testcov: Covariate Significance Test of the Cure Probability
In npcure: Nonparametric Estimation in Mixture Cure Models

Description Usage Arguments Details Value Author(s) References See Also Examples

This function carries out a significance test of covariate effect on the probability of cure.

1	testcov(x, t, d, dataset, bootpars = controlpars())

`x`	If `dataset` is missing, an object giving the covariate values, whose type can be numeric, integer, factor or character. 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.
`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.
`bootpars`	A list of parameters controlling the test. Currently, the only accepted component is `B`, the number of bootstrap resamples. The default is the value returned by the `controlpars` function called without arguments.

The function computes a statistic, based on the method by Delgado and González-Manteiga (2004), to test whether a covariate X has an effect on the cure probability (H_1: cure probability = q(x)) or not (H_0: cure probability = q). Since the cure rate can be written as the regression function E(nu | X = x) = q(x), where nu is the cure indicator, the procedure is carried out as a significance test in nonparametric regression. The challenge of the test is that the cure indicator nu is only partially observed due to censoring: for the uncensored observations nu = 0, but it is unknown if a censored individual will be eventually cured or not (nu unknown). The approach consists in expressing the cure rate as a regression function with another response, not observable but estimable, say eta. The estimated values of eta depend on suitable estimates of the conditional distribution of the censoring variable and tau(x), an unknown time beyond which a subject could be considered as cured; see López-Cheda (2018). For the computation of the values of eta, the censoring distribution is estimated unconditionally using the Kaplan-Meier product-limit estimator. The time tau is estimated by the largest uncensored time.

The test statistic is a weighted mean of the difference between the observations of eta_i and the values of the conditional mean of eta under the null hypothesis:

T_{n(x) = 1/n sum(eta_i - mean(eta)) I(x_i <= x)}

For a qualitative covariate there is no natural way to order the values x_i. In principle, this makes impossible to compute the indicator function in the test statistic. The problem is solved by considering all the possible combinations of the covariate values, and by computing the test statistic for each 'ordered' combination. T_{n(x)} is taken as the largest value of these test statistics.

The distribution of the test statistic under the null hypothesis is approximated by bootstrap, using independent naive resampling.

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

`type`	The constant character string c("test", "Covariate").
`x`	The name of the covariate.
`CM`	The result of the Cramer-von Mises test: a list with components `statistic`, the test statistic, and `pvalue`, the p-value.
`KS`	The result of the Kolmogorov-Smirnov test: a list with components `statistic`, the test statistic, and `pvalue`, the p-value.

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

Delgado M. A., González-Manteiga W. (2001). Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics, 29: 1469-1507.

López-Cheda A. (2018). Nonparametric Inference in Mixture Cure Models. PhD dissertation, Universidade da Coruña. Spain.

controlpars

## 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)

## Test of the significance of the covariate 'x'
testcov(x, t, d, data)

## Test carried out with 1999 bootstrap resamples (the default is 999)
testcov(x, t, d, data, bootpars = controlpars(B = 1999))


## How to apply the test repeatedly when there is more than one
## covariate... 
## ... 'y' is another continuous covariate of the data frame 'data'
data$y <- runif(n, -1, 1)
namecovar <- c("x", "y")
## ... testcov is called from a 'for' loop
for (i in 1:length(namecovar)) {
   result <- testcov(data[, namecovar[i]], data$t, data$d)
   print(result)
}

## In the previous example, testcov() was called without using the
## argument 'dataset'. To use it, the 'for' must be avoided
testcov(x, t, d, data)
testcov(y, t, d, data)

## Non-numeric covariates can also be tested...
## ... 'z' is a nominal covariate of the data frame 'data'
data$z <- rep(factor(letters[1:5]), each = n/5)
testcov(z, t, d, data)

## Example with the dataset 'bmt' in the 'KMsurv' package
## to study the effect on the probability of cure of...
## ... (a) a continuous covariate (z1 = age of the patient)
data("bmt", package = "KMsurv")
set.seed(1) ## Not needed, just for reproducibility.
testcov(z1, t2, d3, bmt, bootpars = controlpars(B = 4999))

## ... (b) a qualitative covariate (z3 = patient gender)
set.seed(1) ## Not needed, just for reproducibility.
testcov(z3, t2, d3, bmt, bootpars = controlpars(B = 4999))