Estimator nelson aalen estimator 4 comparing survival

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Unformatted text preview: approach Kaplan-Meier (product limit) estimator 1 Introduction 2 Censoring and truncation Censoring Truncation Likelihood for censored and truncated data 3 Estimating the lifetime distribution: non-parametric approach Preliminaries Maximum Likelihood with censoring The likelihood as a function of the discrete hazard function Kaplan-Meier (product limit) estimator Nelson-Aalen estimator 4 Comparing survival functions Introductory example A hypothesis test Special cases 19/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator Kaplan-Meier estimator Given what we have above, the KM estimator is obvious and given by ∧ ∧ F (t ) = 1 − 1 − λj j :tj ≤t or alternatively ∧ S (t ) = 1 1− j :tj ≤t if t < t1 dj nj t1 ≤ t ≤ tmax . Note the KM estimator is also called Product-limit estimator. 19/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator 0.0 0.2 0.4 0.6 0.8 1.0 An example of the plot of K-M estimate of a survival function 0 20/42 5 10 15 20 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator The Kaplan-Meier estimator is well defined for time points less than tmax : if t < t1 1 ∧ ∧ S (t ) if tj ≤ t < tj +1 , j = 1, · · · , k − 1 S (t ) = ∧ j S (tk ) if tk ≤ t < tmax . For estimator of the survival function beyond tmax : If tmax corresponds to a death time and there is no censoring at tmax , the estimated survive curve is ZERO beyond tmax . If tmax = tkck , the value of S (t ) for t > tmax is undetermined. Two extreme views: If assuming that the survivors at time tmax would have died ˆ immediately after tmax , S (t ) = 0 for t > tmax . If assuming that the survivors at time tmax would die at ∞, ˆ ˆ...
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