Assume independent individuals and independent censoring conditional on their explanatory variables, with the censoring mechanism carrying no parameter

θ

. A failure contributes its failure-time probability density function

f_{i} (x_{i}; θ)

, while a censored observation contributes the survival function

F_{i} (x_{i}; θ)

because its failure time exceeds

x_{i}

. Terms from the censoring distribution then factor out of the survival-parameter likelihood. Thus the survival likelihood and its logarithm are

L (θ) \propto \prod_{i = 1}^{n} f_{i} (x_{i}; θ)^{v_{i}} F_{i} (x_{i}; θ)^{1 - v_{i}},

ℓ (θ) = i = 1 \sum n {v_{i} lo g f_{i} (x_{i}; θ) + (1 - v_{i}) lo g F_{i} (x_{i}; θ)} + constant .

The individual subscripts allow the same parameter vector to act through different covariate values.

ii

Solution

Insert

f_{i} = h_{i} F_{i}

and

F_{i} = e^{- H_{i}}

into the survival likelihood. Each individual's log contribution simplifies to

v_{i} lo g h_{i} (x_{i}; θ) + lo g F_{i} (x_{i}; θ) .

Therefore

ℓ (θ) = i = 1 \sum n {v_{i} lo g h_{i} (x_{i}; θ) - H_{i} (x_{i}; θ)} + constant .

Failures contribute a log hazard function term as well as exposure, whereas both failures and censored subjects contribute the negative cumulative hazard function over their observed follow-up.

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