probcure {npcure}

Compute Nonparametric Estimator of the Conditional Probability of Cure

Description

This function computes the nonparametric estimator of the conditional probability of cure proposed by Xu and Peng (2014).

Usage

probcure(x, t, d, dataset, x0, h, local = TRUE, conflevel = 0L,
bootpars = if (conflevel == 0 && !missing(h)) NULL else 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 d are interpreted. If it is missing, x, t and d must be objects of the workspace.

x0

A numeric vector of covariate values where the estimates of cure probability will be computed.

h

A numeric vector of bandwidths. If it is missing the default is to use the local bootstrap bandwidth computed by the probcurehboot function.

local

A logical value, TRUE by default, specifying whether local or global bandwidths are used.

conflevel

A value controlling whether bootstrap confidence intervals (CI) of the cure probability are to be computed. With the default value, 0L, the CIs are not computed. If a numeric value between 0 and 1 is passed, it specifies the confidence level of the CIs.

bootpars

A list of parameters controlling the bootstrap when computing either the CIs of the cure probability or the bootstrap bandwidth (if h is missing): B, the number of bootstrap resamples, and nnfrac, the fraction of the sample size that determines the order of the nearest neighbor used for choosing a pilot bandwidth. The default is the value returned by the controlpars function called without arguments. If the CIs are not computed and h is not missing the default is NULL.

Details

The function computes the nonparametric estimator of the conditional cure probability q(X = x_0) \equiv 1 - p(X = x_0)= P(Y=\infty | X=x_0) proposed by Xu and Peng (2014), and also studied by López-Cheda et al (2017). It is only available for a continuous covariate X.

Value

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

estimate

The constant string "cure".

local

The value of the local argument.

h

The value of the h argument, unless this is missing, in which case its value is that of the bootstrap bandwidth.

x0

The value of the x0 argument.

q

A list with the estimates of the probability of cure.

conf

A list of components lower and upper giving the lower and the upper limits of the confidence intervals, respectively.

Author(s)

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

References

López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144–165. https://doi.org/10.1016/j.csda.2016.08.002.

Xu, J., Peng, Y. (2014). Nonparametric cure rate estimation with covariates. The Canadian Journal of Statistics 42: 1-17. https://doi.org/10.1002/cjs.11197.

See Also

controlpars, probcurehboot

Examples

## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = 0.5 * (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)

## Covariate values where cure probability is estimated 
x0 <- seq(-1.5, 1.5, by = 0.1) 

## Nonparametric estimates of cure probability at 'x0'...

## ... (a) with global bandwidths 1, 1.5, 2  
q1 <- probcure(x, t, d, data, x0 = x0, h = c(1, 1.5, 2), local = FALSE)

#### Plot predicted cure probabilities at 'x0' for each bandwidth in 'h'
#### (the true cure probability is displayed for reference)
plot(q1$x0, q1$q$h1, type = "l", xlab = "Covariate", ylab =
"Cure probability", ylim = c(0, 1))
lines(q1$x0, q1$q$h1.5, lty = 2)
lines(q1$x0, q1$q$h2, lty = 3)
lines(q1$x0, 1 - exp(2*q1$x0)/(1 + exp(2*q1$x0)), col = 2)
legend("topright", c(paste("Estimate, ", c("h = 1", "h = 1.5",
"h = 2")), "True"), lty = c(1, 2, 3, 1), col = c(1, 1, 1, 2))

## ... (b) with local bandwidths (default)
#### (the vectors passed to 'x0' and 'h' must have the same length)
q2 <- probcure(x, t, d, data, x0 = x0, h = seq(1, 2.5, along = x0)) 

## ... (c) with local bootstrap bandwidths (based on 1999 booostrap
#### resamples). Besides, 95% confidence intervals are computed  and
#### smoothed (with a 15-th order moving average)
set.seed(1) ## Not needed, just for reproducibility.
hb <- probcurehboot(x, t, d, data, x0 = x0, bootpars = controlpars(B =
1999, hsmooth = 15))
q3 <- probcure(x, t, d, data, x0 = x0, h = hb$hsmooth, conflevel = .95,
bootpars = controlpars(B = 1999))


## ... (d) If the bandwidth is not specified, the local bootstrap
#### bandwidth is used (same results as in (c))
set.seed(1) ## Not needed, just for reproducibility.
q4 <- probcure(x, t, d, data, x0 = x0, conflevel = .95, bootpars =
controlpars(B = 1999, hsmooth = 15))

#### Plot of the estimated cure probabilities evaluated at 'x0'
#### (true cure rate displayed as reference)
plot (q4$x0, q4$q, type = "l", ylim = c(0, 1), xlab = "Covariate X",
ylab = "Cure probability")
lines(q4$x0, q4$conf$lower, lty = 2)
lines(q4$x0, q4$conf$upper, lty = 2)
lines(q4$x0, 1-exp(2 * q4$x0)/(1 + exp(2 * q4$x0)), col = 2)
legend("topright", c("Estimate", "95% CI limits", "True"),
lty = c(1, 2, 1), col = c(1, 1, 2))

## Example with the dataset 'bmt' in the 'KMsurv' package
## to study the probability of cure as a function of the age (z1).
data("bmt", package = "KMsurv")
x0 <- seq(quantile(bmt$z1, .05), quantile(bmt$z1, .95), length.out = 100)
q.age <- probcure(z1, t2, d3, bmt, x0 = x0, conflevel = .95, bootpars =
controlpars(B = 1999, hsmooth = 10)) 

## Plot of estimated cure probability and confidence intervals
par(mar = c(5, 4, 4, 5) + .1)
plot(q.age$x0, q.age$q, type = "l", ylim = c(0, 1), xlab =
"Patient age (years)", ylab = "Cure probability")
lines(q.age$x0, q.age$conf$lower, lty = 2)
lines(q.age$x0, q.age$conf$upper, lty = 2)
## The estimated density of age (z1) is added for reference
par(new = TRUE)
d.age <- density(bmt$z1)
plot(d.age, xaxt = "n", yaxt = "n", xlab = "", ylab = "", col = 2,
main = "", zero.line = FALSE)
mtext("Density", side = 4, col = 2, line = 3) 
axis(4, ylim = c(0, max(d.age$y)), col = 2, col.axis = 2)
legend("topright", c("Estimate", "95% CI limits", "Covariate density"),
lty = c(1, 2, 1), col = c(1, 1, 2))