34.5 Half-Cycle correction

Real processes occur in continuous time, with transitions and other events occurring throughout an interval of time. However, a Markov process occurs as a discrete sequence of snapshots, which can lead to over counting Markov rewards.

Traditional Markov models accumulate the full cycle's state reward at the beginning of each cycles with transitions understood to occur at the end of each cycle, even though some portion of the cohort will leave the state during the cycle. Expected values will therefore overestimate life expectancy by about half of a cycle (0.5 years in a one-year cycle length model).

Consider the simple example of life expectancy in a simple model with health states Alive and Dead. In each cycle, some portion of the cohort is alive and accumulates 1 life year. In reality, however, deaths will occur halfway through a cycle on average. So, someone that dies during a cycle should lose half of the reward they received at the beginning of the cycle (e.g., -0.5 years of life expectancy in a one-year cycle length model).

However, instead of implementing the half-cycle correction as a toll at each transition to death, it is easier to implement it in an absorbing process simply by subtracting a half-reward from the rewards assigned at the beginning of the process, in cycle 0 — i.e., by setting a state’s initial reward to one-half of its incremental reward. This is the primary, though not only, reason that the state rewards are separated into three parts (Init, Incr and Final).

In a non-absorbing process, in which a significant percentage of the cohort may be alive when the process terminates, cohort members still alive at the end of the process should be given back the half-cycle “death” correction taken from their initial reward at the beginning of the process. This is done by adding on a half-reward after termination in the final reward for all alive states (it does not hurt to always include the initial and final components at every state).

The alternative to HCC is Within-Cycle Correction (WCC). WCC Markov models account for transitions within a cycle by accumulating rewards based on the percentage of the cohort in a health state both at the beginning and the end of each cycle. You can find more details about WCC within Markov Cohort Analysis Output.

To perform half-cycle correction:

  • Select a Markov state node (in the image below this is Local Cancer).

  • Open the Markov View.

  • Enter the initial and incremental rewards. In the images, we have cLocal for Init and Incr Costs.

  • In the Markov View, select/highligh a reward for the appropriate payoff set. In this case we will choose the Cost.

  • Click on the "pencil" icon in the view's toolbar to open the State Reward Dialog.

  • Click the Half-Cycle Correct button in the State Reward Dialog.

The initial and final rewards will be updated as below.

Any reward that is a function of life expectancy (i.e., medication costs that occur gradually over a cycle) is usually corrected in the same way.

In models that calculate quantities other than simple life expectancy, for example quality-adjusted life expectancy, different alive states will have different rewards. This means that a perfect half-cycle correction might require correcting not just for death transitions, but for other kinds of transitions from higher value states to lower value states (i.e., where someone should receive half a cycle of the starting state’s reward and half of the ending state’s reward). Note, however, that Markov approximation errors in two strategies will often cancel each other out in incremental calculations, and reasonable judgment should be used to decide when to use half-cycle correction.