Concept Architecture
Concept
Theoretically, the State Space Model is a mathematical framework for representing dynamic systems in terms of unobserved state variables that evolve over time and generate observable outcomes. It is founded on systems theory, stochastic processes and linear algebra, allowing complex dynamic systems to be modelled through separate state and observation equations. In health economics, state space models are used to represent disease progression, healthcare utilisation, epidemiological processes and other time-dependent phenomena where the underlying system cannot be observed directly.
Mathematically, a state space model consists of two coupled equations: a state equation describing the evolution of the latent state variables and an observation equation linking the latent states to observed data. The framework accommodates deterministic or stochastic systems and is commonly estimated using recursive filtering methods such as the Kalman filter for linear Gaussian models or particle filtering for nonlinear or non-Gaussian systems.
In practice, state space models are implemented by specifying the state and observation equations, estimating unknown parameters from longitudinal data and recursively updating state estimates as new observations become available. In health economics they are applied to disease progression modelling, epidemiological forecasting, healthcare demand prediction, survival modelling, health technology assessment and parameter estimation in dynamic decision models.
Purpose
Used to model and estimate dynamic healthcare systems by representing unobserved disease or system states and updating estimates over time using observed data.
Mathematical Formulae
Primary Formula
State equation:
x? = Ax??? + Bu? + w?
Observation equation:
y? = Cx? + Du? + v?
where:
- x? = state vector
- y? = observation vector
- u? = control input
- A = state transition matrix
- B = control matrix
- C = observation matrix
- D = direct transmission matrix
- w? = process noise
- v? = observation noise
Supporting Formulae
Process noise:
w? ? N(0, Q)
Observation noise:
v? ? N(0, R)
Related Mathematical Methods
- Kalman filtering
- Extended Kalman filtering
- Particle filtering
- Hidden Markov models
- Bayesian state estimation
- Maximum likelihood estimation
- Numerical optimisation
Example
A health economist models the progression of chronic kidney disease using an unobserved disease severity score.
The latent disease state evolves according to
x? = 0.94x??? + w?,
while observed estimated glomerular filtration rate (eGFR) measurements are represented by
y? = x? + v?.
As new laboratory measurements become available, the Kalman filter updates the estimated disease state, enabling improved prediction of future healthcare utilisation, costs and quality-adjusted life years for economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(A2:C4,D2:D4) | Performs matrix multiplication for state prediction. |
| TRANSPOSE | =TRANSPOSE(A2:C4) | Computes matrix transposes used in covariance calculations. |
| MINVERSE | =MINVERSE(A2:C4) | Inverts covariance matrices during state estimation. |
| SUMPRODUCT | =SUMPRODUCT(B2:B4,C2:C4) | Calculates vector products within state and observation equations. |
VBA (Optional)
A VBA procedure can automate recursive state estimation by repeatedly updating the state vector and covariance matrices as new health data become available.
Sources
- Durbin J, Koopman SJ. Time Series Analysis by State Space Methods. Oxford University Press.
- Harvey AC. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
- Kalman RE. A New Approach to Linear Filtering and Prediction Problems. Transactions of the ASME?Journal of Basic Engineering. 1960.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
Related Concepts (3)
Library
Publications
1
A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →
Frequently Asked Questions (7)
What is a state space model?
A modelling framework representing a system's evolution over time through a defined set of possible states and transition probabilities between them.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
What is a state-space model?
A state-space model is a modelling framework that represents a system's evolution over time through a defined set of possible states and the transition probabilities between them. At each point in time the system occupies one of the states, and it moves between states according to the transition probabilities, so its behaviour over time is described by the sequence of states. In health economics, Markov models are state-space models, representing patients moving between health states, and the framework underlies much decision-analytic modelling of disease over time.
Source: Briggs, Claxton & Sculpher 2006
What state-based framework is a state-space model?
A state-space model describes a system through a defined set of mutually exclusive and exhaustive states that it can occupy. At any point the system is in exactly one state, and its behaviour is captured by how it moves between them over time. The complete collection of possible states forms the state space that gives the model its name. This structure underlies Markov models and other state-transition methods used in economic evaluation. Choosing states that are clinically meaningful and distinguishable is central to building a credible model.
Source: Briggs, Claxton and Sculpher 2006
What are the elements of a state-space model?
A state-space model has a defined set of states the system can occupy, transition probabilities specifying the chance of moving from each state to others in a time step, and an initial distribution across the states. Values such as costs and health effects are attached to the states, or to transitions. Together these elements describe how the system moves through the states over time and what outcomes accrue, so the model can compute the expected experience of the system or population.
Source: Briggs, Claxton & Sculpher 2006
How does a state-space model represent time?
A state-space model represents time through transitions between states, with the system moving between states at each step according to the transition probabilities. In a discrete-time model, transitions occur at fixed cycles, and the distribution across states is updated each cycle, while continuous-time versions use rates. Costs and effects accrue according to the states occupied over time. By tracking how the distribution across states evolves, the model represents the passage of time and the changing condition of the system or population.
Source: Briggs, Claxton & Sculpher 2006
How does a state-space model relate to a Markov model?
A Markov model is a state-space model in which the probability of moving to the next state depends only on the current state, not on the history of how it was reached, the Markov assumption. Markov models are the most common state-space models in health economics, representing patients moving between health states with transition probabilities. The state-space framework is the general form, and the Markov model is the widely used case with the memoryless property, on which cohort and individual state-transition models are built.
Source: Briggs, Claxton & Sculpher 2006
What are the limitations of state-space models?
State-space models, particularly Markov models, are limited by the assumption that transitions depend only on the current state, so they have no memory of history unless states are added to encode it, which can multiply the number of states. Defining states that capture the important distinctions, while keeping the model tractable, requires judgement, and transition probabilities must be estimated. The framework represents systems well where the state captures what matters, but where history or individual variation is important, the model must be elaborated or an individual-level approach used.
Source: Briggs, Claxton & Sculpher 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Oct 2025
Content version: 1.0.0
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