Setting Cauchy prior scaling in Bayesian t-test - use related effect size?

James

Hi all,

I'm performing a Bayesian paired-samples t-test in JASP and am trying to choose a prior. I read a few papers (Rouder et al 2009 etc) and some blog posts, and my understanding is that using a Cauchy, the scale can be set to the expected effect size, and I can use the resulting computed JZS Bayes factor. I'm testing two variables, and have a robust effect on one at effect size 0.62. Is it then a principled approach to use that as a scaling factor for the prior of the other, or would it be preferable to use a standard one in the literature e.g. r=0.707 or 1?

I should add that with the second variable I'm interested in asserting the null hypothesis if this is supported, so I should be sufficiently conservative to avoid favouring the null unjustifiably (i.e. avoid too high a scaling factor).

Cheers,
James

EJ

Hi James,

The default Cauchy is centered on 0 and has a scale factor "r" that determines the width. This scale factor happens to equal the interquartile range, such that, when r=0.707 for instance, 50% of the prior mass lies in the interval from -0.707 to +0.707. In my opinion, you use the Cauchy(center=0, r=0.707) as a default of reference analysis, that does not require much subjective input. Note that under this distribution, the most likely values of effect size are near zero, but very large effect sizes are possible too.

If you really have a strong expectation about effect size ("it will be about 0.5, possibly as low as 0.2, possibly as high as 0.8") then you can specify a distribution that is centered on that value. We only implemented this functionality in the latest version of JASP, and it is described here: https://arxiv.org/abs/1704.02479

Cheers,
E.J.