Concept Architecture
Concept
Theoretically, Time-Inhomogeneous Model is a state-transition model in which transition probabilities vary over time. Unlike a time-homogeneous model, the probability of moving between health states depends not only on the current state but also on the point in time within the model, allowing risks to change because of ageing, disease progression, treatment duration or secular trends. In health economics, time-inhomogeneous models are used when clinical evidence indicates that transition risks are not constant over the model horizon.
Mathematically, a time-inhomogeneous model is represented by a sequence of time-dependent transition probability matrices rather than a single fixed matrix. Each model cycle uses the transition matrix corresponding to that point in time. Consequently, state distributions are obtained by multiplying successive time-varying transition matrices rather than repeated powers of a single matrix.
In practice, time-varying transition probabilities are estimated using longitudinal clinical data, survival analyses or regression models incorporating time as a covariate. The transition matrix is updated during each model cycle to reflect changing risks. Time-inhomogeneous models are commonly used to represent ageing populations, waning treatment effects, changing mortality risks or disease processes in which transition probabilities evolve over time.
Purpose
Used to model disease progression when transition probabilities change over time, providing more realistic estimates of long-term clinical and economic outcomes than models assuming constant transition risks.
Mathematical Formulae
Primary Formula
State update:
????? = ?????(t)
where:
- ??? = state distribution at cycle t
- ??(t) = transition probability matrix for cycle t
Supporting Formulae
State distribution after n cycles:
??? = ?????(0)??(1)???(n?1)
Time-dependent transition probability:
p??(t) = 1 ? e?????????
where:
- h??(t) = time-dependent hazard rate from state i to state j
- ?t = cycle length
Related Mathematical Methods
- Time-dependent Markov modelling
- Markov chains
- Matrix algebra
- Survival analysis
- Parametric survival modelling
- Cox proportional hazards regression
- Probabilistic sensitivity analysis
Example
A Markov model evaluating adjuvant cancer therapy assumes that recurrence risk is highest during the first three years after treatment and declines thereafter. Annual hazard rates are estimated using a Weibull survival model, and a different transition probability matrix is calculated for each annual cycle. The resulting time-inhomogeneous model produces more accurate estimates of lifetime costs and quality-adjusted life years than a model using constant transition probabilities.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| INDEX | =INDEX(TransitionTable,Cycle,Column) | Retrieve cycle-specific transition probabilities |
| EXP | =1-EXP(-Hazard*CycleLength) | Convert time-dependent hazards into transition probabilities |
| MMULT | =MMULT(StateVector,TransitionMatrix) | Update cohort distribution using the current cycle's matrix |
| SUMPRODUCT | =SUMPRODUCT(StateVector,CostVector) | Calculate expected cycle costs and outcomes |
VBA (Optional)
Automate updating of transition probability matrices for each cycle using time-varying hazard functions and recalculate model outcomes across 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 (7)
What is a time inhomogeneous model?
A Markov model in which transition probabilities are allowed to vary over the time horizon, such as mortality risk rising with age.
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 a time-inhomogeneous model?
A time-inhomogeneous model is a Markov model in which the transition probabilities are allowed to vary over the time horizon, rather than remaining constant, for example with mortality risk rising as the cohort ages. The probabilities governing movement between states differ by cycle, so a patient's transition risks depend on how far through the model they are. This allows the model to represent risks that change with time or age, making it more realistic than a time-homogeneous model where such changes matter.
Source: Sonnenberg & Beck 1993
What time-dependent formulation is a time-inhomogeneous model?
A time-inhomogeneous model is one in which transition probabilities change as time passes, rather than staying fixed. This allows the model to reflect risks that depend on age, on the duration of a disease, or on time since an event. Capturing such variation usually means specifying transition rates that differ by cycle, which places heavier demands on the data available. The added realism can matter greatly when mortality or relapse risk shifts markedly over the horizon. It stands in contrast to the constant-rate time-homogeneous case.
Source: Briggs, Claxton and Sculpher 2006
Why are time-inhomogeneous models used?
Time-inhomogeneous models are used because many transition risks genuinely change over time, most commonly mortality rising with age, so treating probabilities as constant would misrepresent the disease and bias results, especially over long horizons. By allowing transition probabilities to vary by cycle, the model captures these changes, giving a more accurate representation. Age-related mortality is a frequent reason for using a time-inhomogeneous model, since over a lifetime horizon the assumption of constant mortality would be clearly wrong.
Source: Sonnenberg & Beck 1993
How does a time-inhomogeneous model represent changing risks?
A time-inhomogeneous model represents changing risks by specifying transition probabilities that differ by cycle, so that as the model advances, the probabilities are updated to reflect the changing risk, for instance using age-specific mortality rates that rise each cycle. The model applies the appropriate probabilities for each cycle rather than a single fixed set. This lets risks vary with time or age across the horizon, capturing patterns such as increasing mortality that a constant-probability model could not represent.
Source: Sonnenberg & Beck 1993
What are the data demands of a time-inhomogeneous model?
A time-inhomogeneous model requires data on how transition probabilities vary over time, such as age-specific rates, rather than a single value per transition, so it is more data-demanding than a time-homogeneous model. The variation must be specified across the cycles, drawing on evidence such as life tables for age-related mortality or studies showing how risks change with time. Obtaining and incorporating these time-varying probabilities adds effort, which is the cost of the greater realism the model provides.
Source: Sonnenberg & Beck 1993
When should a time-inhomogeneous model be used?
A time-inhomogeneous model should be used when transition risks change materially over the time horizon, so that assuming constant probabilities would misrepresent the disease, as with mortality rising with age over a long horizon. Where such time dependency in model time or age is important, allowing probabilities to vary is necessary for accuracy. If risks are stable over the horizon, a simpler time-homogeneous model suffices, so the inhomogeneous form is chosen when the changing of risks over time affects the results.
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
- Term code
- HE-EM-MM-024
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