Fits Semiparametric Promotion Time Cure Models, taking into account (using a corrected score approach or the SIMEX algorithm) or not the measurement error in the covariates, using a backfitting approach to maximize the likelihood.
miCoPTCM miCoPTCM
The survival model of interest is the promotion time cure model, i.e. a survival model which takes into account the existence of subjects who will never experience the event. The survival function of \(T\), the survival time, is assumed to be improper: $$S(t|\bm{x}) = P(T>t|\bm{X}=\bm{x}) = \exp\left\{-\theta(\bm{x}) F(t) \right\},$$ where \(F\) is a proper baseline cumulative distribution function, \(\theta\) is a link function with an intercept, here \(\theta(\bm{x}) = \exp(\bm{x}^T \bm{\beta})\), and \(\bm{x}\) is the vector of covariates. We work with the semiparametric version of this model, in which no known distribution is assumed for \(F\). It can be shown that the nonparametric estimator of \(F\) is a step function which increases only at the failure times.
We assume that we have right censoring in our data, so that \(Y=\min(T,C)\) is observed, where \(C\) is the censoring time.
The classical additive error model is assumed for the covariates, so that \(\bm{W}=\bm{X}+\bm{U}\) is observed, where \(\bm{W}\) is the vector of observed covariates and \(\bm{U}\) is the vector of measurement errors. We assume that \(\bm{U}\) is independent of \(\bm{X}\) and \(\bm{U}\) follows a continuous distribution with mean zero and known covariance matrix \(\bm{V}\). It is also assumed that \((T,C)\) and \(\bm{W}\) are independent given \(\bm{X}\).
Three possible estimation methods are available in this package. The corrected score approach of Ma and Yin (2008) is implemented in function PTCMestimBF. It consists in solving, through a backfitting procedure, the score equations in which the terms involving \(\bm{x}\) are replaced by some terms involving \(\bm{w}\) and \(\bm{V}\).
The naive method consists in not taking the measurement error in the covariates into account. The naive estimate is obtained by using function PTCMestimBF with a variance-covariance matrix of the error containing only zeros.
Finally, the SIMEX algorithm applied to the promotion time cure model (Bertrand et al., 2015) is implemented in the function PTCMestimSIMEX. The SIMEX algorithm (Cook and Stefanski, 1994) is a generic and intuitive procedure allowing to estimate and reduce the bias in a model in which the covariates are measured with error. In this implementation, the naive estimator required by the procedure is the one of Ma and Yin (2008).
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