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.

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))

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)

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)))

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))

*This document was compiled with version r bfversion of BayesFactor (r rversion).*

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