Partial Likelihood Function for the Cox Model

The PHREG Procedure

Partial Likelihood Function for the Cox Model

Let z_l denote the vector of (possibly time-dependent) explanatory variables for the lth individual. Let t₁ < t₂ < ... <t_k denote the k distinct, ordered event times. Let d_i denote the multiplicity of failures at t_i; that is, d_i is the size of the set D_i of individuals that fail at t_i. Let s_i be the sum of the vectors z_l over the individuals who fail at t_i; that is, $s_{i}=\sum_{l \in {\cal D}_{i} } z_{l}$ .Using this notation, the likelihood functions used in PROC PHREG to estimate ${\beta}$ are described in the following sections.

Continuous Time Scale

Let R_i denote the risk set just before the ith ordered event time t_i . Let ${\cal R}_{i}^{\ast}$ denote the set of individuals whose event or censored times exceed t_i or whose censored times are equal to t_i.

The exact likelihood is

$L_{1}({{\beta}})=\prod_{i=1}^k \{ \int_{0}^{\infty} \prod_{j=1}^{d_i} [ 1-{\rm ... ...l \in {\cal R}_{i}^{\ast}}{\rm exp}(z'_{l}{\beta})}t } } ) ] {\rm exp}(-t)dt \}$

The Breslow likelihood is

$L_{2}({{\beta}})=\prod_{i=1}^k \frac{ {\rm exp}(s'_{i}{{\beta}})} {[ \raisebox{... ...{\displaystyle \sum_{l \in {\cal R}_{i}} {\rm exp}(z'_{l}{\beta}) } } ]^{d_i} }$

The Efron likelihood is

$L_{3}({{\beta}})=\prod_{i=1}^k\frac{ {\rm exp}(s'_{i}{{\beta}}) } {\displaystyle... ...ta}) - \frac{j-1}{d_i} \sum_{l \in {\cal D}_{i}} \ {\rm exp}(z'_{l}{\beta}) ] }$

Discrete Time Scale

Let Q_i denote the set of all subsets of d_i individuals from the risk set R_i. For each $q \in {\cal Q}_{i}$ , q is a d_i-tuple (q₁ , q₂ , ... , q_{d_i}) of individuals who might have failed at t_i. Let $s_{q}^{\ast} = \sum_{l=1}^{d_i} z_{q_{l}}$ .

The discrete logistic likelihood is

$L_{4}({{\beta}})=\prod_{i=1}^k \frac{ {\rm exp}(s'_{i}{{\beta}}) } {\displaystyle \sum_{q \in {\cal Q}_i} {\rm exp}({s_{q}^{\ast}}'{{\beta}}) }$

When there are no ties on the event times (that is, $d_{i} \equiv 1$ ), all four likelihood functions $L_{1}({{\beta}})$ , $L_{2}({{\beta}})$ , $L_{3}({{\beta}})$ , and $L_{4}({{\beta}})$ reduce to the same expression. In a stratified analysis, the partial likelihood is the product of the partial likelihood functions for the individual strata.

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