Bootstrap confidence bands for sojourn distributions in multistate semi-Markov models with right censoring

Ronald Butler and Douglas Bronson have a new paper in Biometrika. This develops approaches to non-parametric estimation of survival or hitting time distributions in multi-state semi-Markov models with particular emphasis on cases where (multiple) backward transitions are possible. The paper is an extension of the authors' previous JRSS B paper. Here the methodology additionally allows for right-censored observations.

The key property used throughout is that, under a semi-Markov model, the data can be partitioned into separate independent sets relating to the sojourn and destinations for each state.
A straightforward, but computationally intensive, approach to finding bootstrap confidence bands for the survival distribution of a semi-Markov process involves a nested re-sampling scheme, first sampling with replacement from the set of pairs of sojourn times and destination states for each state in the network, and for each set of data simulating a large number of walks through the network (starting from the initial state and ending at the designated 'hitting' state) from the empirical distributions implied from the first layer of re-sampling. An alternative, faster approach proposed involves replacing the inner step of the bootstrap with a step which computes saddlepoint approximations of the empirical sojourn distributions and then using flowgraph techniques to find the implied hitting time distributions. A drawback of both approaches is that they require all the mass of the sojourn distributions to be allocated. Hence, in the presence of right-censoring some convention for the mass at the tail of the distribution must be made. Here, the convention of "redistribute-to-the-right" is used, which effectively re-weights the probability mass of all the observed survival times so that it sums to 1. On the face of it this seems a rash assumption. However, right-censoring in the sample only occurs at the end of a sequence of sojourns in several states. As such, in most cases where this technique would be used, the censoring in any particular state is likely to be quite light, even if the overall observed hitting times are heavily censored. Alternative conventions on the probability mass (for instance assuming a constant hazard beyond some point) could be made, but in all cases would be arbitrary, but hopefully would have little impact on the overall estimates.

Unlike non-parametric estimates under a Markov assumption, for which the overall survival distribution will essentially equal the overall Kaplan-Meier estimate, with increasing uncertainty as time increases, under a (homogeneous) semi-Markov assumption the estimated hazard tends to a constant limit and can hence be estimated to a relatively large degree of precision at arbitrarily high times.

Estimating treatment effects with treatment switching via semicompeting risks models: an application to a colorectal cancer study

Donglin Zeng, Qingxia Chen, Ming-Hui Chen and Joseph Ibrahim have a new paper in Biometrika. This considers the problem of comparing survival times in clinical trials in which there may be an intermediate disease progression event which may cause the treatment (if initially randomized to control) to be switched. Previous approaches to this problem have been proposed, through univariate survival methods. Here, the authors instead consider a multi-state (semi-competing risks) model. Interest lies in determining the survival distribution under a treatment given no switching.

In line with the other recent Biometrika paper, a pattern-mixture type parametrization is adopted in that a logistic regression component is defined for the probability of progression before death and separate conditional hazard function for time to death given no-progression and time to progression given progression, plus time to death from progression. Switching is assumed to be non-informative of outcome given that progression has occurred and the act of switching has a proportional effect on the hazard of death from progression.

Besides rigorous proofs of results, there doesn't seem to be any substantial conceptual advances in the paper, though it does seem to represent a better approach to the specific problem than the previous approaches.

Bayesian analysis of multistate event history data: beta-Dirichlet process prior

Yongdai Kim, Lancelot James and Rafael Weissbach have a new paper in Biometrika. This develops a conjugate prior process suitable for non-parametric and semi-parametric Bayesian modelling of right-censored multi-state Markov data. The model is parametrised in terms of the sum of the intensities out of each state

and instantaneous transition probabilities

A possible choice for a prior process is a Dirichlet distribution but this is not independent in the limit of a continuous time process. Instead the authors propose a new beta-Dirichlet process consisting of a beta distributed part which determines the increment in (between 0 and 1) and a Dirichlet part determining the instantaneous transition probabilities for each particular transition. The authors prove this prior process is conjugate in the continuous limit.

A semi-parametric regression model is proposed, which the authors term as a semi-proportional intensities model. This consists of a proportional intensities model for the all-cause hazard of exiting state h and a multinomial type model for the instantaneous transition probabilities out of state h and bears some resemblance to the vertical modeling parametrization for competing risks regression.

In an aside the authors claim that interval censoring can easily be dealt with by treating the unknown transition time as missing data that can be accounted for in the Gibbs sampling. This only works under the assumption that only one transition can have occurred between examination times. While other authors have made this assumption (e.g. Foucher et al 2007) it is dubious to say the least and likely to result in biased estimates. Similarly, the authors claim right-censoring can be dealt with by treating a censoring event as an additional state. While this will obviously allow the observed process to be modelled, it is not clear how this approach would allow the underlying process (without censoring) is estimated?

Nonparametric inference for competing risks current status data with continuous, discrete or grouped observation times.

Marloes Maathuis and Michael Hudgens have a paper available at arXiv.org. This concerns nonparametric inference for competing risks current status data. They consider a naive estimator, which estimates each cumulative incidence function independently using the pooled-adjacent-violators algorithm. Additionally, the NPMLE is also considered. The naive estimator, while consistent, does not guarantee that the sum of CIFs is less than 1. Moreover, previous work by Groeneboom et al suggests it is less efficient. However, the naive estimator has the advantage (in addition to being computationally simpler) that the limiting distribution of the estimates of each CIF is known (being the same as standard current status survival data) and results regarding the likelihood ratio statistic (Banerjee and Wellner 2005) can be applied to get confidence intervals for the CIFs. These results are in the case of a smooth observation distribution. The authors note that if subjects can only be observed in a (finite) pre-defined grid of time points then obviously the CIFs can only be estimated at these time points but also, since the number of parameters cannot increase indefinitely, standard n^1/2 asymptotics apply.

Related to this work is the R package MLEcens developed by Marloes Maathuis. This computes the NPMLE for bivariate interval censored data. Special cases include competing risks data and standard survival data. Moreover the implementation seems to run considerably faster than the package Icens.

*Update: A video of Marloes Maathuis demonstrating MLEcens is available here.

**Update: The paper is now published in Biometrika.