Concept Architecture
Concept
Theoretically, Antithetic Variates are a variance reduction technique used in Monte Carlo simulation to improve the precision of simulation estimates without increasing the number of simulation runs. The method exploits negative correlation between paired random samples so that random variation tends to cancel when results are averaged. In health economics, antithetic variates are used to improve the computational efficiency of probabilistic sensitivity analysis and stochastic decision models.
Mathematically, antithetic variates are generated by pairing each random sample with its complementary value, typically using uniform random numbers U and (1 ? U). The outputs from the paired simulations are averaged, producing an unbiased estimator with reduced variance whenever the paired outputs are negatively correlated. The effectiveness of the method depends on the strength of this negative correlation.
In practice, antithetic variates are implemented within Monte Carlo simulation algorithms by generating paired random numbers for uncertain model parameters and averaging the corresponding model outputs. They are commonly applied in probabilistic sensitivity analysis, microsimulation and discrete-event simulation to reduce simulation error while maintaining unbiased estimates of expected costs, health outcomes and cost-effectiveness.
Purpose
Used to reduce Monte Carlo simulation variance, improve computational efficiency, increase the precision of probabilistic sensitivity analysis and obtain more stable health economic model estimates without increasing simulation size.
Mathematical Formulae
Primary Formula
??? = (??(U) + ??(1 ? U)) / 2
Where:
??? = antithetic estimator
??(U) = estimate obtained using random sample U
??(1 ? U) = estimate obtained using the complementary sample
Supporting Formulae
Variance of the antithetic estimator:
Var(???) = �[Var(???) + Var(???) + 2Cov(???, ???)]
Variance reduction factor:
VRF = Var(Standard Estimator) / Var(Antithetic Estimator)
Related Mathematical Methods
- Monte Carlo simulation
- Probabilistic sensitivity analysis
- Common random numbers
- Control variates
- Importance sampling
- Quasi-Monte Carlo methods
Example
A probabilistic sensitivity analysis estimates the expected incremental cost-effectiveness ratio using 10,000 simulations. Rather than generating 10,000 independent random numbers, each uniform random value U is paired with its complement (1 ? U). The paired simulation outputs are averaged, reducing simulation variance and producing more stable estimates of expected costs and QALYs without increasing the number of model evaluations.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generates the original uniform random number U. |
| 1?x | =1-A2 | Generates the antithetic random number (1 ? U). |
| AVERAGE | =AVERAGE(B2,C2) | Averages paired simulation outputs to obtain the antithetic estimate. |
| VAR.S | =VAR.S(D2:D10001) | Evaluates the reduction in simulation variance. |
VBA (Optional)
VBA can automatically generate antithetic random number pairs and average paired simulation results during probabilistic sensitivity analysis.
Sources
Glasserman P. Monte Carlo Methods in Financial Engineering. Springer; 2004.
Rubinstein RY, Kroese DP. Simulation and the Monte Carlo Method. 3rd ed. Wiley; 2017.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
ISPOR-SMDM Modeling Good Research Practices Task Force. Model Parameter Estimation and Uncertainty Analysis.
NICE. Health Technology Evaluations: The Manual. National Institute for Health and Care Excellence; 2022.
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 antithetic variates?
A variance reduction technique pairing each sampled value with a negatively correlated complementary value to reduce a simulation's output variance.
Source: Hammersley & Morton 1956
Why pair each random draw with a complementary one?
Antithetic variates pair every random draw with a second draw deliberately chosen to be its opposite, so that when one is unusually high the other tends to be unusually low. Averaging the results of the paired runs lets these opposite deviations cancel, giving a more stable estimate from the same number of draws. The technique works best where the model's output responds smoothly and in one direction to the random input, so the pairing genuinely offsets. It reduces variance without extra runs. Briggs and colleagues (2006) describe such techniques.
Source: Briggs et al. 2006
How do antithetic variates work?
Antithetic variates work by generating pairs of random draws that are negatively correlated, for instance by using a random value and its complement, so that when one draw is high the other is low. Averaging the outputs from each pair tends to cancel some of the random variation, since the negatively correlated inputs produce partly offsetting results. The average over the antithetic pairs then has lower variance than the average over the same number of independent draws, improving the efficiency of the estimate.
Source: Hammersley & Morton 1956
Why are antithetic variates used?
Antithetic variates are used to reduce the variance of Monte Carlo estimates, so that a given precision is achieved with fewer simulation runs, saving computation, or a lower variance is obtained for the same number of runs. Because simulation results carry sampling error that falls slowly with the number of samples, variance reduction techniques like antithetic variates make simulation more efficient. This is valuable where each run is costly or many are needed, as in probabilistic sensitivity analysis of complex models.
Source: Metropolis & Ulam 1949
When are antithetic variates effective?
Antithetic variates are most effective when the output is a monotonic function of the random inputs, so that negatively correlated inputs produce negatively correlated outputs that cancel well when averaged, giving substantial variance reduction. Where the relationship is not monotonic, the technique may reduce variance less or not at all. The effectiveness thus depends on the structure of the simulation, so antithetic variates are applied where the negatively correlated pairs genuinely produce offsetting outputs, and their benefit is checked for the particular problem.
Source: Hammersley & Morton 1956
What are the limitations of antithetic variates?
Antithetic variates reduce variance only when the negatively correlated inputs produce offsetting outputs, which requires the output to respond monotonically to the inputs; where it does not, the technique may give little or no benefit. Implementing it requires generating the antithetic pairs correctly, and its effectiveness varies by problem. It is one of several variance reduction techniques, so its suitability is judged case by case. These limitations mean antithetic variates are applied where they genuinely improve efficiency, not universally.
Source: Hammersley & Morton 1956
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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