title: Theoretical Foundations and Practical Applications of Within-Cycle Correction Methods.
abstract: Modeling guidelines recommend applying a half-cycle correction (HCC) to outcomes from discrete-time state-transition models (DTSTMs). However, there is still no consensus on why and how to perform the correction. The objective was to provide theoretical foundations for HCC and to compare (both mathematically and numerically) the performance of different correction methods in reducing errors in outcomes from DTSTMs.
  
  
  
  We defined 7 methods from the field of numerical integration: Riemann sum of rectangles (left, midpoint, right), trapezoids, life-table, and Simpson's 1/3rd and 3/8th rules. We applied these methods to a standard 3-state disease progression Markov chain to evaluate the cost-effectiveness of a hypothetical intervention. We solved the discrete- and continuous-time (our gold standard) versions of the model analytically and derived expressions for various outcomes including discounted quality-adjusted life-years, discounted costs, and incremental cost-effectiveness ratios.
  
  
  
  The standard HCC method gave the same results as the trapezoidal rule and life-table method. We found situations where applying the standard HCC can do more harm than good. Compared with the gold standard, all correction methods resulted in approximation errors. Contrary to conventional wisdom, the errors need not cancel each other out or become insignificant when incremental outcomes are calculated. We found that a wrong decision can be made with a less accurate method. The performance of each correction method vastly improved when a shorter cycle length was selected; Simpson's 1/3rd rule was the fastest method to converge to the gold standard.
  
  
  
  Cumulative outcomes in DTSTMs are prone to errors that can be reduced with more accurate methods like Simpson's rules. We clarified several misconceptions and provided recommendations and algorithms for practical implementation of these methods.
authors.affiliation: Merck & Co., Inc., Kenilworth, NJ (EHE)
authors.forename: Elamin H
authors.initials: EH
authors.lastname: Elbasha
authors.affiliation: Department of Health Services Research, The University of Texas MD Anderson Cancer Center, Houston TX (JC).
authors.forename: Jagpreet
authors.initials: J
authors.lastname: Chhatwal
mesh_terms: D000465:Algorithms; D003362:Cost-Benefit Analysis / Q000379:methods*; D003657:Decision Making; D006801:Humans; D008390:Markov Chains; D018803:Models, Economic; D008962:Models, Theoretical; D017063:Outcome Assessment, Health Care / Q000379:methods*
keywords: continuous time; discrete time; half-cycle correction; numerical integration; state-transition models