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Germán Rodríguez works on statistical methods for population research. His Princeton teaching and research site connects generalized linear models, survival analysis and demographic questions about event timing. The site's biographical note identifies his training in biostatistics and his work in population research.^[ref]

His Survival Models by Germán Rodríguez explains how survival functions and hazard functions encode the same event-time distribution in different ways. With right censoring, an event observed at $t$ contributes density $f(t)$, whereas an observation ending without an event contributes $S(t)$. Writing the contribution as $S(t)h(t{)}^{\delta }$, with event indicator $\delta \in \{0,1\}$, makes that distinction explicit. These likelihood conventions let incomplete follow-up inform estimation without treating a censored time as an observed failure.^[ref]

Consider independent event times with hazard function $h_i(t)=\lambda (t)e^{{\beta }^T{x}_i}$ and baseline cumulative hazard function $\Lambda (t)={\int }_0^t\lambda (s)ds$. For optimization over $\beta$, first hold $\lambda$ and $\Lambda$ fixed and independent of $\beta$. Under ignorable independent censoring, an observation $(y_i,{\delta }_i)$ has log-likelihood contributionSet ${\mu }_i=\Lambda (y_i)e^{{\beta }^T{x}_i}$. A Poisson regression with response ${\delta }_i$ and log link has log-likelihood contribution ${\delta }_i\log {\mu }_i-{\mu }_i-\log ({\delta }_i!)$, which differs from ${\ell }_i$ only by terms independent of $\beta$. Thus the same $\beta$ maximizes both criteria. The linear predictor is ${\beta }^T{x}_i+\log \Lambda (y_i)$, with $\log \Lambda (y_i)$ a known offset whenever $0<\Lambda (y_i)<\infty$.

For uncensored observations every response is one. This is an equivalence of objective functions: the event indicators need not actually be Poisson counts. A related split-interval construction handles piecewise-constant hazard functions.^[ref]

If the baseline is unspecified, its integrated values are not known offsets; fitting an ordinary generalized linear model as if they were known does not estimate an arbitrary Cox proportional-hazards model. One must estimate the baseline jointly, use a suitable baseline representation, or eliminate it through Cox partial likelihood. Zero exposure also needs separate handling rather than taking the logarithm of zero.

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]