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How does BayesFactor account for random effects?

If I understand correctly from Rouder et al's (2012) paper, the way BayesFactor "deals" with random effects is:

  1. Set a wide prior (r = 1)
  2. No sum-to-zero constraint.

But I see no mention of how the presence of random effect affects the computation of the likelihood of related fixed effects (i.e., is there any difference when an effect is between-subjects vs within-subject).

The use of random factors as random effects vs fixed effects with a wide prior in BayesFactor seems to have little effect:

library(BayesFactor)

data(md_12.1, package = "afex")

# BayesFactor - specify "id" as a fixed effect.
m0_f_BF <- lmBF(rt ~ id, md_12.1, rscaleEffects = c(id = 1))
m1_f_BF <- lmBF(rt ~ angle + id, md_12.1, rscaleEffects = c(id = 1))
BF_f_BF <- unname(as.vector(m1_f_BF / m0_f_BF))

# BayesFactor - specify "id" as a fixed effect.
m0_r_BF <- lmBF(rt ~ id, md_12.1, whichRandom = "id")
m1_r_BF <- lmBF(rt ~ angle + id, md_12.1, whichRandom = "id")
BF_r_BF <- unname(as.vector(m1_r_BF / m0_r_BF))

c(as_fixed = BF_f_BF, as_random = BF_r_BF)
#>  as_fixed as_random 
#>  909.1889  900.1979

However the differences are much larger with other methods (below I use the BIC approx. for simplicity, but stan-based methods also produce differences that BayesFactor does not):

library(lmerTest)

BIC_BF <- function(m0,m1){
  d <- (BIC(m0) - BIC(m1)) / 2
  exp(d)
}

# BIC approx - specify "id" as a fixed effect.
m0_f_lm <- lm(rt ~ id, md_12.1)
m1_f_lm <- lm(rt ~ angle + id, md_12.1)
BF_f_lm <- BIC_BF(m0_f_lm, m1_f_lm)


# BIC approx - specify "id" as a random effect.
m0_r_lm <- lmer(rt ~ (1|id), md_12.1)
m1_r_lm <- lmer(rt ~ angle + (1|id), md_12.1)
BF_r_lm <- BIC_BF(m0_r_lm, m1_r_lm)


c(as_fixed = BF_f_lm, as_random = BF_r_lm)
#>    as_fixed   as_random 
#>    5281.736 3920528.548


Might this be the root of the somewhat common question here in the forum regarding differences between frequentist and Bayesian rmANOVAs in JASP?

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