EM algorithms for fitting multistate cure models

Abstract

Multistate cure models are multistate models in which transitions into one or more of the states cannot occur for a fraction of the population. In the study of cancer, multistate cure models can be used to identify factors related to the rate of cancer recurrence, the rate of death before and after recurrence, and the probability of being cured by initial treatment. However, the previous method for fitting multistate cure models requires substantial custom programming, making these valuable models less accessible to analysts. In this article, we present an Expectation-Maximization (EM) algorithm for fitting the multistate cure model using maximum likelihood. The proposed algorithm makes use of a weighted likelihood representation allowing it to be easily implemented with standard software and can incorporate either parametric or non-parametric baseline hazards for the state transition rates. A common complicating feature in cancer studies is that the follow-up times for recurrence and death may differ. Additionally, we may have missingness in the covariates. We propose a Monte Carlo EM (MCEM) algorithm for fitting the multistate cure model in the presence of covariate missingness and/or unequal follow-up of the two outcomes, we describe a novel approach for obtaining standard errors, and we provide some software. Simulations demonstrate good algorithmic performance as long as the modeling assumptions are sufficiently restrictive. We apply the proposed algorithm to a study of recurrence and death in patients with head and neck cancer.

Keywords: Cure models; EM algorithm; Monte Carlo EM algorithm; Multistate models.

Fig. 1.

Diagram of the multistate cure model. States 1 and 2 present baseline cure status. State 3 (recurrence) can only be reached from baseline State 1. State 4 (death) can be reached from all other states.

Fig. 2.

Bias and coverage of multistate model estimates with ten covariates. The plot on the left shows the bias (x100) of the multistate cure model parameters across 500 simulations, and the corresponding 95% confidence interval coverage rates are shown in the right plot. State 1 is the non-cured baseline state, State 2 is the cured baseline state, State 3 is the recurrence state, and State 4 is the death state.

Fig. 3.

Results of applying MCEM algorithm to head and neck data. The corresponding 95% confidence intervals for each part of the multistate cure model are shown for each model covariate. Stars indicate confidence intervals that are significant at 0.05. State 1 is the non-cured baseline state, State 2 is the cured baseline state, State 3 is the recurrence state, and State 4 is the death state.