13_AS_2_lec_a

# Chi squared random variable with 1 degree of freedom

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Unformatted text preview: the pooled sample d1j is the number of deaths that occur in Group 1 at time tj n1j is number at risk prior to time tj in Group 1 nj is the total number at risk prior to time tj 31/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions A hypothesis test Typically the weights are of the form w (tj ) = w (tj ) n1j such that the statistic becomes k w (tj ) d1j − n1j Z1 = j =1 dj nj In this particular case, one can obtain k (w (tj ))2 var (Z1 ) = j =1 32/42 n1j nj 1− n1j nj nj − dj nj − 1 dj Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions A hypothesis test An α level test Under the null hypothsis, the statistic χ2 = 2 Z1 var (Z1 ) is a Chi-squared random variable (with 1 degree of freedom) for large samples This means we will reject the null hypothesis H0 : h1 (t ) = h2 (t ) ; ∀t ≤ τ Z2 if var (1Z1 ) is larger than the αth upper percentage point of the Chi-squared distribution with 1 degree of freedom. 33/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Special cases 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 34/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Special cases Log-rank test: w (t ) ≡ 1 for all t In this particular case, k LR Z1 (d1j − e1j ) = j =1 Note: e1j = n1j dj nj the sum is over all unique failure times in both groups this works best when hazard rates in the diﬀerent populations are proportional to each other 34/42 Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Special cases Under the null hypothesis, k j =1 (d1j − e1j ) 2 LR var (Z1 ) is a Chi-square statistic (with 1 degree of freedom), where k LR var (Z1 ) = j =1 35/42 n1j nj 1− n1j nj nj − dj nj − 1 dj . Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions Special cases Example 2.3 Consider an experiment for which we are interested in the eﬀects of a particular drug (2 types) Survival times f...
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## This document was uploaded on 04/03/2014.

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