THEORETICAL FOUNDATION AND PRACTICAL APPLICATIONS OF HALF-CYCLE CORRECTION METHODS
Purpose: The ISPOR-SMDM Modeling Good Research Practices Task Force recommends the use of half-cycle correction (HCC) to model outcomes such as costs and effectiveness calculated with discrete-time state-transition models (DTSTM). However, there is still no consensus in the modeling community on why and how to perform the correction. In addition, published studies did not use the true gold standard.
Objective: To provide a theoretical foundation of HCC and compare the performance of different HCC methods in reducing errors in DTSTM outcomes both mathematically and numerically.
Method: We defined six half-cycle correction methods from numerical integration field: Riemann sum of rectangles (left, midpoint, right) or trapezoids (trapezoidal rule), life-table, Simpson's composite 1/3rd and 3/8th rules. We applied these methods to a standard three-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 cumulative incidence of disease, life expectancy, discounted quality-adjusted life years, discounted disease costs, incremental cost-effectiveness ratio.
Results: The basis for the currently recommended HCC method of correcting by half of the reward in the first cycle and in the final cycle is a trapezoidal rule. Because the standard HCC method was based on a comparison with the trapezoidal rule rather than a comparison with the gold standard method, there will always be an error. We also found situations where applying the standard HCC can do more harm than good. The performance of each method depends on the function that needs to be integrated. Contrary to conventional wisdom, the approximation errors need not cancel each other out or become insignificant when incremental outcomes are calculated. We found that a wrong decision can be made if the more accurate method is not applied (Fig. 1A). The size of the error was vastly reduced when a shorter cycle length was selected; Simpson's 1/3rd rule was fastest method to converge to the gold standard (Fig. 1B).
Conclusion: 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.