Simulation Based Confidence Intervals for Functions with Complicated Derivatives

Micha Mandel has a new paper in the American Statistician. This concerns the simulation based Delta method approach to obtaining asymptotic confidence intervals which has been used quite extensively for obtaining confidence intervals of transition probabilities in multi-state models. This essentially involves assuming and constructing confidence intervals for by simulating and then considering the empirical distribution of

A formal justification for the approach is laid out and some simulations of its performance compared to the standard Delta method in some important cases is given. It is stated that the utility of the simulation method is not in situations when the Delta method fails, but in situations where the calculating the derivatives needed for the Delta method is difficult. In particular, it still requires the functional g to be differentiable. This seems to down play the simulation method slightly. One advantage of the simulation method is that it is not necessary to linearize g around the MLE. To take a pathological example consider data that are . Suppose we are interested in estimating . When the coverage of the Delta method confidence interval will be anti-conservative because has a point of inflection at . The simulation based will work fine in this situation (as obviously would just constructing a confidence interval for and just cubing the upper and lower limits!).

The gradient function as an exploratory goodness-of-fit assessment of the random-effects distribution in mixed models

Geert Verbeke and Geert Molenberghs have a new paper in Biostatistics. The paper proposes the use of the gradient function (or equivalently the directional derivatives) of the marginal likelihood with respect to the random effects distribution, as a way of assessing goodness-of-fit in a mixed model. They concentrate on cases related to standard longitudinal data analysis using linear (or generalized linear) mixed models, however the method can be extended to other mixed models, such as clustered multi-state models with multivariate (log)-normal random effects.

If we consider data from units i with observations x_i, given a mixing distribution G, we can say the marginal density is given by .

The gradient function is then taken as where N is the total number of independent clusters

The use of the gradient function stems from finite mixture models and in particular the problem of finding the non-parametric maximum likelihood estimate of the mixing distribution. At the NPMLE the gradient function has a supremum of 1. If instead we assume there is a parametric mixing distribution, under correct specification the gradient function should be close to 1 across all values of u. Verbeke and Molenberghs use this property to construct an informal graphical diagnostic of the appropriateness of the proposed random effects distribution.

An advantage of the approach is that essentially no additional calculations are required to compute the measure, above and beyond those already needed for estimation of the parametric mixture model itself. A current limitation of the approach is that there is no formal test to assess whether the observed deviation is statistically significant. It is stated that this is ongoing work. It seems reasonably straightforward to show that the gradient function will tend to a Gaussian process with mean 1 but with a quite complicated covariance structure. Obtaining some nice asymptotics for a statistic based either on the maximum deviation from 1 or some weighted integral of the distance from 1 therefore seems unlikely. However, it may be possible to obtain a simulation based p-value by simulating from the limiting Gaussian process.

Book Review of: Competing Risks and Multistate Models with R.

Ross Maller has written a book review of Beyersmann, Schumacher and Allignol's recent Springer book on Competing Risks and Multistate Models with R, published in Australian & NZ Journal of Statistics. This is primarily a rant against the cause-specific hazard approach to modelling competing risks. For instance cause specific hazards "do not explicitly take into account the obvious mixture of distributions inherent in the data." Moreover, the fact that assuming proportionality in cause-specific-hazards (CSHs) can lead to non-proportional, even crossing, relationships for cumulative incidence functions (CIFs), is painted as a terminal weakness to the approach.

Maller's main contribution to survival analysis is through models for cure fractions (see e.g. Maller and Zhou 1995) an approach that he is evidently very taken with. Apparently the correct approach to take in modelling competing risks data is to assume a finite mixture model, such that individuals in a particular class are only at risk of one particular failure rate. Moreover, the problem of covariates is claimed to be entirely solved by allowing proportional hazards within failure types, which Maller says is the approach taken by Fine and Gray (1999).

The entire nature of survival and event history analysis is in modelling the dynamics of the process. In most circumstances it is much more useful to be able to describe the process at time t given no event has occurred by time t than to describe the process conditional on a latent class membership. Moreover, in the vast majority of competing risks data, at least in medical contexts, all patients are at risk of all event types until experiencing an event. A mixture model could therefore only ever be viewed as a mathematical convenience. The fact that in practice a CSH method is actually substantially more convenient, particularly if a non- or semi-parametric approach is to be adopted, hardly aids the case for mixture models.

Maller is also misrepresenting the Fine-Gray approach which does not assume proportional hazards within failure types. The Larson-Dinse (1985) paper that Maller also cites does involves that approach. But that can lead to the same crossing cumulative incidence curves Maller takes issue with in the context of CSH. Fine-Gray assumes proportionality of the sub-distribution hazard for a particular cause. This does allow proportionality for that cause's corresponding CIF but, as a consequence, is unable to provide a covariate model for other CIFs that is guaranteed to lead to a feasible set of CIFs for all covariate values (ie. we can fit a Fine-Gray model to each cause of failure but the resulting models will be contradictory).

Fundamentally, whatever model that is postulated, we can find the implied cause-specific hazards. Assuming proportionality of the cause-specific hazards is obviously only a modelling assumption but in nearly all cases it will be a better starting point than assuming the existence of cure fractions.

Survival analysis with time varying covariates measured at random times by design

Stephen Rathbun, Xiao Song, Benjamin Neustiftler and Saul Shiffman have a new paper in Applied Statistics (JRSS C). This considers estimation of a proportional hazards survival model in the presence of time dependent covariates which are only intermittently sampled at random time points. Specifically they are interested in examples relating to ecological momentary assessment where data collection may be via electronic devices like smart phones and the decision on sampling times can be automated. They consider a self-correcting point-process sampling design, where the intensity of the sampling process depends on the past history of sampling times, which allows an individual to have random sampling times that are more regular than would be achieved from a Poisson process.

The proposed method of estimation is to use inverse intensity weighting to obtain an estimate of an individual's integrated hazard up to the event time. Specifically the estimator is for a individual with sampling times at times and point process sampling intensity . This then replaces the integrated hazard in an approximate log-likelihood.

In part of the simulation study and in an application the exact point process intensity is unknown and taken from empirical estimates from the sample. Estimating the sampling intensity didn't seem to have major consequences on the integrity of the model estimates. This seems to suggest the approach might be applicable in other survival models where the covariates are sampled longitudinally in a subject specific manner, provided a reasonable model for sampling can be devised.

Drawbacks of the method seem to be that covariates measured at baseline (which is not a random time point) cannot be incorporated in the estimate and that it seems that the covariates must be measured at the event time which may not be the case in medical contexts. The underlying hazard also needs to be specified parametrically, but as stated flexible spline modelled can be used.

Estimating parametric semi-Markov models from panel data using phase-type approximations

Andrew Titman has a new paper in Statistics and Computing. This extends previous work on fitting semi-Markov models to panel data using phase-type distributions. Here, rather than assume a model in which each state is assumed to have a Coxian phase-type sojourn distribution, the model assumes a standard parametric sojourn distribution (e.g. Weibull or Gamma). The computational tractability of phase-type distributions is exploited by approximating the parametric semi-Markov model by a model in which each state has a 5-phase phase-type distribution. In order to achieve this, a family of approximations to the parametric distribution, with scale parameter 1, is developed by solving a relatively large one-off optimization assuming the optimal phase-type parameters for given shape parameters evolve as B-spline functions. The resulting phase-type approximations can then be scaled to give an approximation for any combination of scale and shape parameter and then embedded into the overall semi-Markov process. The resulting approximate likelihood appears to be very close to the exact likelihood, both in terms of shape and magnitude.

In general a 2-phase Coxian phase-type model is likely to give similar results to a Weibull or Gamma model. The only advantages of the Weibull or Gamma model are that the parameters are identifiable under the null (Markov) model so standard likelihood ratio tests can be used to compare models (unlike for the 2-phase model). Also the Weibull or Gamma model requires fewer parameters so may be useful for smaller sample sizes.

Constrained parametric model for simultaneous inference of two cumulative incidence functions

Haiwen Shi, Yu Cheng and Jong-Hyeon Jeong have a new paper in Biometrical Journal. This paper is somewhat similar in aims to the pre-print by Hudgens, Li and Fine in that it is concerned with parametric estimation in competing risks models. In particular, the focus is on building models for the cumulative incidence functions (CIFs) but ensuring that the CIFs sum to less than 1 at the asymptote as time tends to infinity. Hudgens, Li and Fine dealt with interval censored data but without covariates. Here, the data are assumed to be observed up to right-censoring but the emphasis is on simultaneously obtaining regression models directly for each CIF in a model with two risks.

The approach taken in the current paper is to assume that the CIFs will sum to 1 at the asymptote, to model the cause 1 CIF using a modified three-parameter logistic function with covariates via an appropriate link function. The CIF for the second competing risk is assumed to also have a three-parameter logistic form, but covariates only affect this CIF through the probability of this risk ever occurring.

When a particular risk in a competing risks model is of primary interest, the Fine-Gray model is attractive because it makes interpretation of the covariate effects straightforward. The model of Shi et al seems to be for cases where both risks are considered important, but still seems to require that one risk be considered more important. The main danger of the approach seems to be that the model for the effect of covariates on the second risk may be unrealistic, but will have an effect on the estimates for the first risk. If we only care about the first risk the Fine-Gray model would be a safer bet. If we care about both risks it might be wiser to choose a model based on the cause-specific hazards, which are guaranteed to induce a model with well behaved CIFs albeit at the expense of some interpretability of the resulting CIFs.

Obtaining a model with a direct CIF effect for each cause seems an almost impossible task because, if we allow a covariate to effect the CIF in such a way that a sufficiently extreme covariate leads to a CIF arbitrarily close to 1, it must be having a knock-on effect on the other CIF. The only way around this would be to have a model that assigns maximal asymptote probabilities to the CIFs at infinity that are independent of any covariates e.g. where are increasing functions taking values in [0,1] and . The need to restrict the to be independent of covariates would make the model quite inflexible however.

A semi-Markov model for stroke with piecewise-constant hazards in the presence of left, right and interval censoring

Venediktos Kapetanakis, Fiona Matthews and Ardo van den Hout have a new paper in Statistics in Medicine. This develops a progressive three-state illness-death model for strokes with interval-censored data. The proposed model is a time non-homogeneous Markov model. The main approach to computation is to assume a continuous effect of age (time) on transition intensities but to use a piecewise constant approximation to actually fit it. The intensity to death from the stroke state additionally depends on the time since entry into the state (i.e. age at stroke) and since the exact time is typically unknown, it is necessary to numerically integrate over the possible range of transition times (here using Simpson's rule).

The data include subjects whose time to stroke is left-censored because they have already suffered a stroke before the baseline measurement. The authors state that they cannot integrate out the unknown time of stroke because the left-interval (i.e. the last age at which the subject is known to have been healthy) is unknown. They then proceed to propose a seemingly unnecessary ad-hoc EM-type approach based on estimating the stroke age for these individuals, which requires the arbitrary choice of an age at which it can be assumed the subject was stroke free. However, surely if we can assume a , we can just use as the lower limit in the integral for the likelihood?

The real issue seems to be that all subjects are effectively left-truncated at the time of entry into the study (in the sense that they are only sampled due to not having died before their current age). For subjects who are healthy at baseline this left-truncation is accounted for by just integrating the hazards of transition out of state 1 from their age at baseline rather than age 0. For subjects who have already had a stroke things are more complicated because the fact they have survived provides information on the timing of the stroke (e.g. if stroke increases hazard of death, the fact they have survived implies the stroke occurred sooner than one would assume if no information on survival were known). Essentially the correct likelihood is conditional on survival to time and so the unconditional probability of the observed data needs to be divided through by the unconditional probability of survival to time . For instance, in their notation, a subject in state 2 at baseline censored at time should have likelihood contribution: The authors claim that their convoluted EM-approach has "bypassed the problem of left truncation". In reality, they have explicitly corrected for left-truncation (because the expected transition time is conditional on being in state 2 at baseline) but in a way that is seemingly much more computationally demanding than directly computing the left-truncated likelihood would be.

Mixture distributions in multi-state modelling: Some considerations in a study of psoriatic arthritis

Aidan O'Keeffe, Brian Tom and Vern Farewell have a new paper in Statistics in Medicine. This considers random effects models for clustered multi-state models, specifically considering the psoriatic arthritis example considered in their previous paper. The particular emphasis in the current paper is comparing models using a continuous gamma frailty term with an extended model that additionally allows a "stayer" component. The latter can be thought of as a joining of the continuous random effects model (e.g. Cook et al 2004) and the "mover-stayer" model (e.g. Cook et al 2002). In a discussion on random effects, particularly where there is a question of finite mass points, it is strange there is no mention of the non-parametric mixing distribution (Laird, JASA 1978). While this hasn't really been considered for continuous time processes (the nearest example is Frydman's model) it wouldn't be particularly hard to implement at least the (not entirely reliable) EM algorithm type approach used in discrete-time by Maruotti and Rocci. The apparent presence or absence of a "stayer" component is likely to be heavily dependent on the parametric assumptions made about the rest of the mixing distribution. A very small random effect is indistinguishable from a zero random effect. The authors do emphasize the need to consider several possible mover-stayer models and consider the biological plausibility of them. It is also worth mentioning that all these issues will also hinge on the appropriateness of other assumes e.g. conditionally time homogeneous Markov processes.

Bayesian analysis of a disability model for lung cancer survival

Armero, Cabras, Castellanos, Perra, Quirós, Oruezábal and Sánchez-Rubio have a new paper in Statistical Methods in Medical Research. This develops a Bayesian three-state illness-death type model to the progression and survival of lung cancer patients. The data considered are assumed to be complete up to right censoring (in reality there may be some interval censoring but the authors argue the patients can be considered `quasi-continuously' followed. A Weibull semi-Markov model is assumed for the transition intensities and covariates are accommodated via an accelerated failure time model (it's worth noting that for a Weibull distribution the proportional hazard and accelerated failure time models are equivalent up to reparameterization). A feature of the dataset used is the rather small sample size (35 patients) which is perhaps the strongest reason for taking a parametric and Bayesian approach in this case.

Fitting and Interpreting Continuous-Time Latent Markov Models for Panel Data

Jane Lange and Vladimir Minin have a paper currently available as a University of Washington Working paper. This considers the computational aspects of fitting a latent continuous time Markov model to multi-state interval censored or panel data. The latent continuous time Markov model proposed is essentially a generalization of the phase-type semi-Markov models considered by Titman and Sharples. It is assumed that for each observable state r, there exist latent states: Observation in state r can correspond to occupation into any one of these latent states. This induces a type of hidden Markov model and can allow the observed process to be semi-Markov. Lange and Minin propose an EM algorithm to fit the model and compare this with other possible optimization techniques, concluding that their EM algorithm generally performs better, both in terms of reliability and speed of convergence. On this issue, it might have been worth considering the performance of, say, L-BFGS-B (or other constrained optimization methods) using the natural [0,Inf) range for the transition intensities, because many convergence problems for BFGS are due to cases where the true value is near (or at) zero and the exponential parameterization means it never gets there. **Update: This paper is now published in Statistics in Medicine.

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.

Intermittent observation of time-dependent explanatory variables: a multistate modelling approach

Brian Tom and Vern Farewell have a new paper in Statistics in Medicine. This considers the problem of estimating the effect of a time dependent covariate on a multi-state process when both the disease process of interest and the time dependent covariate are only intermittently observed. The most common existing approach to dealing with this problem is to assume that the time dependent covariate is constant between observations, taking the last observed value. The authors instead jointly model the two processes as an expanded multi-state model, if the disease process had n states and the covariate process m states the resulting process will have states. An additional assumption, that movements in the covariate process are not directly affected by the state of the disease process, is also made.

A simulation study is performed which shows that the approach of assuming the time dependent covariate is constant leads to biased estimates, particularly when there is a bias in the trend of the covariate process (e.g. much more likely to decrease than increase in value).

The overall approach taken by the authors is to model their way out of difficulty. They assume that both the disease process of interest and the covariate process are jointly time homogeneous Markov and the validity of the results will depend on these assumptions being correct. As noted by the authors, if the covariate can take more than a small number of values the approach becomes unattractive because of the large number of nuisance parameters required. A point not really emphasized, but related to the analogous approach taken by Cortese and Andersen for continuously observed competing risks data (bizarrely not referenced in this paper despite massive relevance!), is that having modelled the time dependent covariate, the model can then be used to make overall predictions.

One could argue that the convention of following forward the covariate value observed from the previous period is a way of allowing a prediction to be made about the trajectory in the next period. A fairer comparison in some cases might therefore be to look at the bias in estimating the transition probabilities to time given a covariate variate value of at time . While we would expect these estimates still to be biased, the amount of bias is likely to be less than found by looking at the regression coefficients directly.

An open problem seems to be the development of methods that do not require strong assumptions, or else are robust to misspecification, to deal with intermittently observed time dependent covariates.

Impact of delayed diagnosis time in estimating progression rates to hepatitis C virus-related cirrhosis and death

Bo Fu, Wenbin Wang and Xin Shi have a new paper in Statistical Methods in Medical Research. The paper is interested in the estimation of rates of progression from hepatitis C infection to cirrhosis and from cirrhosis to death. The primary complication in the model is that the time of cirrhosis development is interval censored, since it is only diagnosed at clinic examination times.

The model consists of a simple unidirectional 3-state model. A parametric Weibull approach is taken, where the time to cirrhosis from infection has a standard Weibull distribution and the time from cirrhosis to death has a Weibull distribution where the scale parameter is a function of the time between infection and cirrhosis.

Unsurprisingly, they show that a full likelihood approach which takes into account the interval censoring is much less biased than an approach that ignores the problem of interval censoring.

A criticism of the model, which the authors acknowledge, is that the possibility of death before cirrhosis is not accommodated. Potentially, even under their current model there might also be cases where death occurs after cirrhosis but before diagnosis of cirrhosis - a scenario which isn't accommodated in the three cases listed in their likelihood development. The rate of increase of the hazard of cirrhosis from infection is observed to increase after 30 years since infection. It is not clear whether this is a real effect, whether it is due to a small number of people at risk or whether it is an artifact of not accommodating the possibility of death without cirrhosis. Given the long follow-up time in the study it might have been more sensible to consider age rather than either time since cirrhosis or time to infection as the primary timescale, at least for time to death.

Finally, whilst the approach might be novel for HPV patients, the 3-state approach is clearly widely used in other contexts, most specifically in HIV/AIDS studies. For instance the penalised likelihood approach of Joly and Commenges would be highly applicable.

A multi-state model for the analysis of changes in cognitive scores over a fixed time interval

Arnold Mitnitski, Nader Fallah, Charmaine Dean and Kenneth Rockwood have a new paper in Statistical Methods in Medical Research. The paper develops a model to describe the trajectory of cognitive function test data. The responses are test scores out of 100, but the authors choose to group responses in 12 states. Additionally subjects may die before the following assessment. Assessments occur at (roughly) equally spaced intervals so a discrete-time model is adopted. Essentially the data are then ordinal longitudinal data.

The novel aspect of the model is to assume that, conditional on survival between time j-1 and j, the state at time j follows a truncated Poisson distribution on , with the mean of the Poisson distribution taken as a linear function of the state at time j and covariates (including age and/or time since baseline measurement). A separate logistic regression model is applied to the deaths. This avoids having to pretend the data are continuous as one might if a linear mixed model were used, and also avoids there being a very large number of unknown parameters, as there would be if a general discrete-time Markov model were applied. However, the truncated Poisson distribution model makes strong assumptions about the conditional distribution of the states, which may or may not be well supported by the data. The authors note that the model fits better if 12 states are used rather than 16. Whether accommodating the proposed model should be a criterion for choosing the number of states to use in the model is questionable.

It is not clear a multi-state model is particularly appropriate for modelling a response with such a large number of responses. It might be better to follow a latent trait approach where for some Normally distributed latent variable that evolves with time deterministically with the addition of stationary Gaussian noise (not necessarily independent), where the are boundary values to be estimated. Similarly existing approaches to modelling MMSE based on a much simpler classification into cognitive-normal and cognitive-impaired with the possibility of misclassification considered (see e.g. Van den Hout and Matthews) are likely to give more meaningful results, even if data from raw MMSE scores is sacrificed.

Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial

Ralitza Gueorguieva, Robert Rosenheck and Haiqun Lin have a new paper in JRSS A. The paper concerns the joint modelling of a longitudinal outcome and an interval censored competing risks outcome that explains drop-out. As is common with these joint longitudinal and survival types of models the two processes are linked via a normally distributed vector of random effects. The novelty of the paper is in the survival part is a competing risks process and the event time is interval censored. The authors adopt a parametric model for the competing risks, using the family of distributions proposed by Sparling et al (Biostatistics, 2006). This makes inference somewhat more straightforward than it would be if a non-parametric baseline cause-specific hazards were used. As recently noted, parametric treatment of competing risks data is surprisingly rare. One problem faced by the authors is that the hazard family of Sparling, while allowing closed form expressions for interval censored univariate survival data, do not result in closed form expressions for interval censored competing risks data (except in special cases). Instead a numerical integral has to be competed. The presence of the overall random effects would mean the likelihood requires nested integration. To avoid this problem the authors adopt an approximation to the true likelihood for competing risks data. If a patient is known to have had a failure of type j in the interval [t0,t1] the authors assume that the patient is censored of all risks except risk j at time t0. It is clear that this approximation will lead to systematic bias as the time at risk from each failure type will be underestimated so the hazards will tend to be overestimated. The amount of bias will depend on the typical length of the intervals [t0,t1].

For the CATIE data example the proposed approximation is probably not an issue. The drop out (competing risks) part of the model is not the primary focus of the inference, and it is really the relative hazards of different types of drop out rather than their absolute values that is important in determining the trajectories of the longitudinal measure without drop out. For instance the estimates for simulated data of a similar type are close to unbiased.
However in extreme cases like current status competing risks data the approximation will do extremely badly.

Parametric inference for time-to-failure in multi-state semi-Markov models: A comparison of marginal and process approaches

Yang Yang and Vijayan Nair have a new paper in Canadian Journal of Statistics as part of the Kalbfleisch and Lawless special issue. The paper considers inference for progressive semi-Markov processes for complete, right-censored and interval-censored data for parametric models with Gamma or inverse Gamma sojourn distributions. The advantage of considering Gamma or inverse Gamma distributions is that their convolutions have a closed form meaning the overall failure (absorption) time distribution is available in closed form. Inference using only the time-to-failure data is therefore relatively straightforward. The main focus of the paper is investigating the loss in efficiency of only using time-to-failure data when additional data on intermediate states (either through continuous observation or panel data) are available. The authors show that the loss in efficiency can be rather substantial, particularly if interest lies in making predictions about time to failure given an existing process history rather than just mean or median survival. It is therefore suggested that information on intermediate states should be incorporated where possible despite this presenting computationally difficulties in the case of panel data.

Combined survival analysis of cardiac patients by a Cox PH model and a Markov chain

Michal Shauly, Gad Rabinowitz, Harel Gilutz and Yisrael Parmet have a new paper in Lifetime Data Analysis. This considers methods for modelling the effect of a mixture of time dependent and time constant covariates on overall survival, with the complication that the time dependent covariates are only observed at a discrete set of time points. They propose to firstly fit a Cox proportional hazard model assuming all covariates are fixed at their baseline values. The main modelling approach is to assume a discrete-time homogeneous Markov model with states corresponding to the combinations of the time dependent covariates (which are categorical or need to be categorized) and death. Transitions between all covariates states are assumed to be possible between each time point. The Cox model is used to determine which of the constant covariates should be considered in the Markov model. For this the authors propose to again categorize the covariates and consider a separate Markov model for each level of the covariates. Having obtained the estimates from the Markov models, it is then possible to calculate expected survival times for patients conditional on their baseline characteristics.

In general the approach proposed is reasonably sensible. However, there is panel data available for the time dependent covariates. It therefore seems possible to fit a time continuous Markov model to the data using methods appropriate for panel data (e.g. Kalbfleisch and Lawless, 1985). This approach has the advantage that the exact time of the death events can still be used.

The authors rely on categorization throughout. While this seems necessary for the time dependent covariates, there seems scope for using multinomial logit models for other covariates. Similarly, by allowing a different mortality probability for each combination of covariates they are effectively fitting covariate models with interactions (i.e. the effect of being in covariate level 2 compared to 1, is different depending on which level(s) of the other time dependent covariate(s) a subject is in). While such interactions may be necessary, it might be better to allow simpler models where only the evolution of the time dependent covariates is kept general. This is another advantage of a continuous time model since covariate effects could remain on the hazard (transition intensity) scale as in the Cox PH model.

Finally, the authors give a partial justification of the use of a time homogeneous Markov model through the Cox PH model having an approximately constant baseline hazard. It should be noted that a time homogeneous Markov model does not imply a constant absorption hazard (unless the model begins in the quasi-stationary distribution). Conversely, while a constant hazard might suggest homogeneity more than non-homogeneity, it is nevertheless possible to construct non-homogeneous processes with constant (or near constant) marginal absorption hazards. The authors do however report a statistic which gives a better justification of homogeneity.

On inference from Markov chain macro-data using transforms

Martin Crowder and David Stephens have a new paper in Journal of Statistical Planning and Inference. This considers estimation of a discrete-time homogeneous Markov chain from aggregate data (which they term macro-data), ie. where only the overall state counts for N patients are known at time j, while the transition counts are unknown. The likelihood for such data is intractable because it involves summing over the vast number of possible transitions that are consistent with the observed aggregate counts. As a result, inference methods have focused on moment based estimation (see e.g. Kalbfleisch, Lawless and Vollmer, Biometrics 1983) which are reasonably effective in practice.

Crowder and Stephens note that the probability generating function for the observed aggregate counts has a fairly simple form. This motivates an estimation procedure based on trying to match the quantities with their expectations (i.e. the pgfs). An obvious practical issue is the choice of vectors to use to compare the closeness between the pgf and sample quantities. The authors propose choices that avoid computational problems for large sample sizes.

Through a series of simulations, the authors demonstrate that an improvement in efficiency compared to methods based on second moments is possible for small sample sizes (e.g. n <= 25). A comparison is made with the efficiency for micro-data (i.e. where transition counts are known). However, for such small samples computation of the full likelihood for the aggregate data must become a viable option. The practical issues are whether the pgf based approach gives any real improvement in efficiency compared to second moment approaches for say n=100 (the second moment approaches are asymptotically efficient with appropriate weights) or whether the pgf method outperforms (or matches) the full likelihood for small samples sizes (i.e. n<=25) where the full likelihood is calculable. These issues aren't really addressed in the paper.

Progression of liver cirrhosis to HCC: an application of hidden Markov model.

Nicola Bartolomeo, Paolo Trerotoli and Gabriella Serio have a new paper in BMC Medical Research Methodology. This applies a three state hidden Markov model to data on the progression of liver cirrhosis to Hepatocellular carcinoma. A time homogeneous continuous time progressive model is fitted with death as the absorbing state. Covariate effects are included via a proportional intensities model.
Schoenfeld residuals, which are appropriate for right censored data, are applied here as a test of proportionality. It isn't made clear precisely how this is done here. If time of death is known exactly then a Schoenfeld type residual could be defined for the times of death replacing the standard formulation



with



where If the times of death are interval censored then this approach is inappropriate.

On a more trivial level the matrix of misclassification probabilities is missing a 1 in the third row corresponding to the absorbing state.