hat{\eta}_i)$	$\Phi(\hat{\eta}_i +- z_{\alpha/2}\hat{\sigma}({\hat{\eta}}_i))$
CLOGLOG	$1-e^{-e^{\hat{\eta}_i}}$	$1-e^{-e^{\hat{\eta}_i+- z_{\alpha/2}\hat{\sigma}({\hat{\eta}}_i)}}$

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