Signature
SE_anti = s_anti / sqrt(m)
| Inputs | Definition | Unit |
|---|---|---|
s_anti | Sample standard deviation of the m pair means, each pair mean being (f(X_i) + f(X~_i))/2 | output unit |
m | Number of antithetic pairs, half the number of model evaluations | count of pairs |
SE_anti | Monte Carlo standard error of the antithetic estimate of the expected model output | output unit, for example years |
|---|
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 standard error from a column of pair means
With the pair means in a range named PairMeans, the formula returns the standard error. STDEV.S is the sample standard deviation with divisor m minus 1.
=STDEV.S(PairMeans)/SQRT(COUNT(PairMeans))
Assumptions
Pairs as independent observations for the antithetic standard error
The pairs are independent of each other, while the two runs within a pair are dependent. The pair means are therefore an independent sample of size m, and the usual standard error of a mean applies to them rather than to the individual runs.
Enough pairs for a stable antithetic standard error
The number of pairs is large enough for the sample standard deviation of the pair means to be a stable estimate. With very few pairs, as in the two-pair illustration, the standard error is itself imprecise and serves only to check the calculation.
Worked examples
Antithetic standard error for 500 survival pairs
In the survival example each pair mean has variance 100 × 0.3551 / 2, about 17.755, so its standard deviation is about 4.2137 years. With 500 pairs the standard error is 0.188 years, matching the article.
s_anti = 4.2137; m = 500; SE_anti = 0.188
Antithetic standard error from the article's two sampled pairs
The two pairs in the article, drawn at U = 0.20 and U = 0.90, have pair means of 9.163 and 12.040 years, an estimate of about 10.601 years and a sample standard deviation of about 2.034 years. Two pairs give a standard error of about 1.438 years, far too few for a decision but enough to check an implementation.
s_anti = 2.034; m = 2; SE_anti = 1.438
Common errors
Treating antithetic runs as independent for the standard error
Computing the standard error from all 1,000 runs as if they were independent gives about 10 divided by the root of 1,000, or 0.316 years, in the survival example instead of 0.188, hiding the gain. When the within-pair correlation is positive the same mistake gives a standard error that is too small, which is falsely reassuring.
Dividing the antithetic pair-mean standard deviation by the root of n
The divisor is the root of the number of pairs, not of the number of evaluations. In the survival example, 4.2137 divided by the root of 1,000 gives about 0.133 years instead of 0.188, understating the standard error by about 29%.
Sources
Pair-mean variance estimate for antithetic sampling
Owen AB. Monte Carlo Theory, Methods and Examples. 2013. Chapter 8, section 8.2, which notes that values within pairs are dependent and estimates the variance of the antithetic estimate from the m = n/2 pair means, using their sample variance divided by m.
Canonical Identity
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