# Concordance index for right censored survival time data In survival analysis, a pair of patients is called concordant if the risk of the event predicted by a model is lower for the patient who experiences the event at a later timepoint. The concordance probability (C-index) is the frequency of concordant pairs among all pairs of subjects. It can be used to measure and compare the discriminative power of a risk prediction models. The function provides an inverse of the probability of censoring weigthed estimate of the concordance probability to adjust for right censoring. Cross-validation based on bootstrap resampling or bootstrap subsampling can be applied to assess and compare the discriminative power of various regression modelling strategies on the same set of data. ```r cindex( object, formula, data, eval.times, pred.times, cause, lyl = FALSE, cens.model = "marginal", ipcw.refit = FALSE, ipcw.args = NULL, ipcw.limit, tiedPredictionsIn = TRUE, tiedOutcomeIn = TRUE, tiedMatchIn = TRUE, splitMethod = "noPlan", B, M, model.args = NULL, model.parms = NULL, keep.models = FALSE, keep.residuals = FALSE, keep.pvalues = FALSE, keep.weights = FALSE, keep.index = FALSE, keep.matrix = FALSE, multiSplitTest = FALSE, testTimes, confInt = FALSE, confLevel = 0.95, verbose = TRUE, savePath = NULL, slaveseed = NULL, na.action = na.fail, ... ) ``` ## Arguments - `object`: A named list of prediction models, where allowed entries are (1) R-objects for which a predictSurvProb method exists (see details), (2) a `call` that evaluates to such an R-object (see examples), (3) a matrix with predicted probabilities having as many rows as `data` and as many columns as `times`. For cross-validation all objects in this list must include their `call`. - `formula`: A survival formula. The left hand side is used to finde the status response variable in `data`. For right censored data, the right hand side of the formula is used to specify conditional censoring models. For example, set `Surv(time,status)~x1+x2` and `cens.model="cox"`. Then the weights are based on a Cox regression model for the censoring times with predictors x1 and x2. Note that the usual coding is assumed: `status=0` for censored times and that each variable name that appears in `formula` must be the column name in `data`. If there are no covariates, i.e. `formula=Surv(time,status)~1` the `cens.model` is coerced to `"marginal"` and the Kaplan-Meier estimator for the censoring times is used to calculate the weights. If `formula` is `missing`, try to extract a formula from the first element in object. - `data`: A data frame in which to validate the prediction models and to fit the censoring model. If `data` is missing, try to extract a data set from the first element in object. - `eval.times`: A vector of timepoints for evaluating the discriminative ability. At each timepoint the c-index is computed using only those pairs where one of the event times is known to be earlier than this timepoint. If `eval.times` is `missing` then the largest uncensored event time is used. - `pred.times`: A vector of timepoints for evaluating the prediction models. This should either be exactly one timepoint used for all `eval.times`, or be as long as `eval.times`, in which case the predicted order of risk for the jth entry of `eval.times` is based on the jth entry of `pred.times` corresponding - `cause`: For competing risks, the event of interest. Defaults to the first state of the response, which is obtained by evaluating the left hand side of `formula` in `data`. - `lyl`: If `TRUE` rank subjects accoring to predicted life-years-lost (See Andersen due to this cause instead of predicted risk. - `cens.model`: Method for estimating inverse probability of censoring weigths: `cox`: A semi-parametric Cox proportional hazard model is fitted to the censoring times `marginal`: The Kaplan-Meier estimator for the censoring times `nonpar`: Nonparametric extension of the Kaplan-Meier for the censoring times using symmetric nearest neighborhoods -- available for arbitrary many strata variables on the right hand side of argument `formula` but at most one continuous variable. See the documentation of the functions `prodlim` and `neighborhood` from the prodlim package. `aalen`: The nonparametric Aalen additive model fitted to the censoring times. Requires the timereg package maintained by Thomas Scheike. - `ipcw.refit`: If `TRUE` the inverse probability of censoring weigths are estimated separately in each training set during cross-validation. - `ipcw.args`: List of arguments passed to function specified by argument `cens.model`. - `ipcw.limit`: Value between 0 and 1 (but no equal to 0!) used to cut for small weights in order to stabilize the estimate at late times were few individuals are observed. - `tiedPredictionsIn`: If `FALSE` pairs with identical predictions are excluded, unless also the event times are identical and uncensored and `tiedMatchIn` is set to `TRUE`. - `tiedOutcomeIn`: If `TRUE` pairs with identical and uncensored event times are excluded, unless also the predictions are identical and `tiedMatchIn` is set to `TRUE`. - `tiedMatchIn`: If `TRUE` then pairs with identical predictions and identical and uncensored event times are counted as concordant pairs. - `splitMethod`: Defines the internal validation design: `none/noPlan`: Assess the models in the give `data`, usually either in the same data where they are fitted, or in independent test data. `BootCv`: Bootstrap cross validation. The prediction models are trained on `B` bootstrap samples, that are either drawn with replacement of the same size as the original data or without replacement from `data` of the size `M`. The models are assessed in the observations that are NOT in the bootstrap sample. `Boot632`: Linear combination of AppCindex and OutOfBagCindex using the constant weight .632. - `B`: Number of bootstrap samples. The default depends on argument `splitMethod`. When `splitMethod` in c("BootCv","Boot632") the default is 100. For `splitMethod="none"` `B` is the number of bootstrap simulations e.g. to obtain bootstrap confidence limits -- default is 0. - `M`: The size of the bootstrap samples for resampling without replacement. Ignored for resampling with replacement. - `model.args`: List of extra arguments that can be passed to the `predictSurvProb` methods. The list must have an entry for each entry in `object`. - `model.parms`: Experimental. List of with exactly one entry for each entry in `object`. Each entry names parts of the value of the fitted models that should be extracted and added to the value. - `keep.models`: Logical. If `TRUE` keep the models in object. Since fitted models can be large objects the default is `FALSE`. - `keep.residuals`: Experimental. - `keep.pvalues`: Experimental. - `keep.weights`: Experimental. - `keep.index`: Logical. If `FALSE` remove the bootstrap or cross-validation index from the output list which otherwise is included in the method part of the output list. - `keep.matrix`: Logical. If `TRUE` add all `B` prediction error curves from bootstrapping or cross-validation to the output. - `multiSplitTest`: Experimental. - `testTimes`: A vector of time points for testing differences between models in the time-point specific Brier scores. - `confInt`: Experimental. - `confLevel`: Experimental. - `verbose`: if `TRUE` report details of the progress, e.g. count the steps in cross-validation. - `savePath`: Place in your filesystem (directory) where training models fitted during cross-validation are saved. If `missing` training models are not saved. - `slaveseed`: Vector of seeds, as long as `B`, to be given to the slaves in parallel computing. - `na.action`: Passed immediately to model.frame. Defaults to na.fail. If set otherwise most prediction models will not work. - `...`: Not used. ## Returns Estimates of the C-index. ## Details Pairs with identical observed times, where one is uncensored and one is censored, are always considered usuable (independent of the value of `tiedOutcomeIn`), as it can be assumed that the event occurs at a later timepoint for the censored observation. For uncensored response the result equals the one obtained with the functions `rcorr.cens` and `rcorrcens` from the `Hmisc` package (see examples). ## Examples ```r # simulate data based on Weibull regression library(prodlim) set.seed(13) dat <- SimSurv(100) # fit three different Cox models and a random survival forest # note: low number of trees for the purpose of illustration library(survival) cox12 <- coxph(Surv(time,status)~X1+X2,data=dat,x=TRUE,y=TRUE) cox1 <- coxph(Surv(time,status)~X1,data=dat,x=TRUE,y=TRUE) cox2 <- coxph(Surv(time,status)~X2,data=dat,x=TRUE,y=TRUE) # # compute the apparent estimate of the C-index at a single time point # A1 <- pec::cindex(list("Cox X1"=cox1), formula=Surv(time,status)~X1+X2, data=dat, eval.times=10) # # compute the apparent estimate of the C-index at different time points # ApparrentCindex <- pec::cindex(list("Cox X1"=cox1, "Cox X2"=cox2, "Cox X1+X2"=cox12), formula=Surv(time,status)~X1+X2, data=dat, eval.times=seq(1,15,1)) print(ApparrentCindex) plot(ApparrentCindex) # # compute the bootstrap-crossvalidation estimate of # the C-index at different time points # set.seed(142) bcvCindex <- pec::cindex(list("Cox X1"=cox1, "Cox X2"=cox2, "Cox X1+X2"=cox12), formula=Surv(time,status)~X1+X2, data=dat, splitMethod="bootcv", B=5, eval.times=seq(1,15,1)) print(bcvCindex) plot(bcvCindex) # for uncensored data the results are the same # as those obtained with the function rcorr.cens from Hmisc set.seed(16) dat <- SimSurv(30) dat$staus=1 fit12 <- coxph(Surv(time,status)~X1+X2,data=dat,x=TRUE,y=TRUE) fit1 <- coxph(Surv(time,status)~X1,data=dat,x=TRUE,y=TRUE) fit2 <- coxph(Surv(time,status)~X2,data=dat,x=TRUE,y=TRUE) Cpec <- pec::cindex(list("Cox X1+X2"=fit12,"Cox X1"=fit1,"Cox X2"=fit2), formula=Surv(time,status)~1, data=dat) p1 <- predictSurvProb(fit1,newdata=dat,times=10) p2 <- predictSurvProb(fit2,newdata=dat,times=10) p12 <- predictSurvProb(fit12,newdata=dat,times=10) if (requireNamespace("Hmisc",quietly=TRUE)){ library(Hmisc) harrelC1 <- rcorr.cens(p1,with(dat,Surv(time,status))) harrelC2 <- rcorr.cens(p2,with(dat,Surv(time,status))) harrelC12 <- rcorr.cens(p12,with(dat,Surv(time,status))) harrelC1[["C Index"]]==Cpec$AppCindex[["Cox.X1"]] harrelC2[["C Index"]]==Cpec$AppCindex[["Cox.X2"]] harrelC12[["C Index"]]==Cpec$AppCindex[["Cox.X1.X2"]] } # # competing risks # library(riskRegression) library(prodlim) set.seed(30) dcr.learn <- SimCompRisk(30) dcr.val <- SimCompRisk(30) pec::cindex(CSC(Hist(time,event)~X1+X2,data=dcr.learn),data=dcr.val) fit <- CSC(Hist(time,event)~X1+X2,data=dcr.learn) cif <- predictRisk(fit,newdata=dcr.val,times=3,cause=1) pec::cindex(list(fit),data=dcr.val,times=3) ``` ## References TA Gerds, MW Kattan, M Schumacher, and C Yu. Estimating a time-dependent concordance index for survival prediction models with covariate dependent censoring. Statistics in Medicine, Ahead of print:to appear, 2013. DOI = 10.1002/sim.5681 Wolbers, M and Koller, MT and Witteman, JCM and Gerds, TA (2013) Concordance for prognostic models with competing risks Research report 13/3. Department of Biostatistics, University of Copenhagen Andersen, PK (2012) A note on the decomposition of number of life years lost according to causes of death Research report 12/2. Department of Biostatistics, University of Copenhagen Paul Blanche, Michael W Kattan, and Thomas A Gerds. The c-index is not proper for the evaluation of-year predicted risks. Biostatistics, 20(2): 347--357, 2018. ## Author(s) Thomas A Gerds tag@biostat.ku.dk

cindex function