Concept Architecture
Concept
Theoretically, Common Random Numbers is a variance reduction technique used in stochastic simulation whereby identical streams of random numbers are applied to two or more competing simulation models. The method induces positive correlation between model outputs so that random variation attributable to sampling is reduced, allowing differences between interventions to be estimated more precisely. In health economics, Common Random Numbers are widely used in Monte Carlo simulation to improve the efficiency of probabilistic comparisons between alternative healthcare strategies.
Mathematically, Common Random Numbers reduce the variance of the estimated difference between competing model outputs by exploiting covariance. When identical random number streams are used, positive covariance between simulation outputs decreases the variance of the incremental estimate, thereby improving statistical efficiency without increasing the number of simulation replications.
In practice, Common Random Numbers are implemented by assigning identical random number seeds or synchronised random number streams across competing simulation models. The technique is routinely applied in individual-level simulation, discrete event simulation and microsimulation models to improve the precision of incremental costs, health outcomes and cost-effectiveness estimates.
Purpose
Used to reduce simulation variance and improve the precision of incremental comparisons between competing health economic models without increasing computational effort.
Mathematical Formulae
Primary Formula
Var(X ? Y) = Var(X) + Var(Y) ? 2Cov(X, Y)
Supporting Formulae
If Cov(X, Y) > 0, then:
Var(X ? Y) < Var(X) + Var(Y)
Estimated incremental outcome:
? = X ? Y
Related Mathematical Methods
- Monte Carlo Simulation
- Variance Reduction Techniques
- Antithetic Variates
- Control Variates
- Microsimulation
- Discrete Event Simulation
Example
A health economist compares two screening programmes using patient-level microsimulation. Both models use the same sequence of random patient characteristics and event times by applying identical random number seeds. The resulting positive covariance between model outputs reduces the variance of the estimated incremental cost-effectiveness ratio, allowing more precise estimates with the same number of simulation runs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generate a common sequence of random numbers for competing simulations when controlled by a fixed seed through VBA or external software. |
| NORM.S.INV | =NORM.S.INV(RAND()) | Generate identical standard normal random values for competing stochastic models. |
| COVARIANCE.S | =COVARIANCE.S(A2:A1001,B2:B1001) | Estimate covariance between paired simulation outputs. |
| VAR.S | =VAR.S(C2:C1001) | Estimate the variance of incremental simulation outcomes. |
VBA (Optional)
VBA can initialise identical random number seeds across multiple simulation models to ensure common random number streams are maintained throughout probabilistic analyses.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Law AM. Simulation Modeling and Analysis.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
- NICE. Health Technology Evaluation Manual.
Related Concepts (7)
Library
Tools & Resources
1
SAVI — Sheffield Accelerated Value of Information — Mark Strong, Jeremy Oakley & Penny Breeze (University of Sheffield), Web application ed., 2024 (University of Sheffield)
A free, open-access web calculator that computes value-of-information measures (EVPI, partial EVPI/EVPPI and EVSI) directly from a model’s probabilistic sensitivity analysis output — no need to re-run the model. Also reports payer strategy-specific and uncertainty burden.
Web Tool (R Shiny)View source →
Frequently Asked Questions (6)
What are common random numbers?
A variance reduction technique using the same sequence of random numbers across simulations of different alternatives, reducing the variance of their difference.
Source: Law & Kelton 2000
Why use the same random numbers across compared alternatives?
When two alternatives are simulated separately, part of the difference between their results is random noise from the different draws each happened to use, which obscures the true difference. Using the same sequence of random numbers for both means they face identical chance events, so the noise affects them alike and cancels when their results are subtracted. What remains is a cleaner estimate of the genuine difference between the options. This sharpens the incremental comparison that a cost-effectiveness analysis needs. Briggs and colleagues (2006) describe this technique.
Source: Briggs et al. 2006
How do common random numbers reduce variance?
Common random numbers reduce variance by inducing positive correlation between the outcomes of the alternatives being compared: when the same random draws drive both, a random draw that raises one alternative's result tends to raise the other's too, so the difference between them is less affected by the shared randomness. Since the variance of a difference falls when its components are positively correlated, using common random numbers lowers the variance of the estimated difference, sharpening the comparison without needing more simulation runs.
Source: Law & Kelton 2000
When are common random numbers used?
Common random numbers are used when the quantity of interest is the difference between alternatives evaluated by simulation, such as the incremental cost or effect between two strategies in an individual-level model, where reducing the variance of the difference improves precision. Applying the same random stream to each alternative aligns their random conditions. This is valuable in comparisons where the alternatives share structure and random inputs, so that the induced correlation reduces the noise in their difference, which is the focus of the analysis.
Source: Law & Kelton 2000
How are common random numbers implemented?
Common random numbers are implemented by controlling the random number streams so that each alternative uses the same sequence of draws for the same purposes, often by fixing the random seed and synchronising how draws are assigned, so that corresponding random events match across alternatives. Care is needed to keep the streams aligned, since if the alternatives consume draws differently the synchronisation, and hence the benefit, can be lost. Proper implementation ensures the alternatives experience the same randomness, giving the intended reduction in the variance of their difference.
Source: Law & Kelton 2000
What are the limitations of common random numbers?
Common random numbers reduce variance only when they induce positive correlation between the alternatives' outcomes; if the technique instead induces negative or negligible correlation, it may not help or could increase the variance of the difference. Keeping the random streams synchronised across alternatives can be difficult when they use draws differently. The benefit varies with the model. These limitations mean the technique is applied where the alternatives are similar enough for shared randomness to align their outcomes, with the streams carefully synchronised.
Source: Law & Kelton 2000
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Oct 2025
Content version: 1.0.0
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