Goodness-of-fit tests for the cure rate in a mixture cure model

Description

The aim of this function is to test whether the cure rate p, as a function of the covariates, satisfies a certain parametric model.

Usage

goft(
  x,
  time,
  delta,
  model = c("logit", "probit", "cloglog"),
  theta0 = NULL,
  nsimb = 499,
  h = NULL
)

Arguments

Details

We want to test wether the cure rate p, as a function of covariates, satisfies a certain parametric model, such as, logistic, probit or cloglog model. The hypothesis are:

\mathcal{H}_0 : p = p_{\theta} \quad \text{for some} \quad \theta \in \Theta \quad \text{vs} \quad \mathcal{H}_1 : p \neq p_{\theta} \quad \text{for all} \quad \theta \in \Theta,

where \Theta is a finite-dimensional parameter space and p_{\theta} is a known function up to the parameter vector \theta.

The test statistic is based on a weighted L_2 distance between a nonparametric estimator \hat{p}(x) and a parametric estimator p_{\hat{\theta}}(x) under \mathcal{H}_0, as proposed by Müller and Van Keilegom (2019):

\mathcal{T}_n = n h^{1/2} \int \left(\hat{p}(x) - p_{\hat{\theta}}(x)\right)^2 \pi(x) dx,

where \pi(x) is a known weighting function, often chosen as the covariate density f(x).

A practical empirical version of the statistic is given by:

\tilde{\mathcal{T}}_n = n h^{1/2} \frac{1}{n} \sum_{i = 1}^n \left(\hat{p}(x_i) - p_{\hat{\theta}}(x_i)\right)^2,

where the integral is replaced by a sample average.

Value

A list with the following components:

statistic

Numeric value of the test statistic.

p.value

Numeric value of the bootstrap p-value for testing the null hypothesis.

bandwidth

The bandwidth used.

References

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

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)

goft(x, t, d, model = "logit")

Martingale Difference Correlation (MDC)

Description

mdc computes the squared martingale difference correlation between a response variable Y and explanatory variable(s) X, measuring conditional mean dependence. X can be either univariate or multivariate.

Usage

mdc(X, Y, center = "U")

Arguments

Value

Returns the squared martingale difference correlation of Y given X.

References

Shao, X., and Zhang, J. (2014). Martingale difference correlation and its use in high-dimensional variable screening. Journal of the American Statistical Association, 109(507), 1302-1318. doi:10.1080/01621459.2014.887012.

See Also

mdd, mdc_test

Examples

# Generate example data
set.seed(123)
n <- 50
x <- matrix(rnorm(n * 5), nrow = n)  # multivariate data with 5 variables
y <- rbinom(n, 1, 0.5)               # binary covariate

# Compute MDC with U-centering
mdc(x, y, center = "U")

# Compute MDC with double-centering
mdc(x, y, center = "D")

MDC-Based Dependence Tests Between Multivariate Data and a Covariate

Description

Computes dependence between a multivariate dataset x and a univariate covariate y using different variants of the MDC (martingale difference correlation) test.

Usage

mdc_test(x, y, method, permutations = 999, parallel = TRUE, ncores = -1)

Arguments

Value

A list containing the test results and p-values.

References

Shao, X., and Zhang, J. (2014). Martingale difference correlation...

Examples

set.seed(123)
x <- matrix(rnorm(50 * 5), nrow = 50)
y <- rbinom(50, 1, 0.5)
mdc_test(x, y, method = "FMDCU")

Martingale Difference Divergence (MDD)

Description

mdd computes the squared martingale difference divergence (MDD) between response variable(s) Y and explanatory variable(s) X, measuring conditional mean dependence.

Usage

mdd(X, Y, center = "U")

Arguments

Value

Returns the squared Martingale Difference Divergence of Y given X.

References

Shao, X., and Zhang, J. (2014). Martingale difference correlation and its use in high-dimensional variable screening. Journal of the American Statistical Association, 109(507), 1302-1318. doi:10.1080/01621459.2014.887012.

Examples

# Generate example data
set.seed(123)
n <- 50
x <- matrix(rnorm(n * 5), nrow = n)  # multivariate explanatory variables
y_vec <- rbinom(n, 1, 0.5)           # univariate response
y_mat <- matrix(rnorm(n * 2), nrow = n)  # multivariate response

# Compute MDD with vector Y and U-centering
mdd(x, y_vec, center = "U")

# Compute MDD with matrix Y and double-centering
mdd(x, y_mat, center = "D")

Plot Cure Probability: A Comparison of Nonparametric and Parametric Estimation

Description

This function generates a plot comparing nonparametric and parametric estimations of cure probability in a univariate setting. The nonparametric estimate is displayed with 95% confidence bands, while the parametric estimate is based on a logit, probit or complementary log-log link. An optional covariate density curve can be added as a secondary axis.

Usage

plotCure(
  x,
  time,
  delta,
  main.title = NULL,
  title.x = NULL,
  model = "logit",
  theta = NULL,
  legend.pos = "bottom",
  density = TRUE,
  hsmooth = 10,
  npoints = 100
)

Arguments

Details

The function estimates the cure probability nonparametrically using the probcure function and overlays it with a parametric estimate obtained from a logistic regression model. Confidence intervals (95%) are included for the nonparametric estimate. Optionally, the density of the covariate can be shown as a shaded area with a secondary y-axis.

Value

A ggplot object representing the cure probability plot.

See Also

probcure

Partial Martingale Difference Correlation (pMDC)

Description

pmdd measures conditional mean dependence of Y given X, adjusting for the dependence on Z.

Usage

pmdc(X, Y, Z)

Arguments

Value

Returns the squared partial martingale difference correlation of Y given X, adjusting for the dependence on Z.

References

Park, T., Shao, X., and Yao, S. (2015). Partial martingale difference correlation. Electronic Journal of Statistics, 9(1), 1492-1517. doi:10.1214/15-EJS1047.

Examples

# Generate example data
set.seed(123)
n <- 50
x <- matrix(rnorm(n * 5), nrow = n)  # explanatory variables
y <- matrix(rnorm(n), nrow = n)      # response variable
z <- matrix(rnorm(n * 2), nrow = n)  # conditioning variables

# Compute partial MDD
pmdd(x, y, z)

Partial Martingale Difference Divergence (pMDD)

Description

pmdd measures conditional mean dependence of Y given X, adjusting for the dependence on Z.

Usage

pmdd(X, Y, Z)

Arguments

Value

Returns the squared partial martingale difference divergence of Y given X, adjusting for the dependence on Z.

References

Park, T., Shao, X., and Yao, S. (2015). Partial martingale difference correlation. Electronic Journal of Statistics, 9(1), 1492-1517. doi:10.1214/15-EJS1047.

Examples

# Generate example data
set.seed(123)
n <- 50
x <- matrix(rnorm(n * 5), nrow = n)  # explanatory variables
y <- matrix(rnorm(n), nrow = n)      # response variable
z <- matrix(rnorm(n * 2), nrow = n)  # conditioning variables

# Compute partial MDD
pmdd(x, y, z)

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

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:

• test_results: A list with the results (e.g., test statistics and p-values) of the selected test(s).

• nu_hat: A numeric vector of estimated cure probabilities.

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)

Hypothesis test for association between covariate and cure indicator adjusted by a second covariate

Description

Performs a permutation-based test assessing the association between a primary covariate (x) and the cure indicator, while adjusting for a secondary covariate (z). The test calculates the p-value via permutation using the partial martingale difference correlation.

Usage

testcov2(x, time, z, delta, P = 999, H = NULL)

Arguments

Details

In order to test if the cure rate depends on the covariate \boldsymbol{X} given it depends on the covariate \boldsymbol{Z}. The hypotheses are

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

The proxy of the cure rate under the null hypothesis \mathcal{H}_0 is obtained by:

\mathbb{I}(T > \tau) + (1-\delta)\mathbb{I}(T \leq \tau) \, \frac{1 - p(\boldsymbol{Z})}{1 - p(\boldsymbol{Z}) + p(\boldsymbol{Z})S_0(T|\boldsymbol{X,Z})}.

The statistic for testing the covariate hypothesis is based on partial martingale difference correlation and it is given by:

\text{pMDC}_n(\hat{\nu}_{\boldsymbol{H}}|\boldsymbol{X,Z})^2.

The null distribution is approximated using a permutation test.

Value

List with components:

statistic

Numeric. The test statistic value.

p.value

Numeric. The permutation p-value assessing the null hypothesis of no association between x and the latent cure indicator, adjusting for z.

References

Park, T., Saho, X. & Yao, S. (2015). Partial martingale difference correlation. Electronic Journal of Statistics, 9, 1492–1517. doi:10.1214/15-EJS1047

See Also

pmdc for the partial martingale difference correlation, pmdd for the partial martingale difference divergence, testcov for the test for one covariate.