Let us use the term failure time to indicate the time of the event of interest in either a survival analysis or reliability analysis.

Transcription

1 10.2 Product-Limit (Kaplan-Meier) Method Let us use the term failure time to indicate the time of the event of interest in either a survival analysis or reliability analysis. Let T be a continuous random variable denoting failure time, and F (t) be the cdf of T. The reliability function R(t) or survival function S(t) is defined to be R(t) = 1 F (t) or S(t) = 1 F (t). Thus, R(t) and S(t) represent the probability that the event will occur after time t. In the course notes, I will use R(t) for a reliability study. Just replace R(t) with S(t) for a survival study. If we have complete data (no censoring) in a study of n subjects or items, then estimates of F (t) and R(t) are the empirical cdf F (t) and the empirical reliability function R(t). That is, F (t) = number of items failing by time t n R(t) = 1 F number of items not failing (surviving) at time t (t) = n Consider the following complete data set where t i = the i th failure time n i = the number of surviving items just prior to t i = the number of items at risk at time t i d i = the number of failures at t i n i d i i ( ) nk d k i t i n i d i R(ti ) n i n k k= /10 = /10 = /10 =.9 9/10 1*(9/10) = /10 =.8 8/ /10 =.7 7/ /10 =.6 6/ /10 =.5 5/ /10 =.3 3/ /10 =.1 1/ /10 =

2 Note that R(t i ) = i ( ) nk d k. Why? By definition: k=0 n k R(t i ) = P (T > t i ) Note that P (T > t i ) P (T > t i 1 ) can be estimated by P (T > t i ) P (T > t i 1 ) = (number of observations > t i)/n (number of observations > t i 1 )/n = number of observations > t i number of observations > t i 1 = Now we substitute to get R(t i ) = P (T > t i ) P (T > t i 1 ) R(t i 1 ) = n i d i n i R(ti 1 ) Recursively applying the formula yields R(ti ) = for any event time t i. To find the standard error se( R(t)), consider the distribution of X t = the number of items surviving at time t. Then X t Binomial(n, p) where p = R(t)). Therefore, V ar(x t ) = V ar ( Xt n ) = and sd( R(t)) = R(t) (1 R(t). n Substitution of R(t) yields se( R(t)) R(t) (1 = n which can be used to generate approximate confidence intervals R(t) ± z se( R(t)) for R(t). 251

3 The Kaplan-Meier Method (or product-limit method) generalizes the product form for calculating R(t i ) when censoring does exist. The goal is to estimate R(t) when we have censored observations. Consider the following censored data set where t 1 < t 2 < < t k be the failure times of the uncensored items. d i = the number of failures at times t i n i = the number of items at risk at time t i (i = 1, 2,..., k). = the number of censored and uncensored items surviving just prior to t i By convention, t 0 =, d 0 = 0, and n 0 = n. The Kaplan-Meier estimate of R(t) is R(t i ) = for i = 1, 2,..., k For any time t that is not an event time, find the largest t i < t. Then That is, R(t) remains constant between event times. An estimate of the variance of R(t) (Lawless 1982)) is: ( R(t) ) = R(t i ). V ar( R(t i )) = [ ] 2 d k R(ti ) k n k (n k d k ) where the summation is taken over all uncensored times t k such that t k t i. which can be used to generate the following approx- Taking the square root yields se( R(t i )) imate confidence interval for R(t i ): R(t i ) ± z se( R(t i )) Let µ T = E(T ) = be the mean time to a failure event (or, the mean reliability or survival time). The estimated mean is k µ T = R(t i ) (t i t i 1 ) wherelt 0 is defined to be 0. i=1 If the last observation is censored, then µ T is biased and underestimates µ T. The estimated variance of µ T is where A i = k 1 j=i V ar( µ T ) = R(t i ) (t j+1 t j ) and m = m m 1 k d i. i=1 Thus, an approximate confidence interval for µ T is k i=1 µ T ± z V ar( µ T ) A 2 i d i n i s i 252

4 The following example contains data from a product lifetime study of industrial grinders. Twenty grinders were tested and the time each grinder failed or the time it was removed due to censoring was recorded. + implies a censoring time. i t i n i d i n i d i n i i ( ) nk d k k= /20 = / / / / / / / / / / / / n k 253

5 Plot of Kaplan-Meier R(t) for the Grinder Data 254

6 SAS output for Grinder Example RELIABILITY STUDY OF GRINDER LIFETIMES The LIFETEST Procedure Product-Limit Survival Estimates Survival FAILTIME Survival Failure Standard Error Number Failed Number Left * * * * * * * * NOTE: The marked survival times are censored observations. Summary Statistics for Time Variable FAILTIME Quartile Estimates Point 95% Confidence Interval Percent Estimate Transform [Lower Upper) LOGLOG LOGLOG LOGLOG Mean Standard Error NOTE: The mean survival time and its standard error were underestimated because the largest observation was censored and the estimation was restricted to the largest event time. Summary of the Number of Censored and Uncensored Values Percent Total Failed Censored Censored

7 RELIABILITY STUDY OF GRINDER LIFETIMES 95% CONFIDENCE INTERVALS (Lower,Upper) Obs FAILTIME _CENSOR_ SURVIVAL Lower Upper SAS code for Grinder Example **************************************; *** Multiply censored data example ***; **************************************; DATA EXAMPLE2; DO ITEM = 1 TO 20; INPUT FAILTIME STATUS CENSORED = (STATUS= Y ); OUTPUT; END; LABEL FAILTIME = TIME TO FAILURE IN HOURS ; TITLE RELIABILITY STUDY OF GRINDER LIFETIMES ; CARDS; 42.1 Y 77.8 N 83.3 Y 88.7 N N N N N N N Y N N Y Y Y N Y N Y ; PROC LIFETEST DATA= EXAMPLE2 PLOTS=(S) OUTSURV=SURVIVE; TIME FAILTIME*CENSORED(1); PROC PRINT DATA=SURVIVE; RUN; 256

8 10.3 Proc Lifetest and Proc Reliability in SAS 10.3 Using SAS Proc Reliability and Proc Lifetest

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12 10.4 Plotting Methods for Exponential and Weibull Distributions Plot to check for an exponential distribution: Plot to check for an Weibull distribution: Example: Appliance Cycle Data Set: The following table shows the cycles (number of times used) to failure of a component in a small appliance. The engineering group wanted to estimate the percentage failing during the warranty (500 cycles) and an estimate of the median life. For the 54 tested appliances, failure times are unmarked, while censored times are indicated by

13 Example 1 Appliance Cycle Data Example Check for Exponential Distribution Check for Weibull Distribution