The PHREG Procedure

Robust Sandwich Variance Estimate

For the ith subject, i equals 1 comma ellipsis comma n, let upper X Subscript i, w Subscript i, and bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis be the observed time, weight, and the covariate vector at time t, respectively. Let normal upper Delta Subscript i be the event indicator and let upper Y Subscript i Baseline left-parenthesis t right-parenthesis equals upper I left-parenthesis upper X Subscript i Baseline greater-than-or-equal-to t right-parenthesis. Let

upper S Superscript left-parenthesis r right-parenthesis Baseline left-parenthesis bold-italic beta comma t right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts w Subscript j Baseline upper Y Subscript j Baseline left-parenthesis t right-parenthesis normal e Superscript bold-italic beta prime bold upper Z Super Subscript j Superscript left-parenthesis t right-parenthesis Baseline bold upper Z Subscript j Superscript circled-times r Baseline left-parenthesis t right-parenthesis comma r equals 0 comma 1

Let ModifyingAbove bold upper Z With bar left-parenthesis bold-italic beta comma t right-parenthesis equals StartFraction upper S Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis bold-italic beta comma t right-parenthesis Over upper S Superscript left-parenthesis 0 right-parenthesis Baseline left-parenthesis bold-italic beta comma t right-parenthesis EndFraction. The score residual for the ith subject is

bold upper L Subscript i Baseline left-parenthesis bold-italic beta right-parenthesis equals normal upper Delta Subscript i Baseline StartSet bold upper Z Subscript i Baseline left-parenthesis upper X Subscript i Baseline right-parenthesis minus ModifyingAbove bold upper Z With bar left-parenthesis beta comma upper X Subscript i Baseline right-parenthesis EndSet minus sigma-summation Underscript j equals 1 Overscript n Endscripts normal upper Delta Subscript j Baseline StartFraction w Subscript j Baseline upper Y Subscript i Baseline left-parenthesis upper X Subscript j Baseline right-parenthesis normal e Superscript bold-italic beta prime bold upper Z Super Subscript i Superscript left-parenthesis upper X Super Subscript j Superscript right-parenthesis Baseline Over upper S Superscript left-parenthesis 0 right-parenthesis Baseline left-parenthesis bold-italic beta comma upper X Subscript j Baseline right-parenthesis EndFraction StartSet bold upper Z Subscript i Baseline left-parenthesis upper X Subscript j Baseline right-parenthesis minus ModifyingAbove bold upper Z With bar left-parenthesis bold-italic beta comma upper X Subscript j Baseline right-parenthesis EndSet

For TIES=EFRON, the computation of the score residuals is modified to comply with the Efron partial likelihood. For more information, see the section Residuals.

The robust sandwich variance estimate of ModifyingAbove bold-italic beta With caret derived by Binder (1992), who incorporated weights into the analysis, is

ModifyingAbove bold upper V With caret Subscript s Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis equals script upper I Superscript negative 1 Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis left-bracket sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis w Subscript j Baseline bold upper L Subscript j Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis right-parenthesis Superscript circled-times 2 Baseline right-bracket script upper I Superscript negative 1 Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis

where script upper I left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis is the observed information matrix, and bold a Superscript circled-times 2 Baseline equals bold a bold a prime. Note that when w Subscript i Baseline identical-to 1,

ModifyingAbove bold upper V With caret Subscript s Baseline left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis equals bold upper D prime bold upper D

where bold upper D is the matrix of DFBETA residuals. This robust variance estimate was proposed by Lin and Wei (1989) and Reid and Crépeau (1985).

Last updated: December 09, 2022