Estimating survival of dental fillings on the basis of interval-censored data and multi-state models

Pierre Joly, Thomas Gerds, Vibeke Qvist, Daniel Commenges and Niels Keiding have a new paper in Statistics in Medicine. This considers the estimation of survival times of dental fillings from interval censored data. A particular feature of the data is that there is inherent clustering in the form of multiple fillings from the same child.

Bizarrely the authors claim "we are not aware of any paper combining multi-state models for clustered data with interval censoring" implying they (and presumably the referees as well) are unaware of both Cook, Yi, Lee and Gladman (Biometrics, 2004) and
Sutradhar and Cook (JRSS C, 2008), the latter having "clustered", "multistate" and "interval-censored" all in the title!

A progressive four-state model is assumed for each filling, with the states consisting of Treatment, Filling failure, endodontic complication and exfoliation (an absorbing state).
For exfoliation, age of child is taken as the time scale meaning that the time of treatment is taken as a left-truncation time. For transitions from treatment to the other two states, time since treatment is taken as the time scale. Weibull transition intensities are assumed. However, monitoring ended once any filling event (filling failure or endodontic complication) had occurred. Because the exact time of an endodontic complication is known if it occurred during the monitoring period, the transition intensity from endodontic complication to exfoliation is not relevant to the likelihood. The authors also argue that it is necessary to assume that the intensity from filling failure to exfoliation and from treatment to exfoliation is the same, due to never being able to observe a filling failure to exfoliation transition. Strictly speaking, under the assumption of non-informative observation times, there should be some information in the data to estimate something about the separate intensities based on the proportion of cases where a filling was observed compared to the proportion where exfoliation occurred without observing a filling. Indeed in the research report by Frydman et al (2008), using a subset of the data in the current paper and a three-state verison of the model, a discrete-time NPMLE for the intensities was developed.

Random effects are incorporated into the model in a hierarchical way, with a dentist level random effect that affects the intensity to filling failure or endodontic complication and correlated child level random effects determining the correlation to time to failures (thus affecting the transitions to filling failure and endodontic complication) and time to exfoliation (thus affecting only the exfoliation transition intensity). The random effects are taken to be multivariate Normal with a log-additive effect on the intensities. Calculating the likelihood requires numerical integration, which here is achieved via Gauss-Hermite quadrature. 30 quadrature points were used - this seems a rather small number for a multi-dimensional integral. Cook et al (2004) avoided attempting to get a strict approximation to the multivariate Normal by formally restricting the random effects to have a discrete distribution. Sutradhar and Cook (2008) used an MCEM in order to apply a continuous random effects distribution. That approach is likely to be more computationally intensive that Gauss-Hermite quadrature (on 30pts) but more accurate. The recent suggestion by Putter and van Houwelingen to use a simple two-component mixture frailty could also be adaptable for this situation.

Two filling types, amalgam and glass ionomer are compared in terms of probability of surviving without complication, with amalgam performing somewhat better.

Modeling hospital length of stay by Coxian phase-type regression with heterogeneity

Xiaoqin Tang, Zhehui Luo and Joseph Gardiner have a new paper in Statistics in Medicine. This considers modelling right-censored length-of-stay data by using Coxian phase-type distributions, which are distributions defined by the times to absorption of a class of acyclic finite-state time homogeneous Markov processes. Coxian phase-type distributions have been used quite extensively to model right-censored survival data , particularly length-of-stay, see e.g. Marshall and Zenga. The novelty of the current paper is in the estimation of the model. Phase-type distributions suffer from being over parameterized and as a result suffer identifiability issues that in turn cause poor behaviour of optimization procedures. A further issue is the choice of the number of phases of the phase-type distribution. In the current paper a Bayesian reversible jump MCMC approach is taken.

Any acyclic Markov chain can be represented by a Coxian distribution in which the absolute values of the diagonal of the subgenerator matrix are decreasing. If parameters are unrestricted then there are inherent identifiability problems which would hamper the MCMC procedure. To avoid this the authors parameterize based on the first diagonal element and then the ratio (between 0 and 1) of the second element to the first, and so on, with the hazard of absorption from each phase being determined by a proportion (again between 0 and 1) of the diagonal element.

A mixed non-homogeneous hidden Markov model for categorical data, with application to alcohol consumption

Antonello Maruotti and Roberto Rocci have a new paper in Statistics in Medicine. This develops a hidden Markov model for modelling longitudinal data on alcohol consumption in discrete time. The observed data are taken to consist of a three-level ordinal variable denoting whether no drinking (0 drinks), light drinking (1–c drinks), and intense drinking (c+ drinks) occurred in that period of time. The model considered is both time non-homogeneous and mixed, in the sense that there is additional patient level heterogeneity after accounting for covariates. Rather than specifying a continuous distribution for the random effects, the authors adopt the non-parametric mixing distribution approach. Computationally, a finite mixture random effect is much simpler than having a continuous random effect if the random effect is multi-dimensional. However, computation of the full non-parametric maximum likelihood estimate of the mixing distribution is not in itself straightforward. The authors adopt the approach of Aitkin (Statistics and Computing, 1996) which is essentially to work up from a small number of components, performing an EM-algorithm at the fixed level of mixture components. EM based approaches to obtaining the NPMLE of a mixing distribution are known to perform badly and approaches using directional derivatives are preferred (see for instance Wang 2007, JRSS B). The best model, with m components, is assumed to have been reached once taking m+1 components does not produce a better model in terms of AIC or BIC. The main issue with this approach is that the EM algorithm is typically very sensitive to the initial parameter values chosen and prone to fail to find a global maximum. A further danger with these models is to ascribe too great a physical significance to the mixture components estimated.

To reach the final model, choices have to be made regarding: the categorization for the responses (observed level of drinking), the latent Markov states for the HMM (e.g. 2, 3 or 4 latent states), the number of mixture components for the random effect (how many "archetypes" of longitudinal behavior) and the degree of time non-homogeneity of transition probabilities. As a result, while the model is likely to explain the observed data reasonably well, a leap of faith is required to believe the model is an accurate representation of the process of binge drinking/alcoholism.