The LOGISTIC Procedure

Linear Predictor, Predicted Probability, and Confidence Limits

Subsections:

Binary and Cumulative Response Models
Generalized Logit Model

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 $\mb {x}$ , the linear predictor

$\eta _ i= g(\Pr (Y\leq i~ |~ \mb {x})) = \alpha _{i}+\mb {x}’\bbeta \quad 1 \leq i \leq k$

is estimated by

$\hat{\eta }_ i={\widehat{\alpha }_{i}}+\mb {x}’{\widehat{\bbeta }}$

where ${\widehat{\alpha }_{i}}$ and ${\widehat{\bbeta }}$ are the MLEs of $\alpha _{i}$ and $\bbeta$ . 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 $(1, {\mb {x}}^\prime ){\widehat{\bV }_{\mb {b}}}(1, \mb {x}^\prime )^\prime$ , where $\widehat{\bV }_{\mb {b}}$ is the estimated covariance matrix of the parameter estimates. The asymptotic $100(1-\alpha )\%$ confidence interval for ${\eta }_{i}$ is given by

$\hat{\eta }_ i\pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)$

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

The predicted probability and the $100(1-\alpha )\%$ confidence limits for $\pi _ i=\Pr (Y\leq i~ |~ \mb {x})$ are obtained by back-transforming the corresponding measures for the linear predictor, as shown in the following table:

Link	Predicted Probability	100(1– $\bm {\alpha }$ )% Confidence Limits
LOGIT	$1/(1+\exp (-\hat{\eta }_ i))$	$1/(1+\exp (-\hat{\eta }_ i \pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)))$
PROBIT	$\Phi (\hat{\eta }_ i)$	$\Phi (\hat{\eta }_ i \pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i))$
CLOGLOG	$1-\exp (-\exp (\hat{\eta }_ i))$	$1-\exp (-\exp (\hat{\eta }_ i\pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)))$

The CONTRAST statement also enables you to estimate the exponentiated contrast, ${e}^{{\hat{\eta }}_ i}$ . The corresponding standard error is $e^{\hat{{\eta }_ i}}\hat{\sigma }({\hat{\eta }}_ i)$ , and the confidence limits are computed by exponentiating those for the linear predictor: $\exp \{ \hat{\eta }_ i\pm z_{\alpha /2}\hat{\sigma }({\hat{\eta }}_ i)\}$ .

Generalized Logit Model

For a vector of explanatory variables $\mb {x}$ , define the linear predictors $\eta _ i={\alpha _{i}}+\mb {x}’{\bbeta _{i}}$ , and let $\pi _ i$ denote the probability of obtaining the response value i:

$\pi _ i = \left\{ \begin{array}{ll} \pi _{k+1} {e}^{\eta _ i} & 1\le i\le k \\ \displaystyle \frac{1}{1+\sum _{j=1}^{k} {e}^{\eta _ j}} & i=k+1 \end{array} \right.$

By the delta method,

$\sigma ^2({\pi }_ i) = \biggl ( \frac{\partial \pi _ i}{\partial \bbeta } \biggr )’ \bV ({\bbeta }) \frac{\partial \pi _ i}{\partial \bbeta }$

A 100(1 $-\alpha$ )% confidence level for $\pi _ i$ is given by

${\widehat{\pi }}_ i \pm z_{\alpha /2} \hat{\sigma }({\widehat{\pi }}_ i)$

where ${\widehat{\pi }}_ i$ is the estimated expected probability of response i, and $\hat{\sigma }({\widehat{\pi }}_ i)$ is obtained by evaluating $\sigma ({\pi }_ i)$ at $\bbeta ={\widehat{\bbeta }}$ .

Note that the contrast ${{\hat{\eta }}_ i}$ and exponentiated contrast ${e}^{{\hat{\eta }}_ i}$ , their standard errors, and their confidence intervals are computed in the same fashion as for the cumulative response models, replacing $\bbeta$ with $\bbeta _{i}$ .