NONPARAMETRIC SUMMARY CURVES FOR COMPETING RISKS IN R

Transcription

1 NONPARAMETRIC SUMMARY CURVES FOR COMPETING RISKS IN R By Pawel Paczuski 1 University of Michigan November 19, 2012 Abstract In survival analysis, when a subject may fail due to one of K 2 causes, we have a situation of competing risks. An example is cause-specific mortality, where a death may occur as a result of cancer, heart disease, or other causes. Two estimators may be used to perform competing risks analysis: 1) the complement of the Kaplan-Meier estimator and 2) the cumulative incidence function. In this paper, we show how to implement these methods in R using an example dataset. These brief explanations closely follow Klein and Moeschberger (2003) Section 4.7. Contents 1. Introduction" First estimator: the complement of the Kaplan-Meier estimator" Second estimator: the cumulative incidence function" 2 2. Example and implementation in R: BMT data" The cumulative incidence function in R" Interaction plot in R" KM Table 4.8 reconstruction" Multiple groups example" 8 3. Final notes" 9 References" 10 Appendix: R Code" 11 Competing Risks in R: Summary Handout" 14 1 Send suggestions about this paper to pbpacz at umich dot edu

2 Competing Risks in R" 2/14 1. Introduction Competing risks occur when a subject may fail due to more than one cause. Klein and Moeschberger (2003), Section 4.7, describe three methods for summarizing competing risks data: 1) the complement of the Kaplan-Meier estimator, 2) the cumulative incidence function, and 3) the conditional probability function for competing risk. In this paper, we focus on the implementation of the first two using the statistical package R. For a more thorough treatment, we refer the reader to Klein and Moeschberger (2003), Sections 2.7 and 4.7. The example dataset that will be used is the Bone Marrow Transplant for Leukemia dataset referenced in Klein and Moeschberger (2003), Section 1.3. In this study, treatment failure is defined as death in remission or relapse, whichever comes first. Here, we treat both as competing risks, and are interested in the likelihood of these events happening over time. 1.1 First estimator: the complement of the Kaplan-Meier estimator One method to study competing risks uses the complement of the Kaplan-Meier estimator. When we consider one event, E, we treat occurrences of the other event(s) as censored observations, and attempt to estimate the probability of E. This estimator may be interpreted as the probability of event E occurring by time t if the risk of other event(s) is removed. We can think of it as the probability of E occurring in a hypothetical world where the other event(s) are not possible. This is rarely of clinical interest. However, this estimator is related to the cumulative incidence function, which we describe next. 1.2 Second estimator: the cumulative incidence function The cumulative incidence function, CIF, is defined in Klein and Moeschberger (4.7.1) as: where t i, i=1:k are distinct times when one of the competing risks occurs. At time t i : Y i is the number of subjects at risk, r i the number of subjects with an occurrence of the event of interest at this time, and d i the number of subjects with an occurrence of any of the other events of interest at this time.

3 Competing Risks in R" 3/14 Note that: 1) d i + r i is the number of subjects with an occurrence of any of the competing risks at t i 2) independent random censoring is not counted as one of the competing risks and only affects Y i 3) for t t 1, the CIF is: where Ŝ(t i -) is the Kaplan-Meier estimator, evaluated at just before t i, obtained by treating any one of the competing risks as an event. Thus, it may be shown that the sum of the cumulative incidences for all competing risks is 1 Ŝ(t). The CIF estimates the probability that the event of interest occurs before time t and that it occurs before any of the competing causes of failure. The variance of the CIF is estimated by (KM 4.7.2): Confidence intervals may be constructed in the standard manner. 2. Example and implementation in R: BMT data We will illustrate both estimators using the ALL group in the BMT data, as is done in KM 4.7. R code is also provided in the appendix. 2 The data is right-censored, and the two competing risks of interest are death in remission (TRM) and relapse. The package cmprsk allows us to perform the necessary calculations. The dataset is part of the package KMsurv. library(cmprsk) library(kmsurv) data(bmt) #also loads library(survival) #load bmt data ## select group 1 only (ALL patients) bmt1 <- subset(bmt, bmt$group==1) 2 As well as at

4 Competing Risks in R" 4/14 bmt1 <- bmt1[order(bmt1$t2),] In this dataset, we have: t2: Disease Free Survival Time (Time To Relapse, Death or End of Study) d1: Death Indicator d2: Relapse Indicator d3: Disease Free Survival Indicator (0 if alive, 1 if death or relapse) We create a single variable of the censored observations and the two competing risks, necessary for our implementation: bmt1$type <- bmt1$d1+bmt1$d2 > table(bmt1$type) Now, bmt1$type is 0 for the originally censored observations, 1 for TRM, and 2 for relapse. 2.1 The cumulative incidence function in R We obtain the cumulative incidence function with the cuminc() function in cmprsk. Usage is: ## cuminc(ftime, fstatus, group, cencode=0,...) where ## ftime is failure time, fstatus is failure status, group is optional group indicator ## cencode is the code for censored observations in fstatus fit.1 <- cuminc(bmt1$t2, bmt1$type, cencode=0) This obtains the fit for some default timepoints, but we can obtain it for our given timepoints with the timepoints() function ## using timepoints(w, times) ## w is a cuminc() object, times is vector of time fits.c <- timepoints(fit.1,bmt1$t2) > fits.c $est

5 Competing Risks in R" 5/ $var The result is a list. We can easily: ## plot competing risks plot(fit.1) But: ## to customize, use custom function from package > plot.cuminc(fit.1, col=1:2, curvlab=c("trm", "Relapse"), xlim=c (0,1000),xlab="Days Post Transplant", ylab="probability", main="competing Risks: \ncumulative Incidence for ALL group")

6 Competing Risks in R" 6/14 This plots the cumulative incidence for TRM and for relapse from Figures 4.12 and 4.13 in KM. We can generate a table of values with code such as: ## convert list to df, format nicely df <- as.data.frame(cbind(t(fits.c$est),t(fits.c$var))) names(df) <- c("trm CI", "relapse CI", "var(trm CI)", "var(relapse CI)") row.names(df) <- c(1:length(df[[1]])) ## can add times df$time <- sort(unique(bmt1$t2)) ## rearrange order of display and subset df <- df[,c(5,1,2)] > head(df) time trm CI relapse CI This is part of KM Table 4.8. We used unique() above when inserting the times because when obtaining model fits using cuminc() (or even survfit() and other functions), only unique times are outputted. 2.2 Interaction plot in R A very useful comparison between all competing risks is an interaction plot, which appears in KM Figure It plots the relapse CIF and the sum of the relapse and TRM CIFs to show how changes in the likelihood of one event cause changes in the probabilities of the other events. It may be generated with: ## for interaction plot, first add the sum of TRM and relapse df$sum <- df$trm+df$relapse ## interaction plot (fig 4.14) > plot(df$time, df$relapse, type="s", ylim=c(0,1), xlim=c(0,1000),xlab="days Post Transplant", ylab="probability") > lines(df$time, df$sum, type="s") > text(700, 0.1, "Relapse Probability", cex=0.7) > text(700, 0.4, "Death in Remission Probability", cex=0.7) > text(300, 0.8, "Disease Free Survival Probability", cex=0.7)

7 Competing Risks in R" 7/14 > title("interaction between the relapse and\n death in remission (KM Fig 4.14)") 2.3 KM Table 4.8 reconstruction To obtain other output from KM Table 4.8, the following code may be used: ## add 1-KME for both risks ## KME = Kaplan Meier Estimate ## TRM bmt1$death <- ifelse(bmt1$type==1,1,0) km.death <- survfit(surv(bmt1$t2, bmt1$death)~1) ## Relapse bmt1$relapse <- ifelse(bmt1$type==2,1,0) km.relapse <- survfit(surv(bmt1$t2, bmt1$relapse) ~ 1) trm.1mkm <- 1-km.death$surv relapse.1mkm <- 1-km.relapse$surv df2 <- cbind(df$time, trm.1mkm, relapse.1mkm, df$trm, df$relapse)

8 Competing Risks in R" 8/14 df2 <- as.data.frame(df2) names(df2) <- c("time", "TRM 1-KME", "Relapse 1-KME", "TRM CI", "Relapse CI") row.names(df2) <- c(1:dim(df2)[1]) ## now showing that sum of CumInc is 1-KME for original treatment failure ## find sum df2$cisum <- df2[,4]+df2[,5] kme.all <- survfit(surv(bmt1$t2, bmt1$d3) ~ 1) kme.comp <- 1-kme.all$surv ## add to dataframe df2$kme.comp <- kme.comp > head(df2) time TRM 1-KME Relapse 1-KME TRM CI Relapse CI CIsum kme.comp Multiple groups example We can also compare risks between multiple groups. The original bmt dataset has three groups. ## first, rename the groups bmt$group <- factor(bmt$group, levels=c(1:3), labels=c("all", "AML lo", "AML hi")) bmt$type <- bmt$d1+bmt$d2 > table(bmt$group, bmt$type) ALL AML lo AML hi ## obtain CIFs fit.3g <- cuminc(bmt$t2, bmt$type, bmt$group, cencode=0) ## additional output appears: > fit.3g Tests: stat pv df

9 Competing Risks in R" 9/ The test (see Gray (1988)) compares each risk between all groups. Here, we see that the TRM (type=1) CIFs do not vary significantly between the three groups, while the relapse (type=2) CIFs are significantly different among the three groups. This may be confirmed by looking at a plot: ## TRM (1) by group in black, Relapse (2) by group in red > plot.cuminc(fit.3g, col=c(1,1,1,2,2,2), xlab="days Post Transplant", ylab="probability", main="all three groups compared.\nblack: TRM by group. Red: Relapse by group") 3. Final notes Scrucca L et al (2007) implement a single function to plot and compute the CIFs. See Regression models are described in Scheike and Zhang (2008).

10 Competing Risks in R" 10/14 References 1. Gray RJ (1988). A class of K-sample tests for comparing the cumulative incidence of a competing risk, ANNALS OF STATISTICS, 16: Klein JP, Moeschberger, ML (2003). Survival analysis: Techniques for censored and truncated data. New York: Springer. 3. Scheike TH, Zhang MJ, Gerds TA (2008). Predicting Cumulative Incidence Probability by Direct Binomial Regression. BIOMETRIKA, 95: Scrucca L, Santucci A, Aversa F (2007). Competing risk analysis using R: an easy guide for clinicians. BONE MARROW TRANSPLANT, 40(4):381 7.

11 Competing Risks in R" 11/14 Appendix: R Code Also available at ## Competing Risks in R ## UM BIOS 2012 library(cmprsk) library(kmsurv) data(bmt) #also loads library(survival) #load bmt data ## select group 1 only (ALL patients) bmt1 <- subset(bmt, bmt$group==1) bmt1 <- bmt1[order(bmt1$t2),] ## in bmt1: ## d1: death indicator ## d2: relapse indicator ## d3: =0 if alive, =1 if (death or relapse) ## create single column for event types so that: ## 0 is censored ## 1 is death ## 2 is relapse [add d1 and d2 values] bmt1$type <- bmt1$d1+bmt1$d2 #length = 38 ## obtain cumulative incidence fit fit.1 <- cuminc(bmt1$t2, bmt1$type, cencode=0) ## this does not repeat times, so length = 37 fits.c <- timepoints(fit.1,bmt1$t2) fits.c ## plot competing risks plot(fit.1) ## to customize, use custom function from package plot.cuminc(fit.1, col=1:2, curvlab=c("trm", "Relapse"), xlim=c(0,1000),xlab="days Post Transplant", ylab="probability", main="competing Risks:\nCumulative Incidence for ALL group")

12 Competing Risks in R" 12/14 ## convert list to df, format nicely ## dim(df) = 37x4 df <- as.data.frame(cbind(t(fits.c$est),t(fits.c$var))) names(df) <- c("trm CI", "relapse CI", "var(trm CI)", "var(relapse CI)") row.names(df) <- c(1:length(df[[1]])) ## can add times ## unique b/c cuminc() does it this way df$time <- sort(unique(bmt1$t2)) ## rearrange order of display df <- df[,c(5,1,2)] ## for intxn plot, first add the sum of trm and relapse df$sum <- df$trm+df$relapse ## interaction plot (fig 4.14) plot(df$time, df$relapse, type="s", ylim=c(0,1), xlim=c(0,1000),xlab="days Post Transplant", ylab="probability") lines(df$time, df$sum, type="s") text(700, 0.1, "Relapse Probability", cex=0.7) text(700, 0.4, "Death in Remission Probability", cex=0.7) text(300, 0.8, "Disease Free Survival Probability", cex=0.7) title("interaction between the relapse and\n death in remission (KM Fig 4.14)") ## add 1-KME for both risks ## KME = Kaplan Meier Estimate ## TRM bmt1$death <- ifelse(bmt1$type==1,1,0) km.death <- survfit(surv(bmt1$t2, bmt1$death)~1) ## Relapse 1-KME bmt1$relapse <- ifelse(bmt1$type==2,1,0) km.relapse <- survfit(surv(bmt1$t2, bmt1$relapse) ~ 1) trm.1mkm <- 1-km.death$surv relapse.1mkm <- 1-km.relapse$surv

13 Competing Risks in R" 13/14 df2 <- cbind(df$time, trm.1mkm, relapse.1mkm, df$trm, df$relapse) df2 <- as.data.frame(df2) names(df2) <- c("time", "TRM 1-KME", "Relapse 1-KME", "TRM CI", "Relapse CI") row.names(df2) <- c(1:dim(df2)[1]) ## now showing that sum of CumInc is 1-KME for original treatment failure ## find sum df2$cisum <- df2[,4]+df2[,5] kme.all <- survfit(surv(bmt1$t2, bmt1$d3) ~ 1) kme.comp <- 1-kme.all$surv ## add to dataframe df2$kme.comp <- kme.comp ## we can also run all groups ## rename bmt$group <- factor(bmt$group, levels=c(1:3), labels=c("all", "AML lo", "AML hi")) bmt$type <- bmt$d1+bmt$d2 table(bmt$group, bmt$type) fit.3g <- cuminc(bmt$t2, bmt$type, bmt$group, cencode=0) ## and discuss test results fit.3g Tests: stat pv df For type=1 (TRM), the 3 groups are not significantly different For type=2 (Relapse), the 3 groups are significanlty different ##this may be confirmed by looking at the plot ## TRM (1) by group in black, Relapse (2) by group in red plot.cuminc(fit.3g, col=c(1,1,1,2,2,2), xlab="days Post Transplant", ylab="probability", main="all three groups compared.\nblack: TRM by group. Red: Relapse by group")

14 Competing Risks in R: Summary Handout The CIF estimates the probability that the event of interest occurs before time t and that it occurs before any of the competing causes of failure. Cumulative incidence function library(cmprsk) thebestfit <- cuminc(ftime, fstatus, group, cencode=0,...) where: ftime is failure time / fstatus is failure status (competing risks, initially censored observations) / group is optional group indicator / cencode is the code for censored observations in fstatus Custom times timepoints(thebestfit, times) where: times is a vector of times Test for equality of each risk among groups (if > 1): thebestfit$tests Plot using custom function plot.cuminc(thebestfit, main="fancy title", curvlab, ylim, xlim, wh=2,lty=1:length(x), color=1,...) Create interaction plot for two competing risks plot(ftime, risk1,...) lines(ftime, <vector of sum of risk1 and risk2>,...)