3.1 The likelihood with censored data

Each subject contributes a pair \((t_i, \delta _i)\). If the event was observed (\(\delta _i = 1\)), the subject contributes the density \(f(t_i)\), the probability (density) of an event at exactly that time. If the time is censored (\(\delta _i = 0\)), all we know is that \(T_i > t_i\), which has probability \(S(t_i)\). With independent subjects and non-informative censoring, the likelihood is therefore \[ L(\theta ) = \prod _{i = 1}^nf(t_i)^{\delta _i}S(t_i)^{1 - \delta _i} = \prod _{i = 1}^nh(t_i)^{\delta _i}S(t_i) , \] using \(f = hS\) (Theorem 2.3.2). Taking logarithms and \(S = e^{-H}\), \[ \ell (\theta ) = \sum _{i = 1}^n\bigl (\delta _i\log h(t_i) - H(t_i)\bigr ) . \] Every subject, censored or not, contributes \(-H(t_i)\), the hazard accumulated over the time it was observed; only the subjects with events contribute the log hazard at their event time. The non-informative assumption is what allows the censoring mechanism to be left out of the likelihood: its own contribution does not involve \(\theta \).

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