An expectation-maximization algorithm for including oncological COVID-19 deaths in survival analysis

We address the problem of how COVID-19 deaths observed in an oncology clinical trial can be consistently taken into account in typical survival estimates. We refer to oncological patients since there is empirical evidence of strong correlation between COVID-19 and cancer deaths, which implies that COVID-19 deaths cannot be treated simply as non-informative censoring, a property usually required by the classical survival estimators. We consider the problem in the framework of the widely used Kaplan-Meier (KM) estimator. Through a counterfactual approach, an algorithmic method is developed allowing to include COVID-19 deaths in the observed data by mean-imputation. The procedure can be seen in the class of the Expectation-Maximization (EM) algorithms and will be referred to as Covid-Death Mean-Imputation (CoDMI) algorithm. We discuss the CoDMI underlying assumptions and the convergence issue. The algorithm provides a completed lifetime data set, where each Covid-death time is replaced by a point estimate of the corresponding virtual lifetime. This complete data set is naturally equipped with the corresponding KM survival function estimate and all available statistical tools can be applied to these data. However, mean-imputation requires an increased variance of the estimates. We then propose a natural extension of the classical Greenwood's formula, thus obtaining expanded confidence intervals for the survival function estimate. To illustrate how the algorithm works, CoDMI is applied to real medical data extended by the addition of artificial Covid-death observations. The results are compared with the estimates provided by the two naive approaches which count COVID-19 deaths as censoring or as deaths by the disease under study. In order to evaluate the predictive performances of CoDMI an extensive simulation study is carried out. The results indicate that in the simulated scenarios CoDMI is roughly unbiased and outperforms the estimates obtained by the naive approaches. A user-friendly version of CoDMI programmed in R is freely available.