The SURVEYPHREG Procedure

Partial Likelihood Function for the Cox Model

Let t Subscript left-parenthesis 1 right-parenthesis Baseline less-than t Subscript left-parenthesis 2 right-parenthesis Baseline less-than ellipsis less-than t Subscript left-parenthesis upper K right-parenthesis denote the K distinct, ordered event times. Let d Subscript k denote the multiplicity of failures at t Subscript left-parenthesis k right-parenthesis; that is, d Subscript k is the size of the set script upper D Subscript k of individuals that fail at t Subscript left-parenthesis k right-parenthesis. Let w Subscript h i j be the weight associated with the jth observation unit in the ith cluster in stratum h. Using this notation, the pseudo-likelihood functions used in PROC SURVEYPHREG to estimate bold-italic beta Subscript upper N are described in the following sections.

Let script upper R Subscript k denote the risk set just before the kth ordered event time t Subscript left-parenthesis k right-parenthesis.

The Breslow likelihood is expressed as

upper L Subscript Breslow Baseline left-parenthesis bold-italic beta right-parenthesis equals product Underscript k equals 1 Overscript upper K Endscripts StartFraction exp left-parenthesis bold-italic beta prime sigma-summation Underscript script upper D Subscript k Baseline Endscripts w Subscript h i j Baseline bold upper Z Subscript h i j Baseline left-parenthesis t right-parenthesis right-parenthesis Over StartSet sigma-summation Underscript script upper R Subscript k Baseline Endscripts w Subscript h i j Baseline exp left-parenthesis bold-italic beta prime bold upper Z Subscript h i j Baseline left-parenthesis t right-parenthesis right-parenthesis EndSet Superscript sigma-summation Underscript script upper D Super Subscript k Superscript Endscripts w Super Subscript h i j Superscript Baseline EndFraction

The Efron likelihood is expressed as

upper L Subscript Efron Baseline left-parenthesis bold-italic beta right-parenthesis equals product Underscript k equals 1 Overscript upper K Endscripts StartFraction exp left-parenthesis bold-italic beta prime sigma-summation Underscript script upper D Subscript k Baseline Endscripts w Subscript h i j Baseline bold upper Z Subscript h i j Baseline left-parenthesis t right-parenthesis right-parenthesis Over StartSet phi left-parenthesis bold-italic beta comma bold upper Z comma bold w comma k right-parenthesis EndSet Superscript StartFraction 1 Over d Super Subscript k Superscript EndFraction sigma-summation Underscript script upper D Super Subscript k Superscript Endscripts w Super Subscript h i j Superscript Baseline EndFraction

where phi left-parenthesis bold-italic beta comma bold upper Z comma bold w comma k right-parenthesis is

phi left-parenthesis bold-italic beta comma bold upper Z comma bold w comma k right-parenthesis equals product Underscript l equals 1 Overscript d Subscript k Baseline Endscripts StartSet sigma-summation Underscript script upper R Subscript k Baseline Endscripts w Subscript h i j Baseline exp left-parenthesis bold-italic beta prime bold upper Z Subscript h i j Baseline left-parenthesis t right-parenthesis right-parenthesis minus StartFraction l minus 1 Over d Subscript k Baseline EndFraction sigma-summation Underscript script upper D Subscript k Baseline Endscripts w Subscript h i j Baseline exp left-parenthesis bold-italic beta prime bold upper Z Subscript h i j Baseline left-parenthesis t right-parenthesis right-parenthesis EndSet
Last updated: December 09, 2022