#Bayesian t-test Example

# In this example we explore differences between males and females 
# on anxiety
female_anx <- c(51, 57, 42, 41, 36, 42)
male_anx <- c(34, 38, 37, 27, 29, 33)

#Let's start with an ordinary t-test on the data, and we will use
#alpha = .05
t.test(male_anx, female_anx)

#Next we are going to use the BEST package to run a Bayesian
#t-test

#There are several "parameters" in the model, 
#including a mean for each group, a sd for each 
#mean, a sd for each each group, a sd for each 
#sd, a degrees of freedom for the distribution shape (t dist),
#and a sd for the degrees of freedom 

#For each parameter we can explicitly define a prior, 
#or use the default distribution for that prior

#First, let's run the model with all the default priors
library(BEST)
mod1 <- BESTmcmc(male_anx, female_anx, parallel=FALSE)
plot(mod1)

#Next, let's assume we have prior information regarding 
#the means of the groups (e.g., from a past study). 
#Specifically, males have a prior mean of 35 and 
#females have a prior mean of 45 (both with standard
#deviations of 5)

#muM is the argument for the mean parameter, muSD 
#is the argument for the sd of of the mean parameter
priors <- list(muM = c(35,45), muSD = c(5,5))
mod2 <- BESTmcmc(male_anx, female_anx, priors=priors, 
                 parallel=FALSE)
plot(mod2)

#Here is a full list of priors, most set to defaults
priors2 <- list(muM = c(35,45), muSD = c(5,5), 
               sigmaMode = c(sd(male_anx),sd(female_anx)), 
               sigmaSD = c(sd(male_anx)*5,sd(female_anx)*5), 
               nuMean = 30, nuSD = 5)
mod3 <- BESTmcmc(male_anx, female_anx, priors=priors2)
plot(mod3)