Multi-state Markov models for disease progression in the presence of informative examination times: an application to hepatitis C

Uniformization and bounded Taylor series in Newton-Raphson method improves computational performance for a multistate transition model estimation and inference

Multistate transition models (MSTMs) are valuable tools depicting disease progression. However, due to the complexity of MSTMs, larger sample size and longer follow-up time in real-world data, the computation of statistical estimation and inference for MSTMs becomes challenging. A bounded Taylor series in Newton-Raphson procedure is proposed which leverages the uniformization technique to derive maximum likelihood estimates and corresponding covariance matrix. The proposed method, namely uniformization Taylor-bounded Newton-Raphson, is validated in three simulation studies, which demonstrate the accuracy in parameter estimation, the efficiency in computation time and robustness in terms of different situations. This method is also illustrated using a large electronic medical record data related to statin-induced side effects and discontinuation.

Markov-modulated marked Poisson processes for modeling disease dynamics based on medical claims data

We explore Markov-modulated marked Poisson processes (MMMPPs) as a natural framework for modeling patients' disease dynamics over time based on medical claims data. In claims data, observations do not only occur at random points in time but are also informative, that is, driven by unobserved disease levels, as poor health conditions usually lead to more frequent interactions with the health care system. Therefore, we model the observation process as a Markov-modulated Poisson process, where the rate of health care interactions is governed by a continuous-time Markov chain. Its states serve as proxies for the patients' latent disease levels and further determine the distribution of additional data collected at each observation time, the so-called marks. Overall, MMMPPs jointly model observations and their informative time points by comprising two state-dependent processes: the observation process (corresponding to the event times) and the mark process (corresponding to event-specific information), which both depend on the underlying states. The approach is illustrated using claims data from patients diagnosed with chronic obstructive pulmonary disease by modeling their drug use and the interval lengths between consecutive physician consultations. The results indicate that MMMPPs are able to detect distinct patterns of health care utilization related to disease processes and reveal interindividual differences in the state-switching dynamics.

Obesity in Late-Life as a Protective Factor Against Dementia and Dementia-Related Mortality

OBJECTIVE

We estimated the conversion from cognitively normal to mild cognitive impairment (MCI) to probable dementia and death for underweight, normal, overweight, and obese older adults, where the timing of examinations is associated with the severity of dementia.

METHODS

We analyzed six waves of the National Health and Aging Trends Study (NHATS). Body mass (BMI) was computed from height and weight. Multi-state survival models (MSMs) examined misclassification probability, time-to-event ratios, and cognitive decline.

RESULTS

Participants (n = 6078) were 77 years old, 62% had overweight and/or obese BMI. After adjusting for the effects of cardiometabolic factors, age, sex, and race, obesity was protective against developing dementia (aHR=.44; 95%CI [.29-.67]) and dementia-related mortality (aHR=.63; 95%CI [.42-.95]).

DISCUSSION

We found a negative relationship between obesity and dementia and dementia-related mortality, a finding that has been underreported in the literature. The continuing obesity epidemic might complicate the diagnosis and treatment of dementia.

Ensemble machine learning identifies genetic loci associated with future worsening of disability in people with multiple sclerosis

Limited studies have been conducted to identify and validate multiple sclerosis (MS) genetic loci associated with disability progression. We aimed to identify MS genetic loci associated with worsening of disability over time, and to develop and validate ensemble genetic learning model(s) to identify people with MS (PwMS) at risk of future worsening. We examined associations of 208 previously established MS genetic loci with the risk of worsening of disability; we learned ensemble genetic decision rules and validated the predictions in an external dataset. We found 7 genetic loci (rs7731626: HR 0.92, P = 2.4 × 10^-5; rs12211604: HR 1.16, P = 3.2 × 10^-7; rs55858457: HR 0.93, P = 3.7 × 10^-7; rs10271373: HR 0.90, P = 1.1 × 10^-7; rs11256593: HR 1.13, P = 5.1 × 10^-57; rs12588969: HR = 1.10, P = 2.1 × 10^-10; rs1465697: HR 1.09, P = 1.7 × 10^-128) associated with risk worsening of disability; most of which were located near or tagged to 13 genomic regions enriched in peptide hormones and steroids biosynthesis pathways by positional and eQTL mapping. The derived ensembles produced a set of genetic decision rules that can be translated to provide additional prognostic values to existing clinical predictions, with the additional benefit of incorporating relevant genetic information into clinical decision making for PwMS. The present study extends our knowledge of MS progression genetics and provides the basis of future studies regarding the functional significance of the identified loci.

Prognosis of Chronic Kidney Disease in Patients with Non-Alcoholic Fatty Liver Disease: a Northeastern Taiwan Community Medicine Research Cohort

BACKGROUND

Non-alcoholic fatty liver disease (NAFLD) is associated with incident chronic kidney disease (CKD). We aimed to investigate outcomes and risk factors of CKD progression and regression.

METHODS

This is a longitudinal community-based cohort study of patients with NAFLD. Exclusion criteria included alcoholic liver diseases, sero-positive for hepatitis B surface antigen, sero-positive for hepatitis C virus antibodies, fatty liver index < 60, individuals with only one year of data, missing data for fibrosis-4 score and NAFLD fibrosis score, and advanced CKD at baseline. Main outcomes were stratified according to eGFR and albuminuria categories as state 1 (low risk), state 2 (moderately increased risk), and state 3 (high-risk/very-high risk of progression). The multi-state Markov model was used for outcome analysis.

RESULTS

This study included 1,628 patients with NAFLD with a median follow-up of 3.4 years. State 2 CKD was found in 9.3% of patients at 5 years (95% CI, 8.1%-10.6%). Most patients with state 2 CKD recovered to state 1 (69%; 95% CI, 63.7%-74%), while 17.6% progressed to state 3 (95% CI, 13.4%- 22.7%). Advanced liver fibrosis was found to be associated with the risk of transitioning from state 1 to state 2 (Fibrosis-4 score ≥1.3; hazard ratio [HR], 1.42; 95% CI, 1.02-2.00), and reduced recovery from state 2 to state 1 (NAFLD fibrosis score; NFS≥-1.455; HR, 0.56; 95% CI, 0.34-0.91).

CONCLUSIONS

NAFLD severity is associated with CKD, which may be reversible before becoming high-risk. Controlling metabolic risk factors and preventing advanced liver fibrosis are recommended.

Missing at random: a stochastic process perspective

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing-data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness-at-random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory can be extended easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

Partially hidden multi-state modelling of a prolonged disease state defined by a composite outcome

For rheumatic diseases, Minimal Disease Activity (MDA) is usually defined as a composite outcome which is a function of several individual outcomes describing symptoms or quality of life. There is ever increasing interest in MDA but relatively little has been done to characterise the pattern of MDA over time. Motivated by the aim of improving the modelling of MDA in psoriatic arthritis, the use of a two-state model to estimate characteristics of the MDA process is illustrated when there is particular interest in prolonged periods of MDA. Because not all outcomes necessary to define MDA are measured at all clinic visits, a partially hidden multi-state model with latent states is used. The defining outcomes are modelled as conditionally independent given these latent states, enabling information from all visits, even those with missing data on some variables, to be used. Data from the Toronto Psoriatic Arthritis Clinic are analysed to demonstrate improvements in accuracy and precision from the inclusion of data from visits with incomplete information on MDA. An additional benefit of this model is that it can be extended to incorporate explanatory variables, which allows process characteristics to be compared between groups. In the example, the effect of explanatory variables, modelled through the use of relative risks, is also summarised in a potentially more clinically meaningful manner by comparing times in states, and probabilities of visiting states, between patient groups.

Analysis of conversion of Alzheimer’s disease using a multi-state Markov model

With rapid aging of world population, Alzheimer’s disease is becoming a leading cause of death after cardiovascular disease and cancer. Nearly 10% of people who are over 65 years old are affected by Alzheimer’s disease. The causes have been studied intensively, but no definitive answer has been found. Genetic predisposition, abnormal protein deposits in brain, and environmental factors are suspected to play a role in the development of this disease. In this paper, we model progression of Alzheimer’s disease using a multi-state Markov model to investigate the significance of known risk factors such as age, apolipoprotein E4, and some brain structural volumetric variables from magnetic resonance imaging scans (e.g., hippocampus, etc.) while predicting transitions between different clinical diagnosis states. With the Alzheimer’s Disease Neuroimaging Initiative data, we found that the model with age is not significant (p = 0.1733) according to the likelihood ratio test, but the apolipoprotein E4 is a significant risk factor, and the examination of apolipoprotein E4-by-sex interaction suggests that the apolipoprotein E4 link to Alzheimer’s disease is stronger in women. Given the estimated transition probabilities, the prediction accuracy is as high as 0.7849.

A state transition framework for patient-level modeling of engagement and retention in HIV care using longitudinal cohort data

The human immunodeficiency virus (HIV) care cascade is a conceptual model used to outline the benchmarks that reflects effectiveness of HIV care in the whole HIV care continuum. The models can be used to identify barriers contributing to poor outcomes along each benchmark in the cascade such as disengagement from care or death. Recently, the HIV care cascade has been widely applied to monitor progress towards HIV prevention and care goals in an attempt to develop strategies to improve health outcomes along the care continuum. Yet, there are challenges in quantifying successes and gaps in HIV care using the cascade models that are partly due to the lack of analytic approaches. The availability of large cohort data presents an opportunity to develop a coherent statistical framework for analysis of the HIV care cascade. Motivated by data from the Academic Model Providing Access to Healthcare, which has provided HIV care to nearly 200,000 individuals in Western Kenya since 2001, we developed a state transition framework that can characterize patient-level movements through the multiple stages of the HIV care cascade. We describe how to transform large observational data into an analyzable format. We then illustrate the state transition framework via multistate modeling to quantify dynamics in retention aspects of care. The proposed modeling approach identifies the transition probabilities of moving through each stage in the care cascade. In addition, this approach allows regression-based estimation to characterize effects of (time-varying) predictors of within and between state transitions such as retention, disengagement, re-entry into care, transfer-out, and mortality. Copyright © 2017 John Wiley & Sons, Ltd.

A joint model for interval-censored functional decline trajectories under informative observation

Multi-state models are useful for modelling disease progression where the state space of the process is used to represent the discrete disease status of subjects. Often, the disease process is only observed at clinical visits, and the schedule of these visits can depend on the disease status of patients. In such situations, the frequency and timing of observations may depend on transition times that are themselves unobserved in an interval-censored setting. There is a potential for bias if we model a disease process with informative observation times as a non-informative observation scheme with pre-specified examination times. In this paper, we develop a joint model for the disease and observation processes to ensure valid inference because the follow-up process may itself contain information about the disease process. The transitions for each subject are modelled using a Markov process, where bivariate subject-specific random effects are used to link the disease and observation models. Inference is based on a Bayesian framework, and we apply our joint model to the analysis of a large study examining functional decline trajectories of palliative care patients.

Kinetics methods for clinical epidemiology problems

Calculating the probability of each possible outcome for a patient at any time in the future is currently possible only in the simplest cases: short-term prediction in acute diseases of otherwise healthy persons. This problem is to some extent analogous to predicting the concentrations of species in a reactor when knowing initial concentrations and after examining reaction rates at the individual molecule level. The existing theoretical framework behind predicting contagion and the immediate outcome of acute diseases in previously healthy individuals is largely analogous to deterministic kinetics of chemical systems consisting of one or a few reactions. We show that current statistical models commonly used in chronic disease epidemiology correspond to simple stochastic treatment of single reaction systems. The general problem corresponds to stochastic kinetics of complex reaction systems. We attempt to formulate epidemiologic problems related to chronic diseases in chemical kinetics terms. We review methods that may be adapted for use in epidemiology. We show that some reactions cannot fit into the mass-action law paradigm and solutions to these systems would frequently exhibit an antiportfolio effect. We provide a complete example application of stochastic kinetics modeling for a deductive meta-analysis of two papers on atrial fibrillation incidence, prevalence, and mortality.

Joint assessment of dependent discrete disease state processes

In multiple sclerosis, the primary clinical measure of disability level is an ordinal score, the expanded disability severity scale score. In relapsing-remitting multiple sclerosis, measures of relapse are additionally of interest. Multiple sclerosis patients are typically assessed with regard to both the expanded disability severity scale and relapse state at each follow-up visit. As both are discrete measures, the two can be viewed as jointly dependent Markov processes. One of the main goals of multiple sclerosis research is to accurately model, over time, both transitions between expanded disability severity scale states and change in relapse state. This objective requires a number of significant modeling decisions, including decisions about whether or not the combination of specific disease states is warranted and assessment of the dependence structure between the two disease processes. Historically, such decisions are often made in an ad hoc manner and are not formally justified. We propose novel use of Bayes factors and Bayesian variable selection in the assessment of jointly dependent Markovian processes in multiple sclerosis. Methods are assessed using both simulated data and data collected from the Partners Multiple Sclerosis Center in Boston, MA.

A model-informed rank test for right-censored data with intermediate states

The generalized Wilcoxon and log-rank tests are commonly used for testing differences between two survival distributions. We modify the Wilcoxon test to account for auxiliary information on intermediate disease states that subjects may pass through before failure. For a disease with multiple states where patients are monitored periodically but exact transition times are unknown (e.g. staging in cancer), we first fit a multi-state Markov model to the full data set; when censoring precludes the comparison of survival times between two subjects, we use the model to estimate the probability that one subject will have survived longer than the other given their censoring times and last observed status, and use these probabilities to compute an expected rank for each subject. These expected ranks form the basis of our test statistic. Simulations demonstrate that the proposed test can improve power over the log-rank and generalized Wilcoxon tests in some settings while maintaining the nominal type 1 error rate. The method is illustrated on an amyotrophic lateral sclerosis data set.

A joint model for multistate disease processes and random informative observation times, with applications to electronic medical records data

Multistate models are used to characterize individuals' natural histories through diseases with discrete states. Observational data resources based on electronic medical records pose new opportunities for studying such diseases. However, these data consist of observations of the process at discrete sampling times, which may either be pre-scheduled and non-informative, or symptom-driven and informative about an individual's underlying disease status. We have developed a novel joint observation and disease transition model for this setting. The disease process is modeled according to a latent continuous-time Markov chain; and the observation process, according to a Markov-modulated Poisson process with observation rates that depend on the individual's underlying disease status. The disease process is observed at a combination of informative and non-informative sampling times, with possible misclassification error. We demonstrate that the model is computationally tractable and devise an expectation-maximization algorithm for parameter estimation. Using simulated data, we show how estimates from our joint observation and disease transition model lead to less biased and more precise estimates of the disease rate parameters. We apply the model to a study of secondary breast cancer events, utilizing mammography and biopsy records from a sample of women with a history of primary breast cancer.

HIV-1 disease progression during highly active antiretroviral therapy: an application using population-level data in British Columbia: 1996-2011

BACKGROUND

Accurately estimating rates of disease progression is of central importance in developing mathematical models used to project outcomes and guide resource allocation decisions. Our objective was to specify a multivariate regression model to estimate changes in disease progression among individuals on highly active antiretroviral treatment in British Columbia, Canada, 1996-2011.

METHODS

We used population-level data on disease progression and antiretroviral treatment utilization from the BC HIV Drug Treatment Program. Disease progression was captured using longitudinal CD4 and plasma viral load testing data, linked with data on antiretroviral treatment. The study outcome was categorized into (CD4 count ≥ 500, 500-350, 350-200, <200 cells/mm, and mortality). A 5-state continuous-time Markov model was used to estimate covariate-specific probabilities of CD4 progression, focusing on temporal changes during the study period.

RESULTS

A total of 210,083 CD4 measurements among 7421 individuals with HIV/AIDS were included in the study. Results of the multivariate model suggested that current highly active antiretroviral treatment at baseline, lower baseline CD4 (<200 cells/mm), and extended durations of elevated plasma viral load were each associated with accelerated progression. Immunological improvement was accelerated significantly from 2004 onward, with 23% and 46% increases in the probability of CD4 improvement from the fourth CD4 stratum (CD4 < 200) in 2004-2008 and 2008-2011, respectively.

CONCLUSION

Our results demonstrate the impact of innovations in antiretroviral treatment and treatment delivery at the population level. These results can be used to estimate a transition probability matrix flexible to changes in the observed mix of clients in different clinical stages and treatment regimens over time.

Description and prediction of the development of metabolic syndrome: a longitudinal analysis using a markov model approach

BACKGROUND

Delineating the natural history of metabolic syndrome (MetS) is prerequisite to prevention. This study aimed to build Markov models to simulate each component's progress and to test the effect of different initial states on the development of MetS.

METHODS

MetS was defined with revised AHA/NHLBI criteria. Each reversible multistate Markov chain consisted of 8 states (no component, five isolated component states, 2-component state, and MetS state). Yearly transition probabilities were calculated from a five-year population-based follow up studywhich enrolled 2,247 individuals with mean aged 32.4 years at study entry.

RESULTS

In men, high BP or a 2-component state was most likely to initiate the progress of MetS. In women, abdominal obesity or low HDL were the most likely initiators. Metabolic components were likely to occur together. The development of MetS was an increasing monotonic function of time. MetS was estimated to develop within 15 years in 12.7% of young men with no component, and 2 components developed in 16.3%. MetS was estimated to develop in 10.6% of women with at the age of 47, and 2 components developed in 14.3%. MetS was estimated to develop in 24.6% of men and 27.6% of women with abdominal obesity, a rate higher than in individuals initiating with no component.

CONCLUSIONS

This modeling study allows estimation of the natural history of MetS. Men tended to develop this syndrome sooner than women did, i.e., before their fifth decade of life. Individuals with 1 or 2 components showed increased development of MetS.

A Correlated Random Effects Model for Non-homogeneous Markov Processes with Nonignorable Missingness

Life history data arising in clusters with prespecified assessment time points for patients often feature incomplete data since patients may choose to visit the clinic based on their needs. Markov process models provide a useful tool describing disease progression for life history data. The literature mainly focuses on time homogeneous process. In this paper we develop methods to deal with non-homogeneous Markov process with incomplete clustered life history data. A correlated random effects model is developed to deal with the nonignorable missingness, and a time transformation is employed to address the non-homogeneity in the transition model. Maximum likelihood estimate based on the Monte-Carlo EM algorithm is advocated for parameter estimation. Simulation studies demonstrate that the proposed method works well in many situations. We also apply this method to an Alzheimer's disease study.

Armitage lecture 2010: Understanding treatment effects: the value of integrating longitudinal data and survival analysis

There is a single-minded focus on events in survival analysis, and we often ignore longitudinal data that are collected together with the event data. This is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses. Understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences. The latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis. The concept of local independence and the associated graphs are useful. This is related to Granger causality, an important method from econometrics that is generally undervalued by statisticians. This causality concept is different from the counterfactual one since it lacks lacks the intervention aspect. The notion that one can intervene at will in naturally occurring processes, which seems to underly much of modern causal inference, is problematic when studying mediation and mechanisms. It is natural to assume a stochastic process point of view when analyzing dynamic relationships. We present some examples to illustrate this. It is not clear how survival analysis must be developed to handle the complex life-history data that are increasingly being collected today. We give some suggestions.

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

Delay of the diagnosis of hepatitis C virus (HCV), and its treatment to avert cirrhosis, is often present sincethe early stage of HCV progression is latent. Current methods to determine the incubation time to HCV-related cirrhosis and the duration time from cirrhosis to subsequent events (e.g. complications or death) used to be based on the time of liver biopsy diagnosis and ignore this delay which led to an interval censoring for the first event time and a double censoring for the subsequent event time. To investigate the impact of this delay in estimating HCV progression rates and relevant estimating bias, we present a correlated two-stage progression model for delayed diagnosis time and fit the developed model to the previously studied hepatitis C cohort data from Edinburgh. Our analysis shows that taking the delayed diagnosis into account gives a mildly different estimate of progression rate to cirrhosis and significantly lower estimated progression rate to HCV-related death in comparison with conventional modelling. We also find that when the delay increases, the bias in estimating progression increases significantly.

Non-homogeneous Markov process models with informative observations with an application to Alzheimer's disease

Identifying risk factors for transition rates among normal cognition, mildly cognitive impairment, dementia and death in an Alzheimer's disease study is very important. It is known that transition rates among these states are strongly time dependent. While Markov process models are often used to describe these disease progressions, the literature mainly focuses on time homogeneous processes, and limited tools are available for dealing with non-homogeneity. Further, patients may choose when they want to visit the clinics, which creates informative observations. In this paper, we develop methods to deal with non-homogeneous Markov processes through time scale transformation when observation times are pre-planned with some observations missing. Maximum likelihood estimation via the EM algorithm is derived for parameter estimation. Simulation studies demonstrate that the proposed method works well under a variety of situations. An application to the Alzheimer's disease study identifies that there is a significant increase in transition rates as a function of time. Furthermore, our models reveal that the non-ignorable missing mechanism is perhaps reasonable.