Modeling nonhomogeneous Markov processes via time transformation

Summary .   Longitudinal studies are a powerful tool for characterizing the course of chronic disease. These studies are usually carried out with subjects observed at periodic visits giving rise to panel data. Under this observation scheme the exact times of disease state transitions and sequence of disease states visited are unknown and Markov process models are often used to describe disease progression. Most applications of Markov process models rely on the assumption of time homogeneity, that is, that the transition rates are constant over time. This assumption is not satisfied when transition rates depend on time from the process origin. However, limited statistical tools are available for dealing with nonhomogeneity. We propose models in which the time scale of a nonhomogeneous Markov process is transformed to an operational time scale on which the process is homogeneous. We develop a method for jointly estimating the time transformation and the transition intensity matrix for the time transformed homogeneous process. We assess maximum likelihood estimation using the Fisher scoring algorithm via simulation studies and compare performance of our method to homogeneous and piecewise homogeneous models. We apply our methodology to a study of delirium progression in a cohort of stem cell transplantation recipients and show that our method identifies temporal trends in delirium incidence and recovery.     

A model to estimate risk of infection with human herpesvirus 8 associated with transfusion from cross-sectional data

In cross-sectional studies of infectious diseases, the data typically consist of: age at the time of study, status (presence or absence) of infection, and a chronology of events possibly associated with the disease. Motivated by a study of how human herpesvirus 8 (HHV-8) is transmitted among children with sickle cell anemia in Uganda, we have developed a flexible parametric approach for combining current-status data with a history of blood transfusions. We model heterogeneity in transfusion-associated risk by a child-specific random effect. We present an extension of the model to accommodate the fact that there is no gold standard for HHV-8 infection and infection status was assessed by a serological assay. The parameters are estimated via maximum likelihood. Finally, we present results from applying various parameterizations of the model to the Ugandan study.     

A bayesian approach to the multistate Jolly-Seber capture-recapture model

This article considers a Bayesian approach to the multistate extension of the Jolly-Seber model commonly used to estimate population abundance in capture-recapture studies. It extends the work of George and Robert (1992, Biometrika79, 677-683), which dealt with the Bayesian estimation of a closed population with only a single state for all animals. A super-population is introduced to model new entrants in the population. Bayesian estimates of abundance are obtained by implementing a Gibbs sampling algorithm based on data augmentation of the missing data in the capture histories when the state of the animal is unknown. Moreover, a partitioning of the missing data is adopted to ensure the convergence of the Gibbs sampling algorithm even in the presence of impossible transitions between some states. Lastly, we apply our methodology to a population of fish to estimate abundance and movement.     

A conditional Markov model for clustered progressive multistate processes under incomplete observation

Clustered progressive chronic disease processes arise when interest lies in modeling damage in paired organ systems (e.g., kidneys, eyes), in diseases manifest in different organ systems, or in systemic conditions for which damage may occur in several locations of the body. Multistate Markov models have considerable appeal for modeling damage in such settings, particularly when patients are only under intermittent observation. Generalizations are necessary, however, to deal with the fact that processes within subjects may not be independent. We describe a conditional Markov model in which the clustering in processes within subjects is addressed by the use of multiplicative random effects for each transition intensity. The random effects for the different transition intensities may be correlated within subjects, but are assumed to be independent for different subjects. We apply the mixed Markov model to a motivating data set of patients with psoriatic arthritis, and characterize the progressive course of damage in joints of the hand. A generalization to accommodate a subpopulation of "stayers" and extensions which facilitate regression are indicated and illustrated.     

Stopover of migrating birds: simultaneous analysis of different marking methods enhances the power of capture–recapture analyses

Analyses of stopover parameters of migrating birds with Cormack–Jolly–Seber (CJS) capture–recapture models often suffer from low precision due to sparse data sets. Low recapture rates result in low power to detect violations of the underlying assumptions and factors influencing stopover behaviour. We studied stopover behaviour of Palearctic migrant passerines in an oasis in Mauritania, West Africa. Using capture–recapture data and systematic observations of colour-ringed birds, we analysed the effect of increased sample size on probability of stay and recapture probability and the influence of a possible trap response on these parameters. We analysed capture–recapture data with the conventional CJS model and compared the results with those from a multistate model using in addition resighting data. The analyses including resighting data resulted in a higher precision of the estimates of the probabilities of stay compared to analyses using only capture–recapture data of the same individuals. Moreover, the power to detect transients was substantially enhanced. Capture had no effect on the estimates of probability to stay and recapture probability; birds did not leave the stopover site or avoid nets as a reaction to capture. The estimates of probability of stay were up to 15.7% higher when resighting data were included, probably due to the higher power to detect transients and the elimination of the bias induced by non-random temporary emigration when both data types are considered. As a consequence, stopover duration would have been underestimated when only the capture–recapture data were available. We conclude that additional resightings of colour ringed birds can produce more accurate results needed for enhancing our understanding of stopover ecology of migrants. Electronic supplementary material Supplementary material is available in the online version of this article at and is accessible for authorized users. From the Swiss Ornithological Institute project on bird migration across the Sahara.     

Bayesian models for multivariate current status data with informative censoring

Multivariate current status data, consist of indicators of whether each of several events occur by the time of a single examination. Our interest focuses on inferences about the joint distribution of the event times. Conventional methods for analysis of multiple event-time data cannot be used because all of the event times are censored and censoring may be informative. Within a given subject, we account for correlated event times through a subject-specific latent variable, conditional upon which the various events are assumed to occur independently. We also assume that each event contributes independently to the hazard of censoring. Nonparametric step functions are used to characterize the baseline distributions of the different event times and of the examination times. Covariate and subject-specific effects are incorporated through generalized linear models. A Markov chain Monte Carlo algorithm is described for estimation of the posterior distributions of the unknowns. The methods are illustrated through application to multiple tumor site data from an animal carcinogenicity study.     

A random time interval approach for analysing the impact of a possible intermediate event on a terminal event

We consider the impact of a possible intermediate event on a terminal event in an illness-death model with states 'initial', 'intermediate' and 'terminal'. One aim is to unambiguously describe the occurrence of the intermediate event in terms of the observable data, the problem being that the intermediate event may not occur. We propose to consider a random time interval, whose length is the time spent in the intermediate state. We derive an estimator of the joint distribution of the left and right limit of the random time interval from the Aalen-Johansen estimator of the matrix of transition probabilities and study its asymptotic properties. We apply our approach to hospital infection data. Estimating the distribution of the random time interval will usually be only a first step of an analysis. We illustrate this by analysing change in length of hospital stay following an infection and derive the large sample properties of the respective estimator.     

Evaluating hypotheses about dispersal in a vulnerable butterfly

Sound management of species requires reliable estimates of dispersal. Indeed, dispersal of individuals among local populations is a key factor in the biology and persistence of local populations and metapopulations. Here, the small-scale dispersal pattern of a vulnerable species, the endemic Sardinian chalk hill blue butterfly, was studied by applying capture–recapture multistate models and a model selection based on AIC values. Model parameters were survival, capture and movement probabilities. The model selection showed that (a) survival probability of individuals varied between sexes, (b) capture probability varied between sexes and among patches, and (c) movement probability varied with direction. The probability of movement among adjacent local populations was generally low and ranged from 0.009 to 0.212. Movement probabilities were subsequently modeled using data on interpatch distance and donor patch population size or area. The ultrastructural biology-based models turned out to be the most appropriate models for inference, showing that dispersal decreases with increasing interpatch distance and increasing donor patch population size or area, and suggesting that butterfly dispersal is affected by patch geometry and the presence of conspecifics. The application of multistate models, the model selection approach, and ultrastructural modeling allowed testing the validity of some general hypotheses related to dispersal in metapopulations and helped elucidate the butterfly small-scale dispersal pattern.     

Nonparametric inference for the cumulative incidence function of a competing risk,with an emphasis on confidence bands in the presence of left‐truncation

Motivated by a study on pregnancy outcome, a computationally simple resampling procedure for nonparametric analysis of the cumulative incidence function of a competing risk is investigated for left‐truncated data. We also modify the original procedure to producing the more desirable Greenwood‐type variance estimates. These approaches are used to construct simultaneous confidence bands of the cumulative incidence functions which is otherwise hampered by the complicated nature of the covariance process. Simulation results and a real data example are provided.     

Survival and movement of naïve juvenile spiny lobsters returned to the wild

Using multistate Arnson-Schwartz (AS) mark-recapture models, we show that naïve (captive reared) juvenile southern rock lobsters (Jasus edwardsii, Hutton 1875) survived as well as wild-caught lobsters when released to an area of coastal reef. Lobsters captured as pueruli were ongrown in tanks for 12 to 18 months where they were fed to satiation in the absence of predators. Lobsters were marked with antennal tags each carrying a unique code, and released to coastal reef along with tagged wild-caught lobsters of similar size. During 8 dive surveys of the release reef and 3 surveys of adjacent reefs over a 28 day period, divers resighted 40.3% of the naïve lobsters and 70.2% of the wild lobsters. We show that this discrepancy is a function of differing movement rates and spatial differences in resighting probability. The probability of naïve lobsters moving from the release reef to adjacent areas in the first 4 days post-release (0.72 ± 0.04 S.E.) was almost twice that of wild-caught lobsters (0.38 ± 0.08 S.E.). This behavioural difference did not influence daily apparent survival (0.98 ± 0.016 S.E.), which was constant between groups and over time. Our results are encouraging for the potential of enhancing spiny lobster stocks by releasing juveniles, and demonstrate the utility of AS mark-recapture models as a tool for evaluating medium-term survival of mobile marine species.     

Semi‐Markov Models with Phase‐Type Sojourn Distributions

Summary Continuous‐time multistate models are widely used for categorical response data, particularly in the modeling of chronic diseases. However, inference is difficult when the process is only observed at discrete time points, with no information about the times or types of events between observation times, unless a Markov assumption is made. This assumption can be limiting as rates of transition between disease states might instead depend on the time since entry into the current state. Such a formulation results in a semi‐Markov model. We show that the computational problems associated with fitting semi‐Markov models to panel‐observed data can be alleviated by considering a class of semi‐Markov models with phase‐type sojourn distributions. This allows methods for hidden Markov models to be applied. In addition, extensions to models where observed states are subject to classification error are given. The methodology is demonstrated on a dataset relating to development of bronchiolitis obliterans syndrome in post‐lung‐transplantation patients.     

Attributable risk estimation for adjusted disability multistate models: Application to nosocomial infections

Attributable risk has become an important concept in clinical epidemiology. In this paper, we suggest to estimate the attributable risk of nosocomial infections using a multistate approach. Recently, a multistate model (called progressive disability model in the literature) has been developed in order to take into consideration both the time‐dependency of the risk factor (e.g., nosocomial infections) and the presence of competing risks (e.g., death and discharge) at each time point. However, this approach does not take into account the possible heterogeneity of the study population. In this paper, we investigate an extension of this model and suggest an adjusted disability multistate model including covariates in each transition. This new multistate model has led us to define the concepts of overall and profiled attributable risk. We use a classical semiparametric approach to estimate the model and the new attributable risk. A simulation study is investigated and we show, in particular, that neglecting the presence of covariates when estimating the model can lead to an important bias. The methodology developed in this paper is applied to data on ventilator‐associated pneumonia in 12 French intensive care units.     

A semiparametric transition model with latent traits for longitudinal multistate data

SUMMARY: We propose a general multistate transition model. The model is developed for the analysis of repeated episodes of multiple states representing different health status. Transitions among multiple states are modeled jointly using multivariate latent traits with factor loadings. Different types of state transition are described by flexible transition-specific nonparametric baseline intensities. A state-specific latent trait is used to capture individual tendency of the sojourn in the state that cannot be explained by covariates and to account for correlation among repeated sojourns in the same state within an individual. Correlation among sojourns across different states within an individual is accounted for by the correlation between the different latent traits. The factor loadings for a latent trait accommodate the dependence of the transitions to different competing states from a same state. We obtain the semiparametric maximum likelihood estimates through an expectation-maximization (EM) algorithm. The method is illustrated by studying repeated transitions between independence and disability states of activities of daily living (ADL) with death as an absorbing state in a longitudinal aging study. The performance of the estimation procedure is assessed by simulation studies.     

Parameter Redundancy in Multistate Capture‐Recapture Models

Multistate capture‐recapture models are a powerful tool to address a variety of biological questions concerning dispersal and/or individual variability in wild animal populations. However, biologically meaningful models are often over‐parameterized and consequently some parameters cannot be estimated separately. Identifying which quantities are separately estimable is crucial for proper model selection based upon likelihood tests or information criteria and for the interpretation of the estimates obtained. We show how to investigate parameter redundancy in multistate capture‐recapture models, based on formal methods initially proposed by Catchpole and his associates for exponential family distributions (Catchpole, Freeman and Morgan, 1996. Journal of the Royal Statistical Society Series B 58, 763–774). We apply their approach to three models of increasing complexity.     

Bayesian modeling of incidence and progression of disease from cross-sectional data

In the absence of longitudinal data, the current presence and severity of disease can be measured for a sample of individuals to investigate factors related to disease incidence and progression. In this article, Bayesian discrete-time stochastic models are developed for inference from cross-sectional data consisting of the age at first diagnosis, the current presence of disease, and one or more surrogates of disease severity. Semiparametric models are used for the age-specific hazards of onset and diagnosis, and a normal underlying variable approach is proposed for modeling of changes with latency time in disease severity. The model accommodates multiple surrogates of disease severity having different measurement scales and heterogeneity among individuals in disease progression. A Markov chain Monte Carlo algorithm is described for posterior computation, and the methods are applied to data from a study of uterine leiomyoma.     

Survival and population growth of a long-lived threatened snake species, Drymarchon couperi (Eastern Indigo Snake)

Demographic data provide a basis for understanding the life history and ecology of species, factors which are vital for informing conservation efforts; however, little is known regarding the population ecology of most snake species, including the threatened Eastern Indigo Snake (Drymarchon couperi). We used 11 years (1999–2009) of capture-mark-recapture (CMR) and 2.5 years (2003–2005) of radiotelemetry data from southeastern Georgia, USA, in a CMR modeling framework to estimate apparent survival, capture and transition probabilities, and evaluate factors influencing these parameters. The model-averaged estimate of overall apparent annual survival probability was 0.700 (±0.030 SE) and is comparable to that obtained from known fate analysis (radiotelemetry) at the same site. Body size positively influenced survival, regardless of sex. Capture probability differed seasonally by sex, suggesting lower capture probability for females in fall and males in winter. There was no evidence for effect of precipitation or site-specific differences in survival. Model averaged estimate of annual adult survival estimated using multistate CMR models was 0.738 ± 0.030 and 0.515 ± 0.189 for subadults. We estimated population growth rate (λ) and elasticity (proportional sensitivity) of λ to vital rates using a stage-structured matrix population model. Population growth rate ranged from 0.96 to 1.03 depending on the value of the probability of transitioning from subadult to adult stage. The λ was proportionally most sensitive to changes in adult survival rate, followed by subadult survival. Our results suggest that protecting adult snakes and their habitats would result in the highest likelihood of long-term population stability and growth.     

A random effects model for multistate survival analysis with application to bone marrow transplants

We present an extension of the non-homogeneous Markov model for a bone marrow transplant recovery process which allows for possible associations between the transition intensities. The associations between intensities are modeled by a correlated gamma frailty model. Based on a parametric model for the conditional transition intensities, we obtain estimates of the model parameters. We use these estimates to make predictions of patient’s eventual prognosis given the current medical history of the patient. Estimates of the uncertainty in our predictions are obtained by a modified bootstrap technique.