Signature
s_(t+1) = s_t P
| Inputs | Definition | Unit |
|---|---|---|
s_t | A row vector containing the proportion of the cohort in each health state at the start of cycle t | proportion of cohort |
P | A square matrix in which each entry gives the probability of moving from one state to another during one cycle | probability per cycle |
s_(t+1) | A row vector containing the proportion of the cohort in each health state at the start of the next cycle | proportion of cohort |
|---|
tThe numbered model cycle (cycle)
Function
Cohort state-transition function
Maps a cohort's current state distribution and transition probabilities to its state distribution in the next model cycle.
Try this function
Implementations
Excel
Update the cohort distribution
Excel uses matrix multiplication to return the next-cycle state vector. CurrentStateVector and TransitionMatrix are named ranges.
=MMULT(CurrentStateVector,TransitionMatrix)
Assumptions
Mutually exclusive and exhaustive states
Every cohort member occupies one and only one modelled health state in each cycle.
Markov property
Transition probabilities depend on the current state unless relevant history is represented through additional states or another model structure.
Fixed cycle length
Every update represents the same stated length of time.
Time-homogeneous probabilities
The transition matrix P remains constant across cycles for this formula variant.
Valid transition matrix
Each transition probability is between 0 and 1, and every row of P sums to 1.
Worked examples
Three-state cohort after one cycle
A cohort begins entirely in Stable disease. After one cycle, 80% remain Stable, 15% are Progressed and 5% are Dead.
s_0 = [1,0,0]; P = [[0.80,0.15,0.05],[0,0.85,0.15],[0,0,1]]; s_1 = [0.80,0.15,0.05]
Common errors
Using probabilities from the wrong time interval
Applying an annual transition probability directly in a monthly-cycle model changes the intended transition process. Convert or estimate probabilities for the model's actual cycle length first.
Sources
State-transition modelling good-practice report
Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812–820.
Practical guide to Markov models
Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Medical Decision Making. 1993;13(4):322–338.
Canonical Identity
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