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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: Nonparametric 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: Nonparametric methods
Comparing survival functions
A hypothesis test An α level test
Under the null hypothsis, the statistic
χ2 = 2
Z1
var (Z1 ) is a Chisquared 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
Chisquared distribution with 1 degree of freedom.
33/42 Actuarial Statistics – Module 2: Nonparametric 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: nonparametric approach
Preliminaries
Maximum Likelihood with censoring
The likelihood as a function of the discrete hazard function
KaplanMeier (product limit) estimator
NelsonAalen estimator
4 Comparing survival functions
Introductory example
A hypothesis test
Special cases
34/42 Actuarial Statistics – Module 2: Nonparametric methods
Comparing survival functions
Special cases Logrank 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: Nonparametric methods
Comparing survival functions
Special cases Under the null hypothesis,
k
j =1 (d1j − e1j ) 2 LR
var (Z1 ) is a Chisquare 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: Nonparametric 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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