Hazard discrimination summary
R package for estimating the hazard discrimination summary (HDS). HDS is a measure of time-varying prognostic performance. It can be thought of as an incident, time-varying extension of the discrimination slope (Yates 1982), which is perhaps better known as an important part of the integrated discrimination improvement (IDI; Pencina et al. 2008). Alternatively, HDS is a risk-based complement to the incident/dynamic time-dependent AUC (Heagerty and Zheng 2005). Under some circumstances, HDS also has interesting connections to the Cox model partial likelihood. For a detailed overview of HDS, see Liang and Heagerty (2016) and the related discussions and rejoinder.
Installation
To install through CRAN, use
install.packages("hds")To install the latest (though not necessarily stable) GitHub version, make sure you have devtools installed and use
devtools::install_github("liangcj/hds")Example
A simple example using the Mayo PBC data from the survival package demonstrating both hds (estimator based on the Cox model) and hdslc (more flexible estimator based on the local-in-time Cox model):
head(hds(times = survival::pbc[1:312, 2],
status = (survival::pbc[1:312, 3]==2)*1,
m = survival::pbc[1:312, 5]))
hdsres <- hds(times=pbc5[,1], status=pbc5[,2], m=pbc5[,3:7])
hdslcres <- hdslc(times = pbc5[,1], status=pbc5[,2], m = pbc5[,3:7], h = 730)
Survt <- summary(survival::survfit(survival::Surv(pbc5[,1], pbc5[,2])~1))
Survtd <- cbind(Survt$time, c(0,diff(1-Survt$surv)))
tden <- density(x=Survtd[,1], weights=Survtd[,2], bw=100, kernel="epanechnikov")
par(mar=c(2.25,2.25,0,0)+0.1, mgp=c(1.25,0.5,0))
plot(c(hdslcres[,1], hdslcres[,1]), c(hdslcres[,2] - 1.96*hdslcres[,3],
hdslcres[,2] + 1.96*hdslcres[,3]),
type="n", xlab="days", ylab="HDS(t)", cex.lab=.75, cex.axis=.75,
ylim=c(-3,15), xlim=c(0,3650))
polygon(x=c(hdsres[,1], hdsres[312:1,1]), col=rgb(1,0,0,.25), border=NA,
fillOddEven=TRUE, y=c(hdsres[,2]+1.96*hdsres[,3],
(hdsres[,2]-1.96*hdsres[,3])[312:1]))
polygon(x=c(hdslcres[,1], hdslcres[312:1, 1]), col=rgb(0,0,1,.25), border=NA,
fillOddEven=TRUE, y=c(hdslcres[,2] + 1.96*hdslcres[,3],
(hdslcres[,2] - 1.96*hdslcres[,3])[312:1]))
lines(hdsres[,1], hdsres[,2], lwd=2, col="red")
lines(hdslcres[,1], hdslcres[,2], lwd=2, col="blue")
abline(h=1, lty=3)
legend(x=1200, y=14, legend=c("Proportional hazards",
"Local-in-time proportional hazards",
"Time density"), col=c("red", "blue", "gray"),
lwd=2, bty="n", cex=0.75)
with(tden, polygon(c(x, x[length(x):1]),
c(y*3/max(y)-3.5, rep(-3.5, length(x))),
col="gray", border=NA, fillOddEven=TRUE))References
HDS paper, discussions, rejoinder
Liang CJ and Heagerty PJ (2016). A risk-based measure of time-varying prognostic discrimination for survival models. Biometrics. doi:10.1111/biom.12628
Gerds TA and Schumacher M (2016). Discussion of “A risk-based measure of time-varying prognostic discrimination for survival models,” by C. Jason Liang and Patrick J. Heagerty. Biometrics. doi:10.1111/biom.12629
Parast L and Rutter CM (2016). Discussion of “A risk-based measure of time-varying prognostic discrimination for survival models,” by C. Jason Liang and Patrick J. Heagerty. Biometrics. doi:10.1111/biom.12630
Michael H and Tian L (2016). Discussion of “a risk-based measure of time-varying prognostic discrimination for survival models,” by C. Jason Liang and Patrick J. Heagerty. Biometrics. doi:10.1111/biom.12631
Liang CJ and Heagerty PJ (2016). Rejoinder to discussions on: A risk-based measure of time-varying prognostic discrimination for survival models. Biometrics. doi:10.1111/biom.12632
Time-dependent ROC and AUC papers
Saha-Chaudhuri P and Heagerty PJ (2013). Non-parametric estimation of a time-dependent predictive accuracy curve. Biostatistics, 14(1), 42-59. doi: 10.1093/biostatistics/kxs021
Heagerty PJ and Zheng Y (2005). Survival model predictive accuracy and ROC curves. Biometrics, 61: 92–105. doi:10.1111/j.0006-341X.2005.030814.x
Heagerty PJ, Lumley T, and Pepe MS (2000). Time-Dependent ROC curves for censored survival data and a diagnostic marker. Biometrics, 56: 337–344. doi:10.1111/j.0006-341X.2000.00337.x
Partial extensions of DS and IDI to survival data
Uno H, Tian L, Cai T, Kohane IS, and Wei LJ (2013). A unified inference procedure for a class of measures to assess improvement in risk prediction systems with survival data. Statistics in Medicine, 32: 2430–2442. doi:10.1002/sim.5647
Chambless LE, Cummiskey CP, and Cui G (2011). Several methods to assess improvement in risk prediction models: Extension to survival analysis. Statistics in Medicine, 30: 22–38. doi:10.1002/sim.4026
Paper that introduced IDI (and also NRI), and original paper that mentioned DS
Pencina MJ, D' Agostino, RB, D' Agostino, RB, and Vasan RS (2008). Evaluating the added predictive ability of a new marker: From area under the ROC curve to reclassification and beyond. Statistics in Medicine, 27: 157–172. doi:10.1002/sim.2929
Yates JF (1982). External correspondence: Decompositions of the mean probability score. Organizational Behavior and Human Performance, 30: 132–156. doi:10.1016/0030-5073(82)90237-9
Empirical processes result (used for development of HDS estimator and asymptotics)
van der Vaart AW and Wellner JA (2007). Empirical processes indexed by estimated functions. IMS Lecture Notes Monograph Series, 55: 234–252. doi:10.1214/074921707000000382
Thorough study of the Cox model with time-varying coefficients (used for development of HDS estimator and asymptotics)
Tian L, Zucker D, and Wei LJ (2005). On the Cox model with time-varying regression coefficients. Journal of the American Statistical Association, 100(469):172-83. doi:10.1198/016214504000000845
Cai Z and Sun Y (2003). Local linear estimation for time-dependent coefficients in Cox’s regression models. Scandinavian Journal of Statistics, 30: 93–111. doi:10.1111/1467-9469.00320