In BayesFactor: Computation of Bayes Factors for Common Designs

Prior checks

options(markdown.HTML.stylesheet = 'extra/manual.css')
library(knitr)
opts_chunk$set(dpi = 200, out.width = "67%") 
library(BayesFactor)
options(BFprogress = FALSE)
bfversion = BFInfo()
session = sessionInfo()[[1]]
rversion = paste(session$version.string," on ",session$platform,sep="")
set.seed(2)

The BayesFactor has a number of prior settings that should provide for a consistent Bayes factor. In this document, Bayes factors are checked for consistency.

Independent-samples t test and ANOVA

The independent samples $t$ test and ANOVA functions should provide the same answers with the default prior settings.

# Create data
x <- rnorm(20)
x[1:10] = x[1:10] + .2
grp = factor(rep(1:2,each=10))

dat = data.frame(x=x,grp=grp)

t.test(x ~ grp, data=dat)

If the prior settings are consistent, then all three of these numbers should be the same.

as.vector(ttestBF(formula = x ~ grp, data=dat))
as.vector(anovaBF(x~grp, data=dat))
as.vector(generalTestBF(x~grp, data=dat))

Regression and ANOVA

In a paired design with an additive random factor and and a fixed effect with two levels, the Bayes factors should be the same, regardless of whether we treat the fixed factor as a factor or as a dummy-coded covariate.

# create some data
id = rnorm(10)
eff = c(-1,1)*1
effCross = outer(id,eff,'+')+rnorm(length(id)*2)
dat = data.frame(x=as.vector(effCross),id=factor(1:10), grp=factor(rep(1:2,each=length(id))))
dat$forReg = as.numeric(dat$grp)-1.5
idOnly = lmBF(x~id, data=dat, whichRandom="id")

summary(aov(x~grp+Error(id/grp),data=dat))

If the prior settings are consistent, these two numbers should be almost the same (within MC estimation error).

as.vector(lmBF(x ~ grp+id, data=dat, whichRandom="id")/idOnly)
as.vector(lmBF(x ~ forReg+id, data=dat, whichRandom="id")/idOnly)

Independent t test and paired t test

Given the effect size $\hat{\delta}=t\sqrt{N_{eff}}$, where the effective sample size $N_{eff}$ is the sample size in the one-sample case, and [ N_{eff} = \frac{N_1N_2}{N_1+N_2} ] in the two-sample case, the Bayes factors should be the same for the one-sample and two sample case, given the same observed effect size, save for the difference from the degrees of freedom that affects the shape of the noncentral $t$ likelihood. The difference from the degrees of freedom should get smaller for a given $t$ as $N_{eff}\rightarrow\infty$.

# create some data
tstat = 3
NTwoSample = 500
effSampleSize = (NTwoSample^2)/(2*NTwoSample)
effSize = tstat/sqrt(effSampleSize)

# One sample
x0 = rnorm(effSampleSize)
x0 = (x0 - mean(x0))/sd(x0) + effSize

t.test(x0)

# Two sample
x1 = rnorm(NTwoSample)
x1 = (x1 - mean(x1))/sd(x1)
x2 = x1 + effSize

t.test(x2,x1)

These (log) Bayes factors should be approximately the same.

log(as.vector(ttestBF(x0)))
log(as.vector(ttestBF(x=x1,y=x2)))

Paired samples and ANOVA

A paired sample $t$ test and a linear mixed effects model should broadly agree. The two are based on different models — the paired t test has the participant effects substracted out, while the linear mixed effects model has a prior on the participant effects — but we'd expect them to lead to the same conclusions.

These two Bayes factors should be lead to similar conclusions.

# using the data previously defined
t.test(x=dat$x[dat$grp==1],y=dat$x[dat$grp==2],paired=TRUE)

as.vector(lmBF(x ~ grp+id, data=dat, whichRandom="id")/idOnly)
as.vector(ttestBF(x=dat$x[dat$grp==1],y=dat$x[dat$grp==2],paired=TRUE))