The qris package implements estimation procedures for a regression model of the quantiles of residual life, remaining lifetime at a specific time, subject to right censoring. For estimation of regression parameters, we consider an induced smoothed method that solves smoothed weighted estimating equations. We also consider the estimation method that solves the original non-smooth weighted estimating equations via a L1 minimization method. To handle data subject to right censoring, inverse probabilities of censoring are incorporated as weights. For standard errors estimation, a robust sandwich-type covariance estimator aided by an efficient resampling method, and a full multiplier bootstrap approach are considered for the induced smoothed estimator (“smooth”) and non-smooth estimator (“nonsmooth”), respectively. Furthermore, an iterative procedure that simultaneously estimates regression parameters and their standard errors is implemented.
You can install the released version of qris from GitHub with:
> ## install.packages("devtools")
> devtools::install_github("Kyuhyun07/qris")
> library(qris)There are two examples to get started. Here is a simulated data.
> data.gen <- function(n) {
+ r0 <- .2 * sqrt(log(2))
+ r1 <- .1 * sqrt(log(2))
+ dat <- data.frame(censoring = runif(n, 0, 24.35),
+ Time0 = sqrt(-log(1 - runif(n))),
+ X = rbinom(n, 1, .5))
+ dat$Time0 <- ifelse(dat$X > 0, dat$Time0 / r1, dat$Time0 / r0)
+ dat$Time <- pmin(dat$Time0, dat$censoring)
+ dat$status <- 1 * (dat$Time0 < dat$censoring)
+ subset(dat, select = c(Time, status, X))
+ }
> library(survival)
> set.seed(1)
> dat <- data.gen(200)
> fm <- Surv(Time, status) ~ X
> fit1 <- qris(fm, data = dat, t0 = 1, Q = 0.5, nB = 200, "smooth", "pmb", c(1,1))
> fit2 <- qris(fm, data = dat, t0 = 1, Q = 0.5, nB = 200, "nonsmooth", "fmb", "rq")
> fit3 <- qris(fm, data = dat, t0 = 1, Q = 0.5, nB = 200, "iterative", "fmb", "rq",
+ control = qris.control(maxit = 20, tol = 1e-3, trace = TRUE))
Step: 1
beta: 1.239291 0.8531057
se: 0.08670517 0.1267661
Step: 2
beta: 1.239761 0.8534891
se: 0.0890501 0.1356901
> summary(fit1)
Call:
qris(formula = fm, data = dat, t0 = 1, Q = 0.5, nB = 200, method = "smooth",
se = "pmb", init = c(1, 1))
qris Estimator
estimate std.Error z.value p.value
(Intercept) 1.2395 0.0889 13.942 < 2.2e-16 ***
X 0.8525 0.1281 6.653 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> summary(fit2)
Call:
qris(formula = fm, data = dat, t0 = 1, Q = 0.5, nB = 200, method = "nonsmooth",
se = "fmb", init = "rq")
qris Estimator
estimate std.Error z.value p.value
(Intercept) 1.2528 0.0927 13.521 < 2.2e-16 ***
X 0.8100 0.1377 5.883 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> summary(fit3)
Call:
qris(formula = fm, data = dat, t0 = 1, Q = 0.5, nB = 200, method = "iterative",
se = "fmb", init = "rq", control = qris.control(maxit = 20,
tol = 0.001, trace = TRUE))
qris Estimator
estimate std.Error z.value p.value
(Intercept) 1.2398 0.0891 13.922 < 2.2e-16 ***
X 0.8535 0.1357 6.290 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> coef(fit1)
(Intercept) X
1.2395248 0.8525343
> summary(fit2)
Call:
qris(formula = fm, data = dat, t0 = 1, Q = 0.5, nB = 200, method = "nonsmooth",
se = "fmb", init = "rq")
qris Estimator
estimate std.Error z.value p.value
(Intercept) 1.2528 0.0927 13.521 < 2.2e-16 ***
X 0.8100 0.1377 5.883 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> vcov(fit3)
(Intercept) X
(Intercept) 0.007929919 -0.008634304
X -0.008634304 0.018411815
> plot(fit1, Qs = 4:6 / 10)
Here is a real data application.
> ## Load "retinopathy" data from R survival package
> library(survival)
> ## Real data application
> data(cancer, package = "survival")
> lung2 <- subset(lung, select = c(time, status, age, sex))
> ## tidy up the data
> lung2$status <- lung2$status - 1
> lung2$sex <- lung2$sex - 1
> fm <- Surv(time, status) ~ age + sex
> fit1 <- qris(fm, data = lung2, t0 = 0, Q = 0.5, nB = 200, "iterative", "pmb", "rq")
> fit2 <- qris(fm, data = lung2, t0 = 30, Q = 0.5, nB = 200, "nonsmooth", "fmb", c(1, 0, 1))
> fit3 <- qris(fm, data = lung2, t0 = 100, Q = 0.5, nB = 200,"smooth", "pmb", "rq")
> summary(fit1)
Call:
qris(formula = fm, data = lung2, t0 = 0, Q = 0.5, nB = 200, method = "iterative",
se = "pmb", init = "rq")
qris Estimator
estimate std.Error z.value p.value
(Intercept) 6.1622 0.4830 12.758 <2e-16 ***
age -0.0090 0.0076 -1.190 0.2340
sex 0.4680 0.1312 3.567 0.0004 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> summary(fit2)
Call:
qris(formula = fm, data = lung2, t0 = 30, Q = 0.5, nB = 200,
method = "nonsmooth", se = "fmb", init = c(1, 0, 1))
qris Estimator
estimate std.Error z.value p.value
(Intercept) 5.6362 0.9355 6.025 <2e-16 ***
age -0.0015 0.0142 -0.103 0.9182
sex 0.4489 0.1979 2.268 0.0233 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> summary(fit3)
Call:
qris(formula = fm, data = lung2, t0 = 100, Q = 0.5, nB = 200,
method = "smooth", se = "pmb", init = "rq")
qris Estimator
estimate std.Error z.value p.value
(Intercept) 8.6285 2.5117 3.435 0.0006 ***
age -0.0601 0.0395 -1.522 0.1279
sex 1.8121 0.6331 2.862 0.0042 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> plot(fit2, Qs = 4:6 / 10)
Kim, K., and Kang, S., (2022). “Smoothed quantile regression for censored residual life”. Upcoming
Chiou, S., Kang, S., and Yan, J. (2014). “Fitting accelerated failure time model in routine survival analysis with R package aftgee”. Journal of Statistical Software, 61(11): 1–23.
Li, R., Huang, X., & Cortes, J. (2016). “Quantile residual life regression with longitudinal biomarker measurements for dynamic prediction”. Journal of the Royal Statistical Society. Series C (Applied Statistics), 755-773.
Jung, S. H., Jeong, J. H., & Bandos, H. (2009). “Regression on quantile residual life”. Biometrics, 65(4), 1203-1212.