Effect of vitamin A deficiency on respiratory infection: Causal inference for a discretely observed continuous time non-stationary Markov process

Mingyuan Zhang and Dylan Small have a paper to appear in The Canadian Journal of Statistics, currently available here. The paper uses a multi-state model approach to obtain estimates of the causal effects of vitamin D deficiency on respiratory infection.

The observed data consist of observations of respiratory infection status, vitamin D deficiency status and whether the child is stunted at time t. Each of these is a binary variable, leading to 8 possible observation patterns.

The data are assumed to be generated from an underlying non-homogeneous Markov chain on 32 states, consisting of a latent 4-level definition of vitamin D deficiency, the stunting variable, the observed respiratory infection status and additionally a counter-factual respiratory infection status defined as the status at time t hat would have occurred had the child maintained the lowest level of vitamin D deficiency from time 0 to time t.

Let represent the observed infection status, the underlying vitamin deficiency and the counter-factual infection status, the assumed relationship between them is given by , and for j>0. then measures the additional risk of having a respiratory infection at a particular time, given a current vitamin D deficiency at level j>0.

The overall model is a pretty innovative use of a hidden Markov model structure to obtain those causal estimates. The true process is assumed to occur in continuous time. However, it is desired that the underlying transition intensities are not time constant. As a result, the authors choose to approximate the process by one in discrete time (with some similarities to the approach of Bacchetti et al 2010).

In practical terms, the weakness of the model seems to be the assumption that the relative effect of a current vitamin deficiency compared to a perfect record of vitamin D levels, is both constant in time and does not depend on the past history of vitamin D deficiency. The latter assumption is essentially the Markov assumption and would be quite difficult to relax. The observed infection status is effectively a misclassified version of the counter-factual infection status. As a result the former assumption could be relaxed by letting the depend on time, perhaps in a piecewise constant fashion.

The definition of the process as initially being continuous is a little artificial and ill-specified in places. For instance, the transition intensities are defined in terms of a logit transformation from the outset. Also, it is stated that the underlying 32 state process (including both of ) is a continuous-time Markov process. However if is defined only through the misclassification equations, there is no limiting intensity for as . Once the process is in discrete time this is not a problem, but it would be more sensible to define the underlying process in continuous time to be 16 states not including and then specify that the observed are observations in a hidden Markov model.

Estimating Discrete Markov Models From Various Incomplete Data Schemes

Alberto Pasanisi, Shuai Fu and Nicolas Bousquet have a new paper in Computational Statistics & Data Analysis. This considers approaches to Bayesian inference for time-homogeneous discrete-time Markov models under incomplete observation. Firstly, they consider where there are missing observations in a sequence of states (considering different missingness assumptions). Secondly, they consider the case of aggregate data where all that is known is the number of subjects in each state at each time. A Bayesian approach is adopted throughout, which the authors claim is the most convenient in this situation.

The part involving incomplete data follows similar ground to Deltour et al (Biometrics, 1999). The problem only becomes non-trivial if the missingness mechanism is non-ignorable.

The treatment of aggregate data is incorrect as the authors state the likelihood as being the product of independent multinomial random variables with probabilities corresponding to the probability of being in a state at time t given the initial state distribution at time 0. As a result they claim the likelihood is proportional to the case of current status data where each subject or unit is only observed once. The reason that likelihood-based inference for aggregate data is so difficult is that we observe all units multiple times but don't know number or nature of the transitions that occurred. Hence, the full likelihood would require summing over all possible transitions consistent with the aggregate counts. Kalbfleisch and Lawless (Canadian Journal of Statistics, 1984) derived the mean and covariance of the aggregated counts across times to establish a least-squares estimation procedure. Pasanisi et al's procedure is only relevant when the data consist of a series of independent cross-sectional surveys at different time points all assumed to come from different units. An MCMC or simulation based approach would be necessary to compute the exact likelihood or posterior distribution in the true aggregate data case, which the authors did not pursue. However, the gain in efficiency compared to the least-squares approach is probably not worth the trouble except for very small counts. Crowder and Stephens (2011) pursued an approach based on matching the coefficients of the probability generating function of the aggregate counts.

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.

Estimating net transition probabilities from cross-sectional data with application to risk factors in chronic disease modeling

van de Kassteele, Hoogenveen, Engelfriet, van Baal and Boshuizen have a new paper in Statistics in Medicine. This considers the estimation of the transition probabilities in a non-homogeneous discrete-time Markov model, when the only available information is cross-sectional data, i.e. for each time (or age) we have only a sample of individuals and their state occupancy from which the prevalence at that time can be estimated. Note this type of observation is more extreme than aggregate data, considered for instance by Crowder and Stephens, where we only have prevalences at a series of times but the state occupation counts correspond to the same set of subjects.

The authors take a novel, if slightly quirky approach, to estimation. They firstly use P-splines to smooth the observed prevalences. Having obtained these they then need to translate them into transition probabilities. This is not straightforward since there are more parameters to estimate than degrees of freedom. To get around this problem the authors restrict their estimate to be the values that minimize a transportation problem. Essentially this assigns a "cost" to transitions, penalizing those to further apart states and giving zero cost to remaining in the same state. So gives a solution that aims to maximize the diagonals of the transition probability matrices whilst constraining the prevalences to take their P-spline smoothed values.

What is absent from the paper is formal justification for the approach. Presumably a similar outcome could be achieved by applying a penalized likelihood approach, possibly formulating the problem in continuous time and setting the penalty to be the magnitude of the transition intensities (and possibly their derivatives). However, this would require some calibration to choose the penalty weights and it is not clear how this would be done (the usual approach of cross-validation would not work here).

A Bayesian Simulation Approach to Inference on a Multi-State Latent Facotr Intensity Model

Chew Lian Chua, G.C. Lim and Penelope Smith have a new paper in the Australian and New Zealand Journal of Statistics. This models the trajectory of credit ratings of companies. While there are 26 possible rating classes, these are grouped to the 4 broad classes: A,B,C,D, Movement between any two classes is deemed possible resulting in 12 possible transition types. The authors develop methods for fitting what is termed a multi-state latent factor intensity model. This is essentially a multi-state model with a shared random effect that affects all the transition intensities, but the random effect is allowed to evolve in time. For instance, the authors assume the random effect is an AR(1) process.

There seems to be some confusion in the formulation of the model as to whether the process is in continuous or discrete time: The process is expressed in terms of intensities which are "informally" defined as
where is a counting process describing the number of s to k transitions that have occurred to time t and is the ith observation time. But if the time between observations were variable, the AR(1) formulation (which takes no account of this) would not seem sensible.

Nevertheless the idea of having a random effect that can vary temporally within a multi-state model (proposed by Koopman et al) is interesting, though obviously presents various computational challenges which is the main focus of Chua, Lim and Smith's paper.

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.

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.

Discrete-time semi-Markov modeling of human papillomavirus persistence

Mitchell, Hudgens, King, Cu-Uvin, Lo, Rompalo, Sobel and Smith have a new paper in Statistics in Medicine. This considers a non-parametric estimator for a 2-state discrete-time semi-Markov process. Following Kang and Lagakos who assume one of the two states (e.g. state 0) is Markov, the process can be characterised by

where is the probability of making a transition from 1 to 0 given i time units spent in state 1,
is the maximum length of state sequence observable in the data and
Extensions to the model, allowing to depend on time in state and to allow an additional disease-free state corresponding to disease-free with no past disease, are also proposed. Missing observations can be dealt with by summing over all possible observed states at the missing times. Estimation of is by maximum likelihood. An acknowledged limitation is the inability to cope with the case of unknown initiation times if either the sojourn distribution of state 0 is non-geometric or observation can start in the disease state.

Of particular interest to the authors is an estimate of 'persistence' of the disease state. This is defined as spending j time units in the disease state, counting single disease free (negative) observations surrounded by positive observations as time spent in the disease. The probability of persistence is just a function of the transition probabilities and so readily estimable.

The authors claim that their discrete time model doesn't not require a "guarantee time" unlike Kang and Lagakos. This is obviously ridiculous, the discrete time model requires a guarantee time of 1 time unit for all transitions! While adopting a discrete time model simplifies the problem of inference to something quite trivial, one has to question how realistic it is to model something that is clearly a continuous time process as discrete time. Bachetti et al's more general approach is along similar lines. Similarly, while the estimation is nominally non-parametric, the discrete time assumption is in many respects more severe than, say, constraining sojourn distributions to be Weibull distributed.

The clinical definition of persistence which makes the assumption that a negative observation between two positive observations counts as a positive is easily accommodated for via the discrete time model. However, a more satisfactory approach would be to adopt a more formal definition, based in continuous time, e.g. persistence if disease free period is less than say 6 months. This would have parallels with the approach taken by Mandel (2010) in defining a hitting time in terms of having a sojourn of more than some length in the disease state. Farewell and Su also dealt with a similar problem but their approach seems to be best avoided.

A Hidden Markov Model for Informative Dropout in Longitudinal Response Data with Crisis States

Spagnoli, Henderson, Boys and Houwing-Duistermaat have a new paper in Statistics and Probability Letters. The paper is concerned with the modelling of discrete-time continuous response longitudinal data in the presence of informative dropout. The process of dropout is modelled by a three-state (potentially non-homogeneous) discrete time Markov model. The three states correspond to stable, crisis and dropout. The crisis state is characterized by a higher probability of
dropout and a shift in the mean response. The shift is random but is fixed for each individual so that repeated visits to the crisis state have a cumulative effect on the mean. The model is motivated by studies in which dropout is more likely to occur after the treatment has been ineffective for a period of time. A linear-mixed model, with random slope and intercept, is taken for the responses, with the response at time m being shifted by a random quantity, d, times by the number of time periods spent in the crisis state.

The authors put the model within the standard Rubin framework of MCAR/MAR/MNAR, making a distinction between observable and latent filtration. They are careful to formulate the model so that the latent mechanism for dropout only depends on the past and not the future.

The model is applied to schizophrenia data and the Leiden 85+ data. In the former case, interest is in the mean response curves in the hypothetical situation of no dropout. However for the Leiden 85+ data dropout is death and thus such curves would have little meaning. For both cases the crisis-state model represents a substantial improvement in likelihood compared to a two-state model.

Selecting a binary Markov model for a precipitation process

Reza Hosseini, Nhu Le and Jim Zidek have a new paper in Environmental and Ecological Statistics. This considers modelling a 0-1 precipitation process (e.g. daily data with 1 if amount of precipitation is above some threshold) as a binary discrete-time Markov process of some fixed order. The main focus of the paper is model selection where the model with the smallest BIC is chosen. Unsurprisingly simulations find that BIC is better than AIC at picking the "true" model where the "true" model is some model with a relatively small number of parameters.

The data example is rainfall in Calgary over 5 year time periods. Model building is detailed somewhat laboriously. Some of this seems unnecessary. For instance, the exploratory analysis shows clear seasonality in marginal probabilities of rain. However, models without seasonality are first considered leading to choosing a first order Markov model with the number of rainy days that month (i.e. a proxy for season) as a covariate. An obvious difficulty with fitting high order Markov models is that the number of parameters in the general case increases exponentially with the order of the process. Once seasonality is included only a first order Markov model is needed.

There is a disappointing lack of any reference to either hidden Markov models or (hidden) semi-Markov models. In particular the work of Hughes et al (1999, Applied Statistics) and Sansom and Thomson (2001, Journal of Applied Probability) are highly relevant. A hidden process perhaps makes more sense from the perspective of rain occurring due to e.g. the passing of a low-pressure system but the presence of such a system not necessarily manifesting itself with rain.

Hidden Markov models with arbitrary state dwell-time distributions

Langrock and Zucchini have a new paper in Computational Statistics and Data Analysis. This develops models methodology for fitting discrete-time hidden semi-Markov models. They demonstrate that hidden semi-Markov models can be approximated through hidden Markov models with aggregate blocks of states. Dwell-time in a particular state is made up of a series of latent states. An individual enters a state in latent state 1. After k steps in a state where k < m , a subject either leaves the state with probability c(k) or progresses to the next latent state with probability 1-c(k). c(k) therefore represents the hazard rate of the dwell time distribution. If time m in the state is reached then the subject may either stay in latent state m with some probability c(m) or else leave the state. Hence the tail of the dwell-time distribution is constrained to be geometric. By choosing m sufficiently large a good approximation to the desired distribution can be found.

Estimation from aggregate data

Gouno, Coutrai and Fredette have a new paper in Computational Statistics and Data Analysis. This proposes a new approach to estimation of transition rates in multi-state models panel observed in the form of aggregate prevalence counts. Kalbfleisch and Lawless dealt with this problem in the 1980s under a time-homogeneous Markov assumption. The full likelihood is very difficult to compute (for large N), but a least-squares approach is quite effective.

Gouno et al's approach is somewhat different. Essentially they make a discrete-time approximation to a continuous time process by assuming that only 1 transition may occur in a time unit and moreover, that transitions between observation times occur at the midpoint of the interval. The first assumption is convenient because it means the transition counts between states can be established from the prevalence counts. The second assumption allows the sojourn time distributions to be characterised as multinomial random variables. "Complete" data of the form of the number of sojourns of different lengths can be obtained via an Expectation or Monte-Carlo Expectation step in an EM or MCEM algorithm. An MCEM algorithm is required

The authors suggest that estimated sojourn distributions in each state could be compared using a log-rank test. A log-rank test would obviously be inappropriate. A likelihood ratio test between the full model and a model where the two sojourn distributions were constrained to be equal might be appropriate. However, an issue not addressed is identifiability. Regardless of the overall number of units from which the aggregate data are taken, the actual degrees of freedom of the data are fixed. The model fitted to example data appears to be saturated.

There is a lack of explanation in places in the paper. For instance, it is unclear why several of the estimated survival functions for the example dataset start from values other than 1. Also it is unclear why the estimated survival in state 1 drops to 0 at 28 time units, when 10 individuals in the original data survive to t=31.

Progressive multi-state models for informatively incomplete longitudinal data

Chen, Yi and Cook have a new paper in the Journal of Statistical Planning and Inference. This covers similar ground to the paper by the same authors in Statistics in Medicine, being concerned with missing not at random (MNAR) data. This article deals with the discrete time case, whereas the other paper modelled the process in continuous time.

Non-Markov Multistate Modeling Using Time-Varying Covariates

Bacchetti et al have a new paper in The International Journal of Biostatistics. This considers modelling progression of liver fibrosis due to hepatitis C following liver transplant using a 5-state progressive multi-state model. The data are panel observed at irregular time points, so it would be most natural to model the data in continuous time. In addition the observed states are subject to classification error. To avoid having to making either Markov or time homogeneity assumptions, the authors adopt a discrete time assumption: assuming 4 time periods per year. They then model the transition probabilities as linear on the log-odds scale, depending on covariates such as medical center, donor age, year of transplant as well as log time since entry into the current state (relaxing the Markov assumption). Potentially, time since transplant could also be included in the model (for non-homogeneity).

The main challenge in fitting the models is to enumerate all possible complete "paths" of true states at the discrete time points that could result in the observed data. Obviously an iterative algorithm is needed for this. The computational complexity will depend on the complexity of the state transition matrix, the misclassification probability matrix and the degree of discretization.

The main drawback of approximating a continuous time process by a discrete-time process is the restriction that only 1 transition may occur between time points. While this can be acceptable for progressive models such as the one considered here, it may be more problematic when backward transitions are allowable.

The authors have developed an R package called mspath, designed to complement the existing package for continuous-time Markov and hidden Markov model msm, to allow non-Markov models through the discrete-time approximation.

Estimating Transition Probabilities for Ignorable Intermittent Missing Data in a Discrete-Time Markov Chain

Yeh, Chan, Symanski and Davis have a new paper in Communications in Statistics - Simulation and Computation. This considers data from a discrete-time Markov model where some observations are missing in an ignorable way. This is a very common situation in longitudinal data. If the transition matrix is P for a single time period, then it is just P^2, P^3 etc. for 2,3,... time periods. For some reason the authors think that not being able to have a closed form expression for the MLE is a problem. They consider a naive approach based on only considering the observed one-step transitions, an EM algorithm approach and what they term a non-linear equations method. This latter approach is essentially computing the maximum likelihood estimate. However, the authors have failed to consider the possibility of boundary maximums. Hence they have sought to find where the gradient is 0 w.r.t. transition probabilities lying between 0 and 1. They run into problems in some cases simply because the MLE is at p=0, meaning there is no solution to the gradient equation within the allowable limits so negative values are suggested.

A further problem with the paper is that they have incorrectly estimated the standard errors under missing data in the EM algorithm, plugging estimates of the marginal counts into the complete-case formula. This is equivalent to using the full likelihood information rather than the observed information. Hence the standard errors will be underestimates.

In topic the paper has some similarities to Deltour et al, Biometrics (1999). However, that paper was considering a non-trivial problem of non-ignorable missing data, where a Stochastic-EM algorithm was proposed.

A nonstationary Markov transition model for computing the relative risk of dementia before death

Yu et al have a new paper in Statistics in Medicine. This is concerned with estimating the risk of dementia before death. A 5 state multi-state model is used, with three transient states representing levels of cognitive impairment, plus two absorbing states dementia and death. Unlike other recent dementia studies using multi-state models, improvements, as well as deteriation, in cognitive ability are assumed possible. A discrete-time approach is taken. This has some advantages in terms of the flexibility in modelling possible in terms of incorporating non-homogeneity over time and a frailty term - the Markov assumption. A drawback of using a discrete-time approach in the current study was that the study did not have equally spaced observation times, but it was necessary to assume this was the case for the discrete-time model. This is likely to cause some bias.

The main theoretical development in the paper is expressions for the mean and variance of the time spent in each transient state before absorption in the non-homogeneous case.

Nonparametric inference and uniqueness for periodically observed progressive disease models

Beth Griffin and Stephen Lagakos have a new paper in Lifetime Data Analysis. They consider panel observed progressive disease model (chain-of-events) data. The NPMLE estimator under a discrete-time semi-Markov assumption was developed by Sternberg and Satten (Biometrics, 1999). For datasets where individuals are observed at different times, some discretization of the data is required. An issue with the NPMLE is that it is not guaranteed to be unique and therefore reporting a single NPMLE may be misleading. The paper develops procedures for determining which components of the NPMLE are unique based on considering various re-parameterizations of the likelihood. The method is demonstrated on three example datasets including one on bronchiolitis obliterans syndrome in post-lung transplantation patients and one on primary HIV infection. In addition, the authors also provide a more intuitive algorithm for obtaining the NPMLE than the self-consistency algorithm of Sternberg and Satten.

Nonparametric estimation in an "illness-death" model when the transition times are interval-censored and one transition is not observed.

Frydman, Gerds, Groen and Keiding have a paper available as a research report from the Department of Biostatistics, Copenhagen. The paper develops previous work on the non-parametric estimation of interval-censored multi-state data. Here the data in question follow a progressive three-state "illness-death" model but the ill to death transition is never observed. This is because the data arise from clinical observation and the trial ceases if a patient is observed to be in the illness state. Such an observation scheme has strong similarities with data considered by Duffy et al relating to breast cancer screening where a three-stage unidirectional model was assumed and the intermediate state was pre-clinical detectable breast cancer. No data on pre-clinical to clinical breast cancer transitions were available as interest was in the natural progression of the disease. Duffy et al analysed the data parametrically, assuming a time homogeneous Markov model. In contrast Frydman et al fit a non-homogeneous Markov model non-parametrically. Since all transitions are interval censored, they model the process in discrete time.
Update: A paper broadly based upon the research report has now been published in Biometrical Journal. The supplementary materials also includes R code to implement the proposed algorithm.

A multistate approach for estimating the incidence of human immunodeficiency virus by using HIV and AIDS French surveillance data

Sommen, Alioum and Commenges have a new paper in Statistics in Medicine. This applies the penalized likelihood approach to multi-state models, used extensively by the INSERM group, to the area of back-calculation for estimating HIV incidence. The penalization parameters are chosen by minimizing an approximate cross-validation score.