EM algorithms for fitting multistate cure models

Summary

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.

1. Introduction

In medical applications, multistate models describe the rates at which individuals move between various health states. Multistate models have many valuable uses in medical research. Firstly, multistate models allow us to incorporate information from multiple event time outcomes (e.g. recurrence and death) in a unified way. Secondly, multistate models allow us to study which patient characteristics are relevant to which aspects of disease progression. Finally, multistate models are useful for making predictions, which can be valuable for medical decision-making.

The illness-death model is a popular multistate model explored in the literature and consists of three states: healthy (or no event), illness, and death (Andersen and Keiding, 2002). All subjects start out in the “healthy” state at baseline and can then move into the illness or death states as they develop illness or die from other causes. Subjects that develop illness can also transition into the death state. One common application of the illness-death model is in the study of cancer recurrence and death. In this setting, the “healthy” state represents subjects who have been treated for their initial cancer. For the remainder of this article, we will focus on the scenario with outcomes cancer recurrence and death, but these may be different types of events in general.

While the illness-death model is useful in many applications, one limitation is that the model implicitly assumes that all subjects can experience the illness event. In the context of cancer recurrence and death, this is equivalent to assuming that all subjects can experience a recurrence. For many types of cancer, however, this may not be a reasonable assumption. In the case of head and neck cancer, it has been well established that some subjects can be completely cured of their initial disease, and these subjects will never experience a recurrence of their primary cancer (Taylor, 1995; Cognetti and others, 2008). We call the set of cured subjects the “cured fraction.”

In this article, we consider a generalization of the illness-death model called the multistate cure model that accounts for the cured fraction. The Bayesian multistate cure model developed in Conlon and others (2013) breaks the “healthy” state of the illness-death model into two baseline states: cured and non-cured. The non-cured subjects can then experience either cancer recurrence or death under an illness-death model. The cured subjects can only experience the death event.

One challenge to fitting the multistate cure model is that cure status is partially latent. Subjects with an observed recurrence are known to be non-cured, but subjects censored for recurrence have unknown cure status. Cure status is unknown for all subjects at baseline. A natural question that arises in the context of cure models is our ability to identify the cured population when the event time distribution in the non-cured subjects may have a long tail (Farewell, 1982; Conlon and others, 2013). We will assume that we have sufficient follow-up after the last observed event time. One indicator of sufficient follow-up in a cure setting is that a Kaplan–Meier estimator applied to the time to recurrence outcome should have a clear plateau, indicating that there is a time-point after which recurrence events are no longer being observed. In the illness-death model setting, we require further follow-up for death before and after recurrence, so a lack of sufficient follow-up may be less of a concern for data sets that are well-suited for illness-death models.

Another problem for illness-death models in general and multistate cure models in particular is that the follow-up may not be the same for both outcomes of interest (Conlon and others, 2013). We call this situation “unequal follow-up.” For example, death status may be more easily obtained through death records, while assessment of cancer recurrence status requires a clinic visit, so it may often be the case that death status is known and recurrence status is unknown at a particular time . Conlon and others (2013) propose a Gibbs-sampling algorithm for fitting the multistate cure model in which values of cure status are drawn using a data augmentation approach, and unequal follow-up is handled through a modification to the likelihood involving an integral. Their proposed algorithm performs well, but it requires substantial custom programming, it requires specification of prior distributions and tuning parameters, and it can take a long time to reach convergence. Additionally, the algorithm discussed in Conlon and others (2013) assumes that covariates are fully observed, which may not be the case in practice.

The EM algorithm is an alternative, maximum likelihood-based method in the literature for fitting models with latent variables or other types of missing data (Dempster and others, 1977). One advantage of the EM algorithm over Bayesian methods is that, in some cases, an EM algorithm can be more readily implemented using standard software. Additionally, the EM algorithm does not require specification of prior distributions or tuning parameters. In some complicated missing data scenarios, however, the conventional EM algorithm can be difficult to implement. The Monte Carlo EM (MCEM) algorithm proposed in Wei and Tanner (1990) provides a convenient, imputation-based approach for handling more complex missing data within a modified EM algorithm. EM and MCEM algorithms have not been previously explored in the context of multistate cure models in the literature and development of such fitting algorithms could make the multistate cure model much more accessible to investigators.

In this article, we first propose a simple EM algorithm for fitting the standard multistate cure model. We then propose a MCEM algorithm for fitting the model in the presence of covariate missingness and/or unequal follow-up of the outcomes. The proposed algorithms can incorporate either parametric or non-parametric baseline hazards for the transitions between states and can incorporate different assumptions about the rate of death from other causes. The proposed EM algorithm makes use of a weighted likelihood representation, allowing it to be easily implemented using standard software. We provide software for implementing the EM and MCEM algorithms. We describe a novel approach for estimating standard errors for the MCEM algorithm. Simulations demonstrate the performance of the EM and MCEM algorithms under different modeling assumptions. We apply the proposed MCEM to a study of cancer recurrence and death of head and neck patients. We then derive expressions for estimating state occupancy probabilities, which can used to make predictions for individual patients.

In Section 2, we present details about the multistate cure model structure. In Sections 3 and 4, we propose an EM and MCEM algorithm for fitting the model. In Section 5, we discuss how to estimate standard errors. In Section 6, we derive state occupancy probabilities. We present a simulation study in Section 7, and we apply the proposed methods to head and neck cancer data in Section 8. In Section 9, we include a discussion.

2. Multistate cure model specification

Suppose we have two semi-competing events: recurrence and death. By semi-competing, we mean that we can observe death after cancer recurrence but cannot observe recurrence after death. We further suppose that there is some subset of the subjects that are cured of their initial cancer and will never experience a cancer recurrence (even with very long follow-up).

Let and be the underlying recurrence and death times for subject . For cured subjects, . Let be the censoring time for recurrence (loss to follow-up) and be the censoring time for death. Initially, we assume that the follow-up for recurrence and death is equal and define . For all subjects, we observe censored recurrence time information, and , and censored death time information, and . Let denote the covariates for subject , which we will initially suppose are fully observed.

We assume that all subjects have been previously treated for their initial cancer and did not have observable cancer at baseline. For non-cured subjects, however, some unobservable cancer cells remain and will grow until they are eventually observable, called cancer recurrence. Subjects with an observed recurrence are known to have been non-cured at baseline, but all other subjects have unknown baseline cure status. Let be a variable indicating baseline non-cure status: if cured, if not cured. While we never know for sure that subjects are cured, we may strongly believe subjects still at risk after some time are cured and assign for these subjects. Assigning some subjects to be cured improves the stability of the proposed methods, and we use this approach in our simulations and data example. Similar assumptions are often implicitly made for standard cure rate models through restrictions on the event rate in the non-cured subjects (Peng and Dear, 2000; Cai and others, 2012).

Figure 1 shows the conceptual structure of the model proposed in Conlon and others (2013). Solid arrows represent potential state transitions given baseline cure status. If we remove State 4, the multistate cure model would reduce to the popular Cox proportional hazards (CPH) mixture cure model (Kuk and Chen, 1992; Sy and Taylor, 2000). We will assume that the underlying transition times between states are conditionally independent given covariates. We will further assume that and are independent of all underlying transition times given covariates.

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.

We model the probability of not being cured by initial treatment using a logistic model: , where is a subset of . We model the transition rate from state to state for all transitions except using proportional hazards model , where is the subset of used in the model for transition . One important decision in multistate modeling is whether we reset time back to zero upon entering a new state (Putter and others, 2007; Meira-Machado and others, 2009). In the model for the transition, we use the “clock reset” method in which time is reset to zero upon entering State 3, and use a proportional hazards regression to model the residual time in State 3 before entering State 4 as follows: . We can incorporate the time spent in state 1, , as a covariate in if desired.

Let and represent the cumulative hazard and cumulative baseline hazard for transition , and let represent the probability of remaining in state at time . We have that , , and for given . We may use a parametric or non-parametric form for the baseline hazards.

We may place additional assumptions on the hazards for the and transitions. Since these two transitions represent typically death from other causes, it may be reasonable to assume that the hazards are identical (). In this case, the multistate cure model reduces to a CPH cure model for recurrence time with two additional regressions for time to death with and without recurrence (Conlon and others, 2013). However, suppose we do not want to assume the hazards are equal. We may instead assume the hazards are proportional () or that the baseline hazards are equal (, and unrestricted), or we may make no equality assumptions ( and unrestricted).

Let represent the set of model parameters. includes , , and possibly baseline hazard parameters (under parametric baselines). We will assume that and are distinct. Let denote the complete data. The complete data log-likelihood for the multistate cure model takes the following form:

(1)

3. EM algorithm

The EM algorithm is an approach for maximum likelihood estimation in the presence of missing data, which in our setting is the partially latent cure status (Dempster and others, 1977). Let represent the observed data and represent the missing data. The goal is to maximize the observed data log-likelihood, , with respect to . The algorithm breaks the problem of maximizing into iterations of two simpler steps: the E-step and the M-step. In the E-step, we calculate the expected value of the complete data log-likelihood conditioning on the observed data and the most recent parameter estimate, , to obtain

In the M-step, we maximize with respect to . We iterate these steps many times until convergence of the estimated to the maximum likelihood estimate.

The EM algorithm is a common estimation method in the literature for models with latent classes. Frydman and Kadam (2004) proposed an EM algorithm for estimation for a continuous time multistate model in which the underlying population is split into movers and stayers such that only the movers are eligible to experience a state transition. This setting is very similar to our setting except that our model allows both the movers and stayers to experience death.

3.1. E-step

In the E-step for fitting the multistate cure model, we take the expectation of (1). Since (1) is linear in , we can obtain by replacing in (1) with when is unknown. Let . For all subjects, we replace with

(2)

In order to use the formula in (2), we need estimates of the baseline hazard functions for the , , and transitions. Under parametric assumptions, the complete form of the baseline hazards are determined in the M-step. When the baseline hazards are non-parametric, we estimate the baseline hazards prior to calculating (2). In the supplementary material available at Biostatistics online, we use the profile likelihood method to derive estimators for the baseline hazards. The form of the estimator depends on the estimate of from the previous iteration and whether we assume that the baseline hazards for the and transitions are equal.

3.2. M-step

In the M-step, we maximize with respect to . After replacing with in (1) and reorganizing the terms, we have that

Since we assume that censoring times for recurrence and death are the same, implies . Additionally, we note that implies . We can therefore rewrite as:

(3)

The terms involving and separate, so we can maximize (3) with respect to and separately. The terms involving resemble the log-likelihood for a logistic model with as the outcome. We can estimate by fitting a logistic regression to using predictors . We can estimate (and perhaps baseline parameters) by maximizing the last four terms of (3) with respect to the parameters. In the supplementary material available at Biostatistics online, we show how this maximization can be implemented by fitting a single, weighted proportional hazards regression model to an augmented version of the data. The proposed method for estimating is similar to the methods used by mstate in R and other multistate modeling software except that it incorporates transition-specific weights and involves a different augmented data structure (de Wreede and others, 2011).

4. Extension to handle additional missing data

The EM algorithm in Section 3 assumes that cure status is the only source of missing data. However, additional missing data often arises in practice. One source of missing data is missingness in the covariates. Another common source of missingness occurs when the follow-up for recurrence is shorter than the follow-up for death. We call this phenomenon “unequal follow-up.” In order to follow-up for recurrence, patients must come into the clinic, while death status can be more easily obtained from death registries. In this case, recurrence status may only be known up to time , while death status may be known up to time . This results in missing information about recurrence status on the interval , which we will treat as missing data. This setting is similar to interval censoring and panel data for illness-death models (Jackson, 2011).

Conlon and others (2013) handles the problem of unequal follow-up by constructing a modified likelihood function involving an integral. Another potential solution is to censor death back to the follow-up time for recurrence for subjects with unequal follow-up. This approach is unappealing since it throws out valuable information about death. A third solution is to modify the conventional EM algorithm so that the E-step takes the expectation over all types of missing data. However, when we have complicated patterns of missing data, these expectations can be difficult to compute. We consider an alternative approach called the MCEM algorithm, which takes an imputation-based approach to dealing with missing data.

4.1. Monte Carlo EM algorithm

The MCEM algorithm proposed in Wei and Tanner (1990) provides a convenient approach to handling complicated missing data within a modified EM algorithm. The strategy is to replace the usual E-step from the EM algorithm with a step in which we obtain imputations of by drawing the missing data from .

The M-step of the MCEM algorithm then involves maximizing the complete data log-likelihood mixed over the imputed values:

Suppose we create a stacked version of the data set, called , obtained by stacking the imputed versions of the data such that each subject appears in the data set times. We then have that . We can estimate by maximizing with respect to . We then iterate between the imputation and M-steps until “convergence,” where successive estimates of fall around the line with noise (Wei and Tanner, 1990). We can estimate by taking the mean parameter estimate across the last few iterations of the MCEM algorithm.

The imputation step involves drawing the missing data times from . Unlike conventional multiple imputation, missing data is drawn from the predictive distribution of the missing data evaluated at a single estimated parameter value, , rather than independent draws of the parameter (Wei and Tanner, 1990; Neath, 2012). Therefore, the imputations produced are “improper” as described in Little and Rubin (2002). In addition to imputing missing covariate or outcome values, the MCEM algorithm will also involve imputing values for the partially latent cure status, and we will impute each type of missing data separately.

4.2. Imputation for unequal follow-up

Unequal follow-up is very common when the outcomes of interest are recurrence and death and occurs in many other semi-competing risks settings. In the supplementary material available at Biostatistics online, we present a derivation of the proposed imputation approach and provide recommendations for implementation. Here, we include a brief description of the general approach.

Let be the censoring time for recurrence and be the censoring time for death, but now assume that and for some subjects, . For all subjects, we observe -censored recurrence information, and , and -censored death information, and . Our goal is to impute values of and that would have been observed if we had followed subjects for recurrence as long as we followed them for death.

Suppose, we treat previously imputed and as known. Values of and are only unknown for subjects with imputed and observed and . Define . We impute missing from a Bernoulli distribution with probability

If imputed , we set . Otherwise, we draw from

4.3. Imputation of cure status

The cure status imputation approach will depend on whether we have unequal follow-up in the outcome. First, we will assume there is equal follow-up for all subjects. In this case, we can draw missing using as shown in equation (2).

Suppose instead that we have unequal follow-up. We perform imputation of conditioning on the observed and but not the imputed values of and , which allows imputations to more easily move between and in successive iterations. We can impute missing from a Bernoulli distribution using probability :

4.4. Imputation of missing covariates

Many methods can be used to perform the covariate imputation. For a detailed discussion of imputation methods, we refer the reader to Little and Rubin (2002). Chained equations is a popular approach in which we impute each covariate one-by-one using regression models (Van Buuren and others, 2006; Raghunathan, 2001). A method in Bartlett and others (2014) uses the structure of the outcome model in addition to a distribution for the covariates to obtain the imputation distributions. We use this approach in our simulations and data example. Let denote the th covariate in and denote all but the th covariate. Under missing at random assumptions (as defined in Little and Rubin, 2002), we impute each with missingness from a distribution proportional to , where is the conditional distribution of given . This expression is viewed as a function of , treating all other imputed variables as fixed and using estimated values of and .

5. Estimating standard errors

The EM and MCEM algorithms provide an estimate of , but they do not provide corresponding standard errors. Many methods have been explored in the literature for estimating standard errors for parameters estimated using an EM algorithm. Some methods involve and its derivatives (e.g. Louis, 1982). Bootstrap methods are also popular. The bootstrap approach is commonly used to estimate standard errors for the CPH cure model (Sy and Taylor, 2000), and we use this approach for estimating standard errors for the EM algorithm in our simulations.

A similar bootstrap approach can be used to estimate the standard errors from the MCEM algorithm, but we do not recommend this approach due to the relative slowness of the MCEM fitting algorithm. The usual approach for estimating standard errors after an MCEM algorithm estimation of is a generalization of Louis’s method proposed in Wei and Tanner (1990) with:

where are the imputed versions of from the last iteration. The estimated covariance matrix for is . This approach is usually implemented using large for the last few iterations of the fitting algorithm. This approach requires us to directly compute first and second derivatives of with respect to , which may not be convenient. Additionally, in the proposed MCEM algorithm, the imputations at a given iteration depend on the imputations from the previous iteration, so we cannot easily change the value of across iterations.

We propose a post-processing method (below) to obtain proper multiple imputations of . After fitting the multistate cure model to each imputed data set separately, we can then use Rubin’s multiple imputation combining rules to obtain standard errors that correctly account for the uncertainty related to the missing data (Little and Rubin, 2002). This approach is convenient because (i) it does not require us to use large for any iterations and (ii) it does not require us to directly compute derivatives of the observed data log-likelihood. In simulations (not shown), we found similar performance under our proposed estimation method and the approach proposed in Wei 1990. However, the post-processing step is important, and skipping the post-processing resulted in undercoverage of multistate cure model parameters.

We propose the following method to obtain proper multiple imputations of using the (improper) multiple imputations obtained within the MCEM algorithm. Our goal is to obtain independent draws from , which are our proper multiple imputations. At the end of the MCEM algorithm, we have independent draws from , where is the estimate of at the last iteration. Let denote the imputations at the final iteration of the MCEM algorithm. We can obtain approximate draws from by performing the following for each . [Step 1]: Estimate on a bootstrap sample of the most recent . Since contains no missingness, this estimation is easy to perform. This results in an approximate draw of from under a flat prior (Little and Rubin, 2002). [Step 2]: Using the draw of , obtain an updated imputation of as described in Section 4. [Step 3]: Repeat Steps 1–2 several times. We can then use these proper multiple imputations of for variance estimation. We provide some theoretical justification for this approach in the supplementary material available at Biostatistics online.

6. Prediction

We can use the multistate cure model fit to estimate the probability that a subject will be in a particular state at time given that subject’s baseline predictors. These probabilities can be useful for predicting prognosis or exploring the potential impact of different treatments. Below, we provide expressions for estimating state occupancy probabilities over time given only baseline covariate information. In the supplementary material available at Biostatistics online, we provide examples and additional expressions for estimating these probabilities incorporating follow-up information post-baseline.

We are interested in estimating quantities related to the unmeasured (or incompletely measured) variables, and . Using the structure of the multistate cure model, we derive the following probabilities, which sum to 1 for a given :

These expressions can be calculated using numerical integration and a multistate cure model fit. We can estimate these probabilities for different values of to create probability curves over time.

7. Simulation study

7.1. Simulation 1: bias, efficiency, and coverage of multistate cure model parameters

We simulate 500 data sets with 2000 subjects each under a multistate cure model with two covariates and Weibull baseline hazards. We simulate event times such that the hazards for the and transitions are equal. All elements of and are set equal to 0.5. The baseline hazards for the event transitions were chosen to mimic a scenario we may expect to encounter in real data. Additional details can be found in the supplementary material available at Biostatistics online.

We consider three simulation scenarios: (i) no covariate missingness or unequal follow-up, (ii) covariate missingness, and (iii) unequal follow-up. For Scenarios 1 and 2, an outcome censoring time was generated from . This provides sufficient follow-up so that a clear plateau can be observed in the Kaplan–Meier estimator applied to time to recurrence, allowing the cure rate to be well estimated. For Scenario 3, censoring time for death was generated from . For all but the first 750 subjects, we impose an earlier censoring time for recurrence. We use these simulated values to determine the observed data for each subject. This leads to roughly of the subjects needing imputation for unequal censoring. For Scenario 2, we impose 30% missing completely at random missingness in . In all scenarios, we assume subjects still at risk for recurrence after time 50 are cured. This value was chosen as to point in which the Kaplan–Meier estimator applied to recurrence shows a clear plateau.

For each simulated data set, we fit a multistate cure model using the proposed EM (Scenario 1) or MCEM algorithm (Scenarios 2 and 3). Within each scenario, we consider different assumptions regarding baseline hazards (Weibull or Cox) and restrictions for the and transition hazards. Variances from the MCEM algorithm are obtained using the Rubin’s rules-based approach described in Section 5 with five iterations of post-processing. We then compute the bias, empirical variance, and coverage rates of the multistate cure model parameter estimates across the 500 data sets. We also record the median run time and the number of simulations with numerical issues (non-converging M-step or difficulty with variance estimation) for each scenario.

Tables 1 and 2 show the results. When we assume under Weibull or Cox baseline hazard assumptions, the proposed algorithms result in essentially unbiased parameter estimates with nominal coverage rates in all scenarios. When we assume , we again see good bias and coverage properties under Weibull or Cox baseline hazards for Scenarios 1 and under Weibull baseline hazards for Scenario 2 and 3. For Scenarios 2 and 3 under Cox baseline hazards, we see increased bias and/or undercoverage for the parameters related to the and transitions and the logistic regression. When we assume , we generally obtain good bias and coverage properties for all failure time model parameters. For the intercept in the logistic model, we tend to see some undercoverage, particularly when we assume Cox baseline hazards and when we have unequal follow-up. When we do not restrict and , we obtain good bias and covariate properties in Scenario 1 under Weibull baseline hazards, but we see increased bias and undercoverage in all other settings. Failing simulations provide additional evidence of numerical instability. Overall, the proposed algorithms can provide good numerical properties in all three scenarios when the assumptions for the and hazards are sufficiently restrictive. When the restrictions are relaxed (particularly when the baseline hazards are not equal), we can run into numerical problems, and these problems tend to be greater under Cox baseline hazard assumptions and when we have unequal follow-up.

Table 1.

Bias, variance, and coverage of failure time model parameters corresponding to and within the multistate cure model

Baseline hazard	2 4, 1 4 Assumption	Failure time models
		1		2		1		3
Scenario 1: no covariate missingness or unequal follow-up
Weibull		0 (0.17) 94	0 (0.15) 97	0 (0.47) 94	0 (0.45) 96	0 (0.47) 94	0 (0.45) 96	0 (0.14) 95	0 (0.15) 95
Weibull		0 (0.18) 95	0 (0.16) 96	0 (1.03) 94	0 (0.97) 96	1 (3.22) 93	0 (3.19) 94	0 (0.14) 94	0 (0.15) 94
Weibull		0 (0.17) 95	0 (0.15) 96	0 (0.48) 95	0 (0.47) 95	0 (0.48) 95	0 (0.47) 95	0 (0.14) 95	0 (0.15) 94
Weibull	None	0 (0.18) 95	0 (0.16) 96	0 (1.12) 95	1 (1.04) 96	0 (4.54) 95	0 (5.30) 95	0 (0.15) 95	0 (0.15) 95
Cox		0 (0.18) 94	0 (0.15) 97	0 (0.48) 94	0 (0.45) 96	0 (0.48) 94	0 (0.45) 96	0 (0.15) 95	0 (0.16) 95
Cox		0 (0.18) 94	0 (0.16) 96	0 (1.05) 94	0 (0.97) 96	1 (3.40) 94	0 (3.24) 93	0 (0.17) 95	0 (0.16) 94
Cox		0 (0.18) 94	0 (0.16) 96	0 (0.48) 94	0 (0.47) 94	0 (0.48) 94	0 (0.47) 94	0 (0.15) 94	0 (0.16) 95
Cox	None	1 (0.83) 70	1 (0.82) 68	6 (5.07) 60	4 (5.19) 58	67 (15.1) 31	65 (16.7) 35	0 (0.15) 95	0 (0.16) 95
Scenario 2: covariate missingness
Weibull		0 (0.17) 94	0 (0.22) 95	0 (0.45) 97	0 (0.63) 97	0 (0.45) 97	0 (0.63) 97	0 (0.15) 95	0 (0.17) 96
Weibull		0 (0.19) 95	0 (0.23) 95	0 (0.97) 96	0 (1.22) 96	1 (3.36) 95	0 (4.20) 93	0 (0.15) 95	0 (0.17) 96
Weibull		0 (0.17) 95	0 (0.22) 96	0 (0.47) 96	0 (0.64) 96	0 (0.47) 96	0 (0.64) 96	0 (0.47) 97	0 (0.64) 96
Weibull	None	0 (0.19) 94	0 (0.24) 94	1 (1.11) 94	0 (1.42) 95	1 (3.81) 72	0 (5.28) 73	0 (0.14) 96	0 (0.18) 95
Cox		0 (0.18) 95	0 (0.22) 95	0 (0.46) 97	0 (0.64) 96	0 (0.46) 97	0 (0.64) 96	0 (0.16) 96	0 (0.19) 97
Cox		0 (0.20) 95	0 (0.24) 96	1 (1.07) 96	3 (1.30) 95	0 (3.52) 94	0 (4.20) 94	0 (0.16) 95	0 (0.21) 96
Cox		0 (0.18) 95	0 (0.23) 95	1 (0.49) 96	2 (0.61) 95	1 (0.49) 96	2 (0.61) 95	0 (0.16) 97	0 (0.19) 96
Cox	None	0 (0.53) 76	1 (0.52) 81	6 (6.13) 53	5 (5.97) 60	15 (28.8) 37	16 (26.7) 44	0 (0.15) 96	0 (0.18) 97
Scenario 3: unequal follow-up
Weibull		0 (0.19) 95	0 (0.21) 94	0 (0.50) 94	0 (0.50) 95	0 (0.50) 94	0 (0.50) 95	0 (0.16) 95	0 (0.16) 95
Weibull		0 (0.21) 94	0 (0.22) 95	0 (1.05) 96	0 (1.03) 95	1 (3.53) 94	1 (3.39) 94	0 (0.16) 96	0 (0.16) 95
Weibull		0 (0.19) 95	0 (0.21) 94	0 (0.50) 95	0 (0.51) 96	0 (0.50) 95	0 (0.51) 96	0 (0.16) 97	0 (0.16) 95
Weibull	None	0 (0.21) 94	0 (0.22) 95	1 (1.23) 95	0 (1.29) 93	2 (4.30) 71	0 (3.63) 76	0 (0.17) 95	0 (0.17) 94
Cox		0 (0.20) 95	0 (0.21) 94	1 (0.46) 94	2 (0.49) 95	1 (0.46) 94	2 (0.49) 95	0 (0.16) 96	0 (0.16) 95
Cox		1 (0.23) 94	1 (0.24) 93	3 (1.39) 94	1 (1.40) 93	3 (3.98) 92	2 (4.15) 91	0 (0.16) 95	0 (0.16) 95
Cox		0 (0.21) 94	1 (0.23) 93	1 (0.55) 95	1 (0.55) 95	1 (0.55) 95	1 (0.55) 95	0 (0.16) 98	0 (0.16) 95
Cox	None	0 (0.65) 70	1 (0.61) 71	4 (6.49) 43	3 (5.75) 51	15 (30.5) 35	16 (25.0) 37	0 (0.16) 95	0 (0.17) 95

States 1 and 2 are the non-cured and cured baseline states. States 3 and 4 represent recurrence and death. Results across 500 simulations are presented using the following notation: bias (empirical variance) coverage of 95% confidence interval, each multiplied by 100.

Table 2.

Bias, variance, and coverage of logistic model parameters within the multistate cure model

Baseline hazard	2 4, 1 4 Assumption	Logistic model			# Failed (out of 500)	Run time (min/sim)
		Intercept
Scenario 1: no covariate missingness or unequal follow-up
Weibull		0 (0.33) 93	0 (0.42) 95	0 (0.44) 94	0	2.02
Weibull		0 (0.34) 94	0 (0.51) 94	0 (0.51) 94	0	2.12
Weibull		0 (0.59) 93	0 (0.41) 95	0 (0.45) 95	0	2.07
Weibull	None	1 (0.71) 93	0 (0.49) 96	0 (0.51) 97	46	2.16
Cox		0 (0.33) 93	0 (0.42) 97	0 (0.44) 95	0	7.68
Cox		0 (0.34) 93	0 (0.51) 95	0 (0.52) 95	0	8.04
Cox		1 (0.89) 84	0 (0.43) 94	0 (0.48) 95	0	7.98
Cox	None	11 (0.37) 50	2 (2.64) 60	2 (2.83) 64	1	8.46
Scenario 2: covariate missingness
Weibull		0 (0.32) 96	0 (0.44) 97	0 (0.60) 95	0	5.65
Weibull		0 (0.33) 96	0 (0.54) 97	0 (0.70) 95	1	5.71
Weibull		0 (0.63) 90	0 (0.47) 97	0 (0.60) 95	0	5.73
Weibull	None	2 (0.72) 89	0 (0.59) 95	0 (0.73) 93	141	5.57
Cox		0 (0.32) 96	0 (0.46) 96	0 (0.60) 95	0	27.5
Cox		0 (0.33) 94	1 (0.58) 96	2 (0.71) 93	0	27.3
Cox		1 (1.59) 81	1 (0.50) 96	3 (0.60) 94	0	27.4
Cox	None	6 (0.63) 80	2 (2.15) 61	2 (2.28) 67	82	26.7
Scenario 3: unequal follow-up
Weibull		0 (0.35) 96	0 (0.51) 95	0 (0.48) 96	0	8.49
Weibull		0 (0.35) 96	0 (0.64) 95	0 (0.63) 94	0	8.61
Weibull		0 (0.63) 92	0 (0.54) 94	0 (0.51) 96	0	8.56
Weibull	None	3 (0.69) 92	0 (0.66) 95	0 (0.64) 93	102	8.58
Cox		2 (0.35) 94	1 (0.53) 93	1 (0.48) 95	0	18.3
Cox		4 (0.43) 87	2 (0.77) 93	2 (0.81) 90	0	18.5
Cox		5 (1.95) 74	2 (0.60) 94	2 (0.57) 94	0	18.5
Cox	None	4 (0.70) 83	1 (2.65) 53	2 (2.44) 60	19	17.8

Results across 500 simulations are presented using the following notation: bias (empirical variance), coverage of 95% confidence interval, each multiplied by 100. The number of simulations (out of 500) with numerical issues and the median run time per simulation are also shown.

7.2. Simulation 2: multistate cure model estimates with more covariates

We simulate 500 data sets as in Scenario 1 in Simulation 1, but now we include 10 covariates in the model for each transition. All covariates are given effects of 0.5, 0.5, or 0 in and for all transitions. Additional details can be found in the supplementary material available at Biostatistics online. We then fit four different multistate cure models to the simulated data with different baseline hazards (Weibull and Cox) and different assumptions about the and transitions.

Figure 2 presents the bias and coverage for the estimated from each model fit. When we assume , we obtain essentially unbiased parameter estimates with nominal coverage under both Weibull and Cox baseline hazard assumptions. When we assume Weibull baseline hazards and no and hazard restrictions, we see some increased bias and overcoverage for estimating parameters, but otherwise we have good bias and coverage properties. However, we do see evidence of model instability as 119 out of the 500 simulations had numerical issues. When we assume Cox baseline hazards and no and hazard restrictions, we see substantial bias and/or undercoverage, particularly in estimating , , and . As in Simulation 1, we see evidence of estimation instability in general when we place no restrictions on the hazards for the and transitions.

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.

8. Application to study of head and neck cancer recurrence and death

We consider data from a cohort study of patients treated for head and neck squamous cell carcinoma (HNSCC). This study was conducted by the University of Michigan Head and Neck Specialized Program of Research Excellence (SPORE) and followed patients who were treated for HNSCC at the University of Michigan Cancer Center, the Henry Ford Hospital in Detroit, and the VA Hospital in Ann Arbor between 2003 and 2014. Details about this study can be found in Duffy and others (2008), Peterson and others (2016), and in the supplementary material available at Biostatistics online.

After treatment, patients were followed for recurrence and death. Covariate information was collected at baseline. We are interested in studying the association between these covariates and the time to HNSCC recurrence and death after treatment. Additionally, it is been well established that some head and neck cancer patients can be cured of their cancer through their primary treatment, and we are interested in identifying factors related to the underlying cure probability (Taylor, 1995; Grau and others, 1997; Cognetti and others, 2008). A previous analysis of an earlier version of these data explores time to recurrence and cure probability in a CPH cure model, but this analysis does not incorporate death information (Beesley and others, 2016).

Missing data, however, poses an additional complication. For many patients (62.3%), follow-up for recurrence was substantially shorter than for death. Human papillomavirus (HPV) status was unavailable for of the subjects, and a small amount of missingness was present in other variables. We restricted our analyzes to subjects with oropharynx, oral cavity, larynx, or hypopharynx cancer. We further restricted analyzes to subjects that appeared to clear their cancer through the initial treatment.

We fit a multistate cure model assuming Weibull baseline hazards and equal and hazards using 100 iterations of the proposed MCEM algorithm. Subjects still at risk for recurrence and death after 80 months were assumed to be cured. Missing data were imputed using the method proposed in Bartlett and others (2014) as described in Section 4.4. Unequal follow-up was handled using the approach described in Section 4.2. Standard errors were estimated by treating the most recent imputations as proper imputations (after post-processing) and estimating variance within each imputed data set using 50 bootstrap samples. Rubin’s rules were then used to obtain the final estimates for the standard errors.

Figure 3 presents the results of the multistate cure model fit to the head and neck cancer data set. Higher cancer stage and HPV negativity were associated with higher rates of recurrence for non-cured subjects. Greater age, higher cancer stage, worse comorbidities, and increased smoking history were associated with higher rates of death from other causes for both cured and non-cured subjects. Higher cancer stage and increased smoking history were associated with higher rates of death after recurrence, and larynx site was associated with lower rates of death after recurrence compared to oral cavity cancer. Higher cancer stage and HPV negativity were associated with lower rates of cure. Additional details about the data set, this model fit, and a comparison to a standard illness-death model fit are included in the supplementary material available at Biostatistics online.

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.

9. Discussion

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. Additionally, multistate cure models can be very useful for prediction. However, the previous method for fitting multistate cure models requires substantial custom programming, making multistate cure models less accessible to analysts. We are interested in developing methods for fitting multistate cure models that can be implemented more easily and can incorporate different modeling assumptions into the fitting procedure.

In this article, we proposed an Expectation–Maximization (EM) algorithm for fitting the multistate cure model using maximum likelihood. The proposed algorithm can be fit using standard software, can incorporate either parametric or non-parametric baseline hazards for the state transition rates, and can integrate parameter restrictions for the transitions to death from other causes for cured and non-cured subjects. We then 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, and we provide some software.

In simulations, the proposed EM and MCEM algorithms demonstrate good bias and coverage properties when the modeling assumptions are sufficiently restrictive. Additionally, we can still see good model fitting performance when we include more covariates in the model. When the and restrictions are relaxed (particularly when the baseline hazards are not restricted to be equal), we can run into numerical problems in some settings, suggesting that care should be taken to make reasonably restrictive modeling assumptions for these transitions. Additional exploration (not shown) suggests that these numerical issues stem from identifiability issues related to the and model parameters that occur when we have weaker restrictions for the two transitions. When applying the multistate cure model to a particular data set, we recommend fitting a model under different assumptions about the and transitions and evaluating convergence properties and goodness of fit to determine if the and hazard restrictions can be reasonably relaxed. See Conlon and others (2013) for a discussion of goodness of fit diagnostics for multistate cure models.

We applied the proposed MCEM algorithm to study cancer recurrence and death rates in subjects with head and neck cancer. Higher age, worse comorbidities, and increased smoking were associated with higher rates of death from other causes for both cured and non-cured subjects. Increased smoking was also associated with higher rates of death after recurrence, and larynx subsite was associated with lower rates of death after recurrence. HPV positive subjects had significantly lower rates of recurrence and higher rates of cure. Higher cancer stage was significantly associated with all transition rates and the probability of being cured.

Given the various types of imputation required, the relative advantages of the MCEM algorithm over the Bayesian MCMC algorithm should be considered. One disadvantage of the Bayesian approach (even if we apply our proposed imputation methods) is that we require accept–reject methods to draw the parameters, which involves careful consideration of parameter tuning, acceptance rates, and mixing. In contrast, the M-step of the MCEM algorithm is very simple to perform. As a result, the MCEM algorithm can perform estimation more quickly than the Bayesian Markov chain monte carlo. However, unlike with Bayesian approaches, the standard errors of using MCEM are not readily available. Additionally, the Bayesian approach allows the user to more directly incorporate prior assumptions into the estimation. The relative merits of the two approaches may depend on the data and the experience of the analyst.

As illustrated by the head and neck cancer example, multistate cure models offer investigators a useful tool for identifying factors involved in different parts of the disease process, and they can be used for prediction and in medical decision-making. Multistate cure models can be applied in a variety of settings and are certainly not limited to the study of cancer. In this article, we developed methods to make multistate cure models easier to fit in practice. Previous work focused on the setting with fully parametric baseline hazards, and our proposed methods allow us to choose parametric or non-parametric baselines and incorporate different assumptions about the transitions to death from other causes. The novel imputation-based approach for dealing with unequal follow-up proposed as part of the MCEM algorithm can be applied in general semi-competing risks settings. Additionally, we propose a novel approach for obtaining standard errors for an MCEM algorithm, which can be applied in other MCEM settings. The proposed methods provide a convenient estimation method for fitting the multistate cure model and increased flexibility in model specification over existing methods.

10. Software

Software in the form of an R package called MultiCure is available on GitHub. This package provides functions for fitting the multistate cure model via the proposed EM and MCEM algorithms and for estimating corresponding standard errors. This package also includes functions for estimating the derived state occupancy probabilities.

Funding

National Institutes of Health (NIH) (CA83654 and CA129102).