testcov {MDCcure}R Documentation

Covariate Hypothesis Test of the Cure Probability based on Martingale Difference Correlation

Description

Performs nonparametric hypothesis tests to evaluate the association between a covariate and the cure probability in mixture cure models. Several test statistics are supported, including martingale difference correlation (MDC)-based tests and an alternative GOFT test.

Usage

testcov(
  x,
  time,
  delta,
  h = NULL,
  method = "FMDCU",
  P = 999,
  parallel = TRUE,
  ncores = -1
)

Arguments

x

A numeric vector representing the covariate of interest.

time

A numeric vector of observed survival times.

delta

A binary vector indicating censoring status: 1 for event and 0 for censored.

h

Bandwidth parameter for kernel smoothing. Either a positive numeric value, NULL, or the character string "bootstrap". If NULL, an optimal bandwidth is selected automatically. If "bootstrap", the bandwidth is selected using the bootstrap method proposed by López-Cheda et al. (2016).

method

Character string specifying the test to perform. One of:

  • "MDCU": Martingale Difference Correlation with U-centering.

  • "MDCV": Martingale Difference Correlation with double-centering.

  • "FMDCU": Fast approximation of MDC with U-centering.

  • "GOFT": Goodness-of-fit test for the cure model.

  • "All": All of the above tests.

Default is "FMDCU".

P

Integer. Number of permutations or bootstrap replications used to compute the null distribution of the test statistic. For methods "MDCU" or "MDCV", this is the number of permutations. For the "GOFT" method, it is the number of bootstrap replications. Defaults to 999.

parallel

Logical. If TRUE, parallel computing is used to speed up computations. Default is TRUE.

ncores

Integer. Number of cores to use for parallel computing. If NULL, it defaults to one less than the number of available cores.

Details

The function computes a statistic, based on the methodology proposed by Monroy-Castillo et al., to test whether a covariate \boldsymbol{X} has an effect on the cure probability.

\mathcal{H}_0 : \mathbb{E}(\nu | \boldsymbol{X}) \equiv 1 - p \quad \text{a.s.} \quad \text{vs} \quad \mathcal{H}_1 : \mathbb{E}(\nu | \boldsymbol{X}) \not\equiv 1 - p \quad \text{a.s.}

The main problem is that the response variable (cure indicator \nu) is partially observed due to censoring. This is addressed by estimating the cure indicator using the methodology of Amico et al. (2021). We define \tau = \sup_x \tau(x), with \tau(x) = \inf\{t: S_0(t|x) = 0\}. We assume \tau < \infty and that follow-up is long enough so that \tau < \tau_{G(x)} for all x. Therefore, individuals with censored observed times greater than \tau are considered cured (\nu = 1).

Four tests are proposed: three are based on the martingale difference correlation (MDC). For the MDCU and MDCV tests, the null distribution is approximated via a permutation procedure. To provide a faster alternative, a chi-squared approximation is implemented for the MDCU test statistic (FMDCU). Additionally, a modified version of the goodness-of-fit test proposed by Müller and Van Keilegom (2019) is included (GOFT). The test statistic is given by:

\widehat{\mathcal{T}}_n = nh^{1/2}\frac{1}{n}\sum_{i = 1}^{n}\left\{\hat{p}_h(X_i) - \hat{p}\right\}^2,

where \hat{p}_h(X_i) denotes the nonparametric estimator of the cure probability under the alternative hypothesis, and \hat{p} denotes the nonparametric estimator of the cure probability under the null hypothesis. The approximation of the critical value for the test is done using the bootstrap procedure given in Section 3 of Müller and Van Keilegom (2019).

Value

A list containing:

References

Amico, M, Van Keilegom, I. & Han, B. (2021). Assessing cure status prediction from survival data using receiver operating characteristic curves. Biometrika, 108(3), 727–740. doi:10.1093/biomet/asaa080

López-Cheda, A., Cao, R., Jácome, M. A., & Van Keilegom, I. (2016). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 100, 490–502. doi:10.1016/j.csda.2016.04.006

Müller, U.U, & Van Keilegom, I. (2019). Goodness-of-fit tests for the cure rate in a mixture cure model. Biometrika, 106, 211-227. doi:10.1093/biomet/asy058

Shao, X., & Zhang, J. (2014). Martingale difference correlation and its use in high-dimensional variable screening. Journal of the American Statistical Association, 105, 144-165. doi:10.1080/01621459.2014.887012

See Also

mdc, mdd, mdc_test, testcov2

Examples


## 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)

testcov(x, t, d)


[Package MDCcure version 0.1.0 Index]