The LOGISTIC Procedure

Linear Predictor, Predicted Probability, and Confidence Limits

Subsections:

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 $\bbeta _{i}$.