Bayesian one proportion test: different results in JASP and BayesFactor package
Hi,
I am trying to calculate the Bayes Factor for a one-proportion test. I have a categorical variable in which participants selected one out of three options. Where p is the proportion of people who selected the second option in this categorical variable, my hypothesis is: H0: p = 0.5 and H1: p > 1/3
I get very different results when running the test in JASP (with default priors) and when using the BayesFactor package in R (proportionBF function). For instance, BF in JASP is 0.038 but 0.30 with the BayesFactor package. Would be grateful for any insight on what might be causing this.
Many thanks!
Lewend
Comments
Dear Lewend,
Can you give the specific numbers for the test? And your H0 is p = 1/3, right?
Cheers,
E.J.
Dear E.J.,
Thank you so much and apologies for my late answer! Sorry, yes, H0 is p = 1/3. Attached is a table that contains the specific numbers. For instance, the second row shows that BF10 = 0.038. When using the BayesFactor package, I get BF10 = 0.30.
Many thanks again!
Kind regards,
Lewend
Hi Lewend,
Well, these are different models. JASP uses the test proposed by Jeffreys -- under H1, theta gets a uniform prior from 0 to 1. You can change that distribution by adjusting the a and b parameters of the beta distribution. In BayesFactor, the prior is a logistic distribution on the logit of theta, centered at 0.3. I like the BayesFactor specification, but it is more difficult to communicate. I do think we ought to implement it, and in such a way that it is easy to see what the specification of the prior on the logit scale means on the probability scale...
Cheers,
E.J.
Dear E.J.,
Thank you so much for the explanation. This really helps!
Kind regards,
Lewend
I'll look into it a bit more.
E.J.
Thank you! I'll stay tuned then!
Kind regards,
Lewend
OK. So first I confirm that in BayesFactor yields BF01 = 3.3 (approximately)
#R code:
library(BayesFactor)
proportionBF(52,173,1/3)
Second, the default Bayesian test in JASP gives BF01 = 7.5 (approximately). See screenshot.
https://forum.cogsci.nl/uploads/046/RWTZ1HESB118.pngNow let's figure out what beta prior comes closest to the prior from the BayesFactor test. To do this, we use the JAGS module. Steps:
First I'm going to deviate slightly from the BayesFactor specification and assign logit(p) a Normal distribution, centered on logit(1/3), which is log(.5). Using JAGS I'll draw from this prior and transform the samples back to the rate scale. Code:
model{
logit.p0 <- log(.5)
logit.p1 ~ dnorm(logit.p0,r)
r <- .707
p1 <- exp(logit.p1)/(1+exp(logit.p1))
}
I am monitoring p1 and the MCMC samples indicate this distribution has a mean of .37 and an SD of .22:
https://forum.cogsci.nl/uploads/530/KBZ0EU559SBD.pngI'll match these to the moments of the beta distribution and this yields a = 1.41 and b = 2.40. So now I'll return to the default test in JASP, but instead of specifying a beta(1,1) prior I'll use a beta(1.41,2.40) prior. The results:
https://forum.cogsci.nl/uploads/960/745BGRLDW2LD.pngSo now we have BF01 = 4.7. closer to 3.3 then before, but I'd like it to be a little closer still. Perhaps the beta does not quite fit to the logit-normal; or perhaps it is the fact that I used a normal instead of a logistic. Or perhaps I made a mistake!
E.J.
Hi E.J.,
Apologies again for the late answer. This is so helpful, thank you so much! Not sure why but I feel like the test in JASP is somewhat more intuitive.
Kind regards,
Lewend
Hi Lewend and E.J.,
A couple of comments on E.J.'s implementation:
1) logit.p1 ~ dnorm(logit.p0,r): JAGS uses the precision parameterization, not the SD parameterization.
2) r <- .707: The default in proportionBF seems to be 0.5.
3) If one wants to match the logistic and normal better, one could use the actual SD of the logistic which is given by: sqrt(r^2 * pi^2 / 3).
When I adjust these things, I obtain a = 2.15 and b = 3.87 which results in BF01 = 3.729. R code below.
library(BayesFactor)
r <- .5
1 / proportionBF(52,173,1/3, rscale = r)
sd.logistic <- sqrt(r^2 * pi^2 / 3) # https://en.wikipedia.org/wiki/Logistic_distribution
logit.p0 <- log(.5)
logit.p1 <- rnorm(1e6, logit.p0, sd.logistic)
p1 <- exp(logit.p1)/(1+exp(logit.p1))
m <- mean(p1)
s <- sd(p1)
print(m)
print(s)
a <- m * (m * (1 - m) / s^2 - 1)
b <- (1 - m) * (m * (1 - m) / s^2 - 1)
print(a)
print(b)
hist(p1, probability = TRUE, breaks = 30)
plot(function(x) dbeta(x, a, b), add = TRUE)
Thanks Quentin!
E.J.
Hi Quentin,
Apologies for the late reply! Thank you so much for your help and the detailed explanation!
Kind regards,
Lewend