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

Abstract

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.

Markov modelling of viral load adjusting for CD4 orthogonal variable and multivariate conditional autoregressive mapping of the HIV immunological outcomes among ART patients in Zimbabwe

“…Therefore, we fitted two proportional hazard multistate models of the generic form: where λ ij , k / Z ( t ) is the transition rate between state i and state j at the time t given a covariate matrix Z and λ ij , (0) is the baseline hazard rate of the model. The two multistate Markov models fitted were: where equation [ 4 ] defines the CD4 cell counts multistate Markov model with λ ij ( CD 4, 0) as the baseline transition intensity for an individual k with positive residual values for the orthogonal effect of VL and the time-varying VL transient levels ( VL states , k = 1( VL < 50 copies / uL )). Similarly, equation [ 5 ] defines the VL multistate Markov model with λ ij ( VL , 0) representing the baseline transition intensities for an individual k with positive residual values for the orthogonal effect CD4 cell counts and the time-varying CD4 cell counts levels ( CD 4 states , k = 1( CD 4 ≥ 500 cells / uL )).…”

“…where equation [4] defines the CD4 cell counts multistate Markov model with λ ij(CD4, 0) as the baseline transition intensity for an individual k with positive residual values for the orthogonal effect of VL ðP Ã VLðkÞ ¼ 0Þ and the time-varying VL transient levels (VL states, k = 1(VL < 50 copies/uL)). Similarly, equation [5] Since the analyses were purely based on existing modelling approached within the "msm" R library, additional covariates like baseline CD4 cell counts, baseline VL values, age, sex and WHO staging could be adjusted for in the model; however, in this study, we encountered a convergence warning as more covariates were added.…”

“…Multistate Markov models are mathematical models which have been used to evaluate HIV disease processes; however, the number of states, state cuts-off points and the number of transitions vary across studies [ 3 , 4 ]. This explains the flexibility of the multistate models’ implementation, but the models become complex as the number of states and transitions increase.…”

Prone position during venovenous extracorporeal membrane oxygenation: survival analysis needed for a time-dependent intervention

“…We used a multistate survival model to address this issue, which is a more realistic model of patient progression through their journey [ 9 ]. Patients in prone positioning can move to the supine state and vice versa, and this transition contributes to the complexity of the model and eventually affects the outcome [ 10 ]. Hazard ratios from the survival model provide an estimate for transitions between states.…”

Modelling the progression of illicit substance use patterns from real‐world evidence

“…The effect of covariates on model transitions can be assessed. 26,27 Time nonhomogeneous transition matrices have been used to model the effects of drug treatments on sleep stages in insomnia. 28 The goal of our work was to integrate differential equation-based…”