Unformatted text preview: t censored
Ordered times of death t1 < t2 < . . . < tk
dj number of deaths occur at time tj (1 ≤ j ≤ k ), (more than
one death can occur at any time)
d1 + d2 + . . . + dk = m
cj : the number of lives that are right censored at a time
belonging to [tj , tj +1 ) (0 ≤ j ≤ k ), t0 = 0 and tk +1 = ∞
c0 + c1 + . . . ck = N − m
The times at which observations are censored within the time
interval [tj , tj +1 ) are tj 1 , tj 2 , . . . tjcj (need not be distinct)
Deﬁne nj as the number of lives alive and at risk at time tj−
(just before time tj ). Then nj = dj + cj + nj +1 .
The largest observed study time tmax = max{tk , tkck }. Actuarial Statistics – Module 2: Nonparametric methods
Estimating the lifetime distribution: nonparametric approach
Preliminaries Assumptions
The Productlimit and the NelsonAalen estimators (introduced in
this module) are based on the following assumptions
Noninformative censoring: Time to censoring is independent
of Time to death
Lives are independent: Time to censoring or time to death
determined independently for each life.
Examples
1 2 11/42 If we believe that more healthy lives will tend to leave and thus
have lighter mortality, does the ﬁrst assumption still hold?
If we believe that being released is unrelated to state of
health, does the ﬁrst assumption still hold? Actuarial Statistics – Module 2: Nonparametric methods
Estimating the lifetime distribution: nonparametric approach
Preliminaries Example 2.1
10 rats injected with a new drug are observed over 20 days:
Day Event
No. of lives under investigation
3
Rat 4 dies from eﬀects N = 10
of drug
No. of drugrelated death m = 4
4
Rat 3 dies from eﬀects Times when drugrelated deaths are
of drug
observed t1 = 3, t2 = 4, t3 = 11,
6
Rat 7 gnaws through Total no. of drugrelated deaths
bars of cage and escapes d1 = 1, d2 = 1, d3 = 2
11 Rat 6 and 9 die from ef No. of lives censored c0 = 0, c1 = 0,
fects of drug
c2 = 1, c3 = 5,
17 Rat 1 killed by other rats Lives censored at times t2,1 = 6,
2...
View
Full Document
 Three '14
 Statistics, Normal Distribution, Maximum likelihood, Estimation theory, Wilcoxon signedrank test, Survival analysis

Click to edit the document details