Signature
VRF = 1 / (1 + rho); n_eq = n / (1 + rho)
| Inputs | Definition | Unit |
|---|---|---|
rho | Correlation between the outputs of the two members of a pair, above -1 | correlation |
n | Number of model evaluations in the antithetic run, twice the number of pairs | count of evaluations |
VRF | Variance of ordinary sampling divided by the variance of antithetic sampling at the same number of model evaluations | ratio |
|---|---|---|
n_eq | Number of independent model evaluations needed to match the precision of n antithetic evaluations | count of evaluations |
Function
Antithetic variates estimator of an expected model output
Estimates the expected value of a simulation model's output, such as expected costs, QALYs or net monetary benefit, by running the model on n/2 independent sets of random inputs and again on their mirror images, then averaging all n results. For uniform random numbers the mirror image of U is 1-U, applied before the numbers are transformed into parameter values or event times; for a normal input it is the reflection about the mean. The estimate is unbiased, and its variance depends on the correlation between the outputs of the two members of each pair, so the method helps when that correlation is negative.
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Implementations
Excel
Antithetic reduction factor and equivalent sample size in two cells
With named cells PairCorr and Evaluations, the first formula returns the variance reduction factor and the second the equivalent number of independent evaluations.
=1/(1+PairCorr); =Evaluations/(1+PairCorr)
Assumptions
Equal cost of original and mirrored antithetic evaluations
Each model evaluation costs the same whether it uses original or mirrored inputs, so precision per evaluation measures efficiency. When generating the inputs is expensive and mirroring them is cheap, the gain per unit of computing time is larger than the factor shows.
Within-pair correlation above -1 for the antithetic efficiency factor
rho is above -1. At rho equal to -1 a single pair gives the exact answer and the factor has no finite value.
Worked examples
Efficiency of antithetic pairs in the survival example
With rho about -0.6449 the factor is about 2.816, so 1,000 antithetic evaluations match about 2,816 independent patients, roughly 2.8 times as many, as in the article.
rho = -0.6449; n = 1000; VRF = 2.816; n_eq = 2816
Efficiency of antithetic pairs for a rare event
For an illustrative event with annual probability 0.02, the within-pair correlation of the event indicators is about -0.0204, so 1,000 antithetic evaluations are worth only about 1,021 independent ones.
rho = -0.0204; n = 1000; VRF = 1.021; n_eq = 1021
Efficiency of antithetic pairs for a symmetric output
In the article's extreme case with rho equal to 1, the factor is 0.5, and 1,000 antithetic evaluations give the precision of only 500 independent ones.
rho = 1; n = 1000; VRF = 0.5; n_eq = 500
Common errors
Reading an antithetic variance cut as the same cut in standard error
A 64% cut in variance, a factor of 0.3551, lowers the standard error by only about 40%, because the standard error scales with the square root of the variance: 0.188 against 0.316 years in the survival example.
Carrying a best-case antithetic gain over to a full model
The 2.8-fold gain in the survival example comes from one smooth monotone input and is close to a best case. A 2010 type 2 diabetes simulation reported that antithetic variates cut the replications needed for predicted QALYs by 53%, which implies a variance factor of roughly 0.47, about a 2.1-fold gain.
Sources
Antithetic variance reduction factor and sampling cost
Owen AB. Monte Carlo Theory, Methods and Examples. 2013. Chapter 8, sections 8.2 and 8.3, which report the variance reduction factor of antithetic sampling as the reciprocal of 1 + rho and compare its efficiency with plain Monte Carlo when generating the inputs and evaluating the function have different costs.
Antithetic variates in a type 2 diabetes simulation model
McEwan P, Bergenheim K, Yuan Y, Tetlow AP, Gordon JP. Assessing the relationship between computational speed and precision: a case study comparing an interpreted versus compiled programming language using a stochastic simulation model in diabetes care. PharmacoEconomics. 2010;28(8):665-674. Abstract, which reports a typical 53% reduction in replications and run time with antithetic variates.
Canonical Identity
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