Abstract
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.
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References
Chhatwal J, Jayasuriya S, Elbasha EH. Changing cycle lengths in state-transition models: challenges and solutions. Med Decis Mak. 2016;36(8):952–64.
Jones E, Epstein D, Garcia-Mochon L. A procedure for deriving formulas to convert transition rates to probabilities for multistate Markov models. Med Decis Mak. 2017;37(7):779–89.
Welton NJ, Ades AE. Estimation of Markov chain transition probabilities and rates from fully and partially observed data: uncertainty propagation, evidence synthesis, and model calibration. Med Decis Mak. 2005;25(6):633–45.
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. Pharmacoeconomics. 2020;38(11):1153–64.
Naversnik K, Rojnik K. Handling input correlations in pharmacoeconomic models. Value Health. 2012;15(3):540–9.
Canadian Agency for Drugs and Technologies in Health. Guidelines for the economic evaluation of health technologies: Canada. 4th ed. Ottawa (ON): The Agency; 2017. p. 76.
Briggs AH, Weinstein MC, Fenwick EA, et al. Model parameter estimation and uncertainty: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6. Value Health. 2012;15(6):835–42.
Edlin R, McCabe C, Hulme C, et al. Cost effectiveness modelling for health technology assessment: a practical course. New York: Springer; 2015.
Briggs A, Sculpher M, Claxton K. Decision modelling for health economic evaluation. Oxford: Oxford University Press; 2006.
Kim HY. Statistical notes for clinical researchers: covariance and correlation. Restor Dent Endod. 2018;43(1): e4.
Wicklin R. Simulating data with SAS. Cary: SAS Institute Inc; 2013.
Venables WN, Ripley BD. Modern applied statistics with S. 4th ed. New York: Springer; 2002.
Genz A, Bretz F. Computation of multivariate normal and t probabilities. Berlin: Springer; 2009.
Gut A. An intermediate course in probability. 2nd ed. New York: Springer; 2009.
TreeAge Pro 2020, R2. TreeAge Software, Williamstown. http://www.treeage.com.
Simoneau G, Levis B, Cuijpers P, et al. A comparison of bivariate, multivariate random-effects, and Poisson correlated gamma-frailty models to meta-analyze individual patient data of ordinal scale diagnostic tests. Biom J. 2017;59(6):1317–38.
Tanner JA, Hensel J, Davies PE, et al. Economic burden of depression and associated resource use in Manitoba, Canada. Can J Psychiatry. 2020;65(5):338–46.
Hanck C, Arnold M, Gerber A, Schmelzer M. Introduction to econometrics with R. Essen: University of Duisburg-Essen; 2021.
Peasgood T, Brennan A, Mansell P, et al. The impact of diabetes-related complications on preference-based measures of health-related quality of life in adults with type I diabetes. Med Decis Mak. 2016;36(8):1020–33.
Ren S, Minton J, Whyte S, et al. A new approach for sampling ordered parameters in probabilistic sensitivity analysis. Pharmacoeconomics. 2018;36(3):341–7.
Kirkwood BR, Sterne JAC. Essential medical statistics. Malden: Blackwell Science; 2003.
Allison P. Survival analysis using SAS: a practical guide. Cary: SAS Institute; 1995.
SAS Institute Inc. SAS/STAT® 14.1 User’s Guide: The PHREG Procedure. Cary: SAS Institute Inc; 2015.
Albers GW, Marks MP, Kemp S, et al. Thrombectomy for stroke at 6 to 16 hours with selection by perfusion imaging. N Engl J Med. 2018;378(8):708–18.
Luengo-Fernandez R, Paul NL, Gray AM, et al. Population-based study of disability and institutionalization after transient ischemic attack and stroke: 10-year results of the Oxford Vascular Study. Stroke. 2013;44(10):2854–61.
Hill-McManus D. The difference method approach for sampling order constrained parameters: an improved implementation and important limitations. Pharmacoeconomics. 2024;42(1):11–8.
Xie X, Tiggelaar S, Guo J, et al. Developing economic models for assessing the cost-effectiveness of multiple diagnostic tests: methods and applications. Med Decis Mak. 2022;42(7):861–71.
Xie X, Wang M, Gajic-Veljanoski O, et al. Examining the correlation between treatment effects in clinical trials and economic modeling. Expert Rev Pharmacoecon Outcomes Res. 2022;22(7):1071–8.
Xie X, Schaink AK, Gao C, et al. Evaluating the correlations of cost and utility parameters from summary statistics for probabilistic analysis in economic evaluations. Expert Rev Pharmacoecon Outcomes Res. 2023;23(8):901–9.
Acknowledgements
The authors thank Jeanne McKane (Medical Editor) from Ontario Health, Quality Division, Toronto, ON, Canada, for her help in editing the manuscript, and health economist Hailey Saunders from Ontario Health’s Health Technology Assessment Program for reviewing the manuscript and the R code. The authors also thank the anonymous reviewers of this manuscript for their constructive comments and suggestions.
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No sources of funding were used to assist in the preparation of this work.
Conflict of interest
Xuanqian Xie, Alexis K. Schaink, Sichen Liu, Myra Wang, Juan David Rios, and Andrei Volodin declare that they have no conflicts of interest.
Data availability
Data of all seven examples are publicly available. Data citations are included in the reference list.
Code availability
The SAS and R code for simulating and analyzing the data is presented in the Appendices.
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Each author below made significant contributions to this manuscript. Xuanqian Xie conceived the study idea. Xuanqian Xie and Alexis K. Schaink designed the study and drafted the manuscript. Xuanqian Xie and Juan David Rios simulated the data and conducted the analyses. Sichen Liu, Myra Wang, and Andrei Volodin provided important intellectual content and revised the draft manuscript.
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Xie, X., Schaink, A.K., Liu, S. et al. Simplified Methods for Modelling Dependent Parameters in Health Economic Evaluations: A Tutorial. Appl Health Econ Health Policy 22, 331–341 (2024). https://doi.org/10.1007/s40258-024-00874-4
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DOI: https://doi.org/10.1007/s40258-024-00874-4