Estimators of Prediction Accuracy for Time-to-Event Data

Maintainer F. Bertrand

https://doi.org/10.32614/CRAN.package.survAUC

The goal of survAUC is to provide a variety of functions to estimate time-dependent true/false positive rates and AUC curves from a set of censored survival data.

Installation

You can install the released version of survAUC from CRAN with:

install.packages("survAUC")

And the development version from GitHub with:

install.packages("devtools")
devtools::install_github("fbertran/survAUC")

or alternatively

# install.packages("pak")
pak::pak("fbertran/survAUC")

Example

This is a basic example which shows you how to solve a common problem:

library(survAUC)

Retrieve the example dataset, split it into a train (TR) and a test (TE) set. Then train a model on the train set and derive predictions for both train (lp and Surv.rsp) and test (lpnew and Surv.rsp.new).

data(cancer,package="survival")
TR <- ovarian[1:16,]
TE <- ovarian[17:26,]
train.fit  <- survival::coxph(survival::Surv(futime, fustat) ~ age,
                    x=TRUE, y=TRUE, method="breslow", data=TR)

lp <- predict(train.fit)
lpnew <- predict(train.fit, newdata=TE)
Surv.rsp <- survival::Surv(TR$futime, TR$fustat)
Surv.rsp.new <- survival::Surv(TE$futime, TE$fustat) 
times <- seq(10, 1000, 10)

Chambless and Diao’s estimator of cumulative/dynamic AUC for right-censored time-to-event data

This function implements the estimator of cumulative/dynamic AUC proposed in Section 3.3 of Chambless and Diao (2006). In contrast to the general form of Chambless and Diao’s estimator, AUC.cd is restricted to Cox regression.

Specifically, it is assumed that lp and lpnew are the predictors of a Cox proportional hazards model. Estimates obtained from AUC.cd are valid as long as the Cox model is specified correctly.

The iauc summary measure is given by the integral of AUC on $[0,max(\textrm{times})]$ (weighted by the estimated probability density of the time-to-event outcome).

AUC_CD <- AUC.cd(Surv.rsp, Surv.rsp.new, lp, lpnew, times)
AUC_CD$iauc
#> [1] 0.8728364

AUC_CD
#> $auc
#>   [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8467091
#>   [7] 0.8467091 0.8467091 0.8467091 0.8467091 0.8467091 0.8625872
#>  [13] 0.8625872 0.8625872 0.8625872 0.8728364 0.8728364 0.8728364
#>  [19] 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364
#>  [25] 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364
#>  [31] 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364
#>  [37] 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364 0.8728364
#>  [43] 0.8728364 0.8719455 0.8719455 0.8719455 0.8736755 0.8795809
#>  [49] 0.8795809 0.8795809 0.8795809 0.8795809 0.8795809 0.8795809
#>  [55] 0.8795809 0.8795809 0.8945198 0.8945198 0.8945198 0.8945198
#>  [61] 0.8945198 0.8945198 0.8945198 0.9078402 0.9078402 0.9078402
#>  [67] 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402
#>  [73] 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402
#>  [79] 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402
#>  [85] 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402
#>  [91] 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402 0.9078402
#>  [97] 0.9078402 0.9078402 0.9078402 0.9078402
#> 
#> $times
#>   [1]   10   20   30   40   50   60   70   80   90  100  110  120
#>  [13]  130  140  150  160  170  180  190  200  210  220  230  240
#>  [25]  250  260  270  280  290  300  310  320  330  340  350  360
#>  [37]  370  380  390  400  410  420  430  440  450  460  470  480
#>  [49]  490  500  510  520  530  540  550  560  570  580  590  600
#>  [61]  610  620  630  640  650  660  670  680  690  700  710  720
#>  [73]  730  740  750  760  770  780  790  800  810  820  830  840
#>  [85]  850  860  870  880  890  900  910  920  930  940  950  960
#>  [97]  970  980  990 1000
#> 
#> $iauc
#> [1] 0.8728364
#> 
#> attr(,"class")
#> [1] "survAUC"

plot(AUC_CD)

plot of chunk unnamed-chunk-8

Hung and Chiang’s estimator of cumulative/dynamic AUC for right-censored time-to-event data

This function implements the estimator of cumulative/dynamic AUC proposed by Hung and Chiang (2010).

The estimator is based on inverse-probability-of-censoring weights and does not assume a specific working model for deriving the predictor lpnew. It is assumed, however, that there is a one-to-one relationship between the predictor and the expected survival times conditional on the predictor.

The iauc summary measure is given by the integral of AUC on $[0, max(\textrm{times})]$ (weighted by the estimated probability density of the time-to-event outcome).

AUC_hc <- AUC.hc(Surv.rsp, Surv.rsp.new, lpnew, times)
AUC_hc$iAUC
#> NULL

plot(AUC_hc)

plot of chunk unnamed-chunk-10

Song and Zhou’s estimators of AUC for right-censored time-to-event data

The sens.sh and spec.sh functions implement the estimators of time-dependent true and false positive rates proposed by Song and Zhou (2008).

The AUC.sh function implements the estimators of cumulative/dynamic and incident/dynamic AUC proposed by Song and Zhou (2008). These estimators are given by the areas under the time-dependent ROC curves estimated by sens.sh and spec.sh.

In case of cumulative/dynamic AUC, the iauc summary measure is given by the integral of AUC on $[0,max(\textrm{times})]$ (weighted by the estimated probability density of the time-to-event outcome).

In case of incident/dynamic AUC, iauc is given by the integral of AUC on $[0, max(\textrm{times})]$ (weighted by 2 times the product of the estimated probability density and the estimated survival function of the time-to-event outcome).

lp <- predict(train.fit)
AUC_sh <- AUC.sh(Surv.rsp, Surv.rsp.new, lp, lpnew, times)
names(AUC_sh)
#> [1] "auc"   "times" "iauc"
AUC_sh$iauc
#> [1] 0.8430845

plot(AUC_sh)

plot of chunk unnamed-chunk-12

Uno’s estimator of cumulative/dynamic AUC for right-censored time-to-event data

The sens.uno and spec.uno functions implement the estimators of time-dependent true and false positive rates proposed in Section 5.1 of Uno et al. (2007).

The AUC.uno function implements the estimator of cumulative/dynamic AUC that is based on the TPR and FPR estimators proposed by Uno et al. (2007).

It is given by the area(s) under the time-dependent ROC curve(s) estimated by sens.uno and spec.uno. The iauc summary measure is given by the integral of AUC on $[0, max(\textrm{times})]$ (weighted by the estimated probability density of the time-to-event outcome).

AUC_Uno <- AUC.uno(Surv.rsp, Surv.rsp.new, lpnew, times)
names(AUC_Uno)
#> [1] "auc"   "times" "iauc"
AUC_Uno$iauc
#> [1] 0.7552083

plot(AUC_Uno)

plot of chunk unnamed-chunk-14

Censoring-adjusted C-statistic by Uno et al.

The UnoC function implements the censoring-adjusted C-statistic proposed by Uno et al. (2011). It has the same interpretation as Harrell’s C for survival data (implemented in the rcorr.cens function of the Hmisc package).

Cstat <- UnoC(Surv.rsp, Surv.rsp.new, lpnew)
Cstat
#> [1] 0.7333333

References

Chambless, L. E. and G. Diao (2006). Estimation of time-dependent area under the ROC curve for long-term risk prediction. Statistics in Medicine, 25, 3474-3486.
Hung, H. and C.-T. Chiang (2010). Estimation methods for time-dependent AUC models with survival data. Canadian Journal of Statistics, 38, 8-26.
Song, X. and X.-H. Zhou (2008). A semiparametric approach for the covariate specific ROC curve with survival outcome. Statistica Sinica, 18, 947-965.
Uno, H., T. Cai, L. Tian, and L. J. Wei (2007). Evaluating prediction rules for t-year survivors with censored regression models. Journal of the American Statistical Association, 102, 527-537.