Concept Architecture
Concept
Theoretically, Control Variates is a variance reduction technique used in stochastic simulation that improves the precision of Monte Carlo estimates by exploiting the known expected value of a correlated auxiliary variable. The method reduces simulation error without increasing the number of simulation replications. In health economics, control variates are used to improve the computational efficiency of probabilistic sensitivity analysis and complex simulation models.
Mathematically, the method adjusts the estimator of the target variable by incorporating the deviation of a control variable from its known expected value. The optimal adjustment coefficient depends on the covariance between the target and control variables and minimises the variance of the adjusted estimator while maintaining unbiasedness.
In practice, control variates are implemented by identifying a variable that is strongly correlated with the simulation output and whose expected value is known or can be accurately estimated. The adjustment is applied during Monte Carlo simulation to improve the precision of estimates of costs, health outcomes and cost-effectiveness while reducing computational requirements.
Purpose
Used to reduce simulation variance and improve the precision of Monte Carlo estimates by incorporating information from correlated variables with known expected values.
Mathematical Formulae
Primary Formula
X* = X ? c(Y ? E(Y))
where:
X* = adjusted estimator
X = original estimator
Y = control variate
E(Y) = expected value of the control variate
c = adjustment coefficient
Supporting Formulae
Optimal adjustment coefficient:
c = Cov(X, Y) / Var(Y)
Variance of the adjusted estimator:
Var(X*) = Var(X) + c�Var(Y) ? 2cCov(X, Y)
Related Mathematical Methods
- Monte Carlo Simulation
- Variance Reduction Techniques
- Common Random Numbers
- Antithetic Variates
- Probabilistic Sensitivity Analysis
- Covariance Analysis
Example
A probabilistic sensitivity analysis estimates the expected lifetime cost of a treatment pathway using Monte Carlo simulation. A secondary outcome with a known expected value is highly correlated with total cost. Applying a control variate adjustment reduces the variance of the estimated mean cost, producing more precise results without increasing the number of simulation iterations.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| COVARIANCE.S | =COVARIANCE.S(A2:A1001,B2:B1001) | Estimate covariance between the simulation output and the control variate. |
| VAR.S | =VAR.S(B2:B1001) | Estimate the variance of the control variate. |
| AVERAGE | =AVERAGE(B2:B1001) | Estimate the expected value of the control variate when required. |
| SUMPRODUCT | =SUMPRODUCT((A2:A1001-C2),(B2:B1001-D2))/COUNT(A2:A1001) | Calculate covariance components for determining the optimal adjustment coefficient. |
VBA (Optional)
VBA can automate control variate adjustments during Monte Carlo simulation to reduce estimator variance and improve computational efficiency.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Law AM. Simulation Modeling and Analysis.
- Glasserman P. Monte Carlo Methods in Financial Engineering.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →
Frequently Asked Questions (6)
What are control variates?
A variance reduction technique exploiting a related quantity with known expected value to reduce the variance of the estimate actually wanted.
Source: Press et al. 2007
Why does a control variate reduce the variance of an estimate?
A control variate is a related quantity that the simulation also produces but whose true expected value is already known. Because the errors in estimating the wanted quantity and the control tend to move together, the known control acts as a reference, so that when the control comes out too high the wanted estimate is likely too high as well and can be adjusted down. Using this relationship removes part of the random error, giving a more precise estimate from the same runs. The stronger the link between them, the greater the gain. Briggs and colleagues (2006) describe this technique.
Source: Briggs et al. 2006
How do control variates reduce variance?
Control variates reduce variance by using a quantity correlated with the target whose true expected value is known: in each simulation, the deviation of the control's estimate from its known mean indicates part of the random error, which is used to correct the target's estimate. The correction, scaled by an optimal coefficient reflecting the correlation, removes the component of the target's error that is explained by the control. The stronger the correlation between target and control, the greater the variance reduction achieved.
Source: Press et al. 2007
What makes a good control variate?
A good control variate is a quantity strongly correlated with the target of interest and with a known expected value, since the variance reduction grows with the strength of the correlation and requires the control's true mean to be known for the correction. It should be computable within the same simulation as the target. A control weakly correlated with the target gives little benefit. Choosing a control that closely tracks the target and whose expectation is known is therefore central to the technique's effectiveness.
Source: Press et al. 2007
When are control variates used?
Control variates are used in Monte Carlo simulation when a quantity related to the target has a known expected value and is correlated with the target, so that its deviation can correct the target's estimate and reduce its variance. This arises where a simplified or analytically tractable version of the problem provides such a control. The technique improves efficiency, achieving a given precision with fewer runs, so it is applied where a suitable, well-correlated control with a known mean is available.
Source: Law & Kelton 2000
What are the limitations of control variates?
Control variates require a related quantity with a known expected value that is correlated with the target, which may not be available, and the variance reduction depends on the strength of that correlation, being small when the control tracks the target poorly. Estimating the optimal coefficient adds a step, and a poor choice can reduce the benefit. The method adds complexity. These limitations mean control variates are applied where a suitable, well-correlated control with a known mean can be identified for the problem.
Source: Press et al. 2007
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 27 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/control-variates
- Term code
- HE-EM-UA-013
Stable URI · Machine-readable · Resolvable · CC BY 4.0