latencyhboot {npcure}

Compute the Bootstrap Bandwidth for the Nonparametric Estimator of the Latency

Description

This function computes the bootstrap bandwidth for the nonparametric estimator of the conditional latency function.

Usage

latencyhboot(x, t, d, dataset, x0, bootpars = controlpars())

Arguments

x

If dataset is missing, a numeric object giving the covariate values. If dataset is a data frame, it is interpreted as the name of the variable corresponding to the covariate in the data frame.

t

If dataset is missing, a numeric object giving the observed times. If dataset is a data frame, it is interpreted as the name of the variable corresponding to the observed times in the data frame.

d

If dataset is missing, an integer object giving the values of the uncensoring indicator. Censored observations must be coded as 0, uncensored ones as 1. If dataset is a data frame, it is interpreted as the name of the variable corresponding to the uncensoring indicator in the data frame.

dataset

An optional data frame in which the variables named in x, t and indicator are interpreted. If it is missing, x, t and indicator must be objects of the workspace.

x0

A numeric vector of covariate values where the local bootstrap bandwidth will be computed.

bootpars

A list of parameters controlling the process of bandwidth selection. The default is the value returned by the controlpars function called without arguments.

Details

The function computes the bootstrap bandwidth selector for the nonparametric estimator of the conditional latency function at the covariate values given by x0. The bootstrap bandwidth is the minimizer of a bootstrap version of the Mean Integrated Squared Error (MISE) of the latency estimator, which is approximated by Monte Carlo by simulating a large number of bootstrap resamples, B. For each value of x0, the bootstrap MISE is the bootstrap expectation of the integrated difference between the value of the latency estimator computed with the bootstrap sample in a grid of bandwidths and its value computed with the original sample and a pilot bandwidth. The bootstrap resamples are generated by using the simple weighted bootstrap resampling method, fixing the covariate. This method is equivalent to the simple weighted bootstrap of Li and Datta (2001). All the parameters typically involved in the bootstrap bandwidth selection process (number of bootstrap resamples, grid of bandwidths, pilot bandwidth, and right boundary of the integration interval for computing the MISE) are typically set through the controlpars function, whose output is passed to the bootpars argument. Also, the bootstrap bandwidths can be smoothed, and, if so, the smoothed bandwidths are returned as a separate component of the output. See the help of controlpars for details.

Value

An object of S3 class 'npcure'. Formally, a list of components:

type

The constant character string c("Bootstrap bandwidth", "latency").

x0

Grid of covariate values.

h

Selected local bootstrap bandwidths.

hsmooth

Smoothed selected local bootstrap bandwidths (optional)

hgrid

Grid of bandwidths used (optional).

Author(s)

Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]

References

Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53: 708–729. https://doi.org/10.1023/A:1014644700806.

López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. TEST, 26: 353–376. https://doi.org/10.1007/s11749-016-0515-1.

See Also

controlpars, latency

Examples

## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)

## A vector of covariate values
vecx0 <- seq(-1.5, 1.5, by = .1)

## Computation of bootstrap local bandwidths at the values of 'vecx0'...
#### ... with the default control parameters
hb1 <- latencyhboot(x, t, d, data, x0 = vecx0)

#### ... changing the default 'bootpars' with 'controlpars()':
#### (a) 'B = 1999' (1999 bootstrap resamples are generated),
#### (b) 'hbound = c(.2, 4)' and 'hl = 50' (a grid of 50 bandwidths
#### between 0.2 and 4 times the standardized interquartile range of
#### the covariate values is built), and
#### (c) 'hsave = TRUE' (the grid bandwidths are saved), and

hb2 <- latencyhboot(x, t, d, data, x0 = vecx0, bootpars =
controlpars(B = 1999, hbound = c(.2, 4), hl = 50, hsave = TRUE))

## Estimates of the conditional latency at the covariate value x0 = 0
## with the selected bootstrap bandwidths
S1 <- latency(x, t, d, data, x0 = 0, h = hb1$h[hb1$x0 == 0])
S2 <- latency(x, t, d, data, x0 = 0, h = hb2$h[hb2$x0 == 0])

## A plot comparing the estimates with bootstrap bandwidths obtained
## with default and non-default 'bootpars'
plot(S1$testim, S1$S$x0, type = "s", xlab = "Time", ylab = "Latency",
ylim = c(0, 1))
lines(S2$testim, S2$S$x0, type = "s", lty = 2)
lines(S1$testim, pweibull(S1$testim, shape = .5*(0 + 4), lower.tail =
FALSE), col = 2)
legend("topright", c("Estimate with 'hb1'", "Estimate with 'hb2'",
"True"), lty = c(1, 2, 1), col = c(1, 1, 2))

## Example with the dataset 'bmt' of the 'KMsurv' package
## to study the survival of the uncured patients aged 25 and 40
data("bmt", package = "KMsurv")
x0 <- c(25, 40)
hb <- latencyhboot(z1, t2, d3, bmt, x0 = x0, bootpars = controlpars(B =
1999, hbound = c(.2, 4), hl = 150, hsave = TRUE))
S0 <- latency(z1, t2, d3, bmt, x0 = x0, hb$h, conflevel = .95)
## Plot of predicted latency curves and confidence bands
plot(S0$testim, S0$S$x25, type = "s", xlab = "Time (days)",
ylab = "Survival", ylim = c(0,1))
lines(S0$testim, S0$conf$x25$lower, type = "s", lty = 2)
lines(S0$testim, S0$conf$x25$upper, type = "s", lty = 2)
lines(S0$testim, S0$S$x40, type = "s", lty = 1, col = 2)
lines(S0$testim, S0$conf$x40$lower, type = "s", lty = 2, col = 2)
lines(S0$testim, S0$conf$x40$upper, type = "s", lty = 2, col = 2)
legend("topright", c("Age 25: Estimate", "Age 25: 95% CI limits",
"Age 40: Estimate", "Age 40: 95% CI limits"), lty = 1:2,
col = c(1, 1, 2, 2))