A review of multistate modelling approaches in monitoring disease progression: Bayesian estimation using the Kolmogorov-Chapman forward equations

2021 ◽  
pp. 096228022199750
Author(s):  
Zvifadzo Matsena Zingoni ◽  
Tobias F Chirwa ◽  
Jim Todd ◽  
Eustasius Musenge
Keyword(s):  
Bayesian Estimation ◽  
Transition Probabilities ◽  
Markov Property ◽  
Likelihood Estimation ◽  
Single Individual ◽  
Transition Rates ◽  
Multistate Models ◽  
Time To Event ◽  
Modeling Framework ◽  
Forward Equations

There are numerous fields of science in which multistate models are used, including biomedical research and health economics. In biomedical studies, these stochastic continuous-time models are used to describe the time-to-event life history of an individual through a flexible framework for longitudinal data. The multistate framework can describe more than one possible time-to-event outcome for a single individual. The standard estimation quantities in multistate models are transition probabilities and transition rates which can be mapped through the Kolmogorov-Chapman forward equations from the Bayesian estimation perspective. Most multistate models assume the Markov property and time homogeneity; however, if these assumptions are violated, an extension to non-Markovian and time-varying transition rates is possible. This manuscript extends reviews in various types of multistate models, assumptions, methods of estimation and data features compatible with fitting multistate models. We highlight the contrast between the frequentist (maximum likelihood estimation) and the Bayesian estimation approaches in the multistate modeling framework and point out where the latter is advantageous. A partially observed and aggregated dataset from the Zimbabwe national ART program was used to illustrate the use of Kolmogorov-Chapman forward equations. The transition rates from a three-stage reversible multistate model based on viral load measurements in WinBUGS were reported.

Victim Number Effects in Charitable Giving: Joint Evaluations Promote Egalitarian Decisions

2021 ◽  
pp. 014616722098273
Author(s):  
Alexander Garinther ◽  
Holly Arrow ◽  
Pooya Razavi
Keyword(s):  
Bayesian Estimation ◽  
Charitable Giving ◽  
Random Order ◽  
Single Individual ◽  
Temporal Window ◽  
Opposite Pattern ◽  
Online Experiments ◽  
Joint Evaluation

Studies of victim number effects in charitable giving consistently find that people care more and help more when presented with an appeal to help an individual compared with an appeal to help multiple people in need. Across three online experiments ( N = 1,348), Bayesian estimation revealed the opposite pattern when people responded to multiple appeals to help targets of different sizes (1, 2, 5, 7, and 12). In this joint evaluation context, participants donated more to larger groups, when appeals were presented in both ascending order (Study 1) and random order (Study 2). The pattern held whether or not participants saw an overview of all appeals at the start of the study and when a single individual was added to the array (Study 3). These results clarify how compassion fade findings typical of separate evaluations may not generalize to contexts in which people encounter multiple appeals within a short temporal window.

scholarly journals Time-to-event analysis in economic evaluations: a comparison of modelling methods to assess the cost-effectiveness of transplanting a marginal quality kidney

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sameera Senanayake ◽  
Nicholas Graves ◽  
Helen Healy ◽  
Keshwar Baboolal ◽  
Adrian Barnett ◽  
...  
Keyword(s):  
Cost Effectiveness ◽  
Time Horizon ◽  
Transition Probabilities ◽  
Discrete Event ◽  
Economic Evaluations ◽  
Event Analysis ◽  
Time To Event ◽  
Modelling Method ◽  
Cycle Lengths ◽  
Modelling Methods

Abstract Background Economic-evaluations using decision analytic models such as Markov-models (MM), and discrete-event-simulations (DES) are high value adds in allocating resources. The choice of modelling method is critical because an inappropriate model yields results that could lead to flawed decision making. The aim of this study was to compare cost-effectiveness when MM and DES were used to model results of transplanting a lower-quality kidney versus remaining waitlisted for a kidney. Methods Cost-effectiveness was assessed using MM and DES. We used parametric survival models to estimate the time-dependent transition probabilities of MM and distribution of time-to-event in DES. MMs were simulated in 12 and 6 monthly cycles, out to five and 20-year time horizon. Results DES model output had a close fit to the actual data. Irrespective of the modelling method, the cycle length of MM or the time horizon, transplanting a low-quality kidney as compared to remaining waitlisted was the dominant strategy. However, there were discrepancies in costs, effectiveness and net monetary benefit (NMB) among different modelling methods. The incremental NMB of the MM in the 6-months cycle lengths was a closer fit to the incremental NMB of the DES. The gap in the fit of the two cycle lengths to DES output reduced as the time horizon increased. Conclusion Different modelling methods were unlikely to influence the decision to accept a lower quality kidney transplant or remain waitlisted on dialysis. Both models produced similar results when time-dependant transition probabilities are used, most notable with shorter cycle lengths and longer time-horizons.

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (01) ◽  
pp. 82-102
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

scholarly journals Stability Analysis for Markovian Jump Neutral Systems with Mixed Delays and Partially Known Transition Rates

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Lianglin Xiong ◽  
Xiaobing Zhou ◽  
Jie Qiu ◽  
Jing Lei
Keyword(s):  
Partial Information ◽  
Transition Probabilities ◽  
Mixed Delays ◽  
Transition Rates ◽  
Neutral Systems ◽  
Markovian Jump ◽  
Markovian Jump System ◽  
Analysis Technique ◽  
Transition Rate Matrix ◽  
Delay Dependent

The delay-dependent stability problem is studied for Markovian jump neutral systems with partial information on transition probabilities, and the considered delays are mixed and model dependent. By constructing the new stochastic Lyapunov-Krasovskii functional, which combined the introduced free matrices with the analysis technique of matrix inequalities, a sufficient condition for the systems with fully known transition rates is firstly established. Then, making full use of the transition rate matrix, the results are obtained for the other case, and the uncertain neutral Markovian jump system with incomplete transition rates is also considered. Finally, to show the validity of the obtained results, three numerical examples are provided.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

scholarly journals Infinitely divisible random transition probabilities with application to dependent Markov chains

1984 ◽  
Vol 25 (4) ◽  
pp. 463-472
Author(s):  
Peter L. Chesson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
White Noise ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Rates ◽  
Infinitely Divisible ◽  
Random Transition ◽  
Transition Probability Matrices ◽  
Probability Matrices

AbstractRandom transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.

Optimal control of random walks, birth and death processes, and queues

1981 ◽  
Vol 13 (01) ◽  
pp. 61-83 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Optimal Control ◽  
Random Walks ◽  
Queue Length ◽  
Transition Probabilities ◽  
Transition Rates ◽  
Average Reward ◽  
Birth And Death Processes ◽  
Optimal Policies ◽  
One Step ◽  
Increasing Functions

This is a study of simple random walks, birth and death processes, and M/M/s queues that have transition probabilities and rates that are sequentially controlled at jump times of the processes. Each control action yields a one-step reward depending on the chosen probabilities or transition rates and the state of the process. The aim is to find control policies that maximize the total discounted or average reward. Conditions are given for these processes to have certain natural monotone optimal policies. Under such a policy for the M/M/s queue, for example, the service and arrival rates are non-decreasing and non-increasing functions, respectively, of the queue length. Properties of these policies and a linear program for computing them are also discussed.

scholarly journals Developing a composite outcome measure for frailty prevention trials – rationale, derivation and sample size comparison with other candidate measures

2020 ◽  
Author(s):  
Miles D. Witham ◽  
James Wason ◽  
Richard M Dodds ◽  
Avan A Sayer
Keyword(s):  
Sample Size ◽  
Transition Probabilities ◽  
Adverse Outcomes ◽  
Transition Rates ◽  
Composite Outcome ◽  
Composite Measure ◽  
Sample Sizes ◽  
Loss To Follow Up ◽  
Prevention Trials

Abstract Introduction Frailty is the loss of ability to withstand a physiological stressor, and is associated with multiple adverse outcomes in older people. Trials to prevent or ameliorate frailty are in their infancy. A range of different outcome measures have been proposed, but current measures require either large sample sizes, long follow-up, or do not directly measure the construct of frailty. Methods We propose a composite outcome for frailty prevention trials, comprising progression to the frail state, death, or being too unwell to continue in a trial. To determine likely event rates, we used data from the English Longitudinal Study for Ageing, collected 4 years apart. We calculated transition rates between non-frail, prefrail, frail or loss to follow up due to death or illness. We used Markov state transition models to interpolate one- and two-year transition rates, and performed sample size calculations for a range of differences in transition rates using simple and composite outcomes. Results The frailty category was calculable for 4650 individuals at baseline (2226 non-frail, 1907 prefrail, 517 frail); at follow up, 1282 were non-frail, 1108 were prefrail, 318 were frail and 1936 had dropped out or were unable to complete all tests for frailty. Transition probabilities for those prefrail at baseline, measured at wave 4 were respectively 0.176, 0.286, 0.096 and 0.442 to non-frail, prefrail, frail and dead/dropped out. Interpolated transition probabilities were 0.159, 0.494, 0.113 and 0.234 at two years, and 0.108, 0.688, 0.087 and 0.117 at one year. Required sample sizes for a two-year outcome were between 1000 and 7200 for transition from prefrailty to frailty alone, 250 to 1600 for transition to the composite measure, and 75 to 350 using the composite measure with an ordinal logistic regression approach. Conclusion Use of a composite outcome for frailty trials offers reduced sample sizes and could ameliorate the effect of high loss to follow up inherent in such trials due to death and illness.

scholarly journals Developing a composite outcome measure for frailty prevention trials – rationale, derivation and sample size comparison with other candidate measures

2019 ◽  
Author(s):  
Miles D. Witham ◽  
James Wason ◽  
Richard M Dodds ◽  
Avan A Sayer
Keyword(s):  
Sample Size ◽  
Transition Probabilities ◽  
Adverse Outcomes ◽  
Transition Rates ◽  
Composite Outcome ◽  
Composite Measure ◽  
Sample Sizes ◽  
Loss To Follow Up ◽  
Prevention Trials

Abstract Introduction Frailty is the loss of ability to withstand a physiological stressor, and is associated with multiple adverse outcomes in older people. Trials to prevent or ameliorate frailty are in their infancy. A range of different outcome measures have been proposed, but current measures require either large sample sizes, long follow-up, or do not directly measure the construct of frailty. Methods We propose a composite outcome for frailty prevention trials, comprising progression to the frail state, death, or being too unwell to continue in a trial. To determine likely event rates, we used data from the English Longitudinal Study for Ageing, collected 4 years apart. We calculated transition rates between non-frail, prefrail, frail or loss to follow up due to death or illness. We used Markov state transition models to interpolate one- and two-year transition rates, and performed sample size calculations for a range of differences in transition rates using simple and composite outcomes. Results The frailty category was calculable for 4650 individuals at baseline (2226 non-frail, 1907 prefrail, 517 frail); at follow up, 1282 were non-frail, 1108 were prefrail, 318 were frail and 1936 had dropped out or were unable to complete all tests for frailty. Transition probabilities for those prefrail at baseline, measured at wave 4 were respectively 0.176, 0.286, 0.096 and 0.442 to non-frail, prefrail, frail and dead/dropped out. Interpolated transition probabilities were 0.159, 0.494, 0.113 and 0.234 at two years, and 0.108, 0.688, 0.087 and 0.117 at one year. Required sample sizes for a two-year outcome were between 1000 and 7200 for transition from prefrailty to frailty alone, 250 to 1600 for transition to the composite measure, and 75 to 350 using the composite measure with an ordinal logistic regression approach. Conclusion Use of a composite outcome for frailty trials offers reduced sample sizes and could ameliorate the effect of high loss to follow up inherent in such trials due to death and illness.

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