Cycle Length

Cycle Length: Concept Architecture

Orientation and learning roadmap

Cycle length turns continuous clinical time into the repeated intervals used by a discrete-time model. This page explains how the interval should reflect event timing, data, transition probabilities, rewards, and discounting; it then shows how to convert time-dependent inputs and test whether the choice materially affects results. The central principle is that cycle length is a structural modelling decision, not a spreadsheet formatting choice.

One cycle is one model update

In a cohort state-transition model, the state vector is updated at discrete time points separated by the cycle length (\Delta t). A model with monthly cycles updates state membership twelve times per year, while a model with annual cycles updates it once. If (\mathbf{s}_t) is the row vector of state occupancy and (\mathbf{P}_t) is the transition matrix for cycle (t), the update is:

[ \mathbf{s}_{t+1}=\mathbf{s}_t\mathbf{P}_t. ]

Cycle length is different from the time horizon

Cycle length specifies the size of each computational interval, whereas the time horizon specifies the total period over which consequences are modelled. A lifetime model can use weekly, monthly, quarterly, or annual cycles, and shortening the cycle does not automatically lengthen the horizon. When the horizon (T) is divisible by the interval, the number of cycles is:

[ N=\frac{T}{\Delta t}. ]

Clinical timing should lead the choice

The interval should be short enough to represent events and changes that matter to costs, health, or treatment decisions. Acute complications, early treatment discontinuation, rapidly changing hazards, and short-lived adverse events may require shorter cycles than slowly evolving chronic disease. A cycle that is longer than an important process can misplace events, obscure sequencing, or prevent clinically possible pathways.

The model structure may impose timing constraints

Standard cohort state-transition models usually allow at most one net state update per cycle, even if several clinical events could occur during that interval. Shorter cycles, tunnel states, temporary states, microsimulation, or continuous-time methods may be needed when within-cycle order matters. The chosen solution should reflect the decision problem rather than mechanically adding complexity.

Data availability does not automatically determine the interval

Trial results or observational estimates may be reported at annual, quarterly, or irregular times that differ from the model cycle. Those data must be transformed with assumptions that preserve the underlying event process as closely as possible. Copying an annual probability into every monthly cycle would multiply risk incorrectly.

Rates and probabilities are not interchangeable

A constant event rate can be converted to a probability over an interval, but a probability is bounded and depends on the interval to which it refers. A valid conversion must preserve the assumed event-free survival over the selected duration. If a constant hazard rate (r) applies for duration (\Delta t), the corresponding probability is:

[ p(\Delta t)=1-\exp(-r\Delta t). ]

The reverse conversion is:

[ r=-\frac{\ln(1-p)}{\Delta t}. ]

Convert a probability by preserving its implied survival

When a probability (p_L) applies over a longer interval (L), a shorter-cycle probability can be derived under a constant-hazard assumption by preserving the cumulative probability of remaining event-free. This transformation is valid only when that assumption and the event definition are appropriate. For a cycle of length (S):

[ p_S=1-(1-p_L)^{S/L}. ]

Worked conversion from annual to monthly risk

Suppose a single-event annual probability is 0.20 and the model uses one-month cycles. Under a constant hazard, the annual probability corresponds to a rate of (-\ln(0.80)=0.223144) per year and a monthly probability of approximately 0.018423. Compounding that monthly probability over twelve cycles returns the annual value:

[ p_{\text{month}}=1-(1-0.20)^{1/12}=0.018423, ]

[ 1-(1-0.018423)^{12}\approx0.20. ]

Competing events require joint treatment

Converting each cause-specific probability independently can make the outgoing probabilities sum to more than one or distort the allocation among competing destinations. When several mutually exclusive events compete within a cycle, cause-specific hazards or an appropriate multi-state method should be converted jointly. Matrix exponentiation may be required to move coherently between continuous-time intensity matrices and discrete-time transition matrices.

Matrix conversion is not element-by-element conversion

For a continuous-time transition-intensity matrix (\mathbf{Q}), the transition-probability matrix over interval (\Delta t) is the matrix exponential. Applying the scalar probability formula separately to each off-diagonal element generally fails to preserve competing-risk structure. The coherent relationship is:

[ \mathbf{P}(\Delta t)=\exp(\mathbf{Q}\Delta t). ]

Time-varying risks must follow model time or state time

Event risks may change with age, calendar time, time since treatment, or time spent in a state. A shorter cycle allows more frequent updates but does not itself create the correct time dependence; the model must index and apply the appropriate function or table. Analysts should distinguish simulation-time effects from state-residence-time effects and verify boundary transitions between intervals.

Rewards must match the interval and timing convention

Costs and health outcomes may accrue continuously while a person occupies a state, once on entering or leaving a state, or at a specific clinical event. State rewards commonly need multiplication by the fraction of a year represented by the cycle, while transition rewards are usually attached to the event itself rather than prorated automatically. Mixing annual utility values, monthly costs, and per-event charges without explicit conversion creates unit errors.

Half-cycle correction addresses within-cycle timing

A cohort model often records transitions at cycle boundaries even though events occur throughout the interval. Half-cycle correction approximates more continuous accrual by shifting state-based rewards toward the middle of each cycle, but it is not a universal repair for poorly chosen cycles or complex event timing. The correction method must be compatible with whether rewards are state based, transition based, or already integrated continuously.

Shorter cycles reduce some discretization error

Reducing (\Delta t) can better approximate event timing and lower bias caused by assigning events to widely spaced boundaries. The improvement may be limited when the transition process is already represented exactly for the interval or when uncertainty in evidence dominates numerical error. Shorter cycles also increase computations, opportunities for implementation errors, and the size of individual-level simulations.

Cycle-length convergence tests numerical stability

A cycle-length sensitivity analysis repeats the model with progressively shorter intervals while preserving equivalent clinical assumptions and time scales. Results should approach a stable limit if discretization error is being reduced coherently. Stability in total costs and QALYs should be assessed alongside incremental outcomes and the decision conclusion.

Discounting must use the correct fraction of a year

When the annual effective discount rate is (d), a value occurring at time (t) years is multiplied by ((1+d)^{-t}). In a model with (m) equal cycles per year, cycle (k) occurs at a time determined by the chosen beginning-, middle-, or end-of-cycle convention. An end-of-cycle discount factor is:

[ DF_k=(1+d)^{-k/m}. ]

Partial final cycles need an explicit rule

The horizon may not contain an exact whole number of cycles, especially when a stopping age or evidence cutoff is used. Rounding the number of cycles can silently shorten or extend the model and miscount rewards. The model should use a partial cycle, align the horizon exactly, or document a justified approximation.

Individual-level simulation changes the implementation, not the principle

Microsimulation may update individuals at fixed cycles, schedule events in continuous time, or combine both approaches. Fixed-step microsimulation still needs an interval fine enough to represent hazards, event order, and changing covariates. Continuous-time simulation avoids some cycle-boundary approximations but introduces its own event-time and integration requirements.

Calendar alignment can matter in budget models

Budget impact models may use monthly, quarterly, or annual cycles to match purchasing, reimbursement, uptake, and reporting periods. Administrative convenience should not override clinical timing when the mismatch could change resource use or outcomes. Where budget and clinical intervals differ, calculations should retain both time scales explicitly.

Excel implementation should make units visible

A transparent worksheet labels the model cycle in years, the number of cycles per year, and the source interval for every input. If annual probability is in B2 and cycles per year in B3, the constant-hazard cycle probability is =1-(1-B2)^(1/B3); an annual rate in B4 converts with =1-EXP(-B4/B3). Reward and discount formulas should reference the same cycle-length control rather than embedding unexplained divisors such as 12 or 52.

Validation should reconcile time, probability, and rewards

Cycle-length validation is strongest when it checks identities and outcomes rather than merely inspecting formulas. The tests should cover the state trace, transition matrix, cumulative incidence, undiscounted totals, discounted totals, and incremental results. A practical sequence is:

1. Define the interval. Record its duration, unit, clinical rationale, and relationship to the time horizon.
2. Map every input. Identify whether each value is a rate, interval probability, annual reward, per-cycle reward, or event reward.
3. Convert coherently. Use survival-based, competing-risk, or matrix methods suited to the underlying process.
4. Check transition rows. Confirm all probabilities are bounded and each row sums to one within tolerance.
5. Reconcile cumulative risk. Verify that repeated cycle probabilities reproduce the intended longer-interval risk under the stated assumptions.
6. Test reward timing. Separate state occupancy, transition events, half-cycle correction, and discount timing.
7. Shorten the cycle. Compare total and incremental outputs against a finer-cycle benchmark.
8. Document the impact. State whether cycle length can change the preferred strategy or interpretation.

Common modelling errors

Most cycle-length errors arise when values with different time bases are combined without conversion or when boundary timing remains implicit. These mistakes can produce plausible tables while materially biasing results. Common failures include:

• Reusing an annual transition probability in every monthly cycle.
• Dividing a probability by twelve instead of preserving implied survival.
• Converting competing transitions independently and producing incoherent rows.
• Applying a scalar formula element by element to a transition-intensity matrix.
• Prorating one-time transition costs as though they were continuous state costs.
• Applying half-cycle correction to rewards already timed explicitly.
• Discounting every monthly cycle as though a full year had elapsed.
• Assuming shorter cycles automatically fix structural or evidence problems.
• Changing the cycle length without changing every dependent probability, reward, and timing formula.

What a complete model report should disclose

A reproducible report states the cycle duration, its clinical rationale, the horizon, event-timing convention, transition conversion method, reward timing, discount timing, and any half-cycle correction. It should also report the shorter-cycle sensitivity analysis and explain any meaningful change in total or incremental results. These details allow reviewers to distinguish a justified time discretization from an arbitrary modelling default.