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Survival Analysis Example 1. Censoring and the Survival Function

By Robert L. Wolpert


We wish to explore models for survival times Ti, some of which are observed (say, Ti = ti) and some of which are right-censored (say, Tj> tj). To begin with we will take the times Ti to be i.i.d. from some continuous parametric probability distribution with density function f(t | θ), θ ∈ Θ, and will explore inference about θ upon observing times {ti: 1 ≤ i ≤ n} and failure indicators {δi: 1 ≤ i ≤ n} — so δi = 1 if we observe Ti = ti, while δi = 0 if we observe Ti> ti. 1.1. Hazard and Survival The probability of surviving at least t is Sθ(t) ≡ Pr[T> t | θ] = [1 − F (t | θ)] = ∫ ∞ t f(s) ds, so the conditional probability of failing within time ɛ, given survival to time t, is Pr[T ≤ t + ɛ | T> t, θ] = Sθ(t) − Sθ(t + ɛ) Sθ(t) where the instantaneous hazard is defined to be ɛf(t | θ

Year: 2011
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