# Compute Product-Limit Estimator of Conditional Survival Function when Cure Status is Partially Known This function computes the nonparametric estimator of the conditional survival function when cure status is partially known proposed by Safari **et al** (2021). ```r prodlim_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 conditional survival function 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 product-limit 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. ## Returns 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 conditional survival function is computed. - **t**: The observed time values, where the conditional survival function is estimated. - **surv**: Estimates of the survival function for each one of the covariate values specified by the `x0` argument and the bandwidths in `h`. It is a matrix of dimension $(n, length(x0))$ if local bandwidths or bootstrap bandwidths are used, or an array of dimension $(n, length(x0), length(h))$ for global bandwidths instead. ## Details Mixture cure model writes the conditional survival function $S(t|x)=P(Y>t|X=x)$ as $S(t|x)=1-p(x)+p(x)S_0(t|x)$. This function computes the generalized product-limit estimator of the conditional survival function $S(t|x)$, using the Nadaraya-Watson weights, when the cure status is partially known, introduced in Safari **et al** (2021). If there are not individuals known to be cured (`xinu=0`), then the usual generalized product-limit estimator in Beran (1981) is computed. The Epanechnikov kernel is used. If the smoothing parameter `h` is not provided, then the bootstrap bandwidth selector in Safari **et al** (2021) is used. The function is available only for one continuous covariate $X$. ## Examples ```r 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) ## Covariate values where the conditional survival function is estimated x0 <- c(0, 0.5) ## Survival estimates for covariate values x0 = c(0, 0.5) ## ... (a) with 3 global bandwidths (0.5, 1, 1.25) ## The survival function S(t|x) is estimated for each value of x0 with the three ## bandwidths (local == FALSE). ## The estimates are saved in an array (n x length(x0) x length(h)) S1 <- prodlim_curepk(x, t, d, xinu, data, x0 = x0, h = c(0.5, 1, 1.25), local = FALSE) ## Plot predicted survival curve for covariate value x0 = 0.5 and bandwidth ## h = 0.5 x0 <- 0.5 plot(S1$t, S1$surv[, 2, 1], type = "s", xlab = "Time", ylab = "Survival probability", ylim = c(0, 1)) ## The true survival curve is included as reference lines(S1$t, 1 - exp(2*x0)/(1 + exp(2*x0)) + exp(2*x0)/(1 + exp(2*x0))* (1 - pweibull(S1$t, shape = 0.5 * (x0 + 4))), lwd = 2) ## Plot predicted survival curve for covariate value x0 = 0.5 and all ## bandwidths plot(S1$t, S1$surv[, 2, 1], type = "s", xlab = "Time", ylab = "Survival probability", ylim = c(0, 1)) lines(S1$t, S1$surv[, 2, 2], type = "s", lwd = 2) lines(S1$t, S1$surv[, 2, 3], type = "s", lwd = 3) # The true survival curve is included as reference lines(S1$t, 1 - exp(2*x0)/(1 + exp(2*x0)) + exp(2*x0)/(1 + exp(2*x0))* (1 - pweibull(S1$t, shape = 0.5 * (x0 + 4))), lwd = 2) ## ... (b) with local bandwidths h = (3, 1) ## The survival function S(t|x) is estimated for each value of x0 with the ## corresponding assigned bandwidth (local == TRUE). ## Note that the length of the vector x0 and the bandwidth h must be the same. ## The estimates are saved in a matrix of dimension (n, length(x0)) x0 <- c(0, 0.5) h <- c(3, 1) S3 <- prodlim_curepk(x, t, d, xinu, data, x0 = x0, h = h, local = TRUE) ## Plot predicted survival curve for covariate value x = 0 and its bandwidth ## (h = 3) plot(S3$t, S3$surv[, 1], type = "s", xlab = "Time", ylab = "Survival probability", ylim = c(0, 1)) ## The true survival curve is included as reference x0 <- 0 lines(S3$t, 1 - exp(2*x0)/(1 + exp(2*x0)) + exp(2*x0)/(1 + exp(2*x0))* (1 - pweibull(S3$t, shape = 0.5 * (x0 + 4))), lwd = 2) ## ... (c) with the bootstrap bandwidth selector in x0 = 0 (the default ## when the bandwidth 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, 2)). x0 <- 0 (S4 <- prodlim_curepk(x, t, d, xinu, data, x0 = x0)) ## Equivalently (S4 <- prodlim_curepk(x, t, d, xinu, data, x0 = x0, bootpars = controlpars(hl = 10, hbound = c(0.1, 2)))) ## Plot predicted survival curve for covariate value x = 0 and the bootstrap ## bandwidth plot(S4$t, S4$surv[, 1], type = "s", xlab = "Time", ylab = "Survival probability", ylim = c(0, 1)) ## The true survival curve is included as reference lines(S4$t, 1 - exp(2*x0)/(1 + exp(2*x0)) + exp(2*x0)/(1 + exp(2*x0))* (1 - pweibull(S4$t, shape = 0.5 * (x0 + 4))), lwd = 2) ## ... (d) with parallel computation (The bootstrap bandwidth is searched with ## b = 100 bootstrap resamples and 2 cores) library(doParallel) (S5 <- prodlim_curepk(x, t, d, xinu, data, x0 = x0, bootpars = controlpars(b = 100, ncores = 2))) ## Plot predicted survival curve for covariate value x = 0 and the bootstrap ## bandwidth plot(S5$t, S5$surv[, 1], type = "s", xlab = "Time", ylab = "Survival probability", ylim = c(0, 1)) ## The true survival curve is included as reference lines(S5$t, 1 - exp(2*x0)/(1 + exp(2*x0)) + exp(2*x0)/(1 + exp(2*x0))* (1 - pweibull(S5$t, shape = 0.5 * (x0 + 4))), lwd = 2) ``` ## References Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical Report. Berkeley, University of California. 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. tools:::Rd_expr_doi("10.1002/bimj.202000173") . ## See Also `controlpars`

prodlim_curepk function