Calculating Cauchy scale from Cohen d confidence interval?
Dear EJ,
I have been reading on previous posts and blogs on informed prior parameters but I could not find info on how to calculate the scale (or width) of the Cauchy distribution for informed priors.
Location is easy, I simply take the effect size (Cohen d for t-test).
However, for scale, it was less clear.
My reading of this blog (http://xeniaschmalz.blogspot.com/2019/09/justifying-bayesian-prior-parameters-in.html) led to understand that scale is here based on a confidence interval around the informed/expected effect size.
So, would it be correct to compute the 95% confidence interval of my previous study's Cohen d effect size and use it to calculate the Cauchy scale?
But then, if so, how should I translate it into the Cauchy scale?
Simply cut in half my half CI. Eg. if CI = 0.257 to 1.091 centered on 0.679 then I use 0.412 as scale?
I am guessing it is more complex that ! Could you tell me how?
Thanks a lot in advance!
Henryk
Comments
Hi Henryk,
I am not 100% on what you want to accomplish. When you use the Cauchy prior, you simply stipulate it; nothing is calculated from data. Perhaps you want to use existing data to construct a prior? In that case, keep in mind that when you start with a Cauchy prior and data come in, the end result is no longer a Cauchy. These papers might be of use: https://link.springer.com/article/10.3758/s13428-018-1092-x and http://www.ejwagenmakers.com/2014/VerhagenWagenmakers2014.pdf.
However, if you nonetheless like to use prior data to determine the location and scale of a Cauchy, then the simplest method is to use a method-of-moments estimator. The location of a Cauchy distribution is its median, and the scale parameter is its interquartile range. So you could just take the median and IQR of the data and plug those in as parameters of the Cauchy. At least that does not seem too unreasonable to me.
Cheers,
E.J.
Thanks a lot, EJ!
I read your paper "Replication Bayes factors from evidence updating".
I like very much the easy trick to compute the BF of Xp2 that takes into account the posterior distribution of Xp1 as prior distribution !
i.e. dividing the global BF (obtained by merging data of Xp1 and Xp2) by the BF of Xp1
My follow-up question is : In your paper (Figure 3) you used a uniform distribution as prior to assess the global BF but this is not possible to enter uniform distribution via JASP (for a independant sample t-test).
Is there a way to enter a uniform distribution as priors (which equates as an absence of prior knowledge, right?)?
Alternatively : could it be that it doesn't matter whether we compute the global BF with the default Cauchy or a uniform distribution? If global BF and BF of Xp1 are both calculated with the same Cauchy prior, I guess it's fine to use the trick?
Thanks again !
Hi Henryk,
Whether or not the uniform distribution is appropriate for testing depends on the scale. So for a correlation coefficient or a probability, a uniform prior may be acceptable. For the t-test, however, it is not. The reason is that effect sizes for the t-test can range from -infinity to +infinity, and the uniform distribution would be improper (i.e., it does not integrate to 1). The "easy trick" works for any legitimate prior distribution, but in the case of the t-test the uniform does not quality as legitimate (at least not for testing). The uniform (or something very wide, say a Cauchy with scale 1000) also does not represent absence of knowledge, because it predicts that the effect is almost certainly humongous.
Cheers,
E.J.
Thanks a lot for clarifying!