stRoc {smoothROCtime}

Smooth Time-dependent ROC curve estimations

Description

Provides smooth estimations of Cumulative/Dynamic (C/D) and Incident/Dynamic (I/D) ROC curves in presence of rigth censorship and the corresponding Areas Under the Curves (AUCs), at a single point of time or a vector of points.

• The function computes two different procedures to obtain smooth estimations of the C/D ROC curve. Both are based on the kernel density estimation of the joint distribution function of the marker and time-to-event variables, provided by funcen function. The first method, to which we will refere as smooth method, is carried out according to the methodology proposed in https://doi.org/10.1177/0962280217740786. The second one uses this estimation of the joint density function of the variables marker and time-to-event for computing the weights or probabilities allocated to censored observations (undefined individuals) in https://doi.org/10.1080/00949655.2016.1175442 and https://doi.org/10.1177/0962280216680239. It will be referred as p-kernel method.

• In case of the I/D ROC curve, a smooth approximation procedure (smooth method) is computed based as well on the kernel density estimation of the joint distribution function of the marker and time-to-event variables proposed in https://doi.org/10.1177/0962280217740786

Usage

stRoc(data, t, H, bw, adj, tcr, meth, ...)

Arguments

data

matrix of data values with three columns: time-to-event, censoring status (0=censored/1=uncensored) and marker.

t

point of time or vector of points where the time-dependent ROC curve is estimated.

H

2x2 bandwidth matrix.

bw

procedure for computing the bandwidth matrix. Most of the methods included at the kde function can be used: Hpi, Hpi.diag, Hlscv, Hlscv.diag, Hbcv, Hbcv.diag, Hscv, Hscv.diag, Hucv and Hucv.diag. Other considered methods are naive.pdf (diag(N^-1/5, N^-1/5)^2) and naive.cdf (diag(N^-1/3, N^-1/3)^2), where N is the sample size.

adj

adjusment parameter for calculating the bandwidth matrix. Default value 1.

tcr

type of time-dependent ROC curve estimation that will be estimated:

• “C” for Cumulative/Dynamic,

• “I” for Incident/Dynamic,

• “B” for Both time-dependent ROC curve estimations.

meth

method for computing the estimation of the C/D ROC curve.The suitable values are:

• “1” for the smooth method,

• “2” for the p-kernel method.

As default value the smooth method is taken.

...

kde function arguments can be used for estimating the bivariate kernel density function.

Details

Function funcen is called from each execution of function stRoc, in order to compute the kernel density estimation of the joint distribution of the (Marker, Time-to-event) variable, therefore, the input parameters in funcen are input parameters as well in stRoc and the same considerations apply.

The matrix of bandwidths can be defined by using H=matrix() or automatically selected by the method indicated in bw.

Given the matrix of bandwidths, H, the argument adj modifies it and the final matrix is adj^2 H.

If H is missing, the naive.pdf method is used.

If tcr is missing the C/D ROC curve estimation will be computed with the method indicated in meth.

If no value has been placed in meth the smooth method will be used. The I/D ROC curve estimation will be always computed with the smooth method.

Value

An object of class sROCt is returned. It is a list with the following values:

th

considered thresholds for the marker.

FP

false-positive rate calculated at each point in th.

TP

true-positive rate estimated at each point in th.

p

points where the time-dependent ROC curve is evaluated.

R

time-dependent ROC curve values computed at p.

t

time/s at which each time-dependent ROC curve estimation is computed. Each point ot time will appear as many times as the length of the vector of points p.

auc

area under the corresponding time-dependent ROC curve estimation. As in the previous case, each value appears as many times as the length of the vector of points p.

tcr

type of time-dependent ROC curve estimation computed,

• “C” - Cumulative/Dynamic.

• “I” - Incident/Dynamic.

For each computed time-dependent ROC curve estimation this value is repeated as many times as the length of p.

Pi

probabilities calculated for the individuals in the sample if the p-kernel method has been used for the estimation of the C/D ROC curve. This element is a matrix with the following columns:

• time - single point of time at which the estimation each the C/D ROC curve has been computed.

• obvt - observed times for the individuals in the sample.

• p - estimations of the probabilities computed and allocated to each subject.

References

P. Martinez-Camblor and J. C. Pardo-Fernandez. Smooth time-dependent receiver operating characteristic curve estimators. Statistical Methods in Medical Research, 27(3):651-674, 2018.https://doi.org/10.1177/0962280217740786.

P. Martinez-Camblor, G. F-Bay?n, and S. P?rez-Fern?ndez. Cumulative/dynamic ROC curve estimation. JOURNAL of Statistical Computation and Simulation, 86(17):3582-3594, 2016. https://doi.org/10.1080/00949655.2016.1175442.

L. Li, T. Green, and B. Hu. A simple method to estimate the time-dependent receiver operating characteristic curve and the area under the curve with right censored data. Statistical Methods in Medical Research, 27(8), 2016. https://doi.org/10.1177/0962280216680239.

T. Duong. Bandwidth matrices for multivariate kernel density estimation. Ph.D. Thesis, University of Western, Australia, 2004. http://www.mvstat.net/tduong.

Examples

library(smoothROCtime)
require(KMsurv)

data(kidtran)

# Preparing data: a logarithmic transformation of the time-to-event variable is made
  DT <- cbind(log(kidtran$time),kidtran$delta,kidtran$age)

# Point of Time
  t5 <- log(5*365.25) # five years in logarithm scale

# Cumulative/dynamic ROC curve estimation
  rcd <- stRoc(data=DT, t=t5, bw="Hpi", tcr="C", meth=2)

# Plot graphic
  plot(rcd$p, rcd$ROC, type="l", lwd=5, main="C/D ROC",xlab="FPR",ylab="TPR")
  lines(c(0,1),c(0,1),lty=2,col="gray")