Transition Matrix

Concept

Theoretically, Transition Matrix is the mathematical representation of the probabilities of moving between health states in a state-transition model or Markov chain during a single model cycle. Each element of the matrix specifies the probability of transitioning from one health state to another, thereby defining the dynamic behaviour of the model. In health economics, the transition matrix forms the core computational structure used to model disease progression, treatment effects and long-term clinical and economic outcomes.

Mathematically, a transition matrix is a square stochastic matrix in which rows correspond to current health states and columns correspond to future health states. Each element represents a transition probability, and the probabilities within each row sum to one. Successive multiplication of the cohort state vector by the transition matrix generates the evolution of the model over time.

In practice, transition matrices are constructed using probabilities estimated from clinical trials, observational studies, survival analyses or epidemiological data. Separate matrices may be used for different patient subgroups or model cycles when transition risks vary over time. The matrix is repeatedly applied throughout the simulation to estimate state occupancy, costs, quality-adjusted life years and other health economic outcomes.

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Purpose

Used to define the probabilities of movement between health states, providing the mathematical basis for simulating disease progression and estimating long-term clinical and economic outcomes.

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Mathematical Formulae

Primary Formula

Transition matrix:

?? =

?p??  p??  ?  p???
?p??  p??  ?  p???
? ?    ?   ?   ? ?
?p??  p??  ?  p???

where:

• p?? = P(X??? = j ? X? = i)

and

????� p?? = 1

for every row i.

Supporting Formulae

State update:

????? = ?????

State distribution after n cycles:

??? = ?????�

Related Mathematical Methods

• Markov chains
• Matrix algebra
• State-transition modelling
• Cohort simulation
• Eigenvalue analysis
• Probabilistic sensitivity analysis

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Example

A three-state chronic disease model consists of Stable Disease, Progressive Disease and Death.

The annual transition matrix is

?? =

?0.85  0.10  0.05?
?0.00  0.80  0.20?
?0.00  0.00  1.00?

The initial cohort is

??? = [1, 0, 0]

After one cycle:

??? = ????? = [0.85, 0.10, 0.05]

The resulting state occupancies are then used to calculate expected costs and quality-adjusted life years for the cycle.

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Excel Implementation

Function	Example Formula	Health Economics Application
MMULT	=MMULT(StateVector,TransitionMatrix)	Calculate state distribution after each cycle
SUM	=SUM(B2:D2)	Verify that each row of the transition matrix sums to one
SUMPRODUCT	=SUMPRODUCT(StateVector,CostVector)	Calculate expected costs after each transition
SUMPRODUCT	=SUMPRODUCT(StateVector,UtilityVector)	Calculate expected QALYs after each transition
MUNIT	=MUNIT(NumberStates)	Create identity matrices for model construction

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VBA (Optional)

Automate repeated application of transition matrices across model cycles and summarise cumulative health and economic outcomes.

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Sources

• Norris JR. Markov Chains. Cambridge University Press.
• Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322?338.
• 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.
• 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.