Microsimulation

What microsimulation does

In health economics, it is commonly used when patient heterogeneity, prior events, time since an event, or treatment history cannot be represented adequately by a simple cohort-average model.

Each simulated individual follows a pathway under defined clinical and economic rules, accumulating costs and health outcomes over time. Population results are obtained by averaging across many simulated lives rather than by moving one aggregate cohort vector through the model.

The simulated individual

Every simulated person begins with a set of baseline attributes that can affect risks, treatment, utilities, and costs. These attributes may be sampled from patient-level data, a joint distribution, or a synthetic population designed to represent the decision population.

• Fixed attributes can include sex, genotype, baseline disease severity, or treatment eligibility.
• Time-varying attributes can include age, biomarkers, adherence, current treatment, or accumulated adverse events.
• State history records past events, durations, previous treatments, and other information needed for future risk.
• Current state identifies the person's health condition or event status at the present model time.
• Accumulators store discounted costs, life-years, QALYs, events, and other outcomes for that person.

The joint distribution of baseline attributes matters. Sampling correlated characteristics independently can create unrealistic individuals and distort both average outcomes and subgroup results.

How state transitions are simulated

In a discrete-time microsimulation, each individual occupies a state during a cycle and faces transition probabilities conditional on current information. A random draw determines the next state according to the cumulative probabilities.

Let $S_{i,t}$ be person $i$'s state at cycle $t$, and let $H_{i,t}$ summarise relevant history. The transition probability can be written as:

$$ P\left(S_{i,t+1}=s'\mid S_{i,t}=s, X_i,H_{i,t},A_{i,t}\right) $$

where $X_i$ contains baseline characteristics and $A_{i,t}$ represents treatment or another time-varying action. The probabilities across all allowable next states must be non-negative and sum to one for every individual and cycle.

Why memory changes the model

A cohort state-transition model is Markovian when future movement depends only on the current state. Microsimulation can retain memory directly, so two individuals in the same current state can have different risks because their ages, prior events, treatment durations, or event counts differ.

Memory is useful only when it is supported by the decision problem and evidence. Adding patient history without a corresponding effect on transitions, outcomes, or decisions increases complexity without changing model behaviour.

Discrete time and continuous event time

Microsimulation does not require one specific treatment of time. A patient-level state-transition model advances through fixed cycles, while an individual-level discrete-event simulation advances directly to the next event time.

Choosing individual-level simulation does not automatically justify a discrete-event structure. The time representation should match event timing, data resolution, computational needs, and the consequences of within-cycle approximation.

Accumulating costs and health outcomes

Each individual accumulates rewards associated with states, events, treatments, and time. Discounting should be applied consistently to the timing of each cost and health outcome.

For person $i$, discounted cost over cycles $t=0,\ldots,T$ can be represented as:

$$ C_i=\sum_{t=0}^{T}\frac{c_{i,t}}{(1+r_C)^{t\Delta}} $$

and discounted QALYs as:

$$ E_i=\sum_{t=0}^{T}\frac{u_{i,t}\Delta}{(1+r_E)^{t\Delta}} $$

where $c_{i,t}$ is the cost assigned in cycle $t$, $u_{i,t}$ is the utility over that cycle, $\Delta$ is the cycle length in years, and $r_C$ and $r_E$ are the relevant annual discount rates. Event costs and short-lived utility decrements require timing rules that reflect when they occur rather than being assigned automatically as full-cycle rewards.

Averaging simulated outcomes

The expected outcome under strategy $a$ is estimated by averaging the simulated individual outcomes. Increasing the number of individuals reduces first-order Monte Carlo error but does not correct structural error or biased parameters.

For $N$ simulated people:

$$ \widehat{E}[C(a)]=\frac{1}{N}\sum_{i=1}^{N}C_i(a), \qquad \widehat{E}[E(a)]=\frac{1}{N}\sum_{i=1}^{N}E_i(a) $$

The incremental estimates are:

$$ \Delta C=\widehat{E}[C(1)]-\widehat{E}[C(0)], \qquad \Delta E=\widehat{E}[E(1)]-\widehat{E}[E(0)] $$

These averages should be accompanied by evidence that simulation noise is acceptably small for the decision outputs. A stable population mean does not guarantee stable subgroup, tail, or rare-event estimates.

First-order and second-order uncertainty

Microsimulation contains at least two conceptually different sources of variation. Keeping them separate is essential because they answer different questions and require different convergence checks.

First-order variation describes who happens to experience an event in the simulation; it is not parameter uncertainty. Probabilistic sensitivity analysis typically draws one coherent parameter set, simulates enough individuals to control inner-loop noise, and repeats this process across outer-loop parameter draws.

Common random numbers

When strategies are compared, using the same underlying random-number streams for corresponding individuals can reduce the variance of incremental results. This variance-reduction method is called common random numbers.

If a person receives a low event draw under one strategy, the matched person receives the corresponding draw under the comparator, subject to the model's transition logic. The technique should preserve independence between simulated individuals and must be implemented carefully when strategies have different event sequences or consume different numbers of random draws.

Common random numbers improve precision of the comparison but do not remove bias. Results should remain reproducible from recorded seeds and stream-management rules.

A worked cost-effectiveness example

Suppose 100,000 matched simulated individuals are evaluated under a new intervention and usual care. The mean discounted results per person are 8.42 QALYs and £42,600 for the intervention, compared with 8.10 QALYs and £37,800 for usual care.

The incremental outcomes are:

$$ \Delta E=8.42-8.10=0.32\text{ QALYs} $$

$$ \Delta C=\text{£}42{,}600-\text{£}37{,}800 =\text{£}4{,}800 $$

The ICER is:

$$ ICER=\frac{\text{£}4{,}800}{0.32\text{ QALYs}} =\text{£}15{,}000\text{ per QALY} $$

At a threshold of £20,000 per QALY, incremental net monetary benefit is:

$$ INMB=(\text{£}20{,}000)(0.32)-\text{£}4{,}800 =\text{£}1{,}600 $$

The intervention is cost-effective in this illustrative deterministic run because INMB is positive. Before using the result, the analyst must still show that simulation noise is small, parameter uncertainty is propagated, and the model structure and inputs are valid.

Measuring Monte Carlo error

Repeated random simulation produces sampling error even when every model parameter is fixed. Convergence should be judged for decision-relevant incremental results rather than from a visually stable total alone.

For simulated individual outcomes $Y_1,\ldots,Y_N$, an ordinary Monte Carlo standard error of the mean is:

$$ MCSE(\bar{Y})=\frac{s_Y}{\sqrt{N}} $$

Matched strategies using common random numbers should assess the standard error of the person-level incremental outcome rather than treating strategy means as independent. Batch means, repeated seeds, or replicate simulations can help diagnose serial dependence, stream errors, and unstable rare-event outputs.

Selecting the number of simulated individuals

There is no universally adequate sample size for a microsimulation. The required number depends on event rarity, outcome variance, subgroup needs, strategy similarity, probabilistic-analysis design, and the decision tolerance.

Simulation size should be increased until Monte Carlo error is small relative to the incremental outcomes and cannot plausibly change the decision at relevant thresholds. Reporting only the number of simulated patients without a convergence assessment does not establish numerical adequacy.

Modelling competing and recurrent events

Individuals may face several events within a cycle or over continuous time, and the order of those events can affect future risks and accumulated outcomes. The model needs explicit rules for competing events, recurrent events, simultaneous events, and absorbing outcomes such as death.

When probabilities are converted from rates, the time interval and competing-risk structure must be preserved. Applying separate marginal event probabilities sequentially can produce order-dependent results or total probability above one unless a coherent joint event process is defined.

Calibrating a microsimulation

Calibration estimates uncertain model parameters by comparing simulated outputs with external targets. Microsimulation makes calibration computationally demanding because the objective function can contain random noise in addition to parameter uncertainty.

Fixed seeds, common random numbers, larger inner samples, emulators, or stochastic optimisation methods can improve numerical behaviour. Calibration targets, goodness-of-fit measures, parameter constraints, accepted sets, and residual discrepancies should be reported, and calibration uncertainty should be carried into decision analysis where material.

Validation at several levels

A patient-level model can produce plausible averages while containing incorrect individual logic. Validation should therefore examine code, pathways, distributions, and aggregate outcomes rather than relying on one final cost-effectiveness result.

• Face validity checks whether states, histories, event rules, and outputs are clinically and economically credible.
• Internal validity checks formulas, probability sums, event ordering, absorbing states, accounting identities, and reproducibility.
• Cross-model validity compares results with an independently implemented model or a simplified cohort version under aligned assumptions.
• External validity compares model outputs with data not used to build or calibrate the model.
• Predictive validity assesses whether the model predicts later observations when a genuine temporal test is available.

Extreme-value and deterministic test cases are especially valuable. Setting probabilities to zero or one, removing treatment effects, eliminating costs, and forcing identical strategies should produce known limiting behaviour.

Microsimulation compared with related models

Several model types may operate at the individual level, but they represent time, interaction, and resource constraints differently. The label should describe the actual mechanics rather than being used as a synonym for any stochastic model.

Microsimulation is justified when individual history or heterogeneity changes expected outcomes or decisions enough to matter. It should not be chosen solely because it appears more sophisticated than a cohort model.

Equity and subgroup outputs

Individual simulation can preserve distributions that aggregate models conceal. It can estimate outcomes by age, deprivation, disease severity, risk, or another prespecified characteristic and can support distributional economic evaluation when suitable equity inputs and value judgements are defined.

Subgroup estimates require sufficient simulated and empirical information. Large synthetic samples reduce Monte Carlo error but do not create evidence for poorly observed interactions or under-represented populations.

Implementation and reproducibility

Microsimulations often exceed the practical limits of ordinary spreadsheet calculation because they require many individuals, cycles, event histories, and probabilistic draws. Whatever software is used, the implementation should remain traceable through parameter tables, pseudocode or diagrams, tests, seeds, version control, and machine-readable outputs.

Parallel processing requires independent or deliberately coordinated random-number streams. Reusing a seed incorrectly across workers can duplicate simulated people and create a false impression of precision.

A practical workflow

A defensible microsimulation begins with a decision problem that genuinely requires individual-level representation. The following workflow connects conceptual structure, data, computation, and validation.

1. Define the population, strategies, perspective, outcomes, time horizon, time representation, and decision rule.
2. Justify why individual heterogeneity or history changes the analysis relative to a simpler cohort model.
3. Specify baseline attributes, states, events, memory variables, treatment rules, and allowed transitions.
4. Estimate transition, event, cost, utility, and treatment-effect inputs with their dependence and uncertainty.
5. Implement event timing, random-number streams, outcome accumulation, discounting, and stopping rules explicitly.
6. Test deterministic limiting cases, invariants, event order, probability sums, and individual trace histories.
7. Validate aggregate and subgroup outcomes internally, externally, and against simplified or independent models where possible.
8. Establish convergence using Monte Carlo error for incremental and rare-event outcomes.
9. Propagate parameter and structural uncertainty without conflating it with first-order variation.
10. Report structure, assumptions, algorithms, seeds, sample sizes, convergence, validation, uncertainty, and limitations.

Interpreting and reporting model results

Microsimulation produces estimates conditional on its population generator, individual rules, inputs, time representation, and random-number design. A large simulated population can make Monte Carlo error tiny while leaving structural and evidential uncertainty substantial.

Common errors include treating first-order variation as parameter uncertainty, sampling correlated attributes independently, using too few individuals for rare events, failing to align random streams across strategies, hiding event-order rules, and assuming individual-level complexity guarantees validity. Results should include mean and distributional outcomes, incremental estimates, convergence evidence, uncertainty analysis, validation findings, and enough implementation detail for independent scrutiny.