The LOGISTIC Procedure

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 bold x, the linear predictor

eta Subscript i Baseline equals g left-parenthesis probability left-parenthesis upper Y less-than-or-equal-to i vertical-bar bold x right-parenthesis right-parenthesis equals alpha Subscript i Baseline plus bold x prime bold-italic beta Baseline 1 less-than-or-equal-to i less-than-or-equal-to k

is estimated by

ModifyingAbove eta With caret Subscript i Baseline equals ModifyingAbove alpha With caret Subscript i Baseline plus bold x prime ModifyingAbove bold-italic beta With caret

where ModifyingAbove alpha With caret Subscript i and ModifyingAbove bold-italic beta With caret are the MLEs of alpha Subscript i and bold-italic beta. The estimated standard error of eta Subscript i is ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis, which can be computed as the square root of the quadratic form left-parenthesis 1 comma bold x Superscript prime Baseline right-parenthesis ModifyingAbove bold upper V With caret Subscript bold b Baseline left-parenthesis 1 comma bold x Superscript prime Baseline right-parenthesis prime, where ModifyingAbove bold upper V With caret Subscript bold b is the estimated covariance matrix of the parameter estimates. The asymptotic 100 left-parenthesis 1 minus alpha right-parenthesis percent-sign confidence interval for eta Subscript i is given by

ModifyingAbove eta With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis

where z Subscript alpha slash 2 is the 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth percentile point of a standard normal distribution.

The predicted probability and the 100 left-parenthesis 1 minus alpha right-parenthesis percent-sign confidence limits for pi Subscript i Baseline equals probability left-parenthesis upper Y less-than-or-equal-to i vertical-bar bold x right-parenthesis are obtained by back-transforming the corresponding measures for the linear predictor, as shown in the following table:

Link Predicted Probability 100(1–bold-italic alpha)% Confidence Limits
LOGIT 1 slash left-parenthesis 1 plus exp left-parenthesis minus ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-parenthesis 1 slash left-parenthesis 1 plus exp left-parenthesis minus ModifyingAbove eta With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-parenthesis right-parenthesis
PROBIT normal upper Phi left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis normal upper Phi left-parenthesis ModifyingAbove eta With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-parenthesis
CLOGLOG 1 minus exp left-parenthesis minus exp left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-parenthesis 1 minus exp left-parenthesis minus exp left-parenthesis ModifyingAbove eta With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-parenthesis right-parenthesis

The CONTRAST statement also enables you to estimate the exponentiated contrast, e Superscript ModifyingAbove eta With caret Super Subscript i. The corresponding standard error is e Superscript ModifyingAbove eta Super Subscript i Superscript With caret Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis, and the confidence limits are computed by exponentiating those for the linear predictor: exp left-brace ModifyingAbove eta With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis right-brace.

Generalized Logit Model

For a vector of explanatory variables bold x, define the linear predictors eta Subscript i Baseline equals alpha Subscript i Baseline plus bold x prime bold-italic beta Subscript i, and let pi Subscript i denote the probability of obtaining the response value i:

pi Subscript i Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column pi Subscript k plus 1 Baseline e Superscript eta Super Subscript i Baseline 2nd Column 1 less-than-or-equal-to i less-than-or-equal-to k 2nd Row 1st Column StartFraction 1 Over 1 plus sigma-summation Underscript j equals 1 Overscript k Endscripts e Superscript eta Super Subscript j Superscript Baseline EndFraction 2nd Column i equals k plus 1 EndLayout

By the delta method,

sigma squared left-parenthesis pi Subscript i Baseline right-parenthesis equals left-parenthesis StartFraction partial-differential pi Subscript i Baseline Over partial-differential bold-italic beta EndFraction right-parenthesis prime bold upper V left-parenthesis bold-italic beta right-parenthesis StartFraction partial-differential pi Subscript i Baseline Over partial-differential bold-italic beta EndFraction

A 100(1negative alpha)% confidence level for pi Subscript i is given by

ModifyingAbove pi With caret Subscript i Baseline plus-or-minus z Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove pi With caret Subscript i Baseline right-parenthesis

where ModifyingAbove pi With caret Subscript i is the estimated expected probability of response i, and ModifyingAbove sigma With caret left-parenthesis ModifyingAbove pi With caret Subscript i Baseline right-parenthesis is obtained by evaluating sigma left-parenthesis pi Subscript i Baseline right-parenthesis at bold-italic beta equals ModifyingAbove bold-italic beta With caret.

Note that the contrast ModifyingAbove eta With caret Subscript i and exponentiated contrast e Superscript ModifyingAbove eta With caret Super Subscript i, their standard errors, and their confidence intervals are computed in the same fashion as for the cumulative response models, replacing bold-italic beta with bold-italic beta Subscript i.

Last updated: December 09, 2022