Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data: Uncertainty Propagation, Evidence Synthesis, and Model Calibration

Simplified Methods for Modelling Dependent Parameters in Health Economic Evaluations: A Tutorial

BACKGROUND

In health economic evaluations, model parameters are often dependent on other model parameters. Although methods exist to simulate multivariate normal (MVN) distribution data and estimate transition probabilities in Markov models while considering competing risks, they are technically challenging for health economic modellers to implement. This tutorial introduces easily implementable applications for handling dependent parameters in modelling.

METHODS

Analytical proofs and proposed simplified methods for handling dependent parameters in typical health economic modelling scenarios are provided, and implementation of these methods are illustrated in seven examples along with the SAS and R code.

RESULTS

Methods to quantify the covariance and correlation coefficients of correlated variables based on published summary statistics and generation of MVN distribution data are demonstrated using examples of physician visits data and cost component data. The use of univariate normal distribution data instead of MVN distribution data to capture population heterogeneity is illustrated based on the results from multiple regression models with linear predictors, and two examples are provided (linear fixed-effects model and Cox proportional hazards model). A conditional probability method is introduced to handle two or more state transitions in a single Markov model cycle and applied in examples of one- and two-way state transitions.

CONCLUSIONS

This tutorial proposes an extension of routinely used methods along with several examples. These simplified methods may be easily applied by health economic modellers with varied statistical backgrounds.

Stochastic modeling of obesity status in United States adults using Markov Chains: A nationally representative analysis of population health data from 2017-2020

Importance

The prevalence of obesity among United States adults has increased from 34.9% in 2013-2014 to 42.8% in 2017-2018. Developing methods to model the increase of obesity over-time is a necessity to know how to accurately quantify its cost and to develop solutions to combat this national public health emergency.

Methods

A cross-sectional cohort study using the publicly available National Health and Nutrition Examination Survey (NHANES 2017-2020) was conducted in individuals who completed the weight questionnaire and had accurate data for both weight at the time of survey and weight 10 years ago. To model the dynamics of obesity, a Markov transition state matrix was created, which allowed for the analysis of weight transitions over time. Bootstrap simulation was incorporated to account for uncertainty and generate multiple simulated datasets, providing a more robust estimation of the prevalence and trends in obesity within the cohort.

Results

Of the 6146 individuals who met the inclusion criteria, 3024 (49%) individuals were male and 3122 (51%) were female. There were 2252 (37%) White individuals, 1257 (20%) Hispanic individuals, 1636 (37%) Black individuals, and 739 (12%) Asian individuals. The average BMI was 30.16 (SD = 7.15), the average weight was 83.67 kilos (SD = 22.04), and the average weight change was a 3.27 kg (SD = 14.97) increase in body weight. A total of 2411 (39%) individuals lost weight, and 3735 (61%) individuals gained weight. 87 (1%) individuals were underweight (BMI <18.5), 2058 (33%) were normal weight (18.5 ≤ BMI <25), 1376 (22%) were overweight (25 ≤ BMI <30) and 2625 (43%) were in the obese category (BMI >30).

Conclusion

United States adults are at risk of transitioning from normal weight to the overweight or obese category. Markov modeling combined with bootstrap simulations can accurately model long-term weight status.

Multi-state network meta-analysis of progression and survival data

Multiple randomized controlled trials, each comparing a subset of competing interventions, can be synthesized by means of a network meta-analysis to estimate relative treatment effects between all interventions in the evidence base. Here we focus on estimating relative treatment effects for time-to-event outcomes. Cancer treatment effectiveness is frequently quantified by analyzing overall survival (OS) and progression-free survival (PFS). We introduce a method for the joint network meta-analysis of PFS and OS that is based on a time-inhomogeneous tri-state (stable, progression, and death) Markov model where time-varying transition rates and relative treatment effects are modeled with parametric survival functions or fractional polynomials. The data needed to run these analyses can be extracted directly from published survival curves. We demonstrate use by applying the methodology to a network of trials for the treatment of non-small-cell lung cancer. The proposed approach allows the joint synthesis of OS and PFS, relaxes the proportional hazards assumption, extends to a network of more than two treatments, and simplifies the parameterization of decision and cost-effectiveness analyses.

A Tutorial on Time-Dependent Cohort State-Transition Models in R Using a Cost-Effectiveness Analysis Example

In an introductory tutorial, we illustrated building cohort state-transition models (cSTMs) in R, where the state transition probabilities were constant over time. However, in practice, many cSTMs require transitions, rewards, or both to vary over time (time dependent). This tutorial illustrates adding 2 types of time dependence using a previously published cost-effectiveness analysis of multiple strategies as an example. The first is simulation-time dependence, which allows for the transition probabilities to vary as a function of time as measured since the start of the simulation (e.g., varying probability of death as the cohort ages). The second is state-residence time dependence, allowing for history by tracking the time spent in any particular health state using tunnel states. We use these time-dependent cSTMs to conduct cost-effectiveness and probabilistic sensitivity analyses. We also obtain various epidemiological outcomes of interest from the outputs generated from the cSTM, such as survival probability and disease prevalence, often used for model calibration and validation. We present the mathematical notation first, followed by the R code to execute the calculations. The full R code is provided in a public code repository for broader implementation.

Impact of Extended Use of Ablation Techniques in Cirrhotic Patients with Hepatocellular Carcinoma: A Cost-Effectiveness Analysis

Simple Summary

The optimal management of non-metastatic hepatocellular carcinoma (HCC) remains debated. The association between HCC and cirrhosis influences prognosis and therapeutic choices between curative and palliative treatments. The goal of our retrospective study was to evaluate the cost-effectiveness of the extended use of ablation for the treatment of HCC with cirrhosis in an expert ablation center when compared to the non-extended use of ablation in equivalent tertiary care centers. In a propensity-score matched cohort of 532 patients with naïve HCC, the extended use of ablation led to better compliance with the Barcelona Clinic Liver Classification (BCLC) guidelines (80% vs. 67%) and was more effective and less expensive than the non-extended use of ablation strategy, particularly at an earlier stage of the disease. The shift from curative to palliative treatments was noted in a considerable percentage of patients; therefore, this needs to be redefined as the wide choice of ablation techniques and technical advances in imaging guidance increase the curative options available to treat a maximum of patients with HCC.

Abstract

Background: To evaluate the cost-effectiveness of the extended use of ablation for the treatment of hepatocellular carcinoma (HCC) with cirrhosis in an expert ablation center when compared to the non-extended use of ablation in equivalent tertiary care centers. Methods: Consecutive cirrhotic patients with non-metastatic HCC, no prior treatment, and referred to three tertiary care centers between 2012 and 2016 were retrospectively identified. The Bondy group, including all of the patients treated at Jean Verdier Hospital, where the extended use of ablation is routinely performed, was compared to the standard of care (SOC) group, including all of the patients treated at the Beaujon and Mondor Hospitals, using propensity score matching. A cost-effectiveness analysis was carried out from the perspective of French health insurance using a Markov model on a lifetime horizon. Results: 532 patients were matched. The Bondy group led to incremental discounted lifetime effects of 0.8 life-years gained (LYG) (95% confidence interval: 0.4, 1.3) and a decrease in lifetime costs of EUR 7288 (USD 8016) (95% confidence interval: EUR 5730 [USD 6303], EUR 10,620 [USD 11,682]) per patient, compared with the SOC group, resulting in a dominant mean incremental cost-effectiveness ratio (ICER). A compliance with the Barcelona Clinic Liver Classification (BCLC) guidelines for earlier stage contributed to the greater part of the ICER. Conclusion: The extended use of ablation in cirrhotic patients with HCC was more effective and less expensive than the non-extended use of the ablation strategy.

Characterization and Valuation of the Uncertainty of Calibrated Parameters in Microsimulation Decision Models

Background: We evaluated the implications of different approaches to characterize the uncertainty of calibrated parameters of microsimulation decision models (DMs) and quantified the value of such uncertainty in decision making.

Methods: We calibrated the natural history model of CRC to simulated epidemiological data with different degrees of uncertainty and obtained the joint posterior distribution of the parameters using a Bayesian approach. We conducted a probabilistic sensitivity analysis (PSA) on all the model parameters with different characterizations of the uncertainty of the calibrated parameters. We estimated the value of uncertainty of the various characterizations with a value of information analysis. We conducted all analyses using high-performance computing resources running the Extreme-scale Model Exploration with Swift (EMEWS) framework.

Results: The posterior distribution had a high correlation among some parameters. The parameters of the Weibull hazard function for the age of onset of adenomas had the highest posterior correlation of −0.958. When comparing full posterior distributions and the maximum-a-posteriori estimate of the calibrated parameters, there is little difference in the spread of the distribution of the CEA outcomes with a similar expected value of perfect information (EVPI) of $653 and $685, respectively, at a willingness-to-pay (WTP) threshold of $66,000 per quality-adjusted life year (QALY). Ignoring correlation on the calibrated parameters’ posterior distribution produced the broadest distribution of CEA outcomes and the highest EVPI of $809 at the same WTP threshold.

Conclusion: Different characterizations of the uncertainty of calibrated parameters affect the expected value of eliminating parametric uncertainty on the CEA. Ignoring inherent correlation among calibrated parameters on a PSA overestimates the value of uncertainty.

Modelling the episodes of care for iron deficiency anemia patients in a secondary-care center using continuous-time multistate Markov chain

BACKGROUND

Despite the high prevalence of gastro-intestinal (GI) cancer in iron deficiency anemia (IDA), some IDA patients do not complete all the necessary GI investigations at the initial referral. As a result, existing cancers are diagnosed at a later referral with worse prognosis. The potential to detect GI cancer early depends on minimizing the delay time spent between the two consecutive referrals, where a patient did not complete investigations at the first referral, but at the second is diagnosed with positive GI cancer. This retrospective longitudinal study aims to highlight the proper methods to model these referrals.

METHODS

Using anonymized data of 168 episodes of care for IDA patients at an IDA clinic in a secondary care setting, continuous-time multi-state Markov chain is employed to determine the transition rates among three observed states for IDA patients at the IDA clinic, "incomplete investigations," "negative GI cancer," and "positive GI cancer" and to estimate the delay time.

RESULTS

Once in the state of incomplete investigations, an estimated mean delay time of 3.1 years (95% CI: 1.2, 5) is spent before being diagnosed with positive GI cancer. The probability that a "positive GI diagnosis" is next after the state of "incomplete investigation" is 17%, compared with 11% when it is followed in the state of negative GI cancer. Defining the survival as the event of not being in the state of "positive GI cancer," the survival rate of IDA patients with negative GI cancer is always higher than those with incomplete investigations. Finally, being diagnosed with positive GI cancer is always preceded by the prediction of being considered "very high risk" at the earlier visit.

CONCLUSION

A baseline model was proposed to represent episodes of care for IDA patients at a secondary care center. Preliminary results highlight the importance of completing the GI investigations, especially in IDA patients, who are at high risk of GI cancer and fit to go through the investigations.

Proportion of incident genital human papillomavirus detections not attributable to transmission and potentially attributable to latent infections: Implications for cervical cancer screening

BACKGROUND

Infections with human papillomaviruses (HPV) may enter a latent state, and eventually become reactivated following loss of immune control. It is unclear what proportion of incident HPV detections are reactivations of previous latent infections versus new transmissions.

METHODS

The HITCH cohort study prospectively followed young newly-formed heterosexual partners recruited between 2005-2011 in Montréal, Canada. We calculated the fraction of incident HPV detections non-attributable to sexual transmission risk factors with a Bayesian Markov model. Results are the median (2.5-97.5 th percentiles) of the estimated posterior distribution.

RESULTS

544 type-specific incident HPV detection events occurred in 849 participants; 33% of incident HPV detections occurred in participants whose HITCH partners were negative for that HPV type and who reported no other sex partners over follow-up. We estimate that 43% (38-48%) of all incident HPV detections in this population were not attributable to recent sexual transmission and might be potentially reactivation of latent infections.

CONCLUSIONS

A positive HPV test result in many cases may be a reactivated past infection, rather than a new infection from recent sexual behaviors or partner infidelity. The potential for reactivation of latent infections in previously HPV-negative women should be considered in the context of cervical cancer screening.

Estimation of Transition Probabilities for State-Transition Models: A Review of NICE Appraisals

State transition models are used to inform health technology reimbursement decisions. Within state transition models, the movement of patients between the model health states over discrete time intervals is determined by transition probabilities (TPs). Estimating TPs presents numerous issues, including missing data for specific transitions, data incongruence and uncertainty around extrapolation. Inappropriately estimated TPs could result in biased models. There is limited guidance on how to address common issues associated with TP estimation. To assess current methods for estimating TPs and to identify issues that may introduce bias, we reviewed National Institute for Health and Care Excellence Technology Appraisals published from 1 January, 2019 to 27 May, 2020. Twenty-eight models (from 26 Technology Appraisals) were included in the review. Several methods for estimating TPs were identified: survival analysis (n = 11); count method (n = 9); multi-state modelling (n = 7); logistic regression (n = 2); negative binomial regression (n = 2); Poisson regression (n = 1); and calibration (n = 1). Evidence Review Groups identified several issues relating to TP estimation within these models, including important transitions being excluded (n = 5); potential selection bias when estimating TPs for post-randomisation health states (n = 2); issues concerning the use of multiple data sources (n = 4); potential biases resulting from the use of data from different populations (n = 2), and inappropriate assumptions around extrapolation (n = 3). These issues remained unresolved in almost every instance. Failing to address these issues may bias model results and lead to sub-optimal decision making. Further research is recommended to address these methodological problems.

Developing a reference protocol for structured expert elicitation in health-care decision-making: a mixed-methods study

BACKGROUND

Many decisions in health care aim to maximise health, requiring judgements about interventions that may have higher health effects but potentially incur additional costs (cost-effectiveness framework). The evidence used to establish cost-effectiveness is typically uncertain and it is important that this uncertainty is characterised. In situations in which evidence is uncertain, the experience of experts is essential. The process by which the beliefs of experts can be formally collected in a quantitative manner is structured expert elicitation. There is heterogeneity in the existing methodology used in health-care decision-making. A number of guidelines are available for structured expert elicitation; however, it is not clear if any of these are appropriate for health-care decision-making.

OBJECTIVES

The overall aim was to establish a protocol for structured expert elicitation to inform health-care decision-making. The objectives are to (1) provide clarity on methods for collecting and using experts' judgements, (2) consider when alternative methodology may be required in particular contexts, (3) establish preferred approaches for elicitation on a range of parameters, (4) determine which elicitation methods allow experts to express uncertainty and (5) determine the usefulness of the reference protocol developed.

METHODS

A mixed-methods approach was used: systemic review, targeted searches, experimental work and narrative synthesis. A review of the existing guidelines for structured expert elicitation was conducted. This identified the approaches used in existing guidelines (the 'choices') and determined if dominant approaches exist. Targeted review searches were conducted for selection of experts, level of elicitation, fitting and aggregation, assessing accuracy of judgements and heuristics and biases. To sift through the available choices, a set of principles that underpin the use of structured expert elicitation in health-care decision-making was defined using evidence generated from the targeted searches, quantities to elicit experimental evidence and consideration of constraints in health-care decision-making. These principles, including fitness for purpose and reflecting individual expert uncertainty, were applied to the set of choices to establish a reference protocol. An applied evaluation of the developed reference protocol was also undertaken.

RESULTS

For many elements of structured expert elicitation, there was a lack of consistency across the existing guidelines. In almost all choices, there was a lack of empirical evidence supporting recommendations, and in some circumstances the principles are unable to provide sufficient justification for discounting particular choices. It is possible to define reference methods for health technology assessment. These include a focus on gathering experts with substantive skills, eliciting observable quantities and individual elicitation of beliefs. Additional considerations are required for decision-makers outside health technology assessment, for example at a local level, or for early technologies. Access to experts may be limited and in some circumstances group discussion may be needed to generate a distribution.

LIMITATIONS

The major limitation of the work conducted here lies not in the methods employed in the current work but in the evidence available from the wider literature relating to how appropriate particular methodological choices are.

CONCLUSIONS

The reference protocol is flexible in many choices. This may be a useful characteristic, as it is possible to apply this reference protocol across different settings. Further applied studies, which use the choices specified in this reference protocol, are required.

FUNDING

This project was funded by the NIHR Health Technology Assessment programme and will be published in full in Health Technology Assessment; Vol. 25, No. 37. See the NIHR Journals Library website for further project information. This work was also funded by the Medical Research Council (reference MR/N028511/1).

Sex- and Type-specific Genital Human Papillomavirus Transmission Rates Between Heterosexual Partners: A Bayesian Reanalysis of the HITCH Cohort

BACKGROUND

It is unclear whether sexual transmission rates of human papillomaviruses (HPV) differ between sexes and HPV types. We estimate updated transmission rates from the final HITCH cohort study and propose an estimation method that accounts for interval-censored data and infection clearance.

METHODS

We enrolled young women 18-24 years old and their male sex partners ≥18 years old in Montréal, Canada, between 2005 and 2011. We followed women over 24 months and men over 4 months. We tested genital samples with Linear Array for HPV DNA detection and genotyping. We calculated infection transmission rates between partners using a multistate Markov model via a Bayesian approach. We report the posterior median and 2.5%-97.5% percentile intervals (95% PI).

RESULTS

We observed 166 type-specific incident HPV transmission events in 447 women and 402 men. The estimated median transmission rate from an HPV-positive to a negative partner was 4.2 (95% PI = 3.1 to 5.3) per 100 person-months. The transmission rate from men-to-women was 3.5 (95% PI = 2.5 to 4.7) and from women-to-men was 5.6 (95% PI = 3.8 to 7.0) per 100 person-months, corresponding to a rate ratio of 1.6 (95% PI = 1.0 to 2.5). Partners reporting always using condoms had a 0.22 (95% PI = 0.05 to 0.61) times lower HPV transmission rate than those reporting never using condoms. HPV16/18 did not have particularly high transmission rates relative to other HPV types.

CONCLUSION

Our updated analysis supports previous research suggesting higher women-to-men than men-to-women HPV transmission rates and a protective effect of condoms in heterosexual partnerships. Our results also suggest that crude incidence rates underestimate HPV transmission rates due to interval-censoring. See video abstract at http://links.lww.com/EDE/B794.

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

Comparison of Decision Modeling Approaches for Health Technology and Policy Evaluation

We discuss tradeoffs and errors associated with approaches to modeling health economic decisions. Through an application in pharmacogenomic (PGx) testing to guide drug selection for individuals with a genetic variant, we assessed model accuracy, optimal decisions, and computation time for an identical decision scenario modeled 4 ways: using 1) coupled-time differential equations (DEQ), 2) a cohort-based discrete-time state transition model (MARKOV), 3) an individual discrete-time state transition microsimulation model (MICROSIM), and 4) discrete event simulation (DES). Relative to DEQ, the net monetary benefit for PGx testing (v. a reference strategy of no testing) based on MARKOV with rate-to-probability conversions using commonly used formulas resulted in different optimal decisions. MARKOV was nearly identical to DEQ when transition probabilities were embedded using a transition intensity matrix. Among stochastic models, DES model outputs converged to DEQ with substantially fewer simulated patients (1 million) v. MICROSIM (1 billion). Overall, properly embedded Markov models provided the most favorable mix of accuracy and runtime but introduced additional complexity for calculating cost and quality-adjusted life year outcomes due to the inclusion of "jumpover" states after proper embedding of transition probabilities. Among stochastic models, DES offered the most favorable mix of accuracy, reliability, and speed.

Multistate models of developmental toxicity: Application to valproic acid-induced malformations in the zebrafish embryo

For the determination of acute toxicity of chemicals in zebrafish (Danio rerio) embryos, the OECD test guideline 236, relative to the Fish Embryo Toxicity Test (FET), stipulates a dose-response analysis of four lethal core endpoints and a quantitative characterization of abnormalities including their time-dependency. Routinely, the data are analyzed at the different observation times separately. However, observations at a given time strongly depend on the previous effects and should be analyzed jointly with them. To solve this problem, we developed multistate models for occurrence of developmental malformations and live events in zebrafish embryos exposed to eight concentrations of valproic acid (VPA) the first five days of life. Observations were recorded daily per embryo. We statistically infer on model structure and parameters using a numerical Bayesian framework. Hatching probability rate changed with time and we compared five forms of its time-dependence; a constant rate, a piecewise constant rate with a fixed hatching time at 48 h post fertilization, a piecewise constant rate with a variable hatching time, as well as a Hill and Gaussian form. A piecewise constant function of time adequately described the hatching data. The other transition rates were conditioned on the embryo body concentration of VPA, obtained using a physiologically-based pharmacokinetic model. VPA impacted mostly the malformation probability rate in hatched and non-hatched embryos. Malformation reversion probability rates were lowered by VPA. Direct mortality was low at the concentrations tested, but increased linearly with internal concentration. The model makes full use of data and gives a finer grain analysis of the teratogenic effects of VPA in zebrafish than the OECD-prescribed approach. We discuss the use of the model for obtaining toxicological reference values suitable for inter-species extrapolation. A general result is that complex multistate models can be efficiently evaluated numerically.

Introducing cell-free DNA noninvasive testing in a Down syndrome public health screening program: a budget impact analysis

BACKGROUND

Non-invasive prenatal testing (NIPT) using cell-free fetal DNA in maternal plasma is a high accurate test for prenatal screening for Down syndrome. Although it has been reported to be cost effective as a contingent test, evidence about its budget impact is lacking.

OBJECTIVE

To evaluate, using computer simulations, the budget impact of implementing NIPT as a contingent test in the Quebec Program of screening for Trisomy 21.

METHODS

A semi-Markov analytic model built to simulate the budget impact of implementing NIPT into the current Quebec Trisomy 21 public Prenatal Screening, Serum Integrated prenatal screening (SIPS). Comparisons were made for a virtual population similar to that of expected Quebec pregnant women in 2015 in terms of size and age. Data input parameters were retrieved from a thorough literature search and in government databases, especially data from Quebec Program of screening for Trisomy 21. The 2015-2016 fiscal year budget impact was estimated from the Quebec healthcare system perspective and was expressed as the difference in the overall costs between the two alternatives (SIPS minus SPS + NIPT).

RESULTS

Our study found that, at a baseline cost for NIPT of CAD$ 795, NIPT as a second-tier test offered to high-risk women identified by current screening program (SIPS + NIPT) may be affordable for Quebec health care system. Compared to the current screening program, it would be implemented at a neutral cost, considering a modest annual savings of $ 80,432 (95% CI $ 79, $ 874-$ 81,462). Results were sensitive to the NIPT costs and the uptake-rate of invasive diagnostic tests.

CONCLUSION

Introducing NIPT as a contingent test in the Quebec Trisomy 21 screening program is an affordable strategy compared to the current practice. Further research is needed to confirm if our results can be reproduced in other healthcare jurisdictions.

Estimating Transition Probabilities from Published Evidence: A Tutorial for Decision Modelers

This tutorial presents practical guidance on transforming various types of information published in journals, or available online from government and other sources, into transition probabilities for use in state-transition models, including cost-effectiveness models. Much, but not all, of the guidance has been previously published in peer-reviewed journals. Our purpose is to collect it in one location to serve as a stand-alone resource for decision modelers who draw most or all of their information from the published literature. Our focus is on the technical aspects of manipulating data to derive transition probabilities. We explain how to derive model transition probabilities from the following types of statistics: relative risks, odds, odds ratios, and rates. We then review the well-known approach for converting probabilities to match the model's cycle length when there are two health-state transitions and how to handle the case of three or more health-state transitions, for which the two-state approach is not appropriate. Other topics discussed include transition probabilities for population subgroups, issues to keep in mind when using data from different sources in the derivation process, and sensitivity analyses, including the use of sensitivity analysis to allocate analyst effort in refining transition probabilities and ways to handle sources of uncertainty that are not routinely formalized in models. The paper concludes with recommendations to help modelers make the best use of the published literature.

Joint synthesis of conditionally related multiple outcomes makes better use of data than separate meta-analyses

BACKGROUND

When there are structural relationships between outcomes reported in different trials, separate analyses of each outcome do not provide a single coherent analysis, which is required for decision-making. For example, trials of intrapartum anti-bacterial prophylaxis (IAP) to prevent early onset group B streptococcal (EOGBS) disease can report three treatment effects: the effect on bacterial colonisation of the newborn, the effect on EOGBS, and the effect on EOGBS conditional on newborn colonisation. These outcomes are conditionally related, or nested, in a multi-state model. This paper shows how to exploit these structural relationships, providing a single coherent synthesis of all the available data, while checking to ensure that different sources of evidence are consistent.

RESULTS

Overall, the use of IAP reduces the risk of EOGBS (RR: 0.03; 95% Credible Interval (CrI): 0.002-0.13). Most of the treatment effect is due to the prevention of colonisation in newborns of colonised mothers (RR: 0.08, 95% CrI: 0.04-0.14). Node-splitting demonstrated that the treatment effect calculated using only direct evidence was consistent with that predicted from the remaining evidence (p = 0.15). The findings accorded with previously published separate meta-analyses of the different outcomes, once these are re-analysed correctly accounting for zero cells.

CONCLUSION

Multiple outcomes should be synthesised together where possible, taking account of their structural relationships. This generates an internally coherent analysis, suitable for decision making, in which estimates of each of the treatment effects are based on all available evidence (direct and indirect). Separate meta-analyses of each outcome have none of these properties.

Optimizing Interventions Across the HIV Care Continuum: A Case Study Using Process Improvement Analysis

UNAIDS' 90-90-90 goal for 2020 is for 90% of HIV-infected people to know their status, 90% of infected individuals to receive antiretroviral therapy (ART), and 90% of those on ART to achieve viral suppression. To achieve these ambitious goals, effective care delivery programs are needed. In this paper we present a case study showing how HIV care can be improved by viewing the patient care process as a production process and applying methods of process improvement analysis. We examine the continuum of HIV care at a hospital-based HIV clinic in Kingston, Jamaica. We perform qualitative analysis to identify key programmatic, personnel, and clinical areas for process improvement. We then perform quantitative analysis. We develop a stochastic model of the care process which we use to evaluate the effects of potential process improvements on the number of patients who receive ART and the number who achieve viral suppression. We also develop a model for optimal investment of a fixed budget among interventions aimed at improving the care cascade and we use the model to determine the optimal investment among three interventions that the clinic could invest in. By viewing the patient care process as a production process and applying qualitative and quantitative process improvement analysis, our case study illustrates how clinics can identify the best ways to maximize clinical outcomes. Our methods are generalizable to other HIV care clinics as well as to clinics that provide care for other chronic conditions (e.g., diabetes, hepatitis B, or opioid use disorder).

Oral splints for patients with temporomandibular disorders or bruxism: a systematic review and economic evaluation

BACKGROUND

Splints are a non-invasive, reversible management option for temporomandibular disorders or bruxism. The clinical effectiveness and cost-effectiveness of splints remain uncertain.

OBJECTIVES

The objectives were to evaluate the clinical effectiveness and cost-effectiveness of splints for patients with temporomandibular disorders or bruxism. This evidence synthesis compared (1) all types of splint versus no/minimal treatment/control splints and (2) prefabricated versus custom-made splints, for the primary outcomes, which were pain (temporomandibular disorders) and tooth wear (bruxism).

REVIEW METHODS

Four databases, including MEDLINE and EMBASE, were searched from inception until 1 October 2018 for randomised clinical trials. The searches were conducted on 1 October 2018. Cochrane review methods (including risk of bias) were used for the systematic review. Standardised mean differences were pooled for the primary outcome of pain, using random-effects models in temporomandibular disorder patients. A Markov cohort, state-transition model, populated using current pain and Characteristic Pain Intensity data, was used to estimate the incremental cost-effectiveness ratio for splints compared with no splint, from an NHS perspective over a lifetime horizon. A value-of-information analysis identified future research priorities.

RESULTS

Fifty-two trials were included in the systematic review. The evidence identified was of very low quality with unclear reporting by temporomandibular disorder subtype. When all subtypes were pooled into one global temporomandibular disorder group, there was no evidence that splints reduced pain [standardised mean difference (at up to 3 months) -0.18, 95% confidence interval -0.42 to 0.06; substantial heterogeneity] when compared with no splints or a minimal intervention. There was no evidence that other outcomes, including temporomandibular joint noises, decreased mouth-opening, and quality of life, improved when using splints. Adverse events were generally not reported, but seemed infrequent when reported. The most plausible base-case incremental cost-effectiveness ratio was uncertain and driven by the lack of clinical effectiveness evidence. The cost-effectiveness acceptability curve showed splints becoming more cost-effective at a willingness-to-pay threshold of ≈£6000, but the probability never exceeded 60% at higher levels of willingness to pay. Results were sensitive to longer-term extrapolation assumptions. A value-of-information analysis indicated that further research is required. There were no studies measuring tooth wear in patients with bruxism. One small study looked at pain and found a reduction in the splint group [mean difference (0-10 scale) -2.01, 95% CI -1.40 to -2.62; very low-quality evidence]. As there was no evidence of a difference between splints and no splints, the second objective became irrelevant.

LIMITATIONS

There was a large variation in the diagnostic criteria, splint types and outcome measures used and reported. Sensitivity analyses based on these limitations did not indicate a reduction in pain.

CONCLUSIONS

The very low-quality evidence identified did not demonstrate that splints reduced pain in temporomandibular disorders as a group of conditions. There is insufficient evidence to determine whether or not splints reduce tooth wear in patients with bruxism. There remains substantial uncertainty surrounding the most plausible incremental cost-effectiveness ratio.

FUTURE WORK

There is a need for well-conducted trials to determine the clinical effectiveness and cost-effectiveness of splints in patients with carefully diagnosed and subtyped temporomandibular disorders, and patients with bruxism, using agreed measures of pain and tooth wear.

STUDY REGISTRATION

This study is registered as PROSPERO CRD42017068512.

FUNDING

This project was funded by the National Institute for Health Research (NIHR) Health Technology Assessment programme and will be published in full in Health Technology Assessment; Vol. 24, No. 7. See the NIHR Journals Library website for further project information.

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.

HIV Disease Progression Among Antiretroviral Therapy Patients in Zimbabwe: A Multistate Markov Model

Background: Antiretroviral therapy (ART) impact has prolonged survival of people living with HIV. We evaluated HIV disease progression among ART patients using routinely collected patient-level data between 2004 and 2017 in Zimbabwe. Methods: We partitioned HIV disease progression into four transient CD4 cell counts states: state 1 (CD4 ≥ 500 cells/μl), state 2 (350 cells/μl ≤ CD4 < 500 cells/μl), state 3 (200 cells/μl ≤ CD4 < 350 cells/μl), state 4 (CD4 < 200 cells/μl), and the absorbing state death (state 5). We proposed a semiparametric time-homogenous multistate Markov model to estimate bidirectional transition rates. Covariate effects (age, gender, ART initiation period, and health facility level) on the transition rates were assessed. Results: We analyzed 204,289 clinic visits by 63,422 patients. There were 24,325 (38.4%) patients in state 4 (CD4 < 200) at ART initiation, and 7,995 (12.6%) deaths occurred by December 2017. The overall mortality rate was 3.9 per 100 person-years. The highest mortality rate of 5.7 per 100 person-years (4,541 deaths) was from state 4 (CD4 < 200) compared to other states. Mortality rates decreased with increase in time since ART initiation. Health facility type was the strongest predictor for immune recovery. Provincial or central hospital patients showed a diminishing dose-response effect on immune recovery by state from a hazard ratio (HR) of 8.30 [95% confidence interval (95% CI), 6.64-10.36] (state 4 to 3) to HR of 3.12 (95% CI, 2.54-4.36) (state 2 to 1) compared to primary healthcare facilities. Immune system for male patients was more likely to deteriorate, and they had a 32% increased mortality risk (HR, 1.32; 95% CI, 1.23-1.42) compared to female patients. Elderly patients (45+ years) were more likely to immune deteriorate compared to 25-34 years age group: HR, 1.35; 95% CI, 1.18-1.54; HR, 1.56; 95% CI, 1.34-1.81 and HR, 1.53; 95% CI, 1.32-1.79 for states 1 to 2, state 2 to 3, and states 3 to 4, respectively. Conclusion: Immune recovery was pronounced among provincial or central hospitals. Male patients with lower CD4 cell counts were at a higher risk of immune deterioration and mortality, while elderly patients were more likely to immune deteriorate. Early therapeutic interventions when the immune system is relatively stable across gender and age may contain mortality and increase survival outcomes. Interventions which strengthen ART services in primary healthcare facilities are essential.

Problematic Internet Use: A longitudinal study evaluating prevalence and predictors

Objective

The purpose of this 4-year study was to assess the prevalence over time and predictors of PIU using the Problematic and Risky Internet Use Screening Scale (PRIUSS). We also identified an Intermediate risk PRIUSS score.

Study design

In this longitudinal cohort study we recruited participants using random selection from 2 colleges, participants completed a yearly PRIUSS. We used multivariate logistic regression analysis to evaluate predictors of PIU. We pursued receiver operating curve (ROC) analysis to identify an Intermediate risk PRIUSS score. Finally, we applied Markov modeling to test the dynamics of moving through PIU risk states over time.

Results

Our 319 participants included 56% females, 58% from the Midwest and 75% Caucasian. PIU prevalence estimates varied between 9% and 11% over the four years. PIU risk status from the previous time period was identified as the main predictor for PIU (OR=24.1, 95% CI: 12.8-45.4, p<0.0001). ROC analysis identified the optimal threshold for defining Intermediate risk was a PRIUSS score of 15.

Conclusion

This longitudinal study of PIU among college students found that risks were present across groups and over time. The most salient predictor of PIU was being at risk at the previous time point. Based on results, we propose a PRIUSS score of 15 as an Intermediate risk cut-off to better identify those at risk of developing PIU.

A technique for approximating transition rates from published survival analyses

Background

Quality-adjusted-life-years (QALYs) are used to concurrently quantify morbidity and mortality within a single parameter. For this reason, QALYs can facilitate the discussion of risks and benefits during patient counseling regarding treatment options. QALYs are often calculated using partitioned-survival modelling. Alternatively, QALYs can be calculated using more flexible and informative state-transition models populated with transition rates estimated using multistate modelling (MSM) techniques. Unfortunately the latter approach is considered not possible when only progression-free survival (PFS) and overall survival (OS) analyses are reported.

Methods

We have developed a method that can be used to estimate approximate transition rates from published PFS and OS analyses (we will refer to transition rates estimated using full multistate methods as true transition rates).

Results

The approximation method is more accurate for estimating the transition rates out of health than the transition rate out of illness. The method tends to under-estimate true transition rates as censoring increases.

Conclusions

In this article we present the basis for and use of the transition rate approximation method. We then apply the method to a case study and evaluate the method in a simulation study.

Electronic supplementary material

The online version of this article (10.1186/s12962-019-0182-7) contains supplementary material, which is available to authorized users.

Alternative Conversion Methods for Transition Probabilities in State-Transition Models: Validity and Impact on Comparative Effectiveness and Cost-Effectiveness

Background. In state-transition models (STMs), decision problems are conceptualized using health states and transitions among those health states after predefined time cycles. The naive, commonly applied method (C) for cycle length conversion transforms all transition probabilities separately. In STMs with more than 2 health states, this method is not accurate. Therefore, we aim to describe and compare the performance of method C with that of alternative matrix transformation methods. Design. We compare 2 alternative matrix transformation methods (Eigenvalue method [E], Schure-Padé method [SP]) to method C applied in an STM of 3 different treatment strategies for women with breast cancer. We convert the given annual transition matrix into a monthly-cycle matrix and evaluate induced transformation errors for the transition matrices and the long-term outcomes: life years, quality-adjusted life-years, costs and incremental cost-effectiveness ratios, and the performance related to the decisions. In addition, we applied these transformation methods to randomly generated annual transition matrices with 4, 7, 10, and 20 health states. Results. In theory, there is no generally applicable correct transformation method. Based on our simulations, SP resulted in the smallest transformation-induced discrepancies for generated annual transition matrices for 2 treatment strategies. E showed slightly smaller discrepancies than SP in the strategy, where one of the direct transitions between health states was excluded. For long-term outcomes, the largest discrepancy occurred for estimated costs applying method C. For higher dimensional models, E performs best. Conclusions. In our modeling examples, matrix transformations (E, SP) perform better than transforming all transition probabilities separately (C). Transition probabilities based on alternative conversion methods should therefore be applied in sensitivity analyses.

Systems biology: perspectives on multiscale modeling in research on endocrine-related cancers

Drawing on concepts from experimental biology, computer science, informatics, mathematics and statistics, systems biologists integrate data across diverse platforms and scales of time and space to create computational and mathematical models of the integrative, holistic functions of living systems. Endocrine-related cancers are well suited to study from a systems perspective because of the signaling complexities arising from the roles of growth factors, hormones and their receptors as critical regulators of cancer cell biology and from the interactions among cancer cells, normal cells and signaling molecules in the tumor microenvironment. Moreover, growth factors, hormones and their receptors are often effective targets for therapeutic intervention, such as estrogen biosynthesis, estrogen receptors or HER2 in breast cancer and androgen receptors in prostate cancer. Given the complexity underlying the molecular control networks in these cancers, a simple, intuitive understanding of how endocrine-related cancers respond to therapeutic protocols has proved incomplete and unsatisfactory. Systems biology offers an alternative paradigm for understanding these cancers and their treatment. To correctly interpret the results of systems-based studies requires some knowledge of how in silico models are built, and how they are used to describe a system and to predict the effects of perturbations on system function. In this review, we provide a general perspective on the field of cancer systems biology, and we explore some of the advantages, limitations and pitfalls associated with using predictive multiscale modeling to study endocrine-related cancers.

A theoretical foundation for state-transition cohort models in health decision analysis

Following its introduction over three decades ago, the cohort model has been used extensively to model population trajectories over time in decision-analytic modeling studies. However, the stochastic process underlying cohort models has not been properly described. In this study, we explicate the stochastic process underlying a cohort model, by carefully formulating the dynamics of populations across health states and assigning probability rules on these dynamics. From this formulation, we explicate a mathematical representation of the system, which is given by the master equation. We solve the master equation by using the probability generation function method to obtain the explicit form of the probability of observing a particular realization of the system at an arbitrary time. The resulting generating function is used to derive the analytical expressions for calculating the mean and the variance of the process. Secondly, we represent the cohort model by a difference equation for the number of individuals across all states. From the difference equation, a continuous-time cohort model is recovered and takes the form of an ordinary differential equation. To show the equivalence between the derived stochastic process and the cohort model, we conduct a numerical exercise. We demonstrate that the population trajectories generated from the formulas match those from the cohort model simulation. In summary, the commonly-used cohort model represent the average of a continuous-time stochastic process on a multidimensional integer lattice governed by a master equation. Knowledge of the stochastic process underlying a cohort model provides a theoretical foundation for the modeling method.

Estimation of annual probabilities of changing disability levels in Australians with relapsing-remitting multiple sclerosis

BACKGROUND

Transition probabilities are the engine within many health economics decision models. However, the probabilities of progression of disability due to multiple sclerosis (MS) have not previously been estimated in Australia.

OBJECTIVES

To estimate annual probabilities of changing disability levels in Australians with relapsing-remitting MS (RRMS).

METHODS

Combining data from Ausimmune/Ausimmune Longitudinal (2003-2011) and Tasmanian MS Longitudinal (2002-2005) studies (n = 330), annual transition probabilities were obtained between no/mild (Expanded Disability Status Scale (EDSS) levels 0-3.5), moderate (EDSS 4-6.0) and severe (EDSS 6.5-9.5) disability.

RESULTS

From no/mild disability, 6.4% (95% confidence interval (CI): 4.7-8.4) and 0.1% (0.0-0.2) progressed to moderate and severe disability annually, respectively. From moderate disability, 6.9% (1.0-11.4) improved (to no/mild state) and 2.6% (1.1-4.5) worsened. From severe disability, 0.0% improved to moderate and no/mild disability. Male sex, age at onset, longer disease duration, not using immunotherapies greater than 3 months and a history of relapse were related to higher probabilities of worsening.

CONCLUSION

We have estimated probabilities of changing disability levels in Australians with RRMS. Probabilities differed between various subgroups, but due to small sample sizes, results should be interpreted with caution. Our findings will be helpful in predicting long-term disease outcomes and in health economic evaluations of MS.

Nonidentifiability in Model Calibration and Implications for Medical Decision Making

BACKGROUND

Calibration is the process of estimating parameters of a mathematical model by matching model outputs to calibration targets. In the presence of nonidentifiability, multiple parameter sets solve the calibration problem, which may have important implications for decision making. We evaluate the implications of nonidentifiability on the optimal strategy and provide methods to check for nonidentifiability.

METHODS

We illustrate nonidentifiability by calibrating a 3-state Markov model of cancer relative survival (RS). We performed 2 different calibration exercises: 1) only including RS as a calibration target and 2) adding the ratio between the 2 nondeath states over time as an additional target. We used the Nelder-Mead (NM) algorithm to identify parameter sets that best matched the calibration targets. We used collinearity and likelihood profile analyses to check for nonidentifiability. We then estimated the benefit of a hypothetical treatment in terms of life expectancy gains using different, but equally good-fitting, parameter sets. We also applied collinearity analysis to a realistic model of the natural history of colorectal cancer.

RESULTS

When only RS is used as the calibration target, 2 different parameter sets yield similar maximum likelihood values. The high collinearity index and the bimodal likelihood profile on both parameters demonstrated the presence of nonidentifiability. These different, equally good-fitting parameter sets produce different estimates of the treatment effectiveness (0.67 v. 0.31 years), which could influence the optimal decision. By incorporating the additional target, the model becomes identifiable with a collinearity index of 3.5 and a unimodal likelihood profile.

CONCLUSIONS

In the presence of nonidentifiability, equally likely parameter estimates might yield different conclusions. Checking for the existence of nonidentifiability and its implications should be incorporated into standard model calibration procedures.

Attributing diseases to multiple pathways: a causal-pie modeling approach

Characterizing the relations between exposures and diseases is the central tenet of epidemiology. Researchers may want to evaluate exposure-disease causation by assessing whether the disease under concern is induced by the various exposures – the so-called “attribution”. In this paper, the authors propose a method to attribute diseases to multiple pathways based on the causal-pie model. The method can also be used to evaluate the potential impact of an intervention strategy and to allocate responsibility in tort-law liability issues.

Comparative quality-adjusted survival analysis between radiation therapy alone and radiation with androgen deprivation therapy in patients with locally advanced prostate cancer: a secondary analysis of Radiation Therapy Oncology Group 85-31 with novel decision analysis methods

Background

Androgen deprivation therapy in addition to radiation therapy (RT + ADT) has shown benefits in local control and progression-free survival compared with RT alone for patients with locally advanced prostate cancer in Radiation Therapy Oncology Group 85-31. However, the survival gain may be diluted with increased toxicity of ADT. The aim of the study is to compare quality-adjusted life years (QALYs) values between two groups.

Methods

We developed “quality-adjusted survival analysis using duration” (QASAD) and “quality-adjusted survival analysis using probability” (QASAP) to estimate the quality-adjusted survival time. The QASAD uses the median duration in each health state to weight the utilities, whereas the QASAP uses the proportional probability of being in each state for weighting. The survival and complication rates were reconstructed based on published Kaplan–Meier survival curves, and the utility values for states were obtained from the previous literature.

Results

QALYs values for RT + ADT were generally higher than those for RT. The QASAD resulted in a QALY value of 4.93 [95% bootstrapped confidence interval (CI) = 4.12–5.71] for RT and of 5.60 (95% CI = 4.30–6.48) for RT + ADT. QASAP resulted in a QALY value of 4.85 (95% CI = 4.16–5.39) for RT and 4.96 (95% CI = 3.73–5.78) for RT + ADT.

Conclusions

We showed that RT + ADT provided slightly better quality-adjusted survival outcome than RT alone. The QASAD and QASAP methods may help the decision of optimal treatment balancing between survival gain and unfavorable quality of life.

Resolve conflicting rankings of outcomes in network meta-analysis: Partial ordering of treatments

Network meta-analysis has evolved into a core method for evidence synthesis in health care. In network meta-analysis, 3 or more treatments for a given medical condition are compared, based on a number of clinical studies, usually randomized controlled trials. Often, many different endpoints are investigated, related to different aspects of the patient's outcome, such as efficacy, safety, acceptability, or costs of a treatment. Different outcomes may lead to different rankings of the treatments. We use the existing theory of partially ordered sets and show how the relations between the treatments in a network meta-analysis can be illustrated by Hasse diagrams, that is, directed graphs showing the partial order relations, and by structured scatter plots and biplots.

Humans and climate change drove the Holocene decline of the brown bear

The current debate about megafaunal extinctions during the Quaternary focuses on the extent to which they were driven by humans, climate change, or both. These two factors may have interacted in a complex and unexpected manner, leaving the exact pathways to prehistoric extinctions unresolved. Here we quantify, with unprecedented detail, the contribution of humans and climate change to the Holocene decline of the largest living terrestrial carnivore, the brown bear (Ursus arctos), on a continental scale. We inform a spatially explicit metapopulation model for the species by combining life-history data and an extensive archaeofaunal record from excavations across Europe with reconstructed climate and land-use data reaching back 12,000 years. The model reveals that, despite the broad climatic niche of the brown bear, increasing winter temperatures contributed substantially to its Holocene decline — both directly by reducing the species’ reproductive rate and indirectly by facilitating human land use. The first local extinctions occurred during the Mid-Holocene warming period, but the rise of the Roman Empire 2,000 years ago marked the onset of large-scale extinctions, followed by increasingly rapid range loss and fragmentation. These findings strongly support the hypothesis that complex interactions between climate and humans may have accelerated megafaunal extinctions.

Using Parameter Constraints to Choose State Structures in Cost-Effectiveness Modelling

BACKGROUND

This article addresses the choice of state structure in a cost-effectiveness multi-state model. Key model outputs, such as treatment recommendations and prioritisation of future research, may be sensitive to state structure choice. For example, it may be uncertain whether to consider similar disease severities or similar clinical events as the same state or as separate states. Standard statistical methods for comparing models require a common reference dataset but merging states in a model aggregates the data, rendering these methods invalid.

METHODS

We propose a method that involves re-expressing a model with merged states as a model on the larger state space in which particular transition probabilities, costs and utilities are constrained to be equal between states. This produces a model that gives identical estimates of cost effectiveness to the model with merged states, while leaving the data unchanged. The comparison of state structures can be achieved by comparing maximised likelihoods or information criteria between constrained and unconstrained models. We can thus test whether the costs and/or health consequences for a patient in two states are the same, and hence if the states can be merged. We note that different structures can be used for rates, costs and utilities, as appropriate.

APPLICATION

We illustrate our method with applications to two recent models evaluating the cost effectiveness of prescribing anti-depressant medications by depression severity and the cost effectiveness of diagnostic tests for coronary artery disease.

CONCLUSIONS

State structures in cost-effectiveness models can be compared using standard methods to compare constrained and unconstrained models.

Current recommendations on the estimation of transition probabilities in Markov cohort models for use in health care decision-making: a targeted literature review

OBJECTIVE

Although Markov cohort models represent one of the most common forms of decision-analytic models used in health care decision-making, correct implementation of such models requires reliable estimation of transition probabilities. This study sought to identify consensus statements or guidelines that detail how such transition probability matrices should be estimated.

METHODS

A literature review was performed to identify relevant publications in the following databases: Medline, Embase, the Cochrane Library, and PubMed. Electronic searches were supplemented by manual-searches of health technology assessment (HTA) websites in Australia, Belgium, Canada, France, Germany, Ireland, Norway, Portugal, Sweden, and the UK. One reviewer assessed studies for eligibility.

RESULTS

Of the 1,931 citations identified in the electronic searches, no studies met the inclusion criteria for full-text review, and no guidelines on transition probabilities in Markov models were identified. Manual-searching of the websites of HTA agencies identified ten guidelines on economic evaluations (Australia, Belgium, Canada, France, Germany, Ireland, Norway, Portugal, Sweden, and UK). All identified guidelines provided general guidance on how to develop economic models, but none provided guidance on the calculation of transition probabilities. One relevant publication was identified following review of the reference lists of HTA agency guidelines: the International Society for Pharmacoeconomics and Outcomes Research taskforce guidance. This provided limited guidance on the use of rates and probabilities.

CONCLUSIONS

There is limited formal guidance available on the estimation of transition probabilities for use in decision-analytic models. Given the increasing importance of cost-effectiveness analysis in the decision-making processes of HTA bodies and other medical decision-makers, there is a need for additional guidance to inform a more consistent approach to decision-analytic modeling. Further research should be done to develop more detailed guidelines on the estimation of transition probabilities.

Bayesian Methods for Calibrating Health Policy Models: A Tutorial

Mathematical simulation models are commonly used to inform health policy decisions. These health policy models represent the social and biological mechanisms that determine health and economic outcomes, combine multiple sources of evidence about how policy alternatives will impact those outcomes, and synthesize outcomes into summary measures salient for the policy decision. Calibrating these health policy models to fit empirical data can provide face validity and improve the quality of model predictions. Bayesian methods provide powerful tools for model calibration. These methods summarize information relevant to a particular policy decision into (1) prior distributions for model parameters, (2) structural assumptions of the model, and (3) a likelihood function created from the calibration data, combining these different sources of evidence via Bayes' theorem. This article provides a tutorial on Bayesian approaches for model calibration, describing the theoretical basis for Bayesian calibration approaches as well as pragmatic considerations that arise in the tasks of creating calibration targets, estimating the posterior distribution, and obtaining results to inform the policy decision. These considerations, as well as the specific steps for implementing the calibration, are described in the context of an extended worked example about the policy choice to provide (or not provide) treatment for a hypothetical infectious disease. Given the many simplifications and subjective decisions required to create prior distributions, model structure, and likelihood, calibration should be considered an exercise in creating a reasonable model that produces valid evidence for policy, rather than as a technique for identifying a unique theoretically optimal summary of the evidence.

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models

For health-economic analyses that use multistate Markov models, it is often necessary to convert from transition rates to transition probabilities, and for probabilistic sensitivity analysis and other purposes it is useful to have explicit algebraic formulas for these conversions, to avoid having to resort to numerical methods. However, if there are four or more states then the formulas can be extremely complicated. These calculations can be made using packages such as R, but many analysts and other stakeholders still prefer to use spreadsheets for these decision models. We describe a procedure for deriving formulas that use intermediate variables so that each individual formula is reasonably simple. Once the formulas have been derived, the calculations can be performed in Excel or similar software. The procedure is illustrated by several examples and we discuss how to use a computer algebra system to assist with it. The procedure works in a wide variety of scenarios but cannot be employed when there are several backward transitions and the characteristic equation has no algebraic solution, or when the eigenvalues of the transition rate matrix are very close to each other.

Α Markov model for longitudinal studies with incomplete dichotomous outcomes

Missing outcome data constitute a serious threat to the validity and precision of inferences from randomized controlled trials. In this paper, we propose the use of a multistate Markov model for the analysis of incomplete individual patient data for a dichotomous outcome reported over a period of time. The model accounts for patients dropping out of the study and also for patients relapsing. The time of each observation is accounted for, and the model allows the estimation of time-dependent relative treatment effects. We apply our methods to data from a study comparing the effectiveness of 2 pharmacological treatments for schizophrenia. The model jointly estimates the relative efficacy and the dropout rate and also allows for a wide range of clinically interesting inferences to be made. Assumptions about the missingness mechanism and the unobserved outcomes of patients dropping out can be incorporated into the analysis. The presented method constitutes a viable candidate for analyzing longitudinal, incomplete binary data.

Validation and calibration of structural models that combine information from multiple sources

INTRODUCTION

Mathematical models that attempt to capture structural relationships between their components and combine information from multiple sources are increasingly used in medicine. Areas covered: We provide an overview of methods for model validation and calibration and survey studies comparing alternative approaches. Expert commentary: Model validation entails a confrontation of models with data, background knowledge, and other models, and can inform judgments about model credibility. Calibration involves selecting parameter values to improve the agreement of model outputs with data. When the goal of modeling is quantitative inference on the effects of interventions or forecasting, calibration can be viewed as estimation. This view clarifies issues related to parameter identifiability and facilitates formal model validation and the examination of consistency among different sources of information. In contrast, when the goal of modeling is the generation of qualitative insights about the modeled phenomenon, calibration is a rather informal process for selecting inputs that result in model behavior that roughly reproduces select aspects of the modeled phenomenon and cannot be equated to an estimation procedure. Current empirical research on validation and calibration methods consists primarily of methodological appraisals or case-studies of alternative techniques and cannot address the numerous complex and multifaceted methodological decisions that modelers must make. Further research is needed on different approaches for developing and validating complex models that combine evidence from multiple sources.

The natural history of Chlamydia trachomatis infection in women: a multi-parameter evidence synthesis

BACKGROUND AND OBJECTIVES

The evidence base supporting the National Chlamydia Screening Programme, initiated in 2003, has been questioned repeatedly, with little consensus on modelling assumptions, parameter values or evidence sources to be used in cost-effectiveness analyses. The purpose of this project was to assemble all available evidence on the prevalence and incidence of Chlamydia trachomatis (CT) in the UK and its sequelae, pelvic inflammatory disease (PID), ectopic pregnancy (EP) and tubal factor infertility (TFI) to review the evidence base in its entirety, assess its consistency and, if possible, arrive at a coherent set of estimates consistent with all the evidence.

METHODS

Evidence was identified using 'high-yield' strategies. Bayesian Multi-Parameter Evidence Synthesis models were constructed for separate subparts of the clinical and population epidemiology of CT. Where possible, different types of data sources were statistically combined to derive coherent estimates. Where evidence was inconsistent, evidence sources were re-interpreted and new estimates derived on a post-hoc basis.

RESULTS

An internally coherent set of estimates was generated, consistent with a multifaceted evidence base, fertility surveys and routine UK statistics on PID and EP. Among the key findings were that the risk of PID (symptomatic or asymptomatic) following an untreated CT infection is 17.1% [95% credible interval (CrI) 6% to 29%] and the risk of salpingitis is 7.3% (95% CrI 2.2% to 14.0%). In women aged 16-24 years, screened at annual intervals, at best, 61% (95% CrI 55% to 67%) of CT-related PID and 22% (95% CrI 7% to 43%) of all PID could be directly prevented. For women aged 16-44 years, the proportions of PID, EP and TFI that are attributable to CT are estimated to be 20% (95% CrI 6% to 38%), 4.9% (95% CrI 1.2% to 12%) and 29% (95% CrI 9% to 56%), respectively. The prevalence of TFI in the UK in women at the end of their reproductive lives is 1.1%: this is consistent with all PID carrying a relatively high risk of reproductive damage, whether diagnosed or not. Every 1000 CT infections in women aged 16-44 years, on average, gives rise to approximately 171 episodes of PID and 73 of salpingitis, 2.0 EPs and 5.1 women with TFI at age 44 years.

CONCLUSIONS AND RESEARCH RECOMMENDATIONS

The study establishes a set of interpretations of the major studies and study designs, under which a coherent set of estimates can be generated. CT is a significant cause of PID and TFI. CT screening is of benefit to the individual, but detection and treatment of incident infection may be more beneficial. Women with lower abdominal pain need better advice on when to seek early medical attention to avoid risk of reproductive damage. The study provides new insights into the reproductive risks of PID and the role of CT. Further research is required on the proportions of PID, EP and TFI attributable to CT to confirm predictions made in this report, and to improve the precision of key estimates. The cost-effectiveness of screening should be re-evaluated using the findings of this report.

FUNDING

The Medical Research Council grant G0801947.

[Economic Effect of Home Health Care Services for Community-dwelling Vulnerable Populations]

PURPOSE

In this study the costs and benefits of a home health care program were examined to evaluate the economic feasibility of the program.

METHODS

The study participants included 349 patients in the community who had been registered at a home health care center for 5 years. The costs and benefits of the program were analyzed using performance data and health data. The benefits were classified as the effects of pressure ulcer care, skin wound care and catheters management. The program effect was evaluated on the change of progress using transition probability. Benefits were divided into direct benefit such as the savings in medical costs and transportation costs, and indirect benefits which included saving in productivity loss and lost future income.

RESULTS

Participants had an average of 1.82 health problems. The input cost was KRW 36.8~153.3 million, the benefit was KRW 95.4~279.7 million. Direct benefits accounted for 53.4%~81.2%, and was higher than indirect benefits. The net benefit was greater than 0 from 2006 to 2009, and then dropped below 0 in 2010.

CONCLUSION

The average net benefit during 5 years was over 0 and the benefit cost ratoi was over 1.00, indicating that the home health care program si economical.

Changing Cycle Lengths in State-Transition Models: Challenges and Solutions

The choice of a cycle length in state-transition models should be determined by the frequency of clinical events and interventions. Sometimes there is need to decrease the cycle length of an existing state-transition model to reduce error in outcomes resulting from discretization of the underlying continuous-time phenomena or to increase the cycle length to gain computational efficiency. Cycle length conversion is also frequently required if a new state-transition model is built using observational data that have a different measurement interval than the model's cycle length. We show that a commonly used method of converting transition probabilities to different cycle lengths is incorrect and can provide imprecise estimates of model outcomes. We present an accurate approach that is based on finding the root of a transition probability matrix using eigendecomposition. We present underlying mathematical challenges of converting cycle length in state-transition models and provide numerical approximation methods when the eigendecomposition method fails. Several examples and analytical proofs show that our approach is more general and leads to more accurate estimates of model outcomes than the commonly used approach. MATLAB codes and a user-friendly online toolkit are made available for the implementation of the proposed methods.

Transition modeling of neuropsychiatric impairment in HIV

Few studies have reported analyses of neuropsychiatric impairment (NPI) data from HIV patients, in a real world clinical setting with the aim of establishing association between anti-retroviral drug concentrations and NPI development and resolution. No study has modeled the effect of efavirenz exposure beyond the pre-steady state period on the frequency and duration of NPI. The data used consists of 196 HIV patients whose efavirenz pharmacokinetic parameters were previously determined. Neuropsychiatric evaluation was done at baseline, week 2 and week 12. Patients were classified into NORMAL and NPI states. The duration of NPI was further classified as transient (NPI at week 2 but not at week 12), persistent (NPI at week 2 and 12) and delayed (NPI at week 12 but not at week 2). The proportion of patients in each duration category out of the total NPI patients was calculated. A continuous time Markov model was developed in NONMEM 7.3 and used to describe the relationship between efavirenz exposure and the duration of NPI. Monte Carlo simulations with the model were used to describe the effect of efavirenz dose reduction from 600mg to 400mg on the duration of NPI. The model adequately described the data. The influence of efavirenz exposure on the rate of development of NPI decayed with a half-life of 8.4 days. Efavirenz dose reduction to 400mg significantly reduces the duration of NPI, but has no impact on delayed NPI symptoms or efficacy.

Continuous time Markov chain approaches for analyzing transtheoretical models of health behavioral change: A case study and comparison of model estimations

Continuous time Markov chain models are frequently employed in medical research to study the disease progression but are rarely applied to the transtheoretical model, a psychosocial model widely used in the studies of health-related outcomes. The transtheoretical model often includes more than three states and conceptually allows for all possible instantaneous transitions (referred to as general continuous time Markov chain). This complicates the likelihood function because it involves calculating a matrix exponential that may not be simplified for general continuous time Markov chain models. We undertook a Bayesian approach wherein we numerically evaluated the likelihood using ordinary differential equation solvers available from the gnu scientific library. We compared our Bayesian approach with the maximum likelihood method implemented with the R package MSM. Our simulation study showed that the Bayesian approach provided more accurate point and interval estimates than the maximum likelihood method, especially in complex continuous time Markov chain models with five states. When applied to data from a four-state transtheoretical model collected from a nutrition intervention study in the next step trial, we observed results consistent with the results of the simulation study. Specifically, the two approaches provided comparable point estimates and standard errors for most parameters, but the maximum likelihood offered substantially smaller standard errors for some parameters. Comparable estimates of the standard errors are obtainable from package MSM, which works only when the model estimation algorithm converges.

Study of Disease Progression and Relevant Risk Factors in Diabetic Foot Patients Using a Multistate Continuous-Time Markov Chain Model

The diabetic foot is a lifelong disease. The longer patients with diabetes and foot ulcers are observed, the higher the likelihood that they will develop comorbidities that adversely influence ulcer recurrence, amputation and survival (for example peripheral arterial disease, renal failure or ischaemic heart disease). The purpose of our study was to quantify person and limb-related disease progression and the time-dependent influence of any associated factors (present at baseline or appearing during observation) based on which effective prevention and/or treatment strategies could be developed. Using a nine-state continuous-time Markov chain model with time-dependent risk factors, all living patients were divided into eight groups based on their ulceration (previous or current) and previous amputation (none, minor or major) status. State nine is an absorbing state (death). If all transitions are fully observable, this model can be decomposed into eight submodels, which can be analyzed using the methods of survival analysis for competing risks. The dependencies of the risk factors (covariates) were included in the submodels using Cox-like regression. The transition intensities and relative risks for covariates were calculated from long-term data of patients with diabetic foot ulcers collected in a single specialized center in North-Rhine Westphalia (Germany). The detected estimates were in line with previously published, but scarce, data. Together with the interesting new results obtained, this indicates that the proposed model may be useful for studying disease progression in larger samples of patients with diabetic foot ulcers.

Dealing with Time in Health Economic Evaluation: Methodological Issues and Recommendations for Practice

Time is an important aspect of health economic evaluation, as the timing and duration of clinical events, healthcare interventions and their consequences all affect estimated costs and effects. These issues should be reflected in the design of health economic models. This article considers three important aspects of time in modelling: (1) which cohorts to simulate and how far into the future to extend the analysis; (2) the simulation of time, including the difference between discrete-time and continuous-time models, cycle lengths, and converting rates and probabilities; and (3) discounting future costs and effects to their present values. We provide a methodological overview of these issues and make recommendations to help inform both the conduct of cost-effectiveness analyses and the interpretation of their results. For choosing which cohorts to simulate and how many, we suggest analysts carefully assess potential reasons for variation in cost effectiveness between cohorts and the feasibility of subgroup-specific recommendations. For the simulation of time, we recommend using short cycles or continuous-time models to avoid biases and the need for half-cycle corrections, and provide advice on the correct conversion of transition probabilities in state transition models. Finally, for discounting, analysts should not only follow current guidance and report how discounting was conducted, especially in the case of differential discounting, but also seek to develop an understanding of its rationale. Our overall recommendations are that analysts explicitly state and justify their modelling choices regarding time and consider how alternative choices may impact on results.

Spatiotemporal progression of metastatic breast cancer: a Markov chain model highlighting the role of early metastatic sites

Background:

Cancer cell migration patterns are critical for understanding metastases and clinical evolution. Breast cancer spreads from one organ system to another via hematogenous and lymphatic routes. Although patterns of spread may superficially seem random and unpredictable, we explored the possibility that this is not the case.

Aims:

Develop a Markov based model of breast cancer progression that has predictive capability.

Methods:

On the basis of a longitudinal data set of 446 breast cancer patients, we created a Markov chain model of metastasis that describes the probabilities of metastasis occurring at a given anatomic site together with the probability of spread to additional sites. Progression is modeled as a random walk on a directed graph, where nodes represent anatomical sites where tumors can develop.

Results:

We quantify how survival depends on the location of the first metastatic site for different patient subcategories. In addition, we classify metastatic sites as “sponges” or “spreaders” with implications regarding anatomical pathway prediction and long-term survival. As metastatic tumors to the bone (main spreader) are most prominent, we focus in more detail on differences between groups of patients who form subsequent metastases to the lung as compared with the liver.

Conclusions:

We have found that spatiotemporal patterns of metastatic spread in breast cancer are neither random nor unpredictable. Furthermore, the novel concept of classifying organ sites as sponges or spreaders may motivate experiments seeking a biological basis for these phenomena and allow us to quantify the potential consequences of therapeutic targeting of sites in the oligometastatic setting and shed light on organotropic aspects of the disease.

Cost-effectiveness analysis of axitinib through a probabilistic decision model

INTRODUCTION

The Oncology field is characterised by a steady increase in demand and a consistent launching of innovative and expensive products. Therefore, cost-effectiveness analysis can contribute as a significant decision-making tool by elucidating the most economically efficient ways to satisfy compelling health needs.

AREAS COVERED

The scope of this study is to estimate the cost-effectiveness of axitinib versus sorafenib, for the second-line treatment of renal cell carcinoma. A literature review for evidence synthesis was performed and a probabilistic Markov Model was employed to simulate disease progression. This study will also assess Value of Information.

EXPERT OPINION

Compared to sorafenib, axitinib resulted in an incremental cost of 87,936 euro per quality adjusted life year. The probability of axitinib to being cost-effective at the willingness-to-pay threshold of 60,000 euro was 13%, while the corresponding probability of being cost-effective at the highest recommended willingness-to-pay threshold of 100,000 euro was 69.9%. Uncertainty was primarily attributed to the price of the product, the utility values, the progression-free survival and to a lesser degree to the overall survival. Axitinib can be considered as a cost-effective therapeutic option for second-line treatment of renal cell carcinoma.