tk dj number of deaths occur at time tj 1 j k more

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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: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Preliminaries Assumptions The Product-limit and the Nelson-Aalen estimators (introduced in this module) are based on the following assumptions Non-informative 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: Non-parametric methods Estimating the lifetime distribution: non-parametric 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 drug-related death m = 4 4 Rat 3 dies from effects Times when drug-related deaths are of drug observed t1 = 3, t2 = 4, t3 = 11, 6 Rat 7 gnaws through Total no. of drug-related 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...
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