Graphical Methods for Survival Distribution Fitting

Transcription

1 Graphcal Methods for Survval Dstrbuton Fttng In ths Chapter we dscuss the followng two graphcal methods for survval dstrbuton fttng: 1. Probablty Plot, 2. Cox-Snell Resdual Method. Probablty Plot: The probablty plot s so constructed that f the theoretcal dstrbuton s adequate for data, the graph of a functon of t versus a functon of the sample cumulatve dstrbuton functon wll be close to a straght lne. Ths s carred out as follow: 1. A theoretcal dstrbuton for the survval tme has to be selected. 2. The sample cumulatve dstrbuton functon s estmated by usng the ordered values. 3. Plot t or a functon of t versus the estmated sample cumulatve dstrbuton or a functon of t. 4. If the plot shows serous departure from straght-lne, the theoretcal dstrbuton for survval tme s rejected.

2 Example: The whte blood cell counts (WBCs) of 23 pedatrc leukema patents s gven n table 8.1 on page 201 of your text book. We can use PROC LIFETEST to get Kaplan-Meer (KM) estmator of survvor functon. data B; nput WBC status; datalnes; proc lfetest data= B Method=KM; We can get a plot of the estmated survvor functon by requestng t n the PROC LIFETEST statement: proc lfetest data=b plots=(s) graphcs; symbol V=none;

3 We can get plots of the survval and hazard estmates by puttng PLOTS=(S,H) n the PROC LIFETEST statement. Suppose we specfy PLOTS =(S, LS, LLS) n the PROC LIFETEST Statement. The S gves the famlar survval curve. LS keyword produces a plot of log S ( t) versus t. For exponental dstrbuton, ths plot should be normal. The LLS keyword produces a plot of log[ log S ( t)] versus log of t and ths plot should be lnear for Webull dstrbuton.. proc lfetest data=b plots=(ls,lls) notable graphcs; symbol V=none;

4 For the log-normal dstrbuton, a plot of Φ 1 [1 S ( t)] versus log t should be lnear, where Φ (.) 1 s the c.d.f. of a standard normal varable and Φ (.) s ts nverse. For a log-logstc dstrbuton, a plot of log[ 1 log S( t)) / S( t)] versus log t should be lnear. proc lfetest data= B outsurv=a; (The SUTSURV opton on the frst lne produces a data set, named a n ths example, that ncludes the KM estmates of the survvor functon n a varable called Survval. To see what contaned n such data set, use Proc prnt data=a; Run; ) data; set a; s=survval; lnorm=probt(1-s); logt=log((1-s)/s); logwbc=log(wbc);

5 proc gplot; symbol1 value=none =jon; plot lnorm*logwbc logt*logwbc; t Note that log S( t) = h( t) = H ( t). Therefore, the above plots were used to determne whether 0 the hazard functon can be accurately descrbed by certan parametrc models. Cox-Snell Resdual Method: One dffculty wth all these plots s that they are based on the assumpton that the sample s drawn from a homogeneous populaton, mplyng that no covarates are related to survval tme. In practce, that means that a model that looks fne on the plots may not ft well when covarates are taken nto account. Smlarly, a model that s rejected on the bass of the plots may be qute satsfactory when survval tme s allowed to depend on covarates. One soluton to ths s to create plots on the resduals from the regresson model ft.

6 Several dfferent knd of resduals have been proposed for survval models, but the one most sutable for ths purpose are Cox-Snell resduals, defned as = log S ( ) where s t observed event tme for ndvdual, and r e e x e s the vector of covarates values for ndvdual (Your book use nstead of.). It can be shown that has (approxmately) an exponental dstrbuton wth mean 1. Therefore, the procedure for usng Cox-Snell resduals can be summarzed as follows: 1. Fnd MLE of the parameters of the selected theoretcal dstrbuton. e 2. Calculate Cox-Snell resduals = log S ( ), where S (t) s the estmated survval functon wth the MLE of parameters. 3. Apply the Kaplan-Meer method to estmate the survval functon of the Cox-Snell resduals obtaned n step 2. e log ( ) e e 4. plot versus S t x. If the plot s closed to a straght lne wth unt slope and zero ntercept, the ftted dstrbuton s approprate. t x Here s an example of how to do ths for a Webull model ftted to the data of the whte blood cell counts (WBCs) of 23 pedatrc leukema patents. proc lfereg data=b; model WBC*status(0)= /dst=webull; output out=c cdf=f; data d; set c; e= -log(1-f); proc lfetest data=d plots=(ls) notable graphcs; tme e*status(0); symboll v=none;

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