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note17 - Chapter 7 Analysis of Censored Data 1 Estimating...

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Chapter 7: Analysis of Censored Data 1. Estimating the Survival Function Censored Data T : the survival time of an experimental unit * the exact value might not be known * sometimes it is only known that T exceeds some threshold suppose that the true life time T 1 , . . . , T n iid F with F continous observations are X i = min ( T i , C i ) where C i is the censoring time Example 1
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Kaplan-Meier Estimate the survival function (the reliability function) R ( t ) = Pr ( T > t ) = 1 - F ( t ) no censoring ˆ R ( t ) = 1 - ˆ F ( t ) where ˆ F ( t ) is the empirical cdf censored data: Kaplan-Meier estimate * data x i : uncensored x + i : censored * notations · t 1 < t 2 < t D : the distinct event times that are observed · n i : number of observations that are known to have survived t i or longer · d i : number of events at t i i = 1 , 2 , . . . , D * Kaplan-Meier estimate ˆ R ( t ) = braceleftBigg 1 if t < t 1 producttext i : t i t parenleftBig 1 - d i n i parenrightBig if t > t 1 * if max( X i ) is an event time, ˆ R (max( X i )) = 0 * if max( X i ) is a censored time, ˆ R Variance of Survival Estimates: Greenwood’s formula hatwidest V ar bracketleftBig ˆ R ( t ) bracketrightBig = parenleftBig ˆ R ( t ) parenrightBig 2 summationdisplay i : t i t d i n i
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  • Spring '08
  • Staff
  • Probability theory, Cumulative distribution function, Survival analysis, survival function, Kaplan-Meier estimate

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