miCoPTCM (version 1.1)

PTCMestimSIMEX: SIMEX approach

Description

Fits a Semiparametric Promotion Time Cure Model with mismeasured covariates, using the SIMEX algorithm based on a backfitting procedure. This approach is introduced in Bertrand et al. (2015).

Usage

# S3 method for default
PTCMestimSIMEX(x, y, errorDistEstim=c("normal","student","chiSquare","laplace"), 
paramDistEstim=NA, varCov=NA,  nBack=10000, eps=1e-8, Nu=c(0,.5,1,1.5,2), B=50, init, 
orderExtrap=2, multMaxTime=2,...)
# S3 method for formula
PTCMestimSIMEX(formula, data=list(),...)
# S3 method for PTCMestimSIMEX
print(x,...)
# S3 method for PTCMestimSIMEX
summary(object,...)

Arguments

x

a numerical matrix containing the explanatory variables as columns (without a column of 1s for the intercept).

y

the response, a survival object returned by the Surv function.

errorDistEstim

the distribution of the measurement error. See Details.

paramDistEstim

a scalar or a vector of length 2 containing the parameter(s) of the measurement error distribution, for non-Gaussian distributions. See Details.

varCov

the square variance-covariance matrix of measurement error, with as many rows as regression parameters (including the intercept), for Gaussian errors.

nBack

an integer specifying the maximal number of iterations in the backfitting procedure.

eps

convergence criterion.

Nu

a numerical vector containing the grid of lambda values, corresponding to the level of added noise.

B

the number of replications for each value in Nu.

init

a numerical vector of initial values for the regression parameters.

orderExtrap

a scalar or a numerical vector containing the degrees of the polynomials used in the extrapolation step.

multMaxTime

a positive number controlling the time allowed, in one iteration of the backfitting procedure, to function nleqslv (used to solve the score equations) to converge.

formula

a formula object, in which the response is a survival object returned by the Surv function.

data

a dataframe containing the variables appearing in the model.

object

an object of class "PTCMestimBF", i.e., a fitted model.

…

not used.

Value

An object of class PTCMestimBF, i.e. a list including the following elements:

coefficients

The estimated values of the regression parameters.

var

The estimated variances of the estimated regression parameters.

classObs

An integer vector of length 3: the number of censored individuals not considered as cured for the estimation, the number of events, and the number of individuals considered as cured for the estimation.

estimNuBF

A matrix with as many rows as elements in Nu, containing, in each row, the average of the B estimates, for each regression parameter (columns).

Details

More than one covariate can be subject to measurement error. However, in this implementation, all the errors must belong to the same family of distribution (specified with the argument errorDistEstim). Non-zero covariances are allowed between errors following a normal distribution. For the student, chi-squared and Laplace distributions, all variances are assumed to be equal (determined from paramDistEstim) and all covariances are assumed to be 0, even if the off-diagonal elements of vcov are not 0.

When using the laplace distribution, only one element in paramDistEstim is needed (if a vector of two elements is given, only the first element will be considered). With the student and chiSquare distributions, two parameters are required, while none is required with the normal distribution. For the laplace distribution, the parameter is the inverse of the rate \(\gamma\), where \(\gamma\) is such that \(f(x)=\frac{1}{2}\gamma e^{-\gamma |x|}\). The first parameter of the Student distribution corresponds to the degrees of freedom, while the second parameter is a multiplicative factor such that the variance is \(parameter_2^2*\frac{parameter_1}{parameter_1-2}\). Similarly, for the chi-squared distribution, the first parameter gives the degrees of freedom, and the second one is a multiplicative factor yielding a variance of \(2\cdot param_2^2 \cdot param_1\).

References

Bertrand A., Legrand C., Carroll R.J., De Meester C., Van Keilegom I. (2015) Inference in a Survival Cure Model with Mismeasured Covariates using a SIMEX Approach. Submitted.

Cook J.R., Stefanski L.A. (1994) Simulation-Extrapolation Estimation in Parametric Measurement Error Models. Journal of the American Statistical Association, 89, 1314-1328. DOI: 10.2307/2290994

Ma, Y., Yin, G. (2008) Cure rate models with mismeasured covariates under transformation. Journal of the American Statistical Association, 103, 743-756. DOI: 10.1198/016214508000000319

Examples

# NOT RUN {
library("survival")
## Data generation
set.seed(123)
n <- 200
varCov <- matrix(nrow=3,ncol=3,0)
varCov[2,2] <- 0.1^1
X1 <- (runif(n)-.5)/sqrt(1/12) 
V <- round(X1 + rnorm(n,rep(0,3),varCov[2,2]),7)# covariate with measurement error
Xc <- round(as.numeric(runif(n)<0.5),7) # covariate without measurement error

 # censoring times: truncated exponential distribution
C <- round(rexp(n,1/5),5) 
Cbin <- (C>30)
while(sum(Cbin)>0)
{
	C[Cbin] <- round(rexp(sum(Cbin),1/5),5)
	Cbin <- (C>30)
}

expb <- exp(0.5+X1-0.5*Xc) 
cure <- exp(-expb) # cure probabilities

 # event times with baseline cdf of a truncated exponential 
U <- runif(n)
d <- rep(NA,n)
T <- round(-6*log( 1+ (1-exp(-20/6))*log(1-(1-cure)*U)/expb ),5) 
T[(runif(n)<cure)] <- 99999 # cured subjects

Tobs <- rep(NA,n)
Tobs <- pmin(C,T) # observed times
Tmax <- max(Tobs[Tobs==T])
d <- (Tobs==T) # censoring indicator
	
Dat <- data.frame(Tobs,d,V,Xc)

## Model estimation
fm <- formula(Surv(Tobs,d) ~ V + Xc)
resSimex <- PTCMestimSIMEX(fm, Dat, errorDistEstim="normal", 
varCov=varCov,  nBack=10000, eps=1e-8, 
Nu=c(0,.5,1,1.5,2), B=50, init=rnorm(3), orderExtrap=1:3, multMaxTime=2)
resSimex
summary(resSimex)
# }