Draft for review · v0.1.0HealthEconomics.wiki · Interactive demonstrator
A transition matrix is the table of probabilities of moving between, or staying in, health states in one cycle of a Markov model, with rows summing to one. Build one, check it, push a cohort through it, and see why a shorter cycle needs a root of the whole matrix rather than a formula applied to each cell.
Source: Transition Matrix, HealthEconomics.wiki concept page, content version 1.0.3.
In a Markov model the transition matrix moves the cohort forward in time, so every cost and QALY the model reports rests on the state occupancy it produces. This explorer uses the three-state progressive cancer model from the concept page: progression-free (PF), progressed disease (PD) and Dead.
Each scenario loads a matrix into step 1. The first is the page's worked example; the others are illustrative variations built to expose a specific error or boundary. None of the figures are clinical estimates.
| From \ To | PF | PD | Dead | Row total |
|---|---|---|---|---|
| PF | ||||
| PD | ||||
| Dead absorbing |
Rows are the state at the start of a cycle; columns the state at its end. Cycle length: one year.
The model's transition structure is progressive: there is no return from PD to PF, so that cell is a structural zero. Death is absorbing, with 1 on the diagonal and 0 elsewhere in its row.
Multiplying the row vector of state proportions by the matrix moves the cohort forward one cycle. Repeating the step builds the cohort trace. With a constant matrix the result after n cycles is a matrix power.
The trace is not shown as a chart. Fix the failed checks in step 1. With an invalid matrix the cohort is not conserved, so proportions can go negative or stop adding to one, as the table shows.
| End of cycle | PF | PD | Dead | Total |
|---|
Entries of Pn are n-cycle transition probabilities summed over every pathway, including those through intermediate states.
Counting costs and QALYs within each cycle, including half-cycle correction, belongs to cohort simulation and leaves the matrix unchanged, so it is not shown here.
Converting each annual probability to a shorter-cycle probability on its own applies only to two-state models. Assuming the same shorter-cycle matrix applies throughout the year, the correct conversion takes the nth root of the whole matrix, which returns P when raised to the power n.
Because this progressive matrix is upper triangular, its eigenvalues are its diagonal entries, so the root has a closed form: r11 = p111/n, r22 = p221/n, r12 = p12(r11 − r22)/(p11 − p22), and the death entries are the row remainders. For n = 2 this is the page's r12 = p12/(r11 + r22).
Conversion needs a valid annual matrix. Fix the failed checks in step 1 first; converting an invalid matrix would only carry its errors into the shorter cycle.
Starting with everyone in PF. The annual model is the reference the shorter-cycle model should reproduce. Differences from it are shown under each value.
The matrix logarithm of P, when it exists, gives a rate (generator) matrix Q with rows summing to zero. Dividing it by n and exponentiating gives the same root. If Q has a negative off-diagonal rate, P is not embeddable.
In probabilistic sensitivity analysis the exits from a state are sampled together, usually from a Dirichlet distribution, so each sampled row still sums to one. Absorbing rows stay fixed and structural zeros stay zero. When the model runs on shorter cycles, the root has to be taken for each sampled matrix, not once for the mean, because whether it is valid depends on the sampled values.
Illustrative parameterisation: each row's Dirichlet parameters are the step 1 probabilities times the effective sample size. Fitting real parameters from counts is covered on the Dirichlet distribution page. Draws use a fixed seed sequence, so results can be reproduced.
Sampling needs a valid annual matrix. Fix the failed checks in step 1 first.
What applies to the matrix as a whole sits here; single entries, survival curves and hazard ratios belong to the transition probability concept.
A model with K mutually exclusive, collectively exhaustive states needs a K × K matrix for each cycle. Rows are starting states and columns end states, so each row accounts for everyone who starts in that state.
Look for: a stated orientation. Some texts outside health economics put starting states in columns, which reverses the multiplication.
Diagonal entries are the probabilities of staying in a state for a cycle. An absorbing state, usually death, has 1 on the diagonal and 0 elsewhere in its row.
Look for: a diagonal filled as a remainder; it can hide a negative value.
Probabilities for disallowed transitions are zero. These zeros encode the model's transition structure, such as no recovery from a progressed cancer.
Look for: zeros that stay zero through conversion and sampling, and a route to death from every living state.
A model can use one matrix for every cycle (a time-homogeneous chain) or a different matrix P1, P2, … for each cycle, for example when age-specific mortality changes over simulation time.
Look for: one constant matrix used where risk clearly changes with age or time.
When risk depends on time already spent in a state, tunnel states change the shape of the matrix rather than its values. Each adds a row and a column, and its row sends the cohort onward rather than letting it stay.
Look for: how much history the model needs; every tunnel state enlarges the matrix.
When evidence gives rates, build a rate matrix Q and convert it in one step: P(Δ) = exp(ΔQ). Applying p = 1 − e−rt to each rate separately is wrong when a state has competing exits, because a person can have more than one event in a cycle.
Look for: rates converted one at a time in a multistate model.
Every entry lies between 0 and 1, every row sums to one, the cohort moves by mt+1 = mtPt, and a shorter cycle uses a root of the whole matrix that reproduces P when raised to the power n.
Two further checks apply to a converted matrix or a long trace: a converted matrix raised to the right power should return the original, and proportions should add to one in every cycle. The ISPOR-SMDM report also suggests setting parameters so the trace can be predicted, for example so that no one enters a given state.
The matrix describes average movement under the Markov property and carries no costs or utilities. A valid root need not be unique, and different roots can lead to different conclusions about cost-effectiveness. Coherent arithmetic does not show that the states and evidence describe the disease well enough for the decision.
Not every matrix is diagonalisable, and a root may contain complex or negative entries. Necessary conditions for a valid rate matrix to exist include a positive determinant no greater than the product of the diagonal entries, and no zero entry for a transition that is reachable through other states. When a root is not stochastic, Chhatwal and colleagues approximate it with the closest stochastic matrix; that approximation is not implemented here.
PBAC (Australia) asks submissions to present every transition probability in the base case with point and interval estimates and justified sources, to adjust values to the model's cycle and structure, to explain the cycle length and to assess correlation between transition probabilities. Guidelines for preparing a submission to the PBAC, version 5.0 (2016), subsections 3A.2.3 and 3A.4.1.
KCE (Belgium) asks for each state to be specified as temporary or absorbing, for transition probabilities to be presented, for example in matrix form, with an indication of whether each is constant or variable, and for the cycle length to be justified. Belgian guidelines for economic evaluations, third edition, KCE Reports 400C (2025), Guideline 14.
All figures are illustrative and none are clinical estimates. The canonical scenario reproduces the worked example on the concept page; the other scenarios are teaching variations created for this demonstrator. The tool is limited to the three-state progressive structure with a constant annual matrix, so the matrix root uses the closed form for an upper-triangular matrix rather than a general eigendecomposition, and it reports the root with non-negative real diagonal entries. It does not model tunnel states, cycle-dependent matrices, half-cycle correction, costs or QALYs.
The proportional split shown in step 3 divides a state's total shorter-cycle exit probability in proportion to its annual exits. It is appropriate for absorbing exits but not universal; the transition probability page discusses it. Dirichlet parameters in the uncertainty panel are scaled from point estimates purely to illustrate the sampling-loop check. This is an educational tool, not modelling software or medical advice.