# 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:

- Set a wide prior (
`r = 1`

) - 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?**

## Comments

I'll forward this to Richard...