The SURVEYLOGISTIC Procedure

Linear Predictor, Predicted Probability, and Confidence Limits

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 normal upper Delta Subscript alpha slash 2 is the 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth percentile point of a standard normal distribution or a t distribution according to the DF= specification:

StartLayout 1st Row normal upper Delta Subscript alpha slash 2 Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column 100 left-parenthesis 1 minus alpha slash 2 right-parenthesis th percentile point of a standard normal distribution z Subscript alpha slash 2 Baseline 2nd Column if DF equals INFINITY 2nd Row 1st Column 100 left-parenthesis 1 minus alpha slash 2 right-parenthesis th percentile point of a t distribution t Subscript alpha slash 2 Baseline 2nd Column otherwise EndLayout EndLayout

Cumulative Response Models

For a row 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 bold-italic beta comma 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 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 normal upper Delta Subscript alpha slash 2 Baseline ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Subscript i Baseline right-parenthesis

The predicted value and the 100 left-parenthesis 1 minus alpha right-parenthesis percent-sign confidence limits for Prleft-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.

Link Predicted Probability 100 left-parenthesis 1 minus alpha right-parenthesis Confidence Limits
LOGIT 1 slash left-parenthesis 1 plus e Superscript minus ModifyingAbove eta With caret Super Subscript i Superscript Baseline right-parenthesis 1 slash left-parenthesis 1 plus e Superscript minus ModifyingAbove eta With caret Super Subscript i Superscript plus-or-minus normal upper Delta Super Subscript alpha slash 2 Superscript ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Super Subscript i Superscript right-parenthesis Baseline 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 normal upper Delta 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 e Superscript minus e Super Superscript ModifyingAbove eta With caret Super Super Subscript i 1 minus e Superscript minus e Super Superscript ModifyingAbove eta With caret Super Super Subscript i Super Superscript plus-or-minus normal upper Delta Super Super Subscript alpha slash 2 Super Superscript ModifyingAbove sigma With caret left-parenthesis ModifyingAbove eta With caret Super Super Subscript i Super Superscript right-parenthesis

Generalized Logit Model

For a vector of explanatory variables bold x, 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 alpha Super Subscript i Superscript plus bold x bold-italic beta 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 alpha Super Subscript j Superscript plus bold x bold-italic beta 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 theta EndFraction right-parenthesis prime bold upper V left-parenthesis bold-italic theta right-parenthesis StartFraction partial-differential pi Subscript i Baseline Over partial-differential bold-italic theta EndFraction

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

ModifyingAbove pi With caret Subscript i Baseline plus-or-minus normal upper Delta 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 theta equals ModifyingAbove bold-italic theta With caret.

Last updated: December 09, 2022