Half-Cycle Correction

Half-Cycle Correction: Concept Architecture

Orientation and learning roadmap

The page starts with the timing error that can arise when a state-transition model counts everyone at a cycle boundary. It then develops endpoint averaging, checks a one-year example, and shows how to treat recurring state rewards separately from one-off events. The closing sections explain when the approximation may be inadequate and how to audit its implementation.

Locate the timing error in a cycle

A discrete-time state-transition model usually records the number or proportion of people in each state at specified cycle boundaries. Health and resource use, however, can accrue continuously while people enter or leave a state during the intervening interval. Crediting the whole interval only to the starting or ending state can overcount or undercount time spent there.

Half-cycle correction commonly approximates the average state occupancy over each cycle from its two endpoint values. This is a within-cycle integration rule, often implemented as the trapezoidal rule for time-dependent state rewards. It does not assert that every individual actually changes state at precisely the midpoint.

Average endpoint occupancy and rewards

For a cycle of length $\Delta$ years with occupancy $n_t$ and $n_{t+1}$ at its ends, approximate the person-years in a state by $\Delta(n_t+n_{t+1})/2$. If the same per-person annual utility $u$ applies throughout that state and interval, multiply this person-time by $u$ to obtain cohort QALYs. For changing utilities, costs, or discount weights, the general trapezoidal calculation averages the endpoint reward rates rather than occupancy alone.

$$ \widehat R_{0:T}=\sum_{t=0}^{T-1} \frac{\Delta_t}{2}\bigl[g(t_t)+g(t_{t+1})\bigr], $$

where $g(t)$ is the appropriate instantaneous or annualised reward rate, including state occupancy, per-person reward, and discount factor if applicable. With constant one-year cycles and undiscounted rewards, this gives half weight to the initial and final endpoint and full weight to each interior endpoint. Apply it to each strategy and relevant state consistently; a finite time horizon still needs its final endpoint handled correctly.

Check a one-year survival example

Suppose 100 people are alive at the start of a one-year cycle and 80 are alive at the end, with utility 0.8 while alive and zero after death. Start-of-cycle counting produces $100(0.8)=80$ cohort QALYs and end-of-cycle counting produces $80(0.8)=64$. Endpoint averaging gives $(100+80)/2=72$ cohort QALYs, or 0.72 QALYs per initial person.

The corrected estimate corresponds to a linear interpolation in cohort occupancy across the year. If all deaths occurred near the first or last day, their actual accumulated person-time would differ substantially despite the same endpoints. Shorter cycles, a justified within-cycle timing model, or continuous-time methods can be preferable when rapid changes matter.

Separate state rewards from transition events

Many state costs and utilities accrue with time spent in a state, making an occupancy-based correction relevant. A one-off cost triggered by a transition, such as a procedure at treatment initiation or an acute admission, is not simply an annual state reward. Its expected number of events should be counted under the transition model, and its timing and any discounting should be handled deliberately.

In the example, if each of 20 deaths carries an illustrative one-off terminal cost of £500, the undiscounted event cost is $20(500)=£10{,}000$. Halving that event count because a half-cycle correction was used for time alive would be wrong. Costs tied to time in the alive state and costs tied to death can both appear in the same model, but they have different accumulation rules.

Decide whether the approximation is adequate

The trapezoidal rule reduces a common bias but is not exact for every process, reward function, or cycle length. Nonlinear occupancy trajectories, early treatment effects, sharp hazards, and discontinuous payments can produce material within-cycle errors. The model should justify its cycle duration and compare results with shorter cycles or a more suitable integration method when the decision is sensitive to timing.

Methodological work has examined alternatives including life-table approaches and higher-order numerical integration. A more elaborate correction is not automatically better when its timing assumptions are unsupported. Whether the incremental cost-effectiveness result changes depends on differences between strategies as well as the absolute correction in each arm.

Audit the spreadsheet or model code

A transparent implementation should show uncorrected occupancy, corrected person-time, state rewards, transition-event counts, and discounting for each cycle. In a simple cohort trace, corrected state person-time should reconcile with the endpoint-average formula before utilities or costs are attached. Check the first and last cycle, where weighting errors are common.

• Verify that the occupancy trace and transition counts are internally consistent for each strategy.
• Reproduce the one-cycle average manually and compare it with spreadsheet or code output.
• Check that recurring state rewards receive one timing correction and one-off event costs are handled on their own basis.
• Match discount factors to the timing convention and avoid an unexplained additional half-cycle shift.
• Compare cycle lengths and plausible event-timing assumptions when they can affect the preferred option.

Sources and technical basis

ISPOR-SMDM state-transition guidance addresses within-cycle correction for costs and effectiveness. Research on numerical integration and timing in economic models explains the trapezoidal interpretation and limitations of a conventional half-cycle rule. The 100-person example is original teaching arithmetic with explicit occupancy and reward assumptions.

• ISPOR-SMDM, State-Transition Modeling, good research practices task force.
• Elbasha and Chhatwal, Theoretical Foundations and Practical Applications of Within-Cycle Correction Methods.
• Naimark and colleagues, The Half-Cycle Correction Revisited.
• Dealing with Time in Health Economic Evaluation: Methodological Issues and Recommendations for Practice.