PTCMestimBF {miCoPTCM}

Corrected score approach

Description

Fits a Semiparametric Promotion Time Cure Model, taking into account (using a corrected score approach) or not the measurement error in the covariates, using a backfitting approach to maximize the likelihood. Both methods were introduced in Ma and Yin (2008).

Usage

## Default S3 method:
PTCMestimBF(x, y, varCov, init, nBack=10000, eps=1e-8, multMaxTime=2,...)
## S3 method for class 'formula'
PTCMestimBF(formula, data=list(), ...)
## S3 method for class 'PTCMestimBF'
print(x,...)
## S3 method for class 'PTCMestimBF'
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.

varCov

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

init

a numerical vector of initial values for the regression parameters.

nBack

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

eps

convergence criterion. Convergence is declared if the euclidian norm of the vector of changes in the estimated parameters and the euclidian norm of the score equations evaluated at these values are smaller than eps.

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.

Details

This method assumes normally distributed measurement error. The diagonal elements of the matrix varCov corresponding to covariates without error (as is the case for the intercept) have to be set to 0.

Value

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

coefficients

The estimated values of the regression parameters.

estimCDF

The estimated baseline cumulative distribution function.

vcov

The estimated variance-covariance matrix 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.

flag

Termination code: 1 if converged, 2 otherwise.

endK

Number of iterations performed in the backfitting procedure.

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.

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

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)
#colnames(Dat) <- c("Tobs","d","V","Xc")


## Model estimation
fm <- formula(Surv(Tobs,d) ~ V + Xc)
resMY <- PTCMestimBF(fm, Dat, varCov=varCov, init=rnorm(3))
resMY
summary(resMY)