On an extension of the promotion time cure model

Abstract

We consider the problem of estimating the distribution of time-to-event data that is subject to censoring and for which the event of interest might never occur, that is, some subjects are cured. To model this kind of data in the presence of covariates, one of the leading semiparametric models is the promotion time cure model (Stochastic Models of Tumor Latency and Their Biostatistical Applications (1996) World Scientific), which adapts the Cox model to the presence of cured subjects. Estimating the conditional distribution results in a complicated constrained optimization problem, and inference is difficult as no closed-formula for the variance is available. We propose a new model, inspired by the Cox model, that leads to a simple estimation procedure and that presents a closed formula for the variance. In this paper, we show (i) that the new model contains as a special case the promotion time cure model with an exponential link, and hence we have a simpler way to estimate the latter model than what is done so far in the literature; (ii) that in the latter special case, both estimators are equal to the partial likelihood estimator under the usual Cox model; (iii) that the estimators under the new model have certain asymptotic properties when the model is correct and when it is misspecified; (iv) that the error of LASSO type estimators is of order $\sqrt{log(\mathit{n}\mathit{d})/\mathit{n}}$ in the case of high-dimensional covariates with dimension d and sample size n. We also study the practical behaviour of our estimation procedure by means of simulations, and we apply our model and estimation method to a breast cancer data set.