prob_curepk {npcurePK}

Compute Estimator of Cure Probability when Cure Status is Partially Known

Description

This function computes the nonparametric estimator of the cure probability when cure status is partially known proposed by Safari et al (2022).

Usage

    prob_curepk(x, t, d, xinu, dataset, x0, h, local = TRUE,
                bootpars = if (!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.

xinu

If dataset is missing, an integer object giving the values of the cure status indicator. Uncensored and unknown censored observations must be coded as 0, known to be cured censored ones as 1. If dataset is a data frame, it is interpreted as the name of the variable corresponding to the cure status indicator in the data frame.

dataset

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

x0

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

h

A numeric vector of bandwidths.

local

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

bootpars

A list of parameters controlling the bootstrap when computing the bootstrap bandwidths of the cure probability estimator. 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. If h is missing the list of parameters is extended to be the same used for computing the bootstrap bandwidth. The default is the value returned by the controlpars function called without arguments.

Details

Mixture cure model writes the conditional survival function S(t\mid x)=P(Y>t\mid X=x) as

S(t\mid x)=1-p(x)+p(x)S_0(t\mid x)

where 1-p(x)=P(Y=\infty\mid X=x) is the probability of cure.

This function computes the kernel estimator of the probability of cure 1-p(x) in Safari et al (2022). It is based on the previous relationship and the generalized product-limit estimator of the conditional survival function S(t\mid x) in Safari et al (2021), using the Nadaraya-Watson weights, when the cure status is partially known. If there are not individuals known to be cured (xinu=0), then the nonparametric estimator of the cure rate in López-Cheda et al (2017) is computed.

The Epanechnikov kernel is used. If the smoothing parameter h is not provided, then the bootstrap bandwidth selector in Safari et al (2022) is used. The function is available only for one continuous covariate X.

Value

A list of components:

h

The numeric vector of bandwidths used in the estimation. If h argument is missing, the bootstrap bandwidth computed with the control parameters in argument bootpars.

x0

The numeric vector of covariate values where the estimate of the cure probability is computed.

prob_cure

The estimate of the cure probability 1-p(x0) with bandwidth h. It is a vector of the same length as x0.

References

Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical Report. Berkeley, University of California.

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 and Data Analysis 105:144-165. doi:10.1016/j.csda.2016.08.002.

Safari, W. C., López-de-Ullibarri I., Jácome, M. A. (2021). A product-limit estimator of the conditional survival function when cure status is partially known. Biometrical Journal, 63(5): 984-1005. doi:10.1002/bimj.202000173.

Safari, W. C., López-de-Ullibarri I., Jácome, M. A. (2022). Nonparametric kernel estimation of the probability of cure in a mixture cure model when the cure status is partially observed. Statistical Methods in Medical Research, 31(11):2164-2188. doi:10.1177/09622802221115880.

See Also

controlpars

Examples

library(npcurePK)
    
## Data-generating function
## n: sample size
## x_cov_range: range of covariate values
## p_knowncure: probability of known cure
data_gen <- function(n, x_cov_range, p_knowncure) {
    ## probability of being susceptible
    p0 <- function(x) exp(2*x)/(1 + exp(2*x))
    ## covariate values
    x <- runif(n, x_cov_range[1], x_cov_range[2])
    ## censoring times
    c <- rexp(n)
    u <- runif(n)
    v <- runif(n)
    data <- data.frame(matrix(0, nrow = n, ncol = 4L,
                              dimnames = list(NULL, c("x", "t", "d", "xinu"))))
    data[, "x"] <- x
    for (i in 1:n) {
        if (u[i] > p0(x[i])) {
            ## Cured individuals (all of them are censored: Yi = infty,
            ## Ti = Ci, delta = 0, nu = 1)
            data[i, "t"] <- c[i]
            if (v[i] < p_knowncure)
                data[i, "xinu"]  <- 1 
        } else {
            ## Uncured individual (Yi < infty, Ti = min(Yi, Ci),
            ## delta = 1(Yi < Ci), nu = 0)
            ## Uncensored individual (d = 1): cure status is
            ## observed (xi = 1), i.e., xinu = 0
            ## Censored individual (d = 0): cure status is
            ## unknown (xi = 0), i.e., xi.nu = 0
            y <- rweibull(1, shape = 0.5 * (x[i] + 4))
            data[i, "t"]  <- ifelse(v[i] < p_knowncure, y, min(y, c[i]))
            if (data[i, "t"] == y) data[i, "d"] <- 1
        }
    }
    return(data)
}

set.seed(123)
data <- data_gen(n = 100, x_cov_range = c(-2, 2), p_knowncure = 0.8)

## Cure rate estimates for one single covariate value x0 = 0 and using ...
## ... (a) one single fixed bandwidth h = 0.5 
p1 <- prob_curepk(x, t, d, xinu, data, x0 = 0, 
                  h = 0.5, local = TRUE)

## ... (b) a vector of bandwidths h = c(0.25, 0.5, 0.75, 1) 
p2 <- prob_curepk(x, t, d, xinu, data, x0 = c(0, 0, 0, 0), 
                  h = c(0.25, 0.5, 0.75, 1), local = TRUE)

## ... (c) a bootstrap bandwidth (the default when the bandwidths
## argument h is not provided).
## The bootstrap bandwidth is searched in a grid of 10 bandwidths (hl = 10)
## between 0.2 and 2 times the standardized interquartile range of the
## covariate values (hbound = c(0.1, 3)).
(p3 <- prob_curepk(x, t, d, xinu, data, x0 = 0))
## Equivalently
    
    (p3 <- prob_curepk(x, t, d, xinu, data, x0 = 0,
                       bootpars = controlpars(hl = 10, hbound = c(0.1, 3))))


## Cure rate estimates for a vector of 20 covariate values and using ...
x0 = seq(from = min(data$x), to = max(data$x), length.out = 15)
## ... (a) one single fixed bandwidth h = 0.5 
p4 <- prob_curepk(x, t, d, xinu, data, x0 = x0, h = 0.5, local = FALSE)
## Plot predicted cure probabilities for covariate values x0 and bandwidths
## h = 0.5
plot(p4$x0, p4$prob_cure, xlab = "Covariate X", type = "l",
     ylab = "Probability of cure", ylim = c(0, 1))
## The true cure rate is included as reference
lines(p4$x0, 1 - exp(2*x0)/(1 + exp(2*x0)), lwd = 2)

## ... (b) a vector of bandwidths h = c(0.5, 0.75, 1) 
p5 <- prob_curepk(x, t, d, xinu, data, x0 = x0, h = c(0.5, 0.75, 1),
                  local = FALSE)
## Plot predicted cure probabilities for covariate values x0 and bandwidths
## h = 0.5
plot(p5$x0, p5$prob_cure[1, ], xlab = "Covariate X", type = "l",
     ylab = "Probability of cure", ylim = c(0, 1))
## The estimates with bandwidth h = 0.75 and h = 1 are added
lines(p5$x0, p5$prob_cure[2, ])
lines(p5$x0, p5$prob_cure[3, ])
## The true cure rate is included as reference
lines(p5$x0, 1 - exp(2*x0)/(1 + exp(2*x0)), lwd = 2)


    ## ... (c) the bootstrap bandwidth
    (p6 <- prob_curepk(x, t, d, xinu, data, x0 = x0,
                       bootpars = controlpars(b = 50, ncores = 2, seed = 123)))
    ## Plot predicted cure probabilities for covariate values x0 and bootstrap
    ## bandwidths
    plot(p6$x0, p6$prob_cure, xlab = "Covariate X", type = "l",
         ylab = "Probability of cure", ylim = c(0, 1))
    ## The true cure rate is included as reference
    lines(p6$x0, 1 - exp(2*x0)/(1 + exp(2*x0)), lwd = 2)