13_AS_2_lec_a

# Methods comparing survival functions introductory

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Unformatted text preview: nctions Introductory example A hypothesis test Special cases 29/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Introductory example 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 29/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Introductory example Based on data representing weeks to death (or censoring) in 51 adults with recurrent gliomas: A=astrocytoma and G=glioblastoma, the following survival functions have been constructed (Example 2.2) Are these two survival functions diﬀerent? 29/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions A hypothesis test 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 30/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions A hypothesis test Comparing survival functions In many applications, one wants to compare two populations. For example: smokers versus non-smokers eﬀect of diﬀerent treatments for a disease As there is a 1-1 relationship between survival function (S ) and hazard rates (µ’s, here denoted h’s), we can test for diﬀerences between hazard rates We will test the hypothesis H0 : h1 (t ) = h2 (t ) ; ∀t ≤ τ vs H1 : At least one of the h1 (t ) diﬀer from h2 (t ) for some t ≤ τ . 30/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions A hypothesis test To test for diﬀerence in hazard rates h1 (t ) and h2 (t ) of two diﬀerent populations for all time t ≤ τ , the general form of the statistic is k dj d1j − , Z1 = w (tj ) n1j nj j =1 where w (tj ) represent a positive weight function t1 < t2 < · · · < tk are the distinct death times in...
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## This document was uploaded on 04/03/2014.

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