Cohort Model

What a cohort model does

Rather than representing each person separately, it stores the proportion or expected number of people in every state and uses transition probabilities to update that distribution at regular model cycles.

This page explains the cohort trace, transition mathematics, costs and health outcomes, timing choices, worked calculations, uncertainty, and validation. It also shows when a cohort model is appropriate and when individual-level simulation is needed instead.

How the cohort is represented

The model begins with a state vector that allocates the cohort across the available health states. Each person must occupy exactly one state at a given model time unless the model explicitly uses a different accounting structure.

For $K$ states, a row-vector representation at cycle $t$ is:

$$ \mathbf{n}t=\begin{bmatrix}n{t,1}&n_{t,2}&\cdots&n_{t,K}\end{bmatrix} $$

where $n_{t,j}$ is the expected number of cohort members in state $j$. The corresponding prevalence vector is $\boldsymbol{\pi}_t=\mathbf{n}_t/N$, where $N$ is the starting cohort size when no new entrants are added.

How transition probabilities move the cohort

A transition matrix specifies the probability of moving from each current state to every state in the next cycle. Under the row-vector convention used here, rows represent the state occupied at the start of the cycle and columns represent the state occupied at the end.

$$ \mathbf{P}t= \begin{bmatrix} p{11,t}&p_{12,t}&\cdots&p_{1K,t}\ p_{21,t}&p_{22,t}&\cdots&p_{2K,t}\ \vdots&\vdots&\ddots&\vdots\ p_{K1,t}&p_{K2,t}&\cdots&p_{KK,t} \end{bmatrix} $$

Every probability must satisfy $0\leq p_{jk,t}\leq1$, and every row must sum to one:

$$ \sum_{k=1}^{K}p_{jk,t}=1 $$

The next state distribution is calculated by matrix multiplication:

$$ \mathbf{n}_{t+1}=\mathbf{n}_t\mathbf{P}_t $$

Some software instead uses column vectors and the transpose of this equation. Either convention is valid, but mixing conventions produces incorrect flows even when every row appears to sum correctly.

The Markov trace

The sequence of state vectors is called the cohort trace or Markov trace. It shows how the expected population distribution changes cycle by cycle and provides the quantities to which state costs and health outcomes are attached.

In a closed cohort with death represented inside the model, the row total should remain equal to $N$ apart from harmless floating-point error. An open cohort can add incident cases or births, but its entry and exit rules must be modelled explicitly rather than treated as unexplained changes in the total.

A worked three-state model

Consider 1,000 people who all begin in a stable state. They can remain stable, progress to disease, or die; people in the disease state can remain there or die, and death is absorbing.

$$ \mathbf{n}_0=\begin{bmatrix}1000&0&0\end{bmatrix} $$

Using the state order Stable, Disease, Death, suppose the one-cycle transition matrix is:

$$ \mathbf{P}= \begin{bmatrix} 0.80&0.15&0.05\ 0&0.75&0.25\ 0&0&1 \end{bmatrix} $$

After one cycle:

$$ \mathbf{n}_1=mathbf{n}_0\mathbf{P} =\begin{bmatrix}800&150&50\end{bmatrix} $$

After a second cycle:

$$ \mathbf{n}_2=mathbf{n}_1\mathbf{P} =\begin{bmatrix}640&232.5&127.5\end{bmatrix} $$

The fractional values are expected counts, not fractional people observed in practice. They arise because the cohort model propagates population averages rather than simulating individual histories.

How the second-cycle calculation can be audited

Every cell in the next state vector is the sum of flows into that destination. Writing out those flows makes the matrix operation transparent and provides a useful implementation check.

$$ n_{2,\text{Stable}}=800(0.80)+150(0)+50(0)=640 $$

$$ n_{2,\text{Disease}}=800(0.15)+150(0.75)+50(0)=232.5 $$

$$ n_{2,\text{Death}}=800(0.05)+150(0.25)+50(1)=127.5 $$

The three results sum to 1,000, confirming conservation of the closed cohort. A flow-level audit can also identify a transposed matrix, an omitted transition, or a probability row that does not sum to one.

Attaching costs and health outcomes

State rewards assign costs, life-years, quality-adjusted life-years, or other outcomes to the time spent in each state. Transition rewards can separately capture one-off events such as surgery, diagnosis, treatment initiation, or death when a state reward would misrepresent their timing.

Let $\mathbf{c}_t$ be a column vector of per-person state costs and $\mathbf{u}_t$ a column vector of per-person health rewards for cycle $t$. With beginning-of-cycle occupancy, undiscounted totals are:

$$ C_t=\mathbf{n}_t\mathbf{c}_t $$

$$ Q_t=\mathbf{n}_t\mathbf{u}_t $$

If Stable costs £500 and yields 0.90 QALYs per year, Disease costs £2,500 and yields 0.50 QALYs, and Death has zero state reward, the end-of-first-cycle distribution gives:

$$ C_1=800(\text{£}500)+150(\text{£}2{,}500)+50(\text{£}0)=\text{£}775{,}000 $$

$$ Q_1=800(0.90)+150(0.50)+50(0)=795\text{ QALYs} $$

This calculation uses end-of-cycle occupancy for illustration. A real model must choose and consistently implement beginning-of-cycle, end-of-cycle, or corrected within-cycle reward timing.

Counting transition rewards

Some consequences depend on movement between states rather than occupancy. Expected flow from state $j$ to state $k$ during cycle $t$ is $n_{t,j}p_{jk,t}$, so a transition-specific reward can be attached directly to that flow.

If $r_{jk,t}$ is the reward for transition $j\rightarrow k$, the total transition reward is:

$$ R_t^{\mathrm{trans}}= \sum_{j=1}^{K}\sum_{k=1}^{K}n_{t,j}p_{jk,t}r_{jk,t} $$

Transition rewards prevent one-time events from being charged repeatedly to everyone remaining in a state. The modeller must also check that the same cost or outcome is not counted in both a state reward and a transition reward.

Cycle length and within-cycle timing

The cycle length should be short enough to represent clinically and economically important changes without adding needless computation. Monthly cycles may be needed for rapidly changing disease or treatment effects, whereas annual cycles may be adequate for slower processes.

A basic cohort trace observes state membership only at cycle boundaries, although real transitions occur throughout the interval. A half-cycle correction approximates average occupancy by giving half weight to the initial and final observations and full weight to intermediate observations:

$$ R_{\mathrm{HCC}}= \frac{1}{2}R_0+\sum_{t=1}^{T-1}R_t+\frac{1}{2}R_T $$

This correction is not automatically appropriate for every model or reward. Event timing, tunnel-state logic, entry timing, and differential state occupancy may require a more exact life-table or integration approach.

Converting rates into probabilities

Transition evidence may be reported as a continuous-time rate rather than a cycle probability. Rates describe instantaneous event intensity, whereas probabilities describe the chance of an event during a defined interval. Under a constant hazard $r$ over cycle length $\Delta t$, the corresponding probability of at least one event is:

$$ p=1-\exp(-r\Delta t) $$

Conversely:

$$ r=-\frac{\ln(1-p)}{\Delta t} $$

Simply dividing a probability to obtain a shorter-cycle probability is generally inaccurate. Competing events require a coherent competing-risks method because converting each cause-specific rate independently and inserting the resulting probabilities can make a row sum exceed one.

Discounting future values

Economic evaluations commonly discount future costs and health outcomes to reflect their timing. Discount rates and conventions depend on the decision context and should be applied to the point in time at which each reward is assumed to occur.

For annual discount rate $d$ and time in years $t$, a common discount factor is:

$$ DF_t=\frac{1}{(1+d)^t} $$

Total discounted costs and outcomes can then be written as:

$$ C=\sum_{t=0}^{T}C_tDF_{C,t}, \qquad Q=\sum_{t=0}^{T}Q_tDF_{Q,t} $$

Costs and health outcomes may use different prescribed rates. Cycle indices must be converted to elapsed years when cycles are shorter or longer than one year.

Time dependence and the memoryless assumption

A simple time-homogeneous cohort state-transition model uses the same matrix in every cycle and assumes that future movement depends only on the current state. This Markov property is restrictive when risk depends on time since diagnosis, treatment history, prior events, age, or number of recurrences.

Time-varying matrices $\mathbf{P}_t$ can represent calendar time, age, or treatment waning. Tunnel states can encode a limited duration since an event, and additional states can encode important history, but excessive state expansion can make the model difficult to estimate and validate.

Tunnel states and state expansion

A tunnel state is a temporary state through which a cohort progresses for a fixed or limited number of cycles. It is useful when costs, risks, or utilities differ according to time since an event but individual-level simulation is unnecessary.

For example, separate post-event states for months 1, 2, and 3 can enforce movement through an acute recovery period before entry into a long-term state. The expanded state space must remain mutually exclusive and collectively exhaustive, and every additional state requires defensible transition and reward inputs.

Cohort models compared with microsimulation

A cohort model efficiently produces expected outcomes for homogeneous groups and is often transparent enough to audit in a spreadsheet. Microsimulation follows individuals and can preserve detailed histories, continuous characteristics, and person-level heterogeneity.

Microsimulation is not inherently more accurate. It is preferable only when individual heterogeneity, event history, interactions, or timing materially affects the decision and cannot be represented credibly in an aggregate state structure.

Closed, open, and incident cohorts

A closed cohort begins with a fixed population and usually allows exits only through states such as death. This structure is common in cost-effectiveness models that compare lifetime outcomes for a defined starting population.

An open or population cohort permits new entrants over time and may allow other forms of exit. It can be more suitable for public-health programmes, vaccination, screening, or budget-impact questions, but it requires explicit population projections, entry distributions, calendar-time effects, and rules for accumulating multiple entry cohorts.

Treatment effects and baseline risk

Treatment effects must be applied to the same effect scale on which they were estimated and then translated into valid transition probabilities. A hazard ratio, risk ratio, and odds ratio are not interchangeable, especially when the baseline risk is large or changes over time.

Treatment may affect only selected transitions, may wane, or may stop after discontinuation. The model should separate baseline transition risk from relative treatment effect so that both can be sourced, tested, and varied without obscuring their meaning.

Competing risks and mortality

When several mutually exclusive events can occur during the same interval, their probabilities must be derived jointly or reconciled so that the state row remains valid. Background mortality and disease-specific mortality can otherwise be double counted.

Mortality should be conditional on being alive at the relevant time and should align with age, sex, calendar year, and jurisdiction when those dimensions matter. If observed all-cause mortality already incorporates disease deaths, adding disease-specific mortality requires a justified excess-hazard or competing-risk construction.

Parameter uncertainty and heterogeneity

Deterministic sensitivity analysis changes selected inputs to reveal influential assumptions. Probabilistic sensitivity analysis draws a coherent parameter set for each iteration and reruns the cohort trace, producing a distribution of total costs and outcomes without individual-level simulation.

Probability rows require joint distributions or transformations that preserve the sum-to-one constraint. Correlated treatment effects, costs, utilities, and baseline risks should retain their estimated dependence rather than being sampled independently for convenience.

Structural uncertainty cannot always be represented by changing parameter values. Alternative state definitions, time horizons, treatment-waning assumptions, cycle lengths, and extrapolation models may require separate scenario analyses.

Building the model in Excel

Excel can implement a small cohort model transparently by placing the transition matrix in a fixed input range and each cycle's state vector on one row. The next-cycle row is obtained with matrix multiplication, while separate columns calculate state and transition rewards.

If the current state vector is in B12:D12 and the transition matrix is in $B$5:$D$7, the next vector can use =MMULT(B12:D12,$B$5:$D$7). In modern Excel the result spills across three cells; older versions require the formula to be entered as an array formula across the destination range.

Row checks such as =SUM(B5:D5) should equal 1, and trace checks such as =SUM(B12:D12) should equal the intended cohort total. Named ranges, locked formula cells, explicit units, and separate input, calculation, results, and checks sections improve auditability.

Validation and debugging checks

Validation should examine the model's conceptual structure, data inputs, code or formulas, calculations, and external behaviour. Agreement with an expected result is stronger when it is supported by several independent checks rather than one aggregate output comparison.

• Transition checks: Every transition probability lies between zero and one, and each matrix row sums to one within a declared tolerance.
• Trace checks: A closed cohort is conserved across cycles, absorbing states never lose members, and prohibited transitions remain zero.
• Extreme-value checks: Setting mortality to zero produces no deaths, setting progression to zero prevents progression, and a probability of one produces certain movement.
• Reward checks: Zeroing all cost or utility inputs produces zero corresponding totals, and one-time rewards are not counted repeatedly.
• Time checks: Cycle conversion, discounting, half-cycle correction, treatment duration, and time-varying inputs use consistent units.
• Replication checks: Hand calculations for early cycles reproduce the spreadsheet or code output before the full horizon is trusted.

External validation compares model outputs with independent clinical, epidemiological, or economic evidence when suitable evidence exists. A mismatch should be investigated rather than removed through undocumented calibration.

Calibration and model fit

Calibration estimates uncertain inputs so selected model outputs reproduce prespecified targets. It can be appropriate when transition parameters are not directly observed, but it introduces uncertainty and potential non-identifiability.

Targets, loss functions, search methods, parameter bounds, convergence criteria, and validation targets should be reported. Data used for calibration should not then be presented as independent validation evidence, and alternative parameter sets with similar fit should be explored when they imply different decision results.

Common modelling errors

Cohort models can appear numerically tidy while containing important conceptual errors. Most failures arise from inconsistent timing, poorly defined states, or evidence translated onto the wrong probability scale.

• A transposed transition matrix moves the cohort according to the wrong origin-destination convention.
• Invalid probability rows create or lose cohort members when row totals differ from one.
• Double-counted events arise when the same consequence appears in both state and transition rewards.
• Naive cycle conversion treats probabilities as if they were rates and misstates shorter-cycle risk.
• Hidden memory occurs when transition risk depends on prior history that the chosen states do not retain.
• Inconsistent timing applies rewards, discounting, and treatment effects at incompatible points in a cycle.
• Over-aggregation masks clinically important heterogeneity or nonlinear treatment effects.
• Premature truncation uses a time horizon too short to capture relevant differences between strategies.

When a cohort model is a good choice

A cohort model is well suited when expected population outcomes are sufficient and people within each state can reasonably share the same future risks and rewards. It is especially useful when the health process has a modest number of meaningful states and the decision needs a transparent, computationally efficient model.

The approach becomes less suitable when outcomes depend strongly on continuous patient characteristics, complex event histories, interactions between people, resource queues, or exact event timing. In those cases, patient-level simulation, discrete-event simulation, dynamic transmission modelling, or another structure may better match the decision problem.

A practical workflow

A reliable cohort model is built by aligning the decision problem, state structure, evidence, timing, and validation before interpreting results. The workflow below keeps those choices visible and creates an auditable path from inputs to decision outputs.

1. Define the decision problem: State the population, interventions, comparators, perspective, outcomes, time horizon, and decision context.
2. Design the state structure: Create mutually exclusive and collectively exhaustive states that retain all decision-relevant history.
3. Choose the time structure: Set the cycle length, horizon, entry timing, reward timing, and any within-cycle correction.
4. Source transition evidence: Match every transition to defensible evidence and convert rates or effects using the correct scale.
5. Assign rewards: Attach costs and health outcomes to states or transitions without omission or double counting.
6. Build the trace: Apply the transition matrix cycle by cycle and preserve transparent intermediate flows.
7. Discount and aggregate: Calculate strategy totals using context-appropriate timing and discount rates.
8. Validate the model: Perform row, trace, extreme-value, hand-calculation, internal, and external checks.
9. Characterise uncertainty: Run deterministic, probabilistic, and structural scenario analyses with justified parameter distributions.
10. Report transparently: Document equations, evidence, assumptions, limitations, validation findings, and decision consequences.

What should be reported

A reproducible report identifies every state and allowable transition, the starting distribution, cycle length, time horizon, transition-matrix convention, reward timing, within-cycle correction, treatment-effect method, mortality construction, discounting, and uncertainty analysis. It also provides the parameter sources, validation checks, structural assumptions, and enough model logic for an independent reviewer to reconstruct key calculations.

Decision results should include total and incremental costs and outcomes for each strategy, not only a cost-effectiveness ratio. The report should explain which assumptions drive the result and whether a different plausible structure changes the decision.