sicure (version 0.1.1)

sicure.v: Estimation of the vector of parameters in a single-index mixture cure model with a vector of covariates

Description

This function provides the estimation of the vector of parameters (\(\boldsymbol{\theta_0}\)) in a single-index mixture cure model with a vector of covariates (see Piñeiro-Lamas, 2024, Section 3.1, pages 37-38).

Usage

sicure.v(x_cov, time, delta, randomsearch = FALSE)

Value

A list with the following components:

par

A numeric vector of the estimated parameters. The last four correspond to the logarithms of the bandwidths.

value

The value of the objective function (negative log-likelihood) at the estimated parameters.

si

The estimated single-index variable.

Arguments

x_cov

A matrix or data frame giving the covariate values. Each row represents an individual and each column corresponds to a variable.

time

A numeric vector giving the observed times.

delta

A numeric vector giving the values of the uncensoring indicator, where 1 indicates that the event of interest has been observed and 0 indicates that the observation is censored.

randomsearch

A logical value, FALSE by default, specifying whether a random search of the initial iterant is considered.

Details

The vector of parameters, \(\boldsymbol{\theta_0}\), is estimated by maximum likelihood, with the likelihood function also depending on four bandwidths. Since the parameters cannot be obtained directly from the likelihood, the estimation is performed using numerical optimization with the Nelder-Mead method. To begin the optimization, an initial iterant is required. To select a starting point for \(\boldsymbol{\theta_0}\), a logistic regression model is fitted using the uncensoring indicator delta as the response variable and the covariates as predictors. Regarding the initial bandwidths, for \(h_3\) the rule of thumb bandwidth selector used to estimate the density of the time variable is considered. For \(h_1\), \(h_2\) and \(h_4\), the rule of thumb bandwidth used to estimate the density of \(\boldsymbol{\theta_{0,ini}}' \boldsymbol{X}\), where \(\boldsymbol{\theta_{0,ini}}\) is the initial estimate of \(\boldsymbol{\theta_0}\) and \(\boldsymbol{X}\) is the vector of covariates. If randomsearch = TRUE, nine additional starting points are generated by adding a Gaussian noise (mean 0, standard deviation 0.25) to the logistic regression-based iterant. Each of these ten starting points is used to estimate the vector of parameters of the single-index mixture cure model, and the one that gives place to the lowest negative log-likelihood value is selected.

References

Piñeiro-Lamas, B. (2024). High dimensional single-index mixture cure models [PhD thesis]. Universidade da Coruña. Available at https://ruc.udc.es/dspace/handle/2183/37035

See Also

sicure.f, sicure.vf

Examples

# Some artificial data
set.seed(123)
n <- 50
mix1a<-rnorm(n,mean=0,sd=1); mix1b<-rnorm(n,mean=0.25,sd=sqrt(2)); alf1<-rbinom(n,1,0.2)
mix2a<-rnorm(n,mean=0,sd=1); mix2b<-rnorm(n,mean=0.25,sd=sqrt(2)); alf2<-rbinom(n,1,0.2)
mix1<-alf1*mix1a+(1-alf1)*mix1b; mix2<-alf2*mix2a+(1-alf2)*mix2b
x_cov<-array(c(mix1,mix2),dim=c(n,2)) # Matrix of covariate values
theta<-c(1,1.2)
Z<-colSums(theta*t(x_cov))
y<-Z+rnorm(n,sd=sqrt(abs(Z))) # True lifetimes
# Probability of being susceptible
p_fun <- function(x){ 0.55 * exp(1.5*x+1.5)/(1+exp(1.5*x+1.5)) + 0.001 }
for (i in 1:n){
   w <- runif(1)
   if (w > p_fun(Z[i])) y[i] <- Inf
}
c<-rexp(n,rate=0.98) # Censoring values
t<-pmin(y,c) # Observed times
d = 1 * (y<=c) # Uncensoring indicator
# \donttest{
suppressWarnings(sicure.v(x_cov, t, d))
# }