Time-to-Event Data, Censoring, and Hazard Functions › Summary

Time-to-event data is the record of a duration combined with a censoring indicator, $Y_i = \min(T_i, C_i)$ and $\Delta_i = \mathbf{1}\{T_i \le C_i\}$. The indicator is not a nuisance. It separates the two kinds of information a survival dataset contains, namely "the event happened at this time" and "the event had not happened by this time." The likelihood that consumes this pair is $\prod_i f_\theta(y_i)^{\delta_i} S_\theta(y_i)^{1 - \delta_i}$, or equivalently $\prod_i h_\theta(y_i)^{\delta_i} e^{-H_\theta(y_i)}$ after rewriting in terms of the hazard and the cumulative hazard.

Three functions describe the same distribution. The survival function $S(t) = P(T > t)$ is the probability of remaining event-free. The density $f(t) = -S'(t)$ is the familiar probability density. The hazard $h(t) = f(t)/S(t)$ is the instantaneous rate of the event given survival so far. The bridge between the survival and cumulative-hazard representations is $S(t) = \exp(-H(t))$, which is what makes hazard-based models algebraically convenient. You should be comfortable with all three and be able to translate among them at will.

Independent censoring with distinct nuisance parameters permits removal of censoring factors from the event-time likelihood. Unmeasured dependence can invalidate that factorization, and additional models, weighting assumptions, or sensitivity analysis may be needed. Competing-event analyses need a declared target. When that target distinguishes event types, retain an event-type indicator; a prespecified composite first-event target can use a binary indicator. Non-informative delayed entry conditions on survival to $L_i$ through $S(L_i)$, and changes exposure even for an exponential lifetime.

The exponential rate estimate is $\hat\lambda=D/\sum_iY_i$ when there is at least one event and positive exposure. In the six-record example it is $4/21$, while counting censorings as events gives $6/21$ and dropping them gives $4/10$. The primitive optimizer agrees with the analytic estimate, and SciPy's censored exponential fit checks the hand example independently. In the larger seeded sample the censoring-aware estimate is near the generating rate; the naive estimates are substantially larger. The next chapters extend the estimators while retaining explicit attention to what each observed record tells us.