Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 2

For a continuous nonnegative event time with hazard function $h$, the cumulative hazard function isIts derivative is the instantaneous hazard wherever the derivative exists.

Writing $F$ for the survival function here, the relation $F^{\prime }(t)=-h(t)F(t)$ with $F(0)=1$ givesThis notation uses $F$ for survival rather than for the cumulative distribution function.

The continuous failure-time probability density function is the negative derivative of the survival function:The minus sign is essential because survivor probability decreases with time.

Assume independent individuals and independent censoring conditional on their explanatory variables, with the censoring mechanism carrying no parameter $\theta$. A failure contributes its failure-time probability density function $f_i(x_i;\theta )$, while a censored observation contributes the survival function $F_i(x_i;\theta )$ 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 areThe individual subscripts allow the same parameter vector to act through different covariate values.

Insert $f_i=h_i{F}_i$ and $F_i=e^{-H_i}$ into the survival likelihood. Each individual's log contribution simplifies toThereforeFailures 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.

For individual $i$, let ${\pi }_i={\pi }_i(\psi )$, $q_i(t)$ be the known baseline hazard function, and $r_i(t;\varphi )$ the extra hazard. PutThe class-specific survival functions are $e^{-Q_i(t)}$ and $e^{-Q_i(t)-R_i(t)}$. Consequently the unconditional survivor and failure density areMixture probabilities must be summed before taking a logarithm; a weighted average of class log-likelihoods would be a different, complete-data calculation. Substitution into the right-censored survival likelihood givesEquivalently, this additive excess-hazard mixture likelihood has contributionHere $q_i$ and $q_i+r_i$ must be nonnegative hazards. The factor $-Q_i(x_i)$ can be dropped when maximizing over $\varphi ,\psi$ because $q_i$ is known. Factoring out censoring presumes a common parameter-free censoring mechanism independent of failure and latent class, conditional on the observed covariates; unmodelled class-specific censoring would change these mixture contributions.