The SURVEYPHREG Procedure

The Multiplicative Hazards Model

Consider a set of n subjects such that the counting process upper N Subscript i Baseline identical-to StartSet upper N Subscript i Baseline left-parenthesis t right-parenthesis comma t greater-than-or-equal-to 0 EndSet for the ith subject represents the number of observed events that are experienced over time t. The sample paths of the process upper N Subscript i are step functions with jumps of size 1, with upper N Subscript i Baseline left-parenthesis 0 right-parenthesis equals 0. Let bold-italic beta denote the vector of unknown regression coefficients. The multiplicative hazards function normal upper Lamda left-parenthesis t comma bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis right-parenthesis for upper N Subscript i is given by

upper Y Subscript i Baseline left-parenthesis t right-parenthesis d normal upper Lamda left-parenthesis t comma bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis right-parenthesis equals upper Y Subscript i Baseline left-parenthesis t right-parenthesis exp left-parenthesis bold-italic beta prime bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis right-parenthesis d normal upper Lamda 0 left-parenthesis t right-parenthesis

where

  • upper Y Subscript i Baseline left-parenthesis t right-parenthesis indicates whether the ith subject is at risk at time t (specifically, upper Y Subscript i Baseline left-parenthesis t right-parenthesis equals 1 if at risk and upper Y Subscript i Baseline left-parenthesis t right-parenthesis equals 0 otherwise)

  • bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis is the vector of explanatory variables for the ith subject at time t

  • normal upper Lamda 0 left-parenthesis t right-parenthesis is an unspecified baseline hazard function

See Fleming and Harrington (1991) and Andersen et al. (1992). The Cox model is a special case of this multiplicative hazards model, where upper Y Subscript i Baseline left-parenthesis t right-parenthesis equals 1 until the first event or censoring, and upper Y Subscript i Baseline left-parenthesis t right-parenthesis equals 0 thereafter.

The partial likelihood for n independent triplets left-parenthesis upper N Subscript i Baseline comma upper Y Subscript i Baseline comma bold upper Z Subscript i Baseline right-parenthesis comma i equals 1 comma ellipsis comma n, has the form

script upper L left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript n Endscripts product Underscript t greater-than-or-equal-to 0 Endscripts StartSet StartFraction upper Y Subscript i Baseline left-parenthesis t right-parenthesis exp left-parenthesis bold-italic beta prime bold upper Z Subscript i Baseline left-parenthesis t right-parenthesis right-parenthesis Over sigma-summation Underscript j equals 1 Overscript n Endscripts w Subscript j Baseline upper Y Subscript j Baseline left-parenthesis t right-parenthesis exp left-parenthesis bold-italic beta prime bold upper Z Subscript j Baseline left-parenthesis t right-parenthesis right-parenthesis EndFraction EndSet Superscript w Super Subscript i Superscript normal upper Delta upper N Super Subscript i Superscript left-parenthesis t right-parenthesis

where w Subscript i is the weight for subject i, normal upper Delta upper N Subscript i Baseline left-parenthesis t right-parenthesis equals 1 if upper N Subscript i Baseline left-parenthesis t right-parenthesis minus upper N Subscript i Baseline left-parenthesis t minus right-parenthesis equals 1, and normal upper Delta upper N Subscript i Baseline left-parenthesis t right-parenthesis equals 0 otherwise.

A CLUSTER statement is necessary in order to appropriately estimate variances from multiplicative hazards models. If your design has a primary sampling unit (PSU), then use the PSU identification in the CLUSTER statement. Otherwise, use the subject identification in the CLUSTER statement.

Last updated: December 09, 2022