Multi-state models are commonly used for intermittent observations of a state over time, but these are generally based on the Markov assumption, that transition rates are independent of the time spent in current and previous states. In a semi-Markov model, the rates can depend on the time spent in the current state, though available methods for this are either restricted to specific state structures or lack general software. One approach uses a “phase-type” distribution for the sojourn time in a state, which expresses a semi-Markov model as a hidden Markov model, allowing the likelihood to be calculated easily for any state structure. While this approach involves a proliferation of latent parameters, identifiability can be improved by restricting the phase-type family to one with similar properties to a simpler distribution such as the Gamma or Weibull. A moment-matching method to obtain this family is proposed, making general semi-Markov models for intermittent data accessible in software for the first time. The method is implemented in a new R package, msmbayes, which implements Bayesian or maximum likelihood estimation for multi-state models with general state structures and covariates. The software is tested through simulation-based calibration, and an application to cognitive function decline illustrates the use of the method in a typical modelling workflow.

Introduction: Health impact assessment models are a key tool for stakeholders to understand the effect of transport on public health. The Integrated Transport and Health Impact Model for Global Cities (ITHIM-Global) is an open-source model developed primarily for low-and middle-income countries to assess the impact of changes in transport on population health at the city level. It models impacts through three different pathways: air pollution, physical activity, and road traffic fatalities; however, the uncertainty in input parameters and model calculations requires further examination. Methods: ITHIM-Global uses Monte Carlo simulation, sampling from predefined probability distributions of inputs to generate credible intervals for model outputs. Value of Information (VoI) analysis identifies which input parameters most influence output uncertainty. Using Bogota as an example, we illustrate this approach with three hypothetical scenarios, each shifting 5% of trips to cycling, public transport, or car journeys. Results: Increases in cycling or public transport consistently improved population health, regardless of input uncertainty, while increased car journeys consistently worsened health outcomes. Perfect knowledge of individual input parameters, such as the number of people with no travel-related physical activity, could reduce the standard deviation of credible intervals for years of life lost by up to 7%. Conclusions: ITHIM-Global can support evidence-based decision-making by demonstrating health impacts of transport scenarios across varied contexts, the practicalities of using VoI analysis and the value of prioritising data collection to reduce model uncertainty.

BackgroundPopulation-adjusted indirect comparison using parametric Simulated Treatment Comparison (STC) has had limited application to survival outcomes in unanchored settings. Matching-Adjusted Indirect Comparison (MAIC) is commonly used but does not account for violation of proportional hazards or enable extrapolations of survival. We developed and applied a novel methodology for STC in unanchored settings. We compared overall survival (OS) and progression-free survival (PFS) of lenvatinib plus pembrolizumab (LEN + PEM) against nivolumab plus ipilimumab (NIVO + IPI), pembrolizumab plus axitinib (PEM + AXI), avelumab plus axitinib (AVE + AXI), and nivolumab plus cabozontanib (NIVO + CABO) in patients with advanced renal cell carcinoma (RCC). Unanchored comparison was necessitated as the control groups differed in their use of PD-1/PD-L1 rescue therapy.MethodsWe fit covariate-adjusted survival models to individual patient data from phase 3 trial of LEN + PEM, including standard parametric distributions and Royston-Parmar spline models with up to 3 knots. We used these models to predict OS and PFS in the population of comparator treatments. The base case model was selected by minimum Akaike Information Criterion (AIC). Treatment effects were measured using difference in restricted mean survival time (RMST), over shortest follow-up of input trials, and hazard ratios at 6, 12, 18, and 24 months.ResultsThe survival model with the lowest AIC was 1-knot spline odds for OS and log-logistic for PFS. Difference in RMST OS was 6.90 months (95% CI: 1.95, 11.36), 5.31 (3.58, 7.28), 5.99 (1.82, 9.42), and 11.59 (8.41, 15.38) versus NIVO + IPI (over 64.8 months follow-up), AVE + AXI (46.7 months), PEM + AXI (64.8 months), NIVO + CABO (53.0 months), respectively. Difference in RMST PFS was 4.50 months (95% CI: 0.92, 8.26), 8.23 (5.60, 10.57), 5.38 (2.06, 9.09), and 4.58 (0.09, 9.44) versus NIVO + IPI (over 57.8 months), AVE + AXI (44.9 months), PEM + AXI (57.8 months), NIVO + CABO (23.8 months), respectively. Hazard ratios indicated strong evidence of greater OS and PFS on LEN + PEM at most timepoints.ConclusionsWe developed and applied a novel methodology for comparing survival outcomes in unanchored settings using STC. Pending investigation with a simulation study or further examples, this methodology could be used for clinical decision-making and, if long-term data are available, inform economic models designed to extrapolate outcomes for the evaluation of lifetime cost-effectiveness.Trial registrationNCT02811861 (registered: 23/06/2016).

Background Health economic modelling indicates that referral to a behavioural weight management programme is cost saving and generates QALY gains compared with a brief intervention. The aim of this study was to conduct a cross-model validation comparing outcomes from this cost-effectiveness analysis to those of a comparator model, to understand how differences in model structure contribute to outcomes. Methods The outcomes produced by two models, the School for Public Health Research diabetes prevention (SPHR) and Health Checks (HC) models, were compared for three weight-management programme strategies; Weight Watchers (WW) for 12 weeks, WW for 52 weeks, and a brief intervention, and a simulated no intervention scenario. Model inputs were standardised, and iterative adjustments were made to each model to identify drivers of differences in key outcomes. Results The total QALYs estimated by the HC model were higher in all treatment groups than those estimated by the SPHR model, and there was a large difference in incremental QALYs between the models. SPHR simulated greater QALY gains for 12-week WW and 52-week WW relative to the Brief Intervention. Comparisons across socioeconomic groups found a stronger socioeconomic gradient in the SPHR model. Removing the impact of treatment on HbA1c from the SPHR model, running both models only with the conditions that the models have in common and, to a lesser extent, changing the data used to estimate risk factor trajectories, resulted in more consistent model outcomes. Conclusions The key driver of difference between the models was the inclusion of extra evidence-based detail in SPHR on the impacts of treatments on HbA1c. The conclusions were less sensitive to the dataset used to inform the risk factor trajectories. These findings strengthen the original cost-effectiveness analyses of the weight management interventions and provide an increased understanding of what is structurally important in the models.

Multistate models provide a useful framework for modelling complex event history data in clinical settings and have recently been extended to the joint modelling framework to appropriately handle endogenous longitudinal covariates, such as repeatedly measured biomarkers, which are informative about health status and disease progression. However, the practical application of such joint models faces considerable computational challenges. Motivated by a longitudinal multimorbidity analysis of large-scale UK health records, we introduce novel Bayesian inference approaches for these models that are capable of handling complex multistate processes and large datasets with straightforward implementation. These approaches decompose the original estimation task into smaller inference blocks, leveraging parallel computing and facilitating flexible model specification and comparison. Using extensive simulation studies, we show that the proposed approaches achieve satisfactory estimation accuracy, with notable gains in computational efficiency compared to the standard Bayesian estimation strategy. We illustrate our approaches by analysing the coevolution of routinely measured systolic blood pressure and the progression of three important chronic conditions, using a large dataset from the Clinical Practice Research Datalink Aurum database. Our analysis reveals distinct and previously lesser-known association structures between systolic blood pressure and different disease transitions.

In this work, we introduce a personalized and age-specific net benefit function, composed of benefits and costs, to recommend optimal timing of risk assessments for cardiovascular disease (CVD) prevention. We extend the 2-stage landmarking model to estimate patient-specific CVD risk profiles, adjusting for time-varying covariates. We apply our model to data from the Clinical Practice Research Datalink, comprising primary care electronic health records from the UK. We find that people at lower risk could be recommended an optimal risk-assessment interval of 5 years or more. Time-varying risk factors are required to discriminate between more frequent schedules for high-risk people.

OBJECTIVES:Bivalent original/BA.4-5 and monovalent XBB.1.5 mRNA boosters were offered to UK healthcare workers (HCWs) in the autumn of 2023. We aimed to estimate booster vaccine effectiveness (VE) and post-infection immunity among the SIREN HCW cohort over the subsequent 6-month period of XBB.1.5 and JN.1 variant circulation. METHODS:Between October 2023 to March 2024, 2867 SIREN study participants tested fortnightly for SARS-CoV-2 and completed symptoms questionnaires. We used multi-state models, adjusted for vaccination, prior infection, and demographic covariates, to estimate protection against mild/asymptomatic and moderate SARS-CoV-2 infection. RESULTS:Half of the participants (1422) received a booster during October 2023 (280 bivalent, 1142 monovalent), and 536 (19%) had a PCR-confirmed infection over the study period. Bivalent booster VE was 15.1% (-55.4 to 53.6%) at 0-2 months and 4.2% (-46.4 to 37.3%) at 2-4 months post-vaccination. Monovalent booster VE was 44.2% (95% CI 21.7 to 60.3%) at 0-2 months, and 24.1% (-0.7 to 42.9%) at 2-4 months. VE was greater against moderate infection than against mild/asymptomatic infection, but neither booster showed evidence of protection after 4 months. Controlling for vaccination, compared to an infection >2 years prior, infection within the past 6 months was associated with 58.6% (30.3 to 75.4%) increased protection against moderate infection and 38.5% (5.8 to 59.8%) increased protection against mild/asymptomatic infection. CONCLUSIONS:Monovalent XBB.1.5 boosters provided short-term protection against SARS-CoV-2 infection, particularly against moderate symptoms. Vaccine formulations that target the circulating variant may be suitable for inclusion in seasonal vaccination campaigns among HCWs.

Value of Information (VOI) methods are used to predict the value of reducing or eliminating uncertainty in the parameters of a decision model. This chapter is about the expected value of perfect information (EVPI) and the expected value of partial perfect information (EVPPI). The EVPI is the expected value of learning the exact values of all model parameters. This is used to give an upper bound for the potential value of any further research, so that research can be judged to be unnecessary if the costs exceed this value. The EVPPI is the expected value of learning specific parameters exactly, and is used to determine which parameters contribute the most to the uncertainty about the decision — hence where further research to reduce this uncertainty should be focused. After an introduction to the theory of VOI methods, this chapter defines the EVPI and EVPPI and how they are calculated in practice. An R package "voi" for doing VOI calculations is introduced, and its use for calculating EVPI and EVPPI is demonstrated. The final section gives a worked example of how EVPPI is calculated, presented and interpreted in the chemotherapy case study introduced in the previous chapter.

A widely-used model for determining the long-term health impacts of public health interventions, often called a "multistate lifetable", requires estimates of incidence, case fatality, and sometimes also remission rates, for multiple diseases by age and gender. Generally, direct data on both incidence and case fatality are not available in every disease and setting. For example, we may know population mortality and prevalence rather than case fatality and incidence. This paper presents Bayesian continuous-time multistate models for estimating transition rates between disease states based on incomplete data. This builds on previous methods by using a formal statistical model with transparent data-generating assumptions, while providing accessible software as an R package. Rates for people of different ages and areas can be related flexibly through splines or hierarchical models. Previous methods are also extended to allow age-specific trends through calendar time. The model is used to estimate case fatality for multiple diseases in the city regions of England, based on incidence, prevalence and mortality data from the Global Burden of Disease study. The estimates can be used to inform health impact models relating to those diseases and areas. Different assumptions about rates are compared, and we check the influence of different data sources.

BACKGROUND:Health policy decisions are often informed by estimates of long-term survival based primarily on short-term data. A range of methods are available to include longer-term information, but there has previously been no comprehensive and accessible tool for implementing these.RESULTS:This paper introduces a novel model and software package for parametric survival modelling of individual-level, right-censored data, optionally combined with summary survival data on one or more time periods. It could be used to estimate long-term survival based on short-term data from a clinical trial, combined with longer-term disease registry or population data, or elicited judgements. All data sources are represented jointly in a Bayesian model. The hazard is modelled as an M-spline function, which can represent potential changes in the hazard trajectory at any time. Through Bayesian estimation, the model automatically adapts to fit the available data, and acknowledges uncertainty where the data are weak. Therefore long-term estimates are only confident if there are strong long-term data, and inferences do not rely on extrapolating parametric functions learned from short-term data. The effects of treatment or other explanatory variables can be estimated through proportional hazards or with a flexible non-proportional hazards model. Some commonly-used mechanisms for survival can also be assumed: cure models, additive hazards models with known background mortality, and models where the effect of a treatment wanes over time. All of these features are provided for the first time in an R package, survextrap, in which models can be fitted using standard R survival modelling syntax. This paper explains the model, and demonstrates the use of the package to fit a range of models to common forms of survival data used in health technology assessments.CONCLUSIONS:This paper has provided a tool that makes comprehensive and principled methods for survival extrapolation easily usable.

The Expected Value of Sample Information (EVSI) describes the economic value of a specific study that aims to reduce uncertainty in the parameters of a decision-analytic model. EVSI can be used to support decision makers to determine (a) whether the available evidence is sufficient for decision making, (b) which studies might provide substantial value for the decision maker and (c) the optimal design of the proposed studies. EVSI has traditionally been very challenging to compute, but methods are now available to estimate EVSI quickly and efficiently, thereby unlocking its potential for research design and prioritisation. This Chapter presents key methods for computing EVSI, discusses their relative advantages and disadvantages, and presents how they can be computed in the R software. The Expected Net Benefit of Sampling (ENBS), a measure of the net monetary benefit of undertaking research, is presented, and an extended running example is implemented to demonstrate how EVSI and ENBS can be used to support research design.

Supplementary methods, figures and tables Table S3: Fractional polynomial functions for age at diagnosis (years), tumour size (mm), number of positive nodes, tumour grade (I, II, III) and mode of detection (screening vs clinical) by ER status (positive, negative). Table S4: Results, for the fixed effects and interaction terms from RJMCMC Logistic model for ER-positive tumours. Table S5: Results, for the fixed effects and interaction terms from RJMCMC Weibull model for ER-positive tumours. Table S6: Results, for the fixed effects and interaction terms from RJMCMC Logistic model for ER-negative tumours. Table S7: Results, for the fixed effects and interaction terms from RJMCMC Weibull model for ER-negative tumours. Figure S1: Marginal Posterior Probability of inclusion in the model for each interaction term, i.e. the relative frequency a covariate was included in the model for (A) ER positive and (B) ER negative tumours. It can be interpreted as the posterior probability an association exists with survival, adjusted for all other covariates in the model. Figure S2: Individual calibration plots of the four models in the validation set with ;7% CI for (A) ER positive and (B) ER negative tumours. Figure S3: Individual net benefit plots of the four models in the validation set for (A) ER positive and (B) ER negative tumours. Shaded area represents ;7% confidence intervals based on 1000 bootstrap replicates.

To help health economic modelers respond to demands for greater use of complex systems models in public health. To propose identifiable features of such models and support researchers to plan public health modeling projects using these models. A working group of experts in complex systems modeling and economic evaluation was brought together to develop and jointly write guidance for the use of complex systems models for health economic analysis. The content of workshops was informed by a scoping review. A public health complex systems model for economic evaluation is defined as a quantitative, dynamic, non-linear model that incorporates feedback and interactions among model elements, in order to capture emergent outcomes and estimate health, economic and potentially other consequences to inform public policies. The guidance covers: when complex systems modeling is needed; principles for designing a complex systems model; and how to choose an appropriate modeling technique. This paper provides a definition to identify and characterize complex systems models for economic evaluations and proposes guidance on key aspects of the process for health economics analysis. This document will support the development of complex systems models, with impact on public health systems policy and decision making.

Healthcare policy makers frequently face decisions about how to allocate scarce resources to maximise health. Although these decisions must be made, they are generally based on imperfect information, often resulting in uncertainty. To guide policy makers, the consequences of different decisions are often estimated using a model-based cost-effectiveness analysis (CEA), or other type of economic evaluation. This allows evaluation of the expected health effects and costs of each decision option, as well as the uncertainty surrounding these expected values and the decision. Understanding the consequences of making a suboptimal decision helps inform the potential need for gathering more evidence to support decision making. A value of information (VOI) analysis plays a central role in this assessment. This chapter introduces the theoretical background for healthcare policy making, and the mathematical and conceptual principles that underpin the methods of VOI analysis. We start with a brief introduction to CEA and the different components and measures that are involved. We thereafter outline the main types of decision-analytic models, and provide a brief overview of the typical kinds of evidence that are used to inform CEAs. Finally, we introduce the idea of uncertainty in model-based CEAs. Specifically, we describe probabilistic analysis: the framework of representing uncertainty by probability distributions for model input parameters. This forms the foundation of VOI analysis. We explain how such probability distributions can be defined based on data, and how probabilistic analysis can be implemented by using Monte Carlo simulation from the joint distribution of model input parameters to propagate uncertainty to model outcomes such as expected costs, health outcomes and net benefits. Finally we describe different measures of decision uncertainty that can be obtained from a probabilistic analysis, which motivates the idea of VOI.

Joint models (JM) for longitudinal and survival data have gained increasing interest and found applications in a wide range of clinical and biomedical settings. These models facilitate the understanding of the relationship between outcomes and enable individualized predictions. In many applications, more complex event processes arise, necessitating joint longitudinal and multistate models. However, their practical application can be hindered by computational challenges due to increased model complexity and large sample sizes. Motivated by a longitudinal multimorbidity analysis of large UK health records, we have developed a scalable Bayesian methodology for such joint multistate models that is capable of handling complex event processes and large datasets, with straightforward implementation. We propose two blockwise inference approaches for different inferential purposes based on different levels of decomposition of the multistate processes. These approaches leverage parallel computing, ease the specification of different models for different transitions, and model/variable selection can be performed within a Bayesian framework using Bayesian leave-one-out cross-validation. Using a simulation study, we show that the proposed approaches achieve satisfactory performance regarding posterior point and interval estimation, with notable gains in sampling efficiency compared to the standard estimation strategy. We illustrate our approaches using a large UK electronic health record dataset where we analysed the coevolution of routinely measured systolic blood pressure (SBP) and the progression of multimorbidity, defined as the combinations of three chronic conditions. Our analysis identified distinct association structures between SBP and different disease transitions.

Background The protection of fourth dose mRNA vaccination against SARS-CoV-2 is relevant to current global policy decisions regarding ongoing booster roll-out. We estimate the effect of fourth dose vaccination, prior infection, and duration of PCR positivity in a highly-vaccinated and largely prior-COVID-19 infected cohort of UK healthcare workers. Methods Participants underwent fortnightly PCR and regular antibody testing for SARS-CoV-2 and completed symptoms questionnaires. A multi-state model was used to estimate vaccine effectiveness (VE) against infection from a fourth dose compared to a waned third dose, with protection from prior infection and duration of PCR positivity jointly estimated. Results 1,298 infections were detected among 9,560 individuals under active follow-up between September 2022 and March 2023. Compared to a waned third dose, fourth dose VE was 13.1% (95%CI 0.9 to 23.8) overall; 24.0% (95%CI 8.5 to 36.8) in the first two months post-vaccination, reducing to 10.3% (95%CI - 11.4 to 27.8) and 1.7% (95%CI -17.0 to 17.4) at 2-4 and 4-6 months, respectively. Relative to an infection >2 years ago and controlling for vaccination, 63.6% (95%CI 46.9 to 75.0) and 29.1% (95%CI 3.8 to 43.1) greater protection against infection was estimated for an infection within the past 0-6, and 6-12 months, respectively. A fourth dose was associated with greater protection against asymptomatic infection than symptomatic infection, whilst prior infection independently provided more protection against symptomatic infection, particularly if the infection had occurred within the previous 6 months. Duration of PCR positivity was significantly lower for asymptomatic compared to symptomatic infection. Conclusions Despite rapid waning of protection, vaccine boosters remain an important tool in responding to the dynamic COVID-19 landscape; boosting population immunity in advance of periods of anticipated pressure, such as surging infection rates or emerging variants of concern. Funding UK Health Security Agency, Medical Research Council, NIHR HPRU Oxford, and others.

BackgroundMultimorbidity, characterised by the coexistence of multiple chronic conditions in an individual, is a rising public health concern. While much of the existing research has focused on cross-sectional patterns of multimorbidity, there remains a need to better understand the longitudinal accumulation of diseases. This includes examining the associations between important sociodemographic characteristics and the rate of progression of chronic conditions.Methods and findingsWe utilised electronic primary care records from 13.48 million participants in England, drawn from the Clinical Practice Research Datalink (CPRD Aurum), spanning from 2005 to 2020 with a median follow-up of 4.71 years (IQR: 1.78, 11.28). The study focused on 5 important chronic conditions: cardiovascular disease (CVD), type 2 diabetes (T2D), chronic kidney disease (CKD), heart failure (HF), and mental health (MH) conditions. Key sociodemographic characteristics considered include ethnicity, social and material deprivation, gender, and age. We employed a flexible spline-based parametric multistate model to investigate the associations between these sociodemographic characteristics and the rate of different disease transitions throughout multimorbidity development. Our findings reveal distinct association patterns across different disease transition types. Deprivation, gender, and age generally demonstrated stronger associations with disease diagnosis compared to ethnic group differences. Notably, the impact of these factors tended to attenuate with an increase in the number of preexisting conditions, especially for deprivation, gender, and age. For example, the hazard ratio (HR) (95% CI; p-value) for the association of deprivation with T2D diagnosis (comparing the most deprived quintile to the least deprived) is 1.76 ([1.74, 1.78]; p < 0.001) for those with no preexisting conditions and decreases to 0.95 ([0.75, 1.21]; p = 0.69) with 4 preexisting conditions. Furthermore, the impact of deprivation, gender, and age was typically more pronounced when transitioning from an MH condition. For instance, the HR (95% CI; p-value) for the association of deprivation with T2D diagnosis when transitioning from MH is 2.03 ([1.95, 2.12], p < 0.001), compared to transitions from CVD 1.50 ([1.43, 1.58], p < 0.001), CKD 1.37 ([1.30, 1.44], p < 0.001), and HF 1.55 ([1.34, 1.79], p < 0.001). A primary limitation of our study is that potential diagnostic inaccuracies in primary care records, such as underdiagnosis, overdiagnosis, or ascertainment bias of chronic conditions, could influence our results.ConclusionsOur results indicate that early phases of multimorbidity development could warrant increased attention. The potential importance of earlier detection and intervention of chronic conditions is underscored, particularly for MH conditions and higher-risk populations. These insights may have important implications for the management of multimorbidity.

Value of information (VoI) is a decision-theoretic approach to estimating the expected benefits from collecting further information of different kinds, in scientific problems based on combining one or more sources of data. VoI methods can assess the sensitivity of models to different sources of uncertainty and help to set priorities for further data collection. They have been widely applied in healthcare policy making, but the ideas are general to a range of evidence synthesis and decision problems. This article gives a broad overview of VoI methods, explaining the principles behind them, the range of problems that can be tackled with them, and how they can be implemented, and discusses the ongoing challenges in the area.

BackgroundFor people with symptomatic COVID-19, the relative risks of hospital admission, death without hospital admission and recovery without admission, and the times to those events, are not well understood. We describe how these quantities varied with individual characteristics, and through the first wave of the pandemic, in Milan, Italy.MethodsA cohort study of 27 598 people with known COVID-19 symptom onset date in Milan, Italy, testing positive between February and June 2020 and followed up until 17 July 2020. The probabilities of different events, and the times to events, were estimated using a mixture multistate model.ResultsThe risk of death without hospital admission was higher in March and April (for non-care home residents, 6%–8% compared with 2%–3% in other months) and substantially higher for care home residents (22%–29% in March). For all groups, the probabilities of hospitalisation decreased from February to June. The probabilities of hospitalisation also increased with age, and were higher for men, substantially lower for healthcare workers and care home residents, and higher for people with comorbidities. Times to hospitalisation and confirmed recovery also decreased throughout the first wave. Combining these results with our previously developed model for events following hospitalisation, the overall symptomatic case fatality risk was 15.8% (15.4%–16.2%).ConclusionsThe highest risks of death before hospital admission coincided with periods of severe burden on the healthcare system in Lombardy. Outcomes for care home residents were particularly poor. Outcomes improved as the first wave waned, community healthcare resources were reinforced and testing became more widely available.