A semi-Markov model for multistate and interval-censored data with multiple terminal events. Application in renal transplantation

Annual transition probabilities of overweight and obesity in older adults: Evidence from World Health Organization Study on global AGEing and adult health

Overweight/obesity is becoming increasingly prevalent in sub-Saharan Africa including Ghana. However, transition probabilities, an essential component to develop cost-effective measures for weight management is lacking in this population. We estimated annual transition probabilities between three body mass index (BMI) categories: normal weight (BMI ≥18.5 and <25.0 kg/m2), overweight (BMI ≥25.0 and <30.0 kg/m2), and obesity (BMI ≥30.0 kg/m2), among older adults aged ≥50 years in Ghana. Data were used from a nationally representative, multistage sample of 1496 (44.3% females) older adults in both Waves 1 (2007/8) and 2 (2014/15) of the Ghana WHO SAGE. A multistage Markov model was used to estimate annual transition probabilities. We further examined the impact of specific socio-economic factors on the transition probabilities. At baseline, 22.8% were overweight and 11.1% were obese. The annual transition probability was 4.0% (95% CI: 3.4%, 4.8%) from normal weight to overweight, 11.1% (95% CI: 9.5%, 13.0%) from overweight to normal weight and 4.9% (95% CI: 3.8%, 6.2%) from overweight to obesity. For obese individuals, the probability of remaining obese, transitioning to overweight and completely reverting to normal weight was 90.2% (95% CI: 87.7%, 92.3%), 9.2% (95% CI: 7.2%, 11.6%) and 0.6% (95% CI: 0.4%, 0.8%) respectively. Being female, aged 50-65 years, urban residence, having high education and high wealth were associated with increased probability of transitioning into the overweight or obese categories. Our findings highlight the difficulty in transitioning away from obesity, especially among females. The estimated transition probabilities will be essential in health economic simulation models to determine sustainable weight management interventions.

ESTIMATING AND COMPARING CANCER PROGRESSION RISKS UNDER VARYING SURVEILLANCE PROTOCOLS

Outcomes after cancer diagnosis and treatment are often observed at discrete times via doctor-patient encounters or specialized diagnostic examinations. Despite their ubiquity as endpoints in cancer studies, such outcomes pose challenges for analysis. In particular, comparisons between studies or patient populations with different surveillance schema may be confounded by differences in visit frequencies. We present a statistical framework based on multistate and hidden Markov models that represents events on a continuous time scale given data with discrete observation times. To demonstrate this framework, we consider the problem of comparing risks of prostate cancer progression across multiple active surveillance cohorts with different surveillance frequencies. We show that the different surveillance schedules partially explain observed differences in the progression risks between cohorts. Our application permits the conclusion that differences in underlying cancer progression risks across cohorts persist after accounting for different surveillance frequencies.

A stochastic estimation procedure for intermittently-observed semi-Markov multistate models with back transitions

Multistate models provide an important method for analyzing a wide range of life history processes including disease progression and patient recovery following medical intervention. Panel data consisting of the states occupied by an individual at a series of discrete time points are often used to estimate transition intensities of the underlying continuous-time process. When transition intensities depend on the time elapsed in the current state and back transitions between states are possible, this intermittent observation process presents difficulties in estimation due to intractability of the likelihood function. In this manuscript, we present an iterative stochastic expectation-maximization algorithm that relies on a simulation-based approximation to the likelihood function and implement this algorithm using rejection sampling. In a simulation study, we demonstrate the feasibility and performance of the proposed procedure. We then demonstrate application of the algorithm to a study of dementia, the Nun Study, consisting of intermittently-observed elderly subjects in one of four possible states corresponding to intact cognition, impaired cognition, dementia, and death. We show that the proposed stochastic expectation-maximization algorithm substantially reduces bias in model parameter estimates compared to an alternative approach used in the literature, minimal path estimation. We conclude that in estimating intermittently observed semi-Markov models, the proposed approach is a computationally feasible and accurate estimation procedure that leads to substantial improvements in back transition estimates.

A multistate additive relative survival semi-Markov model

Medical researchers are often interested to investigate the relationship between explicative variables and times-to-events such as disease progression or death. Such multiple times-to-events can be studied using multistate models. For chronic diseases, it may be relevant to consider semi-Markov multistate models because the transition intensities between two clinical states more likely depend on the time already spent in the current state than on the chronological time. When the cause of death for a patient is unavailable or not totally attributable to the disease, it is not possible to specifically study the associations with the excess mortality related to the disease. Relative survival analysis allows an estimate of the net survival in the hypothetical situation where the disease would be the only possible cause of death. In this paper, we propose a semi-Markov additive relative survival (SMRS) model that combines the multistate and the relative survival approaches. The usefulness of the SMRS model is illustrated by two applications with data from a French cohort of kidney transplant recipients. Using simulated data, we also highlight the effectiveness of the SMRS model: the results tend to those obtained if the different causes of death are known.

Semi-Markov models for interval censored transient cognitive states with back transitions and a competing risk

Continuous-time multi-state stochastic processes are useful for modeling the flow of subjects from intact cognition to dementia with mild cognitive impairment and global impairment as intervening transient cognitive states and death as a competing risk. Each subject's cognition is assessed periodically resulting in interval censoring for the cognitive states while death without dementia is not interval censored. Since back transitions among the transient states are possible, Markov chains are often applied to this type of panel data. In this manuscript, we apply a semi-Markov process in which we assume that the waiting times are Weibull distributed except for transitions from the baseline state, which are exponentially distributed and in which we assume no additional changes in cognition occur between two assessments. We implement a quasi-Monte Carlo (QMC) method to calculate the higher order integration needed for likelihood estimation. We apply our model to a real dataset, the Nun Study, a cohort of 461 participants.

Regression methods for investigating risk factors of chronic kidney disease outcomes: the state of the art

Background

Chronic kidney disease (CKD) is a progressive and usually irreversible disease. Different types of outcomes are of interest in the course of CKD such as time-to-dialysis, transplantation or decline of the glomerular filtration rate (GFR). Statistical analyses aiming at investigating the association between these outcomes and risk factors raise a number of methodological issues. The objective of this study was to give an overview of these issues and to highlight some statistical methods that can address these topics.

Methods

A literature review of statistical methods published between 2002 and 2012 to investigate risk factors of CKD outcomes was conducted within the Scopus database. The results of the review were used to identify important methodological issues as well as to discuss solutions for each type of CKD outcome.

Results

Three hundred and four papers were selected. Time-to-event outcomes were more often investigated than quantitative outcome variables measuring kidney function over time. The most frequently investigated events in survival analyses were all-cause death, initiation of kidney replacement therapy, and progression to a specific value of GFR. While competing risks were commonly accounted for, interval censoring was rarely acknowledged when appropriate despite existing methods. When the outcome of interest was the quantitative decline of kidney function over time, standard linear models focussing on the slope of GFR over time were almost as often used as linear mixed models which allow various numbers of repeated measurements of kidney function per patient. Informative dropout was accounted for in some of these longitudinal analyses.

Conclusions

This study provides a broad overview of the statistical methods used in the last ten years for investigating risk factors of CKD progression, as well as a discussion of their limitations. Some existing potential alternatives that have been proposed in the context of CKD or in other contexts are also highlighted.

Adjusting for mortality when identifying risk factors for transitions to mild cognitive impairment and dementia

Risk factors for mild cognitive impairment (MCI) and dementia are often investigated without accounting for the competing risk of mortality, which can bias results and lead to spurious conclusions, particularly regarding protective factors. Here, we apply a semi-Markov modeling approach to 531 participants in the University of Kentucky Biologically Resilient Adults in Neurological Studies (BRAiNS) longitudinal cohort, over one-third of whom died without transitioning to a cognitively impaired clinical state. A semi-Markov approach enables a statistical study of clinical state transitions while accounting for the competing risk of death and facilitates insights into both the odds that a risk factor will affect clinical transitions as well as the age at which the transition to MCI or dementia will occur. Risk factors assessed in the current study were identified by matching those reported in the literature with the data elements collected on participants. The presence of Type II diabetes at baseline shortens the time it takes cognitively intact individuals to transition to MCI by seven years on average while use of estrogen replacement therapy at enrollment (baseline) decreases the time required to convert from MCI to dementia by 1.5 years. Finally, smoking and being overweight do not promote transitions to impaired states but instead hasten death without a dementia. In contrast, conventional statistical analyses based on Cox proportional hazards models fail to recognize diabetes as a risk, show that being overweight increases the risk of clinical MCI, and that high blood pressure at baseline increases the risk of a dementia.

Fitting and interpreting continuous-time latent Markov models for panel data

Multistate models characterize disease processes within an individual. Clinical studies often observe the disease status of individuals at discrete time points, making exact times of transitions between disease states unknown. Such panel data pose considerable modeling challenges. Assuming the disease process progresses accordingly, a standard continuous-time Markov chain (CTMC) yields tractable likelihoods, but the assumption of exponential sojourn time distributions is typically unrealistic. More flexible semi-Markov models permit generic sojourn distributions yet yield intractable likelihoods for panel data in the presence of reversible transitions. One attractive alternative is to assume that the disease process is characterized by an underlying latent CTMC, with multiple latent states mapping to each disease state. These models retain analytic tractability due to the CTMC framework but allow for flexible, duration-dependent disease state sojourn distributions. We have developed a robust and efficient expectation-maximization algorithm in this context. Our complete data state space consists of the observed data and the underlying latent trajectory, yielding computationally efficient expectation and maximization steps. Our algorithm outperforms alternative methods measured in terms of time to convergence and robustness. We also examine the frequentist performance of latent CTMC point and interval estimates of disease process functionals based on simulated data. The performance of estimates depends on time, functional, and data-generating scenario. Finally, we illustrate the interpretive power of latent CTMC models for describing disease processes on a dataset of lung transplant patients. We hope our work will encourage wider use of these models in the biomedical setting.

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

This paper presents a parametric method of fitting semi-Markov models with piecewise-constant hazards in the presence of left, right and interval censoring. We investigate transition intensities in a three-state illness–death model with no recovery. We relax the Markov assumption by adjusting the intensity for the transition from state 2 (illness) to state 3 (death) for the time spent in state 2 through a time-varying covariate. This involves the exact time of the transition from state 1 (healthy) to state 2. When the data are subject to left or interval censoring, this time is unknown. In the estimation of the likelihood, we take into account interval censoring by integrating out all possible times for the transition from state 1 to state 2. For left censoring, we use an Expectation–Maximisation inspired algorithm. A simulation study reflects the performance of the method. The proposed combination of statistical procedures provides great flexibility. We illustrate the method in an application by using data on stroke onset for the older population from the UK Medical Research Council Cognitive Function and Ageing Study. Copyright © 2012 John Wiley & Sons, Ltd.

Bayesian analysis of a disability model for lung cancer survival

Bayesian reasoning, survival analysis and multi-state models are used to assess survival times for Stage IV non-small-cell lung cancer patients and the evolution of the disease over time. Bayesian estimation is done using minimum informative priors for the Weibull regression survival model, leading to an automatic inferential procedure. Markov chain Monte Carlo methods have been used for approximating posterior distributions and the Bayesian information criterion has been considered for covariate selection. In particular, the posterior distribution of the transition probabilities, resulting from the multi-state model, constitutes a very interesting tool which could be useful to help oncologists and patients make efficient and effective decisions.

Cost-benefit estimation of cadaveric kidney transplantation: the case of a developing country

BACKGROUND

In this paper we have estimated the cost savings for the health care system and quality-of-life improvement for patients from an increased number of kidney transplants in Chile. We compared the present value of dialysis and transplantation costs and quality of life over a 20-year horizon.

METHODS

We used Markov models and introduced some degree of uncertainty in the value of some of the parameters that built the model. Using Monte Carlo simulations, we estimated the confidence intervals for our results.

RESULTS

Our estimates suggested that a kidney transplant showed an expected savings value of US$28,000 for the health care system. If the quality-of-life improvement was also considered, the expected savings rise to US$ 102,000. These results imply that increasing donation rate by 1 donor per million population would achieve an estimated cost saving of US$827,000 per year, or near US$3 million per year considering the effect on the quality of life.

CONCLUSION

These results demonstrated that kidney transplantation along with a better quality of life for patients are a cost-saving decision for developing countries.

Semi-Markov models with phase-type sojourn distributions

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.

Multi-state Markov models in cancer screening evaluation: a brief review and case study

This work presents a brief overview of Markov models in cancer screening evaluation and focuses on two specific models. A three-state model was first proposed to estimate jointly the sensitivity of the screening procedure and the average duration in the preclinical phase, i.e. the period when the cancer is asymptomatic but detectable by screening. A five-state model, incorporating lymph node involvement as a prognostic factor, was later proposed combined with a survival analysis to predict the mortality reduction associated with screening. The strengths and limitations of these two models are illustrated using data from French breast cancer service screening programmes. The three-state model is a useful frame but parameter estimates should be interpreted with caution. They are highly correlated and depend heavily on the parametric assumptions of the model. Our results pointed out a serious limitation to the five-state model, due to implicit assumptions which are not always verified. Although it may still be useful, there is a need for more flexible models. Over-diagnosis is an important issue for both models and induces bias in parameter estimates. It can be addressed by adding a non-progressive state, but this may provide an uncertain estimation of over-diagnosis. When the primary goal is to avoid bias, rather than to estimate over-diagnosis, it may be more appropriate to correct for over-diagnosis assuming different levels in a sensitivity analysis. This would be particularly relevant in a perspective of mortality reduction estimation.

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.