The Analysis of Panel Data under a Markov Assumption

Spatial re-establishment dynamics of local populations of vectors of Chagas disease

Background

Prevention of Chagas disease depends mainly on control of the insect vectors that transmit infection. Unfortunately, the vectors have been resurgent in some areas. It is important to understand the dynamics of reinfestation where it occurs. Here we show how continuous- and discrete-time models fitted to patch-level infestation states can elucidate different aspects of re-establishment. Triatoma infestans, the main vector of Chagas disease, reinfested sites in three villages in northwest Argentina after community-wide insecticide spraying in October 1992.

Methodology/Principal Findings

Different methods of estimating the probabilities of bug establishment on each site were compared. The results confirmed previous results showing a 6-month time lag between detection of a new infestation and dispersal events. The analysis showed that more new bug populations become established from May to November than from November to May. This seasonal increase in bug establishment coincides with a seasonal increase in dispersal distance. In the fitted models, the probability of new bug establishment increases with increasing time since last detected infestation.

Conclusions/Significance

These effects of season and previous infestation on bug establishment challenge our current understanding of T. infestans ecology and highlight important gaps in knowledge. Experiments necessary to close these gaps are discussed.

Chagas disease is transmitted by blood-sucking bugs (vectors) and presents a severe public health threat in the Americas. Worldwide there are approximately 10 million people infected with Chagas disease, a disease for which there is currently no effective cure. Vector suppression is the main strategy to control the spread of this disease. Unfortunately, the vectors have been resurgent in some areas. It is important to understand the dynamics of reinfestation where it occurs. Here we show how different models fitted to patch-level bug infestation data can elucidate different aspects of re-establishment dynamics. Our results demonstrated a 6-month time lag between detection of a new infestation and dispersal events, seasonality in dispersal rates and effects of previous vector infestation on subsequent vector establishment rates. In addition we provide estimates of dispersal distances and the effect of insecticide spraying on rates of vector re-establishment. While some of our results confirm previous findings, the effects of season and previous infestation on bug establishment challenge our current understanding of T. infestans ecology and highlight important gaps in our knowledge of T. infestans dispersal.

Assessing the Natural Course of Diabetic Retinopathy: A Population-Based Study in Kinmen, Taiwan

PURPOSE

To explore the natural course of diabetic retinopathy among type 2 diabetics using the indirect ophthalmoscope and single-field fundus photographs in Kinmen, Taiwan.

METHODS

A screening program for diabetic retinopathy was carried out by a panel of ophthalmologists, who employed the ophthalmoscope and 45-degree retinal color photographs to examine the fundus after pupil dilation. Screening, which was conducted between 1999 and 2002, involved 971 patients diagnosed with type 2 diabetes. A multi-state Markov model was used to assess the natural course of diabetic retinopathy among type 2 diabetics.

RESULTS

Among the 725 diabetes patients who attended at least two ophthalmological fundus check-ups and were screened, the overall response rate was about 75%. The mean duration of the disease states mild nonproliferative diabetic retinopathy, moderate nonproliferative diabetic retinopathy, severe nonproliferative diabetic retinopathy, and proliferative diabetic retinopathy were 4.05 [95% confidence interval (CI): 3.28-5.32], 4.18 (95% CI: 3.18-6.06), 2.52 (95% CI: 1.78-4.27), and 4.22 (95% CI: 2.88-7.81) years, respectively. Compared to controls, the incidence of blindness reduction for annual, biennial, 3-year, 4-year, and 5-year screenings of diabetic retinopathy were approximately 94.4% (95% CI: 91.6%-96.3%), 83.9% (95% CI: 83.6%-84.2%), 70.2% (95% CI: 69.8%-70.7%), 57.2% (95% CI: 56.7%-57.7%), and 45.6% (95% CI: 45.0%-46.1%), respectively.

CONCLUSIONS

In conclusion, the average time for the development of diabetic retinopathy from nonexistence to blindness was approximately 26.5 years. The present recommendation for annual screening in type 2 diabetics with nonproliferative diabetic retinopathy should be retained only for the mild form, not for the moderate or severe forms.

The disablement process: Factors associated with progression of disability and recovery in French elderly people

PURPOSE

To study the factors associated with progression, recovery and death from different grades of disability in elderly people.

METHOD

The sample included 3198 participants of the PAQUID ('Personnes Agées QUID') cohort, aged 65 and over and community-dwellers at baseline. Subjects were re-interviewed 1, 3, 5, 8 and 10 years after baseline. A five-state Markov model was used to estimate transition intensities between four grades of disability and toward death. We used a hierarchic scale of disability, which combines basic and instrumental activities of daily living and mobility. Several explanatory variables were investigated: medical, personal and environmental factors.

RESULTS

The factors associated with progression and/or no recovery of disability were cardiovascular diseases, stroke and diabetes, low cognition, visual impairment and dyspnoea (for pathologies and impairments), older age, female gender, low educational level (for risk factors), depression (for intra-individual factor) and being married, recent hospitalization and number of drugs (for extra-individual factors). Older age, male gender, tobacco consumption and living in an urban area were associated with mortality.

CONCLUSIONS

These findings confirm the independent contribution of each group of variables in the disablement process and stress their different impact on progression of disability or on recovery from different grades of disability.

Semiparametric analysis of panel count data with correlated observation and follow-up times

This paper discusses regression analysis of panel count data that often arise in longitudinal studies concerning occurrence rates of certain recurrent events. Panel count data mean that each study subject is observed only at discrete time points rather than under continuous observation. Furthermore, both observation and follow-up times can vary from subject to subject and may be correlated with the recurrent events. For inference, we propose some shared frailty models and estimating equations are developed for estimation of regression parameters. The proposed estimates are consistent and have asymptotically a normal distribution. The finite sample properties of the proposed estimates are investigated through simulation and an illustrative example from a cancer study is provided.

The natural history of trachoma infection and disease in a Gambian cohort with frequent follow-up

Background

The natural history of ocular Chlamydia trachomatis infections in endemic communities has not been well characterised and is an important determinant of the effectiveness of different mass treatment strategies to prevent blindness due to trachoma.

Methodology/Principal Findings

A multistate hidden Markov model was fitted to data on infection and active disease from 256 untreated villagers in The Gambia who were examined every 2 weeks over a 6-month period. Parameters defining the natural history of trachoma were estimated, and associations between these parameters, demographic and baseline immune measurements examined. The median incubation period following infection was estimated at 17 days (95% confidence interval: 11–28). Disease persisted for longer than infection (median 21 (15–32) weeks) versus 17 (12–24) weeks), with an estimated median duration of post-infection inflammation of 5 (3–8) weeks. The duration of active disease showed a significant decline with age even after accounting for lower rates of re-infection and disease at older ages (p = 0.004). Measurements of levels of baseline IgA to epitopes in the major outer membrane protein of Chlamydia trachomatis were not significantly correlated with protection or more rapid clearance of infection.

Conclusions

The average duration of infection with Chlamydia trachomatis especially at younger ages is long. This contributes to the persistence and gradual return of trachoma after community-wide treatment with antibiotics.

Trachoma is an infectious disease of the eye that causes blindness in many of the poorest parts of the world. In this paper, we use a novel statistical approach to estimate the characteristics of this disease among people living in The Gambia who were examined every 2 weeks over a 6-month period. We found that the typical duration of infection with Chlamydia trachomatis and of clinically active disease were significantly longer than previously estimated. We tested different hypotheses about the natural history of trachoma that explain the relationship between infection and disease observed in the field. We also confirmed that disease lasts significantly longer among young children under 5 years old compared with older children and adults, even after accounting for high rates of re-infection in this age group, consistent with the development of immunity with age. The long duration of infection, especially among younger children, contributes to the persistence and gradual return of trachoma after community-wide treatment with azithromycin. This implies the need for high treatment coverage if infection is to be eliminated from a community, even where the return of infection after treatment is seen to be slow.

A general goodness-of-fit test for Markov and hidden Markov models

Markov models are a convenient and useful method of estimating transition rates between levels of a categorical response variable, such as a disease stage, which changes over time. In medical applications the response variable is typically observed at irregular intervals. A Pearson-type goodness-of-fit test for such models was proposed by Aguirre-Hernandez and Farewell (Statist. Med. 2002; 21:1899-1911), but this test is not applicable in the common situation where the process includes an absorbing state, such as death, for which the time of entry is known precisely nor when the data include censored state observations. This paper presents a modification to the Pearson-type test to allow for these cases. An extension of the method, to allow for the class of hidden Markov models where the response variable is subject to misclassification error, is given. The method is applied to data on cardiac allograft vasculopathy in post-heart-transplant patients.

A flexible semi-Markov model for interval-censored data and goodness-of-fit testing

Multi-state approaches are becoming increasingly popular to analyse the complex evolution of patients with chronic diseases. For example, the evolution of kidney transplant recipients can be broken down into several clinical states. With this application in mind, we present a flexible semi-Markov model. The distribution functions are fitted to the durations in states and the relevance of the generalised Weibull distribution is shown. The corresponding likelihood function allows for interval censoring, i.e. the times of transitions and the sequences of states are not available during the elapsed times between two visits. The explanatory variables are introduced through the Markov chain and through the probability density functions of durations. A goodness-of-fit test is also defined to examine the stationarity of the semi-Markov model.

Multi-state models for the analysis of time-to-event data

The experience of a patient in a survival study may be modelled as a process with two states and one possible transition from an "alive" state to a "dead" state. In some studies, however, the "alive" state may be partitioned into two or more intermediate (transient) states, each of which corresponding to a particular stage of the illness. In such studies, multi-state models can be used to model the movement of patients among the various states. In these models issues, of interest include the estimation of progression rates, assessing the effects of individual risk factors, survival rates or prognostic forecasting. In this article, we review modelling approaches for multi-state models, and we focus on the estimation of quantities such as the transition probabilities and survival probabilities. Differences between these approaches are discussed, focussing on possible advantages and disadvantages for each method. We also review the existing software currently available to fit the various models and present new software developed in the form of an R library to analyse such models. Different approaches and software are illustrated using data from the Stanford heart transplant study and data from a study on breast cancer conducted in Galicia, Spain.

Testing transition probability matrix of a multi-state model with censored data

In this paper, we develop procedures to test hypotheses concerning transition probability matrices arising from certain nonhomogeneous Markov processes. It is assumed that the data consist of sample paths, some of which are observed until a certain terminal state, and the other paths are censored. Problems of this type arise in the context of multi-state models relevant to Health Related Quality of Life (HRQoL) and Competing Risks. The test statistic is based on the estimator for the associated intensity matrix. We show that the asymptotic null distribution of the proposed statistic is Gaussian, and demonstrate how the procedure can be adopted for HRQoL studies and competing risks model using real data sets. Finally, we establish that the test statistic for the HRQoL has greatest local asymptotic power against a sequence of proportional hazards alternatives converging to the null hypothesis.

A piecewise-constant Markov model and the effects of study design on the estimation of life expectancies in health and ill health

Multi-state models are frequently applied to describe transitions over time between three states: healthy, not healthy and death. The three-state model can be used to estimate life expectancies in health and ill health. In this article, continuous-time Markov models are specified for the transitions between the three states. Transition intensities are regressed on age as a time-dependent covariate. The covariate is handled in a piecewise-constant fashion where the time interval between two consecutive observations is divided into subintervals of fixed and equal lengths. Study design choices such as sample size, length of follow-up, and time intervals between observations are investigated in a simulation study. The effects on parameter estimation are discussed as well as the effects on the estimation of life expectancies. In addition, data taken from the UK Cognitive Functioning and Ageing Study are analysed.

Estimating time-to-event from longitudinal ordinal data using random-effects Markov models: application to multiple sclerosis progression

Longitudinal ordinal data are common in many scientific studies, including those of multiple sclerosis (MS), and are frequently modeled using Markov dependency. Several authors have proposed random-effects Markov models to account for heterogeneity in the population. In this paper, we go one step further and study prediction based on random-effects Markov models. In particular, we show how to calculate the probabilities of future events and confidence intervals for those probabilities, given observed data on the ordinal outcome and a set of covariates, and how to update them over time. We discuss the usefulness of depicting these probabilities for visualization and interpretation of model results and illustrate our method using data from a phase III clinical trial that evaluated the utility of interferon beta-1a (trademark Avonex) to MS patients of type relapsing-remitting.

A Markov regression random-effects model for remission of functional disability in patients following a first stroke: a Bayesian approach

Few attempts have been made to model the dynamics of stroke-related disability. It is possible though, using panel data and multi-state Markov regression models that incorporate measured covariates and latent variables (random effects). This study aimed to model a series of functional transitions (following a first stroke) using a three-state Markov model with or without considering random effects. Several proportional hazards parameterizations were considered. A Bayesian approach that utilizes the Markov Chain Monte Carlo (MCMC) and Gibbs sampling functionality of WinBUGS (a Windows-based Bayesian software package) was developed to generate the marginal posterior distributions of the various transition parameters (e.g. the transition rates and transition probabilities). Model building and comparisons was guided by reference to the deviance information criteria (DIC). Of the four proportional hazards models considered, exponential regression was preferred because it led to the smallest deviances. Adding random effects further improved the model fit. Of the covariates considered, only age, infarct size, and baseline functional status were significant. By using our final model we were able to make individual predictions about functional recovery in stroke patients.

Recovering Transitions From Repeated Cross-Sectional Samples

This paper proposes a dynamic Markov model for the estimation of binary state-to-state transition probabilities from a sequence of independent cross-sectional samples. It discusses parameter estimation and inference using maximum likelihood (ML) methodology. The model is illustrated by the application of a three-wave panel study on pupils’ interest in learning physics. These data encompass more information than what is used to estimate the model, but this surplus information allows us to assess the accuracy and the precision of the transition estimates. Bootstrap and Bayesian simulations are used to evaluate the accuracy and the precision of the ML estimates. To mimic genuine cross-sectional data, samples of independent observations randomly drawn from the panel are also analyzed.

Estimation of progression of multi-state chronic disease using the Markov model and prevalence pool concept

Background

We propose a simple new method for estimating progression of a chronic disease with multi-state properties by unifying the prevalence pool concept with the Markov process model.

Methods

Estimation of progression rates in the multi-state model is performed using the E-M algorithm. This approach is applied to data on Type 2 diabetes screening.

Results

Good convergence of estimations is demonstrated. In contrast to previous Markov models, the major advantage of our proposed method is that integrating the prevalence pool equation (that the numbers entering the prevalence pool is equal to the number leaving it) into the likelihood function not only simplifies the likelihood function but makes estimation of parameters stable.

Conclusion

This approach may be useful in quantifying the progression of a variety of chronic diseases.

Stenseth N. Statistical analysis of the dynamics of antibody loss to a disease-causing agent: plague in natural populations of great gerbils as an example

We propose a new stochastic framework for analysing the dynamics of the immunity response of wildlife hosts against a disease-causing agent. Our study is motivated by the need to analyse the monitoring time-series data covering the period from 1975 to 1995 on bacteriological and serological tests-samples from great gerbils being the main host of Yersinia pestis in Kazakhstan. Based on a four-state continuous-time Markov chain, we derive a generalized nonlinear mixed-effect model for analysing the serological test data. The immune response of a host involves the production of antibodies in response to an antigen. Our analysis shows that great gerbils recovered from a plague infection are more likely to keep their antibodies to plague and survive throughout the summer-to-winter season than throughout the winter-to-summer season. Provided the seasonal mortality rates are similar (which seems to be the case based on a mortality analysis with abundance data), our finding indicates that the immune function of the sampled great gerbils is seasonal.

Comparison of variance estimation approaches in a two-state Markov model for longitudinal data with misclassification

We examine the behaviour of the variance-covariance parameter estimates in an alternating binary Markov model with misclassification. Transition probabilities specify the state transitions for a process that is not directly observable. The state of an observable process, which may not correctly classify the state of the unobservable process, is obtained at discrete time points. Misclassification probabilities capture the two types of classification errors. Variance components of the estimated transition parameters are calculated with three estimation procedures: observed information, jackknife, and bootstrap techniques. Simulation studies are used to compare variance estimates and reveal the effect of misclassification on transition parameter estimation. The three approaches generally provide similar variance estimates for large samples and moderate misclassification. In these situations, the resampling methods are reasonable alternatives when programming partial derivatives is not appealing. With smaller chains or higher misclassification probabilities, the bootstrap method appears to be the best choice.

Sensitivity analysis of Markov models for communities of competing sessile organisms

1. Communities of competing sessile organisms are often modelled using Markov chains. Sensitivity analysis of the stationary distribution of these models tells us how we expect the abundance of each organism to respond to changes in interactions between species. This is important for conservation and management. 2. Markov models for such communities have usually been formulated in discrete time. Each column of the discrete-time transition matrix must sum to 1 (column stochasticity). Sensitivity analysis therefore involves defining a pattern of compensation that maintains column stochasticity as a single transition probability changes. There is little biological theory about the appropriate compensation pattern, but the usual choices involve changing only the elements of a single column of the transition matrix. 3. I argue that if the underlying dynamics occur in continuous time, each transition probability is the net outcome of direct and many indirect interactions. 4. Determining the consequences of changing a single direct interaction will often be of interest. I show how this can be achieved using a continuous-time model. The resulting discrete-time compensation pattern is quite different from those that have been considered elsewhere, with changes occurring in many columns. 5. I also show how to determine which direct interactions are being changed under any discrete-time compensation pattern.

Statistical methods for panel data from a semi-Markov process, with application to HPV

Continuous-time, multistate processes can be used to represent a variety of biological processes in the public health sciences; yet the analysis of such processes is complex when they are observed only at a limited number of time points. Inference methods for such panel data have been developed for time homogeneous Markov models, but there has been little research done for other classes of processes. We develop likelihood-based methods for panel data from a semi-Markov process, where transition intensities depend on the duration of time in the current state. The proposed methods account for possible misclassification of states. To illustrate the methods, we investigate a three- and a four-state models in detail and apply the results to model the natural history of oncogenic genital human papillomavirus infections in women.

A population-based assessment of the prognostic value of the CD19 positive lymphocyte count in B-cell chronic lymphocytic leukemia using Cox and Markov models

No population-based study has assessed the prognostic impact on survival of the CD19 positive lymphocyte count, evaluated by immunophenotyping at diagnosis, in B-cell chronic lymphocytic leukemia (B-CLL). Aiming at addressing this issue, we investigated the clinical outcome of a well-defined population of B-CLL patients. Survival of B-CLL patients, diagnosed between 1990 and 1999 and recorded by the Registry of Hematological Malignancies of the Côte d'Or, was analysed applying Cox's regression model to the 237 included cases and to the 195 Binet stage A patients. To assess simultaneously the predictive value of each parameter on the risk of disease progression and on the risk of death, we completed this analysis by applying a three-states homogeneous Markov model to the whole study population. Analysis of the entire population showed that age (p < 0.001), Binet stage (p = 0.008) and CD19 positive lymphocyte count (p = 0.038) were three independent prognostic factors. However, in stage A patients, only progression into a more advanced stage, analysed as a time-dependent variable, and age had a clear impact on survival (p < 0.001 for both). Markov model revealed that an increased CD19 positive lymphocyte count increased the risk of disease progression in stage A patients (p = 0.002) but did not have direct impact on survival of either stage A patients with stable disease or stage B or C patients. An increased CD19 positive lymphocyte count at diagnosis is a marker of an increased risk of disease progression in stage A patients. Thus, it can be a useful tool for the clinical management of these patients.

Markov Modelling of Immunological and Virological States in HIV-1 Infected Patients

The purpose of this study was to evaluate the evolution of HIV infected patients and to bring out some significant factors associated with this pathology. The main criteria revealing the State of illness is viral load measurement (VL). However the CD4 lymphocytes also represent an important marker as these reflect the State of the immune reservoir. Many studies have been carried out in this field and different models have been proposed with a view to a better understanding of this disease. Multi State Markov models defined in terms of CD4 counts, or in terms of viral load, have proved to be very useful tools for modelling HIV disease progression. The model we have developed in this study is based on both the CD4 lymphocytes counts and VL. Markov models are characterized by transition intensities. In this paper we explored several structures in succession. First, we used a homogeneous continuous time Markov process with four states defined by crossed values of CD4 and VL in a given patient at a given time. Then, the effect of certain covariates on the infection process was introduced into the model via the transition intensity functions, as with a Cox regression model. Since the hypothesis of homogeneity may be unrealistic in certain cases, we also considered piecewise homogeneous Markov models. Finally, the effects of covariates and time were combined in a piecewise homogeneous model with a covariate. We applied these methods to data from 1313 HIV-infected patients included in the NADIS cohort.

A queueing model for chronic recurrent conditions under panel observation

In many chronic conditions, subjects alternate between an active and an inactive state, and sojourns into the active state may involve multiple lesions, infections, or other recurrences with different times of onset and resolution. We present a biologically interpretable model of such chronic recurrent conditions based on a queueing process. The model has a birth-death process describing recurrences and a semi-Markov process describing the alternation between active and inactive states, and can be fit to panel data that provide only a binary assessment of the active or inactive state at a series of discrete time points using a hidden Markov approach. We accommodate individual heterogeneity and covariates using a random effects model, and simulate the posterior distribution of unknowns using a Markov chain Monte Carlo algorithm. Application to a clinical trial of genital herpes shows how the method can characterize the biology of the disease and estimate treatment efficacy.

Non-homogeneous Markov Processes for Biomedical Data Analysis

Some time ago, the Markov processes were introduced in biomedical sciences in order to study disease history events. Homogeneous and Non-homogeneous Markov processes are an important field of research into stochastic processes, especially when exact transition times are unknown and interval-censored observations are present in the analysis. Non-homogeneous Markov process should be used when the homogeneous assumption is too strong. However these sorts of models increase the complexity of the analysis and standard software is limited. In this paper, some methods for fitting non-homogeneous Markov models are reviewed and an algorithm is proposed for biomedical data analysis. The method has been applied to analyse breast cancer data. Specific software for this purpose has been implemented.

Assessment of possible impact of a health promotion program in Korea from health risk trends in a longitudinally observed cohort

BACKGROUND: Longitudinally observed cohort data can be utilized to assess the potential for health promotion and healthcare planning by comparing the estimated risk factor trends of non-intervened with that of intervened. The paper seeks (1) to estimate a natural transition (patterns of movement between states) of health risk state from a Korean cohort data using a Markov model, (2) to derive an effective and necessary health promotion strategy for the population, and (3) to project a possible impact of an intervention program on health status. METHODS: The observed transition of health risk states in a Korean employee cohort was utilized to estimate the natural flow of aggregated health risk states from eight health risk measures using Markov chain models. In addition, a reinforced transition was simulated, given that a health promotion program was implemented for the cohort, to project a possible impact on improvement of health status. An intervened risk transition was obtained based on age, gender, and baseline risk state, adjusted to match with the Korean cohort, from a simulated random sample of a US employee population, where a health intervention was in place. RESULTS: The estimated natural flow (non-intervened), following Markov chain order 2, showed a decrease in low risk state by 3.1 percentage points in the Korean population while the simulated reinforced transition (intervened) projected an increase in low risk state by 7.5 percentage points. Estimated transitions of risk states demonstrated the necessity of not only the risk reduction but also low risk maintenance. CONCLUSIONS: The frame work of Markov chain efficiently estimated the trend, and captured the tendency in the natural flow. Given only a minimally intense health promotion program, potential risk reduction and low risk maintenance was projected.

Evaluating the role of human papillomavirus vaccine in cervical cancer prevention

Persistent genital infection with human papillomavirus (HPV) is a natural candidate as a surrogate marker for cervical cancer because of the strong epidemiologic and molecular evidence that HPV infection is the causative agent for almost all cervical cancers. However, while infection with high-risk types of HPV appears to be necessary for the development of cervical cancer, most infections are controlled by host immune response and do not lead to cancer in the vast majority of infected women. Because diagnostic tests cannot distinguish a persistent infection in the pathogenesis of cervical cancer from a transient infection, it is difficult to describe the disease mechanism as a progressive process based on observations. Therefore, the disease pathogenesis pathway does not fit into the usual surrogate marker framework, raising practical concerns about using HPV infection as a surrogate for a clinical endpoint in vaccine trials. In this paper, we describe the challenges in defining HPV infection as a surrogate endpoint in a HPV vaccine trial that is aimed at reducing cervical cancer rates and examine potential effects of the vaccine. We then outline some issues in the design and analysis of HPV vaccine trials, including the use of operationally defined HPV infection events meant to capture persistent infections. We conclude with a recommendation for a multistate model that uses HPV infection to help explain the mechanisms of vaccine action rather than validate it as an endpoint substitute.

The analysis of asthma control under a Markov assumption with use of covariates

In studies of disease states and their relation to evolution, data on the state are usually obtained at in frequent time points during follow-up. Moreover in many applications, there are measured covariates on each individual under study and interest centres on the relationship between these covariates and the disease evolution. We developed a continuous-time Markov model with use of time-dependent covariates and a Markov model with piecewise constant intensities to model asthma evolution. Methods to estimate the effect of covariates on transition intensities, to test the assumption of time homogeneity and to assess goodness-of-fit are proposed. We apply these methods to asthma control. We consider a three-state model and we discuss in detail the analysis of asthma control evolution.

Assessing progression and efficacy of treatment for diabetic retinopathy following the proliferative pathway to blindness: implication for diabetic retinopathy screening in Taiwan

AIMS

The natural history and treatment efficacy of diabetic retinopathy (DR) play important roles in the evaluation of screening. Therefore, the natural history of DR and rates of transition after treatment (including metabolic control and laser photocoagulation) from no diabetic retinopathy (NDR) to blindness were quantified.

METHODS

We studied a cohort of 795 patients with diabetes mellitus (DM) receiving fundus examination in the ophthalmology out-patient department of one medical centre between 1 January 1990 and 31 December 1992 in Taiwan. Follow-up data until 31 December 1998 were collected by chart review. Two multistate Markov models were proposed to assess the efficacy of the treatment regime in reducing progression to blindness.

RESULTS

The average times spent in states (i) no diabetic retinopathy (NDR), (ii) background diabetic retinopathy (BDR), (iii) preproliferative diabetic retinopathy (PPDR), and (iv) proliferative retinopathy (PDR) were 10.86 years, 8.33 years, 1.67 years, and 2.17 years, respectively. Early detection of PPDR may lead to a 60% reduction in PDR and an 83% reduction in blindness. Simulated results based on these parameters show that an annual screening programme, a biennial screening regime and a 4-yearly screening regime can lead to 54% (95% confidence interval (CI): 44-62%), 51% (95% CI: 41-59%), and 46% (95% CI: 36-54%) reductions in blindness, respectively.

CONCLUSIONS

Assessing the progression of DR following the proliferative pathway in this study suggests that screening for DR is worthwhile and that a 4-year interscreening interval for patients as yet without DR may be justified.

Bias correction of two-state latent Markov process parameter estimates under misclassification

A discretely observed two-state process may misclassify the state of an unobservable continuous-time, two-state Markov process. We examine the behaviour of maximum likelihood transition probability estimates as functions of known misclassification probabilities. Since maximum likelihood estimators are not available in closed form, we provide two alternatives for bias-adjusted estimation. In the case of large samples, the asymptotic bias is quantified and estimators are constructed iteratively using transition counts and specified misclassification probabilities. For finite samples, we provide an approximation based on partial derivatives. Estimators that are bias-adjusted to a first approximation are easily constructed and may serve well when misclassification probabilities are known to be small. Simulation studies reveal the effect of misclassification on estimation. Repeated diagnostic testing data illustrate the approaches.

An illness-death stochastic model in the analysis of longitudinal dementia data

A significant source of missing data in longitudinal epidemiological studies on elderly individuals is death. Subjects in large scale community-based longitudinal dementia studies are usually evaluated for disease status in study waves, not under continuous surveillance as in traditional cohort studies. Therefore, for the deceased subjects, disease status prior to death cannot be ascertained. Statistical methods assuming deceased subjects to be missing at random may not be realistic in dementia studies and may lead to biased results. We propose a stochastic model approach to simultaneously estimate disease incidence and mortality rates. We set up a Markov chain model consisting of three states, non-diseased, diseased and dead, and estimate the transition hazard parameters using the maximum likelihood approach. Simulation results are presented indicating adequate performance of the proposed approach.

Applications of continuous time hidden Markov models to the study of misclassified disease outcomes

Disease progression in prospective clinical and epidemiological studies is often conceptualized in terms of transitions between disease states. Analysis of data from such studies can be complicated by a number of factors, including the presence of individuals in various prevalent disease states and with unknown prior disease history, interval censored observations of state transitions and misclassified measurements of disease states. We present an approach where the disease states are modelled as the hidden states of a continuous time hidden Markov model using the imperfect measurements of the disease state as observations. Covariate effects on transitions between disease states are incorporated using a generalized regression framework. Parameter estimation and inference are based on maximum likelihood methods and rely on an EM algorithm. In addition, techniques for model assessment are proposed. Applications to two binary disease outcomes are presented: the oral lesion hairy leukoplakia in a cohort of HIV infected men and cervical human papillomavirus (HPV) infection in a cohort of young women. Estimated transition rates and misclassification probabilities for the hairy leukoplakia data agree well with clinical observations on the persistence and diagnosis of this lesion, lending credibility to the interpretation of hidden states as representing the actual disease states. By contrast, interpretation of the results for the HPV data are more problematic, illustrating that successful application of the hidden Markov model may be highly dependent on the degree to which the assumptions of the model are satisfied.

A Pearson-type goodness-of-fit test for stationary and time-continuous Markov regression models

Markov regression models describe the way in which a categorical response variable changes over time for subjects with different explanatory variables. Frequently it is difficult to measure the response variable on equally spaced discrete time intervals. Here we propose a Pearson-type goodness-of-fit test for stationary Markov regression models fitted to panel data. A parametric bootstrap algorithm is used to study the distribution of the test statistic. The proposed technique is applied to examine the fit of a Markov regression model used to identify markers for disease progression in psoriatic arthritis.

Estimation of the transition matrix of a discrete-time Markov chain

Discrete-time Markov chains have been successfully used to investigate treatment programs and health care protocols for chronic diseases. In these situations, the transition matrix, which describes the natural progression of the disease, is often estimated from a cohort observed at common intervals. Estimation of the matrix, however, is often complicated by the complex relationship among transition probabilities. This paper summarizes methods to obtain the maximum likelihood estimate of the transition matrix when the cycle length of the model coincides with the observation interval, the cycle length does not coincide with the observation interval, and when the observation intervals are unequal in length. In addition, the bootstrap is discussed as a method to assess the uncertainty of the maximum likelihood estimate and to construct confidence intervals for functions of the transition matrix such as expected survival.

Comparison of smoothing techniques for CD4 data in a Markov model with states defined by CD4: an example on the estimation of the HIV incubation time distribution

Multi-state models defined in terms of CD4 counts are useful for modelling HIV disease progression. A Markov model with six progressive CD4-based states and an absorbing state (AIDS) was used to estimate the cumulative probability of progressing to AIDS in 158 HIV-1 infected haemophiliacs with known seroconversion (SC) dates. A problem arising in such analysis is how to define CD4-based states, since this marker is subject to measurement error and short timescale variability. Four approaches were used: no smoothing, ad hoc smoothing (to move to a later/previous state two consecutive measurements to later/previous states are needed), kernel smoothing and random effects (RE) models. The estimates were compared with the Kaplan-Meier estimate based solely on data concerning time to AIDS. There was an apparent lack of agreement between the Kaplan-Meier and the "no smoothing" estimate. With the exception of the "no smoothing" method, "ad hoc", kernel and RE estimates fell within the range of the 95 per cent CIs of the Kaplan-Meier curve. Simulations demonstrated that the use of raw CD4 counts provides overestimated transition intensities. Compared to the kernel method, ad hoc is easier to implement and overcomes the problem of the choice of bandwidth. The RE approach leads to simple models, since it usually results in very few transitions to previous states, and can handle individuals with sparse data by smoothing their predictions towards the population mean. Ad hoc was the method that performed better, in terms of bias, than the other smoothing approaches.

A transitional model of barrier methods compliance with unbalanced loss to follow-up

Differential loss to follow-up can substantially influence the evaluation of treatment effects on a dichotomous outcome of interest in longitudinal trials. The use of transitional models incorporating loss to follow-up as an additional category of response and the nature of the correlated responses can provide a comprehensive view of a trial with unbalanced loss to follow-up. Under the Markov assumption, transitional models estimate the probability of changing from one outcome to another outcome between follow-up visits. Patterns of the response variable can be described by the estimated transition probabilities. The effects of intervention and covariates on the outcome of interest can also be estimated using a conditional likelihood function or a multinomial logit regression. Data from a randomized barrier method study designed to compare the proportion of participants using barrier methods consistently in two counselling groups are used to illustrate the proposed model.

A mixed model for two-state Markov processes under panel observation

Many chronic medical conditions can be meaningfully characterized in terms of a two-state stochastic process. Here we consider the problem in which subjects make transitions among two such states in continuous time but are only observed at discrete, irregularly spaced time points that are possibly unique to each subject. Data arising from such an observation scheme are called panel data, and methods for related analyses are typically based on Markov assumptions. The purpose of this article is to present a conditionally Markov model that accommodates subject-to-subject variation in the model parameters by the introduction of random effects. We focus on a particular random effects formulation that generates a closed-form expression for the marginal likelihood. The methodology is illustrated by application to a data set from a parasitic field infection survey.

Longitudinal data analysis (repeated measures) in clinical trials

Longitudinal data is often collected in clinical trials to examine the effect of treatment on the disease process over time. This paper reviews and summarizes much of the methodological research on longitudinal data analysis from the perspective of clinical trials. We discuss methodology for analysing Gaussian and discrete longitudinal data and show how these methods can be applied to clinical trials data. We illustrate these methods with five examples of clinical trials with longitudinal outcomes. We also discuss issues of particular concern in clinical trials including sequential monitoring and adjustments for missing data. A review of current software for analysing longitudinal data is also provided. Published in 1999 by John Wiley & Sons, Ltd. This article is a US Government work and is the public domain in the United States.

Timing of human immunodeficiency virus type 1 (HIV-1) transmission from mother to child: bayesian estimation using a mixture

The timing of mother-to-child HIV transmission is not directly observable but influences the infected child's viral and immune status in the neonatal period. A hierarchical model was developed in a Bayesian framework to 'back-calculate' the timing of HIV-1 transmission from mother to child from the virological and immunological kinetics in the infected infant. Joint evolution of viral markers and immune response was modelled as a continuous time Markov process. The modelling of the period from infection to birth was based on a mixture of three distributions taking into account the various mother-to-child transmission pathways: In utero (early or late in gestation) and intrapartum (during the delivery process), integrating the fact that transmission is a continuum during the pregnancy. Gibbs sampling was used to estimate the marginal posterior distributions of the transition intensities between stages of HIV infection and those of the individual times from infection to birth. We applied our model to data on 135 perinatally HIV-1-infected children included in the French Prospective Study on Pediatric HIV infection. The model suggested that transmission occurred late in utero during the last month of pregnancy and that the day of delivery was a particularly critical time in HIV-1 transmission from mother to child. The paper ends with a discussion of model assumptions and a comparison with results obtained using a non-parametric method.

Markov model and markers of small cell lung cancer: assessing the influence of reversible serum NSE, CYFRA 21-1 and TPS levels on prognosis

High serum NSE and advanced tumour stage are well-known negative prognostic determinants of small cell lung cancer (SCLC) when observed at presentation. However, such variables are reversible disease indicators as they can change during the course of therapy. The relationship between risk of death and marker level and disease state during treatment of SCLC chemotherapy is not known. A total of 52 patients with SCLC were followed during cisplatin-based chemotherapy (the median number of tumour status and marker level assessments was 4). The time-homogeneous Markov model was used in order to analyse separately the prognostic significance of change in the state of the serum marker level (NSE, CYFRA 21-1, TPS) or the change in tumour status. In this model, transition rate intensities were analysed according to three different states: alive with low marker level (state 0), alive with high marker level (state 1) and dead (absorbing state). The model analysing NSE levels showed that the mean time to move out of state 'high marker level' was short (123 days). There was a 44% probability of the opposite reversible state 'low marker level' being reached, which demonstrated the reversible property of the state 'high marker level'. The relative risk of death from this state 'high marker level' was about 2.24 times greater in comparison with that of state 0 'low marker level' (Wald's test; P < 0.01). For patients in state 'high marker level' at time of sampling, the probability of death increased dramatically, a transition explaining the rapid decrease in the probability of remaining stationary at this state. However, a non-nil probability to change from state 1 'high marker level' to the opposite transient level, state 0 'low marker level', was observed suggesting that, however infrequently, patients in state 1 'high marker level' might still return to state 0 'low marker level'. Almost similar conclusions can be drawn regarding the three-state model constructed using the tumour response status. For the two cytokeratin markers, the Markov model suggests the lack of a true reversible property of these variables as there was only a very weak probability of a patient returning to state 'low marker level' once having entered state 'high marker level'. In conclusion, The Markov model suggests that the observation of an increase in serum NSE level or a lack of response of the disease at any time during follow-up (according to the homogeneous assumption) was strongly associated with a worse prognosis but that the reversion to a low mortality risk state remains possible.

A Markov model for measuring vaccine efficacy for both susceptibility to infection and reduction in infectiousness for prophylactic HIV vaccines

We use a discrete-time non-homogeneous Markov chain to model data from augmented human immunodeficiency virus (HIV) vaccine trials. For this design, the study population consists of primary participants some of whom have steady sexual partners who are also enrolled to augment the trial. The state space consists of the infection status of primary participants without steady partners and the infection status of both persons in the steady partnerships. The transition probabilities are functions of the two parameters: vaccine efficacy for susceptibility (VES) and infectiousness (VEI). We use likelihood methods to estimate VES and VEI from time-to-event data. We then use stochastic simulations to explore the bias and precision of the estimators under various plausible conditions for HIV vaccine trials. We show that both the VES and VEI are estimable with reasonable precision for the conditions that may exist for planned HIV vaccine trials. We show that exams conducted every six months will likely provide sufficient information to estimate the VE parameters accurately, and that there is little gain in precision for more frequent exams. Finally, we show that joint estimation of the VES and VEI will likely be feasible in a currently planned HIV vaccine trial among injecting drug users in Bangkok, Thailand, if one augments the information about the primary participants in the trial with information about their steady sexual partners.