Survival Analysis IIBios 162Michael G. Hudgens, [email protected]://www.bios.unc.edu/∼mhudgens2005-12-05 14:33BIOS 162 1 Survival Analysis IIGoal• How do we test if two survival functions are differentunder minimal assumptions?• For example: leukemia patients are randomized to treat-ment or placebo. Are the survival functions the samebetween the two groups?• Without censoring, use a rank test (e.g., Wilcoxon ranksum)• In the presence of right censoring, use logrank testBIOS 162 2 Survival Analysis IILogrank test• Data from two samples(Tij, δij)for i = 1, 2 and j = 1, 2, . . . , ni• Want to testH0: S1(t) = S2(t)whereSj(t) = Pr[T∗j> t] for j = 1, 2BIOS 162 3 Survival Analysis IILogrank test• Let t(1), t(2), . . . , t(k)be the ordered failure times in thetwo groups combined• At each time t(j), construct the table:Group Deaths Surviving At risk1 m1jR1(t(j)) − m1jR1(t(j))2 m2jR2(t(j)) − m2jR2(t(j))mjR(t(j)) − mjR(t(j))BIOS 162 4 Survival Analysis IILogrank Test• Under H0, the expected number of deaths in group 1 isE1j= R1(t(j))mjR(t(j))• The hypergeometric variance isV1j=R1(t(j))R2(t(j))mj{R(t(j)) − mj}R2(t(j)){R(t(j)) − 1}BIOS 162 5 Survival Analysis IILogrank Test• The logrank (Mantel-Haenszel) statistic:E1=kXj=1E1j, O1=kXj=1m1j, V1=kXj=1V1j• Under H0: S1(t) = S2(t),X =(O1− E1)2V1∼ χ21BIOS 162 6 Survival Analysis IILogrank Test• Leukemia ExampleTreatment (n = 21) Placebo (n = 21)6, 6, 6, 6+ 6, 6, 6, 67, 9+, 10, 10+ 6, 6, 7, 711+, 13, 16, 17+ 7, 10, 10, 12+, 1319+, 20+, 22, 23 13, 15+, 16, 17+25+, 32+, 32+ 34+, 35+ 22, 23, 23, 23+BIOS 162 7 Survival Analysis IILogrank Test: Leukemia Example0 5 10 15 20 25 30 350.0 0.2 0.4 0.6 0.8 1.0tS(t)TreatmentPlaceboBIOS 162 8 Survival Analysis IIR Code for Two Kaplan-Meier Curveslibrary("survival")fit <- survfit(Surv(t, delta)~rx,conf.type="none")pdf("surv_leuk1.pdf",width=11,height=8.5)plot(fit,xlab="t",ylab="S(t)",lwd=c(1,3))legend(25,1,c("Treatment","Placebo"),lwd=c(3,1))dev.off()BIOS 162 9 Survival Analysis IILogrank test “by hand”: Leukemia Examplet(j)m1jR1(t(j)) m2jR2(t(j)) mjR(t(j)) E1jV1j6 3 21 6 21 9 42 4.50 1.817 1 17 3 15 4 32 2.13 0.9010 1 15 2 12 3 27 1.67 0.6813 1 12 2 9 3 21 1.71 0.6616 1 11 1 6 2 17 1.29 0.4322 1 7 1 4 2 11 1.27 0.4223 1 6 2 3 3 9 2.00 0.509 14.57 5.4BIOS 162 10 Survival Analysis IILogrank test: Leukemia Example• ThereforeX =(9 − 14.57)25.4= 5.75Pr[χ21> 5.75] = 0.0165• R code:Call:survdiff(formula = Surv(t, delta) ~ rx)N Observed Expected (O-E)^2/E (O-E)^2/Vrx=p 21 17 11.4 2.72 5.75rx=t 21 9 14.6 2.13 5.75Chisq= 5.8 on 1 degrees of freedom, p= 0.0165BIOS 162 11 Survival Analysis IILogrank test: Leukemia Example• SAS codeproc lifetest;time t*delta(0);strata trt;Test of Equality over StrataPr >Test Chi-Square DF Chi-SquareLog-Rank 5.7507 1 0.0165Wilcoxon 4.3357 1 0.0373-2Log(LR) 6.0441 1 0.0140BIOS 162 12 Survival Analysis IILogrank test: SASdata;input time group death wt;cards;6 1 1 36 1 0 186 2 1 66 2 0 157 1 1 17 1 0 167 2 1 37 2 0 12...proc freq order=data;tables time*group*death/chisq cmh;weight wt;BIOS 162 13 Survival Analysis IILogrank test: SASThe FREQ ProcedureSummary Statistics for group by deathControlling for timeCochran-Mantel-Haenszel Statistics (Based on Table Scores)Statistic Alternative Hypothesis DF Value Prob---------------------------------------------------------------1 Nonzero Correlation 1 5.7507 0.01652 Row Mean Scores Differ 1 5.7507 0.01653 General Association 1 5.7507 0.0165BIOS 162 14 Survival Analysis
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