Cohort Trace

Cohort Trace: Concept Architecture

Orientation and learning roadmap

The page starts with the states, starting distribution, and cycle that determine each entry in a trace. It then follows two hand-checkable cycles, connects state occupancy to health and cost outcomes, and explains how time dependence and transition rewards change the calculation. The final checks help a modeller find errors that can hide in a plausible-looking table.

Set the states and starting row

A cohort state-transition model assigns a population to mutually exclusive, collectively exhaustive health states at each specified cycle boundary. The first row records the initial distribution before any modeled transition. Subsequent rows record the expected proportions or numbers in each state after applying the transition probabilities for that cycle.

Let $\mathbf m_t$ be a row vector of proportions at the start of cycle $t$, and let $P_t$ have rows indexed by starting state and columns indexed by destination state. The next distribution is obtained by multiplying the row vector by the matrix. For a time-homogeneous model, the same $P$ is used each cycle; otherwise $P_t$ can vary with calendar time, age, or another modeled variable.

$$\mathbf m_{t+1}=\mathbf m_t P_t,\qquad\sum_j m_{t,j}=1.$$

If the trace is expressed as expected counts instead, the row sum is the initial cohort size as long as death remains an included absorbing state and nobody enters or leaves the modeled population. A fraction of a person in a count trace is an expected count, not a simulated fractional individual. Specify whether rows represent start-of-cycle or end-of-cycle occupancy so that rewards and plots are interpreted consistently.

Follow a three-state example

Consider Healthy, Sick, and Dead states with a one-year cycle and $1{,}000$ people initially Healthy. A Dead state is absorbing: those already Dead remain there in later cycles. The illustrative transition matrix places each origin in a row and each destination in a column.

$$P=\begin{pmatrix}0.85&0.10&0.05\\0.20&0.60&0.20\\0&0&1\end{pmatrix},\qquad\mathbf n_0=(1000,0,0).$$

After one cycle, the expected counts are $\mathbf n_1=(850,100,50)$. After another, Healthy contains $850(0.85)+100(0.20)=742.5$, Sick contains $850(0.10)+100(0.60)=145$, and Dead contains $850(0.05)+100(0.20)+50=112.5$. Every row totals $1{,}000$, including the accumulated Dead count.

The recovery transition from Sick to Healthy is permitted here, but another disease process might prohibit it. The values are invented for arithmetic practice and should not be treated as disease-specific estimates. A trace makes the consequences of the assumed state definitions and transition rules visible; it does not establish that those assumptions are clinically valid.

Distinguish occupancy from movement

A trace cell answers how much of the cohort occupies a state at a cycle boundary. It does not directly state how many people first entered that state during the preceding interval. The expected flow from origin $i$ to destination $j$ in cycle $t$ is the starting occupancy multiplied by the corresponding transition probability.

$$F_{ij,t}=n_{i,t}p_{ij,t}.$$

In the example's second cycle, $85$ expected Healthy people move to Sick, while $60$ expected Sick people remain Sick; the Sick occupancy at $t=2$ is their sum, $145$. Calling that $145$ “new cases” would be wrong. A total transition count, new-event count, or resource cost tied to an event needs the relevant flow, not simply the destination's occupancy.

Rows that appear stable can conceal substantial two-way movement. If the model requires a history such as “ever had a complication” or “time since entry,” a single undifferentiated state may discard information that affects future transitions or rewards. Use tunnel states or another suitable structure where that history materially matters.

Attach health and cost consequences

A reward vector can assign a per-cycle utility or cost to time spent in each state. Multiplying occupancy by that vector gives an expected cohort total for the chosen cycle convention, and summing appropriately weighted cycles gives total model outcomes. Per-person results require dividing a count-based cohort total by the initial population size.

$$Q=\sum_{t=0}^{T-1}d_t\,\mathbf m_t\mathbf q_t,\qquad C=\sum_{t=0}^{T-1}d_t\,\mathbf m_t\mathbf c_t,$$

where $\mathbf m_t$ contains proportions, $\mathbf q_t$ contains per-person QALYs for a full cycle in each state, $\mathbf c_t$ contains per-person state costs for that cycle, and $d_t$ is the chosen discount factor. If utility values are annual rates but cycles are shorter, multiply by the cycle's duration in years before summing. The displayed formula assumes start-of-cycle occupancy; a half-cycle correction or more exact within-cycle method changes the reward calculation.

A transition can incur a one-time cost or disutility, such as a hospital admission or treatment start. Such a reward should be attached to $F_{ij,t}$ and counted once, rather than added to every subsequent cycle in the destination state. Some events occur and resolve between cycle boundaries; a simple occupancy trace may need supplementary event tracking or shorter cycles to represent them.

Handle time, strategies, and uncertainty

One trace is normally produced for each strategy and parameter set under the same state definitions and horizon. Differences in probabilities, initial distributions, or state and transition rewards then produce different expected costs and QALYs. Comparing traces can reveal which states and periods drive an incremental result.

Age-dependent mortality or waning treatment effect can make $P_t$ change with cycle time. Time spent in a state can also matter, which a simple clock-time change alone will not capture if people enter the state in different cycles. Document the dependence explicitly and verify that each cycle-specific transition row still sums to one.

In probabilistic sensitivity analysis, draw uncertain parameter values and regenerate complete traces for each draw; the mean of inputs passed through one trace need not equal the mean of outcomes across uncertain inputs. A deterministic cohort trace has no first-order Monte Carlo noise, but it can still be highly uncertain because its probabilities, effects, and rewards are estimated. Keep parameter uncertainty separate from numerical implementation checks.

Audit the trace before interpreting it

Check the initial row, row sums, nonnegative entries, absorbing states, and expected direction of mortality or other irreversible transitions. Recalculate at least one cycle by hand and inspect edge cases such as zero events or all members starting in an absorbing state. Compare occupancy and flows with external clinical or epidemiological evidence when feasible.

For a spreadsheet, place the initial row above the cycle rows and use matrix multiplication such as =MMULT(B2:D2,$H$2:$J$4) for the next three-state row in a spreadsheet supporting dynamic arrays. Verify that the matrix range uses origin rows in the same order as the trace columns, then copy or recalculate for subsequent cycles. Keep costs and QALYs on separate sheets or clearly labelled columns so reward timing is auditable.

An attractive smooth curve is not evidence that the transition matrix, cycle timing, or rewards are correct. Report the state diagram, horizon, initial distribution, matrix sources, time-dependence rules, and occupancy convention together with plotted traces. If the model's observed outputs cannot be reconciled to its trace and transition flows, resolve the discrepancy before using its economic results.

Evidence used for this explanation

ISPOR-SMDM guidance establishes the role and reporting needs of state-transition models, and the original DARTH tutorials show the state vector, transition matrix, and trace in economic evaluation. The numerical three-state example here is original and its row arithmetic is independently checkable. The cited sources support the method rather than the illustrative probabilities.

• ISPOR-SMDM Modeling Good Research Practices Task Force-3, “State-Transition Modeling”.
• Alarid-Escudero F et al., “An Introductory Tutorial on Cohort State-Transition Models in R Using a Cost-Effectiveness Analysis Example”. Medical Decision Making (2023).
• Alarid-Escudero F et al., “A Tutorial on Time-Dependent Cohort State-Transition Models in R Using a Cost-Effectiveness Analysis Example”. Medical Decision Making (2023).