The PHREG Procedure

Partial Likelihood Function for the Cox Model

Let bold upper Z Subscript l Baseline left-parenthesis t right-parenthesis denote the vector explanatory variables for the lth individual at time t. Let t 1 less-than t 2 less-than ellipsis less-than t Subscript k denote the k distinct, ordered event times. Let d Subscript i denote the multiplicity of failures at t Subscript i; that is, d Subscript i is the size of the set script upper D Subscript i of individuals that fail at t Subscript i. Let w Subscript l be the weight associated with the lth individual. Using this notation, the likelihood functions used in PROC PHREG to estimate bold-italic beta are described in the following sections.

Continuous Time Scale

Let script upper R Subscript i denote the risk set just before the ith ordered event time t Subscript i. Let script upper R Subscript i Superscript asterisk denote the set of individuals whose event or censored times exceed t Subscript i or whose censored times are equal to t Subscript i.

Exact Likelihood
upper L 1 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartSet integral Subscript 0 Superscript normal infinity Baseline product Underscript j element-of script upper D Subscript i Baseline Endscripts left-bracket 1 minus exp left-parenthesis minus StartFraction normal e Superscript bold-italic beta prime bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline Over sigma-summation Underscript l element-of script upper R Subscript i Superscript asterisk Baseline Endscripts normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline EndFraction t right-parenthesis right-bracket exp left-parenthesis negative t right-parenthesis d t EndSet

Breslow Likelihood
upper L 2 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartFraction normal e Superscript bold-italic beta prime sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline Over left-bracket sigma-summation Underscript l element-of script upper R Subscript i Baseline Endscripts normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline right-bracket Superscript d Super Subscript i Superscript Baseline EndFraction

Incorporating weights, the Breslow likelihood becomes

upper L 2 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartFraction normal e Superscript bold-italic beta prime sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts w Super Subscript j Superscript bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline Over left-bracket sigma-summation Underscript l element-of script upper R Subscript i Baseline Endscripts w Subscript l Baseline normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline right-bracket Superscript sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts w Super Subscript j Superscript Baseline EndFraction
Efron Likelihood
upper L 3 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartStartFraction normal e Superscript bold-italic beta prime sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline OverOver product Underscript j equals 1 Overscript d Subscript i Baseline Endscripts left-parenthesis sigma-summation Underscript l element-of script upper R Subscript i Baseline Endscripts normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline minus StartFraction j minus 1 Over d Subscript i Baseline EndFraction sigma-summation Underscript l element-of script upper D Subscript i Baseline Endscripts normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline right-parenthesis EndEndFraction

Incorporating weights, the Efron likelihood becomes

upper L 3 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartFraction normal e Superscript bold-italic beta prime sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts w Super Subscript j Superscript bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline Over left-bracket product Underscript j equals 1 Overscript d Subscript i Baseline Endscripts left-parenthesis sigma-summation Underscript l element-of script upper R Subscript i Baseline Endscripts w Subscript l Baseline normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline minus StartFraction j minus 1 Over d Subscript i Baseline EndFraction sigma-summation Underscript l element-of script upper D Subscript i Baseline Endscripts w Subscript l Baseline normal e Superscript bold-italic beta prime bold upper Z Super Subscript l Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline right-parenthesis right-bracket Superscript StartFraction 1 Over d Super Subscript i Superscript EndFraction sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts w Super Subscript j Superscript Baseline EndFraction

Discrete Time Scale

Let script upper Q Subscript i denote the set of all subsets of d Subscript i individuals from the risk set script upper R Subscript i. For each bold q element-of script upper Q Subscript i, bold q is a d Subscript i-tuple left-parenthesis q 1 comma q 2 comma ellipsis comma q Subscript d Sub Subscript i Subscript Baseline right-parenthesis of individuals who might have failed at t Subscript i.

Discrete Logistic Likelihood

upper L 4 left-parenthesis bold-italic beta right-parenthesis equals product Underscript i equals 1 Overscript k Endscripts StartFraction normal e Superscript bold-italic beta prime sigma-summation Underscript j element-of script upper D Super Subscript i Superscript Endscripts bold upper Z Super Subscript j Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline Over sigma-summation Underscript bold q element-of script upper Q Subscript i Baseline Endscripts normal e Superscript bold-italic beta prime sigma-summation Underscript l equals 1 Overscript d Super Subscript i Superscript Endscripts bold upper Z Super Subscript q Super Sub Subscript l Super Subscript Superscript left-parenthesis t Super Subscript i Superscript right-parenthesis Baseline EndFraction

The computation of upper L 4 left-parenthesis bold-italic beta right-parenthesis and its derivatives is based on an adaptation of the recurrence algorithm of Gail, Lubin, and Rubinstein (1981) to the logarithmic scale. When there are no ties on the event times (that is, d Subscript i Baseline identical-to 1), all four likelihood functions upper L 1 left-parenthesis bold-italic beta right-parenthesis, upper L 2 left-parenthesis bold-italic beta right-parenthesis, upper L 3 left-parenthesis bold-italic beta right-parenthesis, and upper L 4 left-parenthesis bold-italic beta right-parenthesis reduce to the same expression. In a stratified analysis, the partial likelihood is the product of the partial likelihood functions for the individual strata.

Last updated: March 08, 2022