0
$\begingroup$

I have the following problem and I wanted to see if somebody with more experience can help me.

I'm doing a Bayesian random-effects meta-analysis in rJAGS with models like these:

model {
for (i in 1:12) {
P[i] <- 1/S.sqr[i]      # Calculate precision
T[i] ~ dnorm(theta[i], P[i]) # study effects
theta[i] ~ dnorm(mu, prec) # random effects

}
mu ~ dnorm(0, 0.5)  # mean difference prior
tau ~ dunif(0,10)   # Uniform on SE
tau.sqr <- tau*tau   # between-study variance
prec<-1/(tau.sqr) # precision of tau
}

Basically, I have about a dozen studies and for each of them I have calculated an effect size in ms, e.g.:

T<- c(7, 12, 6, 23, 15, 9, 17, 20, 15, 11, 25, 9)

I have a theoretically-motivated prior for a null hypothesis (e.g. μ ~ N(0, 0.1)) and an alternative hypothesis (e.g. μ ~ N(15, 0.1)). My question is: is it possible to calculate Bayes factors to quantify the evidence in support of the alternative hypothesis compared to the null hypothesis?

So far, I haven't been able to find a solution to this. I know that the R package "BayesFactor" has a function for a meta-analysis, but it uses t-values from individual studies, which doesn't work for me (it also assumes that the studies estimate the same effect size, which is also not what I want to do).

$\endgroup$
1
$\begingroup$

If you want to know which of the hypotheses is more likely, why don't you calculate the probability that the a posteriori mean $\mu$ is below a certain threshold close to zero, say, $Pr(\mu \le 0.5\, |\, D)$ (where $D$ is your data)? That would allow you to make direct probability statements about the mean of the effect sizes (e.g. "there's an $6\%$ chance that the effect size is, on average, close to zero").

If you are set on using the Bayes factor, a conceptually simple way to calculate it is to set up a model with a random discrete indicator variable whose value decides which prior ($\mu \sim N(0, 0.1)$ or $\mu \sim N(15, 0.1)$) is used in each step. The posterior and prior distributions of this indicator variable can then be used to calculate the Bayes factor, sidestepping the difficult issue of multi-dimensional numerical integration. Unfortunately, I'm not fluent in R, so I can't help you with the implementation.

Cheers!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.