# SIMEX approach 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). ```r ## Default S3 method: 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 class 'formula' PTCMestimSIMEX(formula, data=list(),...) ## S3 method for class 'PTCMestimSIMEX' print(x,...) ## S3 method for class '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. ## 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 $2\cdot param_2^2 \cdot param_1$. ## Returns 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). ## 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 ```r ## 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)

PTCMestimSIMEX function