Effective Sample Size

In statistics, effective sample size is a notion defined for correlated (or weighted) samples from a distribution.[1]


Correlated Samples

Suppose several samples y_{i} are drawn from a distribution with mean \mu and std. deviation \sigma. Then the best estimate for the mean of this distribution is

\hat \mu = \frac{1}{n} \sum_{i=1}^{n} y_{i}

In that case, the variance of \hat \mu is given by

Var(\hat \mu) = \frac{\sigma^{2}}{n}

However, if the samples are correlated, then Var(\hat \mu) is somewhat higher. For instance, if all samples are completely correlated (\rho_{(i,j)} = 1), then Var(\hat \mu) = \sigma^{2} regardless of n.

The effective sample size n_{eff} is the unique value (not necessarily an integer) such that

Var(\hat \mu) = \frac{\sigma^{2}}{n_{eff}}

n_{eff} is a function of the correlation between samples. Suppose that all the correlations are the same, i.e. if i ≠ j, then \rho_{(i,j)} = \rho. In that case, if \rho = 0, then n_{eff} = n. Similarly, if \rho=1 then n_{eff} = 1. More generally,

n_{eff} = \frac{n}{1 + (n-1)\rho}

The case where the correlations are not uniform is somewhat more complicated. Note that if the correlation is negative, the effective sample size may be larger than the actual sample size. Similarly, it is possible to construct correlation matrices that have an n_{eff}>n even when all correlations are positive. Intuitively, n_{eff} may be thought of as the information content of the observed data.

Weighted Samples

If the data has been weighted, then several samples have been pulled from the distribution with effectively 100% correlation with some previous sample. In this case, the effect is known as Kish's Effective Sample Size[2]

n_{eff} = \frac{(\sum_{i=1}^{n} w_{i})^{2}}{\sum_{i=1}^{n} w_{i}^{2}}

References

  1. ↑ Tom Leinster (December 18, 2014). "Effective Sample Size" (html).
  2. ↑ "Design Effects and Effective Sample Size" (html).

Further reading

This article is issued from Wikipedia - version of the Monday, February 15, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.