Nonparametric kernel estimation of the probability of cure in a mixture cure model when the cure status is partially observed

Abstract

Cure models are a class of time-to-event models where a proportion of individuals will never experience the event of interest. The lifetimes of these so-called cured individuals are always censored. It is usually assumed that one never knows which censored observation is cured and which is uncured, so the cure status is unknown for censored times. In this paper, we develop a method to estimate the probability of cure in the mixture cure model when some censored individuals are known to be cured. A cure probability estimator that incorporates the cure status information is introduced. This estimator is shown to be strongly consistent and asymptotically normally distributed. Two alternative estimators are also presented. The first one considers a competing risks approach with two types of competing events, the event of interest and the cure. The second alternative estimator is based on the fact that the probability of cure can be written as the conditional mean of the cure status. Hence, nonparametric regression methods can be applied to estimate this conditional mean. However, the cure status remains unknown for some censored individuals. Consequently, the application of regression methods in this context requires handling missing data in the response variable (cure status). Simulations are performed to evaluate the finite sample performance of the estimators, and we apply them to the analysis of two datasets related to survival of breast cancer patients and length of hospital stay of COVID-19 patients requiring intensive care.

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“…All the methods can be extended to incorporate these characteristics in the estimations as covariates. If only one covariate is to be included, the NP method and the EMNP method can be extended following Safari et al [ 30 , 33 ]. When there are many covariates, the sparseness of data gives rise to the well known “curse of dimensionality”, which implies that massive amounts of data will be required for accurate estimate as the number of covariates increases.…”

“…When estimating the distribution of Z , “time in hospital ward until admission to the ICU”, all patients admitted to ICU ( δ i ( t ) = 1) are uncensored while those who have already been discharged from hospital at a time before t without ICU admission ( ν i ( t ) = 1) are cured from the event `ICU admission´ as they will never be admitted to ICU in the future. For a fixed time t , the NP method estimates the distribution function of Z , F Z ( z ) = p ( Z ≤ z ), nonparametrically using the estimator in Safari et al [ 33 ] and the observations where t 1 ≤⋯≤ t n ( t ) are the sorted observed times in Eq ( 1 ).…”

“…Finally, the probability of requiring ICU from ward is estimated for a fixed time t using the nonparametric estimator in Safari [ 32 ]: …”

“…Note that the first method is biased under the common assumption of independent censoring. Besides, the main disadvantage of the second approach is that, if the patient with the largest observed time is still in hospital ward and did not require ICU (HW set), then the estimator of p WI is not unique, and only upper and lower bounds are provided [ 32 ].…”

“…Nonetheless, that is not the case in our context. MCM when the cured subjects are randomly identified addresses this situation, and it has received much attention in recent years (see [29][30][31][32]).…”