# Required sample sizes for Bayesian analysis

For frequentist hypothesis testing, several power calculators (e.g. G*Power1, powerandsamplesize.com2) are available to determine the minimum sample size needed to detect an effect with a given probability (statistical power). No such calculators are available (to my knowledge) for the Bayesian versions of t-tests, ANOVA, and linear correlations returning a Bayes factor.3 Calculating statistical power for these techniques for given sample sizes cannot be done analytically4 but can be done using simulation, and to save everyone a lot of time I will summarize my findings below.

## Minimum sample sizes needed

### Bayesian t-test

The analyses below show the minimum number of subjects needed per condition as a function of the true effect size in the population. This is based on a Bayes factor threshold of 3, a statistical power of 80% (i.e. you have an 80% probability of finding a BF10 > 3), a noninformative Jeffreys prior placed on the variance of the normal population, and a Cauchy prior (JZS) placed on the standardized effect size with scale $$\sqrt{2}/2$$.5 Note that these sample sizes are somewhat more conservative than a standard (frequentist) t-test.6 In the rightmost column you can find the required sample size for a standard (frequentist) t-test to reject the null hypothesis using a desired significance level of .05.

True effect size7 Cohen’s d Bayesian sample size Frequentist sample size
0.00 (null) 1028 N/A
Very small 0.01 ~425,000 156,656
0.10 3,050 1,567
Small 0.20 667 392
0.30 275 175
0.40 147 98
Medium 0.50 91 63
0.60 62 44
0.70 45 32
Large 0.80 34 25
0.90 27 20
1.00 22 16
Very large 1.20 16 11
Huge 2.00 7 4

### Bayesian test for linear correlation

The analyses below show the minimum number of paired observations (data points) needed as a function of the true correlation $$\rho$$ in the population. This is based on a Bayes factor threshold of 3, a statistical power of 80% (i.e. you have an 80% probability of finding a BF10 > 3), noninformative priors assumed for the population means and variances of the two populations, and a $$\operatorname{Beta}(3,3)$$ prior distribution is assumed for $$\rho$$.5

True effect size9 $$\rho$$ Bayesian sample size Frequentist sample size
0.00 (null) 2458 N/A
Small 0.10 1,300 782
0.20 273 193
Medium 0.30 109 84
0.40 54 46
Large 0.50 32 29
0.60 21 19
0.70 15 13
0.80 11 9
0.90 8 6

## Software used

Simulations and analyses were performed using Richard D. Morey’s BayesFactor package and the cluster computing snowfall package for R.

1. Note that the usefulness of Bayes factors is subject of discussion. Some authors (e.g. Kruschke) argue that precise description of posterior distributions is a better idea.

2. Some would go so far as to argue that statistical power in the sense used in frequentist statistics is meaningless in a Bayesian framework.

3. These are standard, noninformative priors. If you have a reasonable alternative prior, or test against a null interval instead of a point null you could improve power.  2

4. See Rouder, J.N., Speckman, P.L., Sun, D., et al. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16, 225–237. doi:10.3758/PBR.16.2.225

5. Labels taken from Sawilowsky, S. (2009). New effect size rules of thumb. Journal of Modern Applied Statistical Methods, 8, 467–474. doi:10.22237/jmasm/1257035100

6. This concerns the sample size needed for BF01 > 3 in the case where the true effect size is zero.  2

7. Labels taken from Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.)