Concept Architecture
Concept
Theoretically, Time Dependency is the property of a state-transition model in which transition probabilities, event rates or other model parameters vary as a function of time. Time dependency allows the model to represent changing disease risks, treatment effects, ageing, secular trends or duration-related processes that cannot be captured by constant transition probabilities. In health economics, time dependency is used to improve the realism of long-term models when clinical risks or intervention effects evolve over the model horizon.
Mathematically, time dependency is represented by allowing transition probabilities or hazard rates to vary with time. The transition probability matrix therefore becomes a function of time rather than remaining constant across model cycles. Transition probabilities are commonly derived from time-dependent hazard functions or survival models fitted to longitudinal data.
In practice, time dependency is implemented by estimating time-varying transition probabilities from clinical trials, registries or survival analyses. Parametric survival models, time-dependent regression models or age-specific risk equations are frequently used. During simulation, the transition matrix is updated at each cycle to reflect the appropriate risks, allowing more accurate estimation of long-term costs, life years and quality-adjusted life years.
Purpose
Used to represent changing transition risks over time, enabling health economic models to reflect disease progression, ageing, waning treatment effects and other time-varying clinical processes.
Mathematical Formulae
Primary Formula
Time-dependent transition probability:
p(t) = 1 ? e???????
where:
- p(t) = transition probability during cycle beginning at time t
- h(t) = time-dependent hazard rate
- ?t = cycle length
Supporting Formulae
Time-dependent transition matrix:
??(t) = [p??(t)]
State update:
????? = ?????(t)
Related Mathematical Methods
- Time-dependent Markov modelling
- Survival analysis
- Parametric survival modelling
- Cox proportional hazards regression
- Matrix algebra
- State-transition modelling
- Probabilistic sensitivity analysis
Example
A Markov model evaluating breast cancer treatment assumes that the annual recurrence hazard decreases over time following treatment. A Weibull survival model estimates the hazard function, from which annual transition probabilities are calculated. The transition matrix is updated each year to reflect the declining recurrence risk, producing more realistic estimates of lifetime costs and quality-adjusted life years than a model using constant transition probabilities.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =1-EXP(-Hazard*CycleLength) | Convert time-dependent hazards into cycle-specific transition probabilities |
| INDEX | =INDEX(HazardTable,Cycle) | Retrieve cycle-specific hazard rates |
| MMULT | =MMULT(StateVector,TransitionMatrix) | Update state occupancies using the current cycle's transition matrix |
| SUMPRODUCT | =SUMPRODUCT(StateVector,CostVector) | Calculate expected costs and outcomes after each cycle |
VBA (Optional)
Automate updating of cycle-specific transition matrices using time-varying hazard functions throughout the simulation horizon.
Sources
- Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Medical Decision Making. 2012;32(5):690?700.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322?338.
- Latimer NR. Survival analysis for economic evaluations alongside clinical trials. Medical Decision Making. 2013.
- NICE. Health Technology Evaluation Manual.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press.
Related Concepts (3)
Library
Publications
1
State-Transition Modeling: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3 — Siebert, Alagoz, Bayoumi, Jahn, Owens, Cohen & Kuntz, Task Force Report 3 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance for cohort and individual-based state-transition (Markov) models, covering development, analysis, validation and reporting.
Journal ArticleView source →
Frequently Asked Questions (6)
What is time dependency in a Markov model?
A situation where the probability of a health state transition depends on elapsed time, such as time since diagnosis, rather than staying constant.
Source: Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322-338. doi:10.1177/0272989X9301300409.
What is an example of time dependency in a disease?
Time dependency arises when a transition risk changes with elapsed time rather than staying fixed. The risk of dying rises with age, the chance of recurrence is often highest soon after treatment and falls thereafter, and the hazard just after surgery differs from that months later. In each case the probability depends on how much time has passed, either since birth or since entering a state. A constant-probability model cannot capture such change without adaptation. Briggs and colleagues (2006) give these examples.
Source: Briggs et al. 2006
What are the types of time dependency?
There are two main types of time dependency. Dependency on time in the model, such as risks changing with the passage of time or patient age, affects all patients as cycles advance and can be handled by a time-inhomogeneous model with transition probabilities that vary by cycle. Dependency on time in a state, such as risk changing with time since entering a state, depends on individual history and violates the Markov assumption, requiring tunnel states or individual simulation to represent, since the state alone does not capture the elapsed time.
Source: Sonnenberg & Beck 1993
Why does time dependency challenge Markov models?
Time dependency challenges Markov models because the basic model assumes transition probabilities depend only on the current state and, in the time-homogeneous form, are constant, whereas time-dependent risks vary with elapsed time. Dependency on time since entering a state also violates the memoryless assumption, since the state does not record how long a patient has been there. Representing such dependency therefore requires extending the model, either by varying probabilities over cycles or by adding states to encode elapsed time.
Source: Sonnenberg & Beck 1993
How is time dependency handled in a Markov model?
Time dependency is handled in different ways depending on its type. Dependency on model time or age is handled by a time-inhomogeneous model, with transition probabilities that vary by cycle to reflect changing risk. Dependency on time since entering a state, which the memoryless model cannot capture, is handled by tunnel states, a sequence of temporary states representing successive periods in the state, or by individual simulation that tracks each patient's time in state. These approaches let the model represent risks that change with elapsed time.
Source: Sonnenberg & Beck 1993
Why is representing time dependency important?
Representing time dependency is important because many risks genuinely change with time, such as mortality rising with age or event risk being highest soon after an event, so assuming constant transition probabilities would misrepresent the disease and bias the results. A model that ignored real time dependency could substantially misestimate costs and outcomes. Capturing it, through time-inhomogeneous probabilities, tunnel states, or individual simulation, ensures the model reflects how risks actually vary over time, which is often necessary for a valid representation.
Source: Sonnenberg & Beck 1993
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 7 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/time-dependency
- Term code
- HE-EM-MM-022
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