 Survival likelihood

For a continuous event time $T$ with probability density function $f_{\theta }$, survival function $S_{\theta }$ and hazard function $h_{\theta }=f_{\theta }/S_{\theta }$, right censoring records $Y=\min (T,C)$ and $\delta =1_{\{T\le C\}}$. Suppose $T$ and $C$ are conditionally independent given the observed covariates, suppressed here. Write $g_{\eta }$ and $G_{\eta }$ for the censoring-time probability density function and survival function. The joint observed-data likelihood contribution isAn event contributes its event-time probability density function times the probability of remaining uncensored until then. A censored observation contributes its censoring-time probability density function times the probability of remaining event-free until then.

If the censoring model has separate parameters $\eta$ with no constraints linking them to $\theta$, its factor can be omitted when maximizing over $\theta$. The event-model part for independent subjects is thenwhere $H_{\theta }$ is the cumulative hazard function. Independent censoring alone does not permit dropping a censoring term that itself contains the event-model parameters. The familiar event-versus-censoring formula is thus a factorization under an ignorable observation mechanism, rather than an unconditional rule for every missing event time.^[ref]