npcure (version 0.1-4)

testcov: Covariate Significance Test of the Cure Probability

Description

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

Usage

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

Arguments

x

If dataset is missing, an object giving the covariate values, whose type can be numeric, integer, factor or character. Ifdataset 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.

Value

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.

Details

The function computes a statistic, based on the method by Delgado and Gonz<U+00E1>lez-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 L<U+00F3>pez-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:

$$T_{n(x)} = 1/n \sum (\eta_i - \bar{\eta}) I(x_i \leq x)$$.

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.

References

Delgado M. A., Gonz<U+00E1>lez-Manteiga W. (2001). Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics, 29: 1469-1507.

L<U+00F3>pez-Cheda A. (2018). Nonparametric Inference in Mixture Cure Models. PhD dissertation, Universidade da Coru<U+00F1>a. Spain.

See Also

controlpars

Examples

# NOT RUN {
## 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))

# }
# NOT RUN {
## 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))
# }