Simple-CV

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Slide 1
10.1007_s41470-019-00063-y
The Psilocybin-Telomere Hypothesis An empirically falsifiable prediction concerning the beneficial neuropsychopharmacological effects of psilocybinon genetic aging(1)

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Keywords: Statistical computingOpen-source softwareBayesian analysisMarkov chain Monte Carlo methodsData visualisationLogical inferenceHypothesis testingDeductive reasoningAbductionCredibility intervalsProbability theoryDecision algorithmsNew statisticsReplication crisisCreative statistics

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Frequentist inference
Fixed-effects ANOVAGeneral linear models: mixing continuous and categorical covariatesOutput plot as PDFSimple linear regressionSkewness & Kurtosis
data(ToothGrowth)

## Example plot from ?ToothGrowth

coplot(len ~ dose | supp, data = ToothGrowth, panel = panel.smooth,
       xlab = "ToothGrowth data: length vs dose, given type of supplement")
## Treat dose as a factor
ToothGrowth$dose = factor(ToothGrowth$dose)
levels(ToothGrowth$dose) = c("Low", "Medium", "High")

summary(aov(len ~ supp*dose, data=ToothGrowth))


#install.packages("xtable")
library(xtable)
xtable(x, caption = NULL, label = NULL, align = NULL, digits = NULL,
       display = NULL, auto = FALSE, ...)

print(xtable(d), type="html")
print(xtable(d), type="latex") # anova table to latex
#https://cran.r-project.org/web/packages/xtable/index.html
#https://rmarkdown.rstudio.com/


data(ToothGrowth)

# model log2 of dose instead of dose directly
ToothGrowth$dose = log2(ToothGrowth$dose)

# Classical analysis for comparison
lmToothGrowth <- lm(len ~ supp + dose + supp:dose, data=ToothGrowth)
summary(lmToothGrowth)

x<- (1:5)
y<- (1:111)
pdf(file=file.choose())
hist(x)
plot(x, type='o')
dev.off()

Notes
file.choose() is a very handy command which saves the work associated with defining absolute and relative paths which can be quite cumbersome.
list.files for non-interactive selection.
choose.files for selecting multiple files interactively.
More
https://www.rdocumentation.org/packages/base/versions/3.5.2/topics/file.choose

# http://wiki.math.yorku.ca/index.php/VR4:_Chapter_1_summary
x <- seq( 1, 20, 0.5)   # sequence from 1 to 20 by 0.5
x                       # print it
w <- 1 + x/2            # a weight vector
y <- x + w*rnorm(x)     # generate y with standard deviations
                          #    equal to w
 
dum <- data.frame( x, y, w )     # make a data frame of 3 columns
dum                     # typing it's name shows the object
rm( x, y, w )           # remove the original variables


fm <- lm ( y ~ x, data = dum)   # fit a simple linear regression (between x and y)
summary( fm )                   # and look at the analysis



fm1 <- lm( y ~ x, data = dum, weight = 1/w^2 )  # a weighted regression  
summary(fm1)
 
lrf <- loess( y ~ x, dum)    # a smooth regression curve using a
                               #   modern local regression function
attach( dum )                # make the columns in the data frame visible
                               #   as variables (Note: before this command, typing x and
                               #   y would yield errors)
plot( x, y)                  # a standard scatterplot to which we will
                               #   fit regression curves 
lines( spline( x, fitted (lrf)), col = 2)  # add in the local regression using spline
                                             #  interpolation between calculated
                                             #  points
abline(0, 1, lty = 3, col = 3)    # add in the true regression line (lty=3: read line type=dotted;
                                    #   col=3: read colour=green; type "?par" for details)
abline(fm, col = 4)               # add in the unweighted regression line
abline(fm1, lty = 4, col = 5)     # add in the weighted regression line
plot( fitted(fm), resid(fm),
       xlab = "Fitted Values",
       ylab = "Residuals")          # a standard regression diagnostic plot
                                    # to check for heteroscedasticity.
                                    # The data were generated from a
                                    # heteroscedastic process. Can you see
                                    # it from this plot?
qqnorm( resid(fm) )               # Normal scores to check for skewness,
                                    #   kurtosis and outliers.  How would you
                                    #   expect heteroscedasticity to show up?
search()                          # Have a look at the search path
detach()                          # Remove the data frame from the search path
search()                          # Have another look. How did it change?
rm(fm, fm1, lrf, dum)             # Clean up (this is not that important in R 
                                    #    for reasons we will understand later)



library(rgl)
rgl.open()
rgl.bg(col='white')
x <- sort(rnorm(1000))
y <- rnorm(1000)
z1 <- atan2(x,y)
z <- rnorm(1000) + atan2(x, y)
plot3d(x, y, z1, col = rainbow(1000), size = 2)
bbox3d()

skew(x, na.rm = TRUE)
kurtosi(x, na.rm = TRUE)

#x A data.frame or matrix
#na.rm how to treat missing data

## The function is currently defined as
function (x, na.rm = TRUE) 
{
    if (length(dim(x)) == 0) {
        if (na.rm) {
            x <- x[!is.na(x)]
        			}
        
        mx <- mean(x)
         sdx <- sd(x,na.rm=na.rm)
        kurt <- sum((x - mx)^4)/(length(x) * sd(x)^4)  -3
        } else {
    
    kurt <- rep(NA,dim(x)[2])
    mx <- colMeans(x,na.rm=na.rm)
     sdx <- sd(x,na.rm=na.rm)
    for (i in 1:dim(x)[2]) {
    kurt[i] <- sum((x[,i] - mx[i])^4,  na.rm = na.rm)/((length(x[,i]) - sum(is.na(x[,i]))) * sdx[i]^4)  -3
            }
    }
    return(kurt)
  }
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Bayes Factor analysis
ttestBF function (one sample) BFmcmc functionttestBF function (two sample)Bayesian meta analysisBayes factor robustness analysis, one-sidedAnimating Robustness-Check of Bayes Factor PriorsSkewness & KurtosisanovaBF function
#https://richarddmorey.github.io/BayesFactor/
bf = ttestBF(x = diffScores)
## Equivalently:
## bf = ttestBF(x = sleep$extra[1:10],y=sleep$extra[11:20], paired=TRUE)
bf

Comment

ttestBF function, which performs the “JZS” t test described by Rouder, Speckman, Sun, Morey, and Iverson (2009).

#https://richarddmorey.github.io/BayesFactor/
chains2 = recompute(chains, iterations = 10000)
plot(chains2[,1:2])

Comment
The posterior function returns a object of type BFmcmc, which inherits the methods of the mcmc class from the coda package.

#https://richarddmorey.github.io/BayesFactor/
## Compute Bayes factor
bf = ttestBF(formula = weight ~ feed, data = chickwts)
bf
###
chains = posterior(bf, iterations = 10000)
plot(chains[,2])

#https://richarddmorey.github.io/BayesFactor/
## Bem's t statistics from four selected experiments
t = c(-.15, 2.39, 2.42, 2.43)
N = c(100, 150, 97, 99)
###
## Do analysis again, without nullInterval restriction
bf = meta.ttestBF(t=t, n1=N, rscale=1)

## Obtain posterior samples
chains = posterior(bf, iterations = 10000)
plot(chains)

Comment
ttestBF function, which performs the “JZS” t test described by Rouder, Speckman, Sun, Morey, and Iverson (2009).

## This source code is licensed under the FreeBSD license
## (c) 2013 Felix Schönbrodt

#' @title Plots a comparison of a sequence of priors for t test Bayes factors
#'
#' @details
#'
#'
#' @param ts A vector of t values
#' @param ns A vector of corresponding sample sizes
#' @param rs The sequence of rs that should be tested. r should run up to 2 (higher values are implausible; E.-J. Wagenmakers, personal communication, Aug 22, 2013)
#' @param labels Names for the studies (displayed in the facet headings)
#' @param dots Values of r's which should be marked with a red dot
#' @param plot If TRUE, a ggplot is returned. If false, a data frame with the computed Bayes factors is returned
#' @param sides If set to "two" (default), a two-sided Bayes factor is computed. If set to "one", a one-sided Bayes factor is computed. In this case, it is assumed that positive t values correspond to results in the predicted direction and negative t values to results in the unpredicted direction. For details, see Wagenmakers, E. J., & Morey, R. D. (2013). Simple relation between one-sided and two-sided Bayesian point-null hypothesis tests.
#' @param nrow Number of rows of the faceted plot.
#' @param forH1 Defines the direction of the BF. If forH1 is TRUE, BF > 1 speak in favor of H1 (i.e., the quotient is defined as H1/H0). If forH1 is FALSE, it's the reverse direction.
#'
#' @references
#'
#' Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin and Review, 16, 225-237.
#' Wagenmakers, E.-J., & Morey, R. D. (2013). Simple relation between one-sided and two-sided Bayesian point-null hypothesis tests. Manuscript submitted for publication
#' Wagenmakers, E.-J., Wetzels, R., Borsboom, D., Kievit, R. & van der Maas, H. L. J. (2011). Yes, psychologists must change the way they analyze their data: Clarifications for Bem, Utts, & Johnson (2011)

BFrobustplot <- function(
    ts, ns, rs=seq(0, 2, length.out=200), dots=1, plot=TRUE,
    labels=c(), sides="two", nrow=2, xticks=3, forH1=TRUE)
{
    library(BayesFactor)
   
    # compute one-sided p-values from ts and ns
    ps <- pt(ts, df=ns-1, lower.tail = FALSE)   # one-sided test
   
    # add the dots location to the sequences of r's
    rs <- c(rs, dots)
       
    res <- data.frame()
    for (r in rs) {
       
        # first: calculate two-sided BF
        B_e0 <- c()
        for (i in 1:length(ts))
            B_e0 <- c(B_e0, exp(ttest.tstat(t = ts[i], n1 = ns[i], rscale=r)$bf))
   
        # second: calculate one-sided BF
        B_r0 <- c() for (i in 1:length(ts)) { if (ts[i] > 0) {
                # correct direction
                B_r0 <- c(B_r0, (2 - 2*ps[i])*B_e0[i])
            } else {
                # wrong direction
                B_r0 <- c(B_r0, (1 - ps[i])*2*B_e0[i])
            }
        }
   
        res0 <- data.frame(t=ts, n=ns, BF_two=B_e0, BF_one=B_r0, r=r) if (length(labels) > 0) {
            res0$labels <- labels
            res0$heading <- factor(1:length(labels), labels=paste0(labels, "\n(t = ", ts, ", df = ", ns-1, ")"), ordered=TRUE)
        } else {
            res0$heading <- factor(1:length(ts), labels=paste0("t = ", ts, ", df = ", ns-1), ordered=TRUE)
        }
        res <- rbind(res, res0)
    }
   
    # define the measure to be plotted: one- or two-sided?
    res$BF <- res[, paste0("BF_", sides)]

   
    # Flip BF if requested
    if (forH1 == FALSE) {
        res$BF <- 1/res$BF
    }
       
   
    if (plot==TRUE) {
        library(ggplot2)
        p1 <- ggplot(res, aes(x=r, y=log(BF))) + geom_line() + facet_wrap(~heading, nrow=nrow) + theme_bw() + ylab("log(BF)")
        p1 <- p1 + geom_hline(yintercept=c(c(-log(c(30, 10, 3)), log(c(3, 10, 30)))), linetype="dotted", color="darkgrey")
        p1 <- p1 + geom_hline(yintercept=log(1), linetype="dashed", color="darkgreen")
   
        # add the dots
        p1 <- p1 + geom_point(data=res[res$r %in% dots,], aes(x=r, y=log(BF)), color="red", size=2)
   
        # add annotation
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-2.85, label=paste0("Strong~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-1.7 , label=paste0("Moderate~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=-.55 , label=paste0("Anectodal~H[", ifelse(forH1==TRUE,0,1), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=2.86 , label=paste0("Strong~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=1.7  , label=paste0("Moderate~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, size=3, color="black", parse=TRUE)
        p1 <- p1 + annotate("text", x=max(rs)*1.8, y=.55  , label=paste0("Anectodal~H[", ifelse(forH1==TRUE,1,0), "]"), hjust=1, vjust=.5, vjust=.5, size=3, color="black", parse=TRUE)
       
        # set scale ticks
        p1 <- p1 + scale_y_continuous(breaks=c(c(-log(c(30, 10, 3)), 0, log(c(3, 10, 30)))), labels=c("-log(30)", "-log(10)", "-log(3)", "log(1)", "log(3)", "log(10)", "log(30)"))
        p1 <- p1 + scale_x_continuous(breaks=seq(min(rs), max(rs), length.out=xticks))

        return(p1)
    } else {
        return(res)
    }
}

References
Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t-tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin and Review, 16, 225-237

#https://jimgrange.wordpress.com/2015/11/27/animating-robustness-check-of-bayes-factor-priors/
### plots of prior robustness check

# gif created from PNG plots using http://gifmaker.me/

# clear R's memory
rm(list = ls())

# load the Bayes factor package
library(BayesFactor)

# set working directory (will be used to save files here, so make sure
# this is where you want to save your plots!)
setwd <- "D:/Work//Blog_YouTube code//Blog/Prior Robust Visualisation/plots"

#------------------------------------------------------------------------------
### declare some variables for the analysis

# what is the t-value for the data?
tVal <- 3.098

# how many points in the prior should be explored?
nPoints <- 1000

# what Cauchy rates should be explored?
cauchyRates <- seq(from = 0.01, to = 1.5, length.out = nPoints)

# what effect sizes should be plotted?
effSize <- seq(from = -2, to = 2, length.out = nPoints)

# get the Bayes factor for each prior value
bayesFactors <- sapply(cauchyRates, function(x) 
  exp(ttest.tstat(t = tVal, n1 = 76, rscale = x)[['bf']]))
#------------------------------------------------------------------------------

#------------------------------------------------------------------------------
### do the plotting

# how many plots do we want to produce?
nPlots <- 50
plotWidth <- round(seq(from = 1, to = nPoints, length.out = nPlots), 0)

# loop over each plot
for(i in plotWidth){

  # set up the file
  currFile <- paste(getwd(), "/plot_", i, ".png", sep = "")
  
  # initiate the png file
  png(file = currFile, width = 1200, height = 1000, res = 200)
  
  # change the plotting window so plots appear side-by-side
  par(mfrow = c(1, 2))
    
  #----
  # do the prior density plot
  d <- dcauchy(effSize, scale = cauchyRates[i]) 
  plot(effSize, d, type = "l", ylim = c(0, 5), xlim = c(-2, 2), 
       ylab = "Density", xlab = "Effect Size (d)", lwd = 2, col = "gray48", 
       main = paste("Rate (r) = ", round(cauchyRates[i], 3), sep = ""))
  
  #-----
  # do the Bayes factor plot
  plot(cauchyRates, bayesFactors, type = "l", lwd = 2, col = "gray48", 
       ylim = c(0, max(bayesFactors)), xaxt = "n", 
                xlab = "Cauchy Prior Width (r)", ylab = "Bayes Factor (10)")
  abline(h = 0, lwd = 1)
  abline(h = 6, col = "black", lty = 2, lwd = 2)
  axis(1, at = seq(0, 1.5, 0.25))
    
  # add the BF at the default Cauchy point
  points(0.707, 9.97, col = "black", cex = 1.5, pch = 21, bg = "red")
    
  # add the BF for the Cauchy prior currently being plotted
  points(cauchyRates[i], bayesFactors[i], col = "black", pch = 21, cex = 1.3, 
         bg = "cyan")
    
  # add legend
  legend(x = 0.25, y = 3, legend = c("r = 0.707", paste("r = ", 
                                                      round(cauchyRates[i], 3), 
                                                            sep = "")),
         pch = c(21, 21), lty = c(NA, NA), lwd = c(NA, NA), pt.cex = c(1, 1),
         col = c("black", "black"), pt.bg = c("red", "cyan"), bty = "n")

  # save the current plot
  dev.off()
  
}
#------------------------------------------------------------------------------


skew(x, na.rm = TRUE)
kurtosi(x, na.rm = TRUE)

#x A data.frame or matrix
#na.rm how to treat missing data

## The function is currently defined as
function (x, na.rm = TRUE) 
{
    if (length(dim(x)) == 0) {
        if (na.rm) {
            x <- x[!is.na(x)]
        			}
        
        mx <- mean(x)
         sdx <- sd(x,na.rm=na.rm)
        kurt <- sum((x - mx)^4)/(length(x) * sd(x)^4)  -3
        } else {
    
    kurt <- rep(NA,dim(x)[2])
    mx <- colMeans(x,na.rm=na.rm)
     sdx <- sd(x,na.rm=na.rm)
    for (i in 1:dim(x)[2]) {
    kurt[i] <- sum((x[,i] - mx[i])^4,  na.rm = na.rm)/((length(x[,i]) - sum(is.na(x[,i]))) * sdx[i]^4)  -3
            }
    }
    return(kurt)
  }

#https://richarddmorey.github.io/BayesFactor/
data(ToothGrowth)

## Example plot from ?ToothGrowth

coplot(len ~ dose | supp, data = ToothGrowth, panel = panel.smooth,
       xlab = "ToothGrowth data: length vs dose, given type of supplement")
bf = anovaBF(len ~ supp*dose, data=ToothGrowth)
bf
plot(bf[3:4] / bf[2])

#reduce number of comparisons (alpha inflation)
bf = anovaBF(len ~ supp*dose, data=ToothGrowth, whichModels="top")
bf

bfMainEffects = lmBF(len ~ supp + dose, data = ToothGrowth)
bfInteraction = lmBF(len ~ supp + dose + supp:dose, data = ToothGrowth)
## Compare the two models
bf = bfInteraction / bfMainEffects
bf
##
newbf = recompute(bf, iterations = 500000)
newbf

# posterior function
## Sample from the posterior of the full model
chains = posterior(bfInteraction, iterations = 10000)
## 1:13 are the only "interesting" parameters
summary(chains[,1:13])
#lot posterior of selected effects
plot(chains[,4:6])

Comment
Reduce number of comparisons by specifyig the xact model of interest a priori.

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Markov chain Monte Carlo methods
MCMC 1GBEST RBEST - Bayesian MCMC power analysis
# MODIFIED FROM BEST.R FOR ONE GROUP INSTEAD OF TWO.
# Version of Dec 02, 2015.
# John K. Kruschke  
# johnkruschke@gmail.com
# http://www.indiana.edu/~kruschke/BEST/
#
# This program is believed to be free of errors, but it comes with no guarantee! 
# The user bears all responsibility for interpreting the results. 
# Please check the webpage above for updates or corrections.
#
### ***************************************************************
### ******** SEE FILE BEST1Gexample.R FOR INSTRUCTIONS ************
### ***************************************************************

source("openGraphSaveGraph.R") # graphics functions for Windows, Mac OS, Linux

BEST1Gmcmc = function( y , numSavedSteps=100000, thinSteps=1, showMCMC=FALSE) { 
  # This function generates an MCMC sample from the posterior distribution.
  # Description of arguments:
  # showMCMC is a flag for displaying diagnostic graphs of the chains.
  #    If F (the default), no chain graphs are displayed. If T, they are.
  
  require(rjags)

  #------------------------------------------------------------------------------
  # THE MODEL.
  modelString = "
  model {
    for ( i in 1:Ntotal ) {
      y[i] ~ dt( mu , tau , nu )
    }
    mu ~ dnorm( muM , muP )
    tau <- 1/pow( sigma , 2 )
    sigma ~ dunif( sigmaLow , sigmaHigh )
    nu ~ dexp(1/30)
}
  " # close quote for modelString
  # Write out modelString to a text file
  writeLines( modelString , con="BESTmodel.txt" )
  
  #------------------------------------------------------------------------------
  # THE DATA.
  # Load the data:
  Ntotal = length(y)
  # Specify the data in a list, for later shipment to JAGS:
  dataList = list(
    y = y ,
    Ntotal = Ntotal ,
    muM = mean(y) ,
    muP = 0.000001 * 1/sd(y)^2 ,
    sigmaLow = sd(y) / 1000 ,
    sigmaHigh = sd(y) * 1000 
  )
  
  #------------------------------------------------------------------------------
  # INTIALIZE THE CHAINS.
  # Initial values of MCMC chains based on data:
  mu = mean(y) 
  sigma = sd(y) 
  # Regarding initial values in next line: (1) sigma will tend to be too big if 
  # the data have outliers, and (2) nu starts at 5 as a moderate value. These
  # initial values keep the burn-in period moderate.
  initsList = list( mu = mu , sigma = sigma , nu = 5 )
  
  #------------------------------------------------------------------------------
  # RUN THE CHAINS

  parameters = c( "mu" , "sigma" , "nu" )     # The parameters to be monitored
  adaptSteps = 500               # Number of steps to "tune" the samplers
  burnInSteps = 1000
  nChains = 3 
  nIter = ceiling( ( numSavedSteps * thinSteps ) / nChains )
  # Create, initialize, and adapt the model:
  jagsModel = jags.model( "BESTmodel.txt" , data=dataList , inits=initsList , 
                          n.chains=nChains , n.adapt=adaptSteps )
  # Burn-in:
  cat( "Burning in the MCMC chain...\n" )
  update( jagsModel , n.iter=burnInSteps )
  # The saved MCMC chain:
  cat( "Sampling final MCMC chain...\n" )
  codaSamples = coda.samples( jagsModel , variable.names=parameters , 
                              n.iter=nIter , thin=thinSteps )
  # resulting codaSamples object has these indices: 
  #   codaSamples[[ chainIdx ]][ stepIdx , paramIdx ]
  
  #------------------------------------------------------------------------------
  # EXAMINE THE RESULTS
  if ( showMCMC ) {
    openGraph(width=7,height=7)
    autocorr.plot( codaSamples[[1]] , ask=FALSE )
    show( gelman.diag( codaSamples ) )
    effectiveChainLength = effectiveSize( codaSamples ) 
    show( effectiveChainLength )
  }

  # Convert coda-object codaSamples to matrix object for easier handling.
  # But note that this concatenates the different chains into one long chain.
  # Result is mcmcChain[ stepIdx , paramIdx ]
  mcmcChain = as.matrix( codaSamples )
  return( mcmcChain )

} # end function BESTmcmc

#==============================================================================

BEST1Gplot = function( y , mcmcChain , compValm=0.0 , ROPEm=NULL , 
                       compValsd=NULL , ROPEsd=NULL , 
                       ROPEeff=NULL , showCurve=FALSE , pairsPlot=FALSE ) {
  # This function plots the posterior distribution (and data).
  # Description of arguments:
  # y is the data vector.
  # mcmcChain is a list of the type returned by function BEST1G.
  # compValm is hypothesized mean for comparing with estimated mean.
  # ROPEm is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the mean.
  # ROPEsd is a two element vector, such as c(14,16), specifying the limit
  #   of the ROPE on the difference of standard deviations.
  # ROPEeff is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the effect size.
  # showCurve is TRUE or FALSE and indicates whether the posterior should
  #   be displayed as a histogram (by default) or by an approximate curve.
  # pairsPlot is TRUE or FALSE and indicates whether scatterplots of pairs
  #   of parameters should be displayed.
  mu = mcmcChain[,"mu"]
  sigma = mcmcChain[,"sigma"]
  nu = mcmcChain[,"nu"]
  
  if ( pairsPlot ) {
    # Plot the parameters pairwise, to see correlations:
    openGraph(width=7,height=7)
    nPtToPlot = 1000
    plotIdx = floor(seq(1,length(mu),by=length(mu)/nPtToPlot))
    panel.cor = function(x, y, digits=2, prefix="", cex.cor, ...) {
      usr = par("usr"); on.exit(par(usr))
      par(usr = c(0, 1, 0, 1))
      r = (cor(x, y))
      txt = format(c(r, 0.123456789), digits=digits)[1]
      txt = paste(prefix, txt, sep="")
      if(missing(cex.cor)) cex.cor <- 0.8/strwidth(txt)
      text(0.5, 0.5, txt, cex=1.25 ) # was cex=cex.cor*r
    }
    pairs( cbind( mu , sigma , log10(nu) )[plotIdx,] ,
           labels=c( expression(mu) ,  
                     expression(sigma) , 
                     expression(log10(nu)) ) , 
           lower.panel=panel.cor , col="skyblue" )
  }

  # Define matrix for storing summary info:
  summaryInfo = NULL
  
  source("plotPost.R")
  # Set up window and layout:
  openGraph(width=6.0,height=5.0)
  layout( matrix( c(3,3,4,4,5,5, 1,1,1,1,2,2) , nrow=6, ncol=2 , byrow=FALSE ) )
  par( mar=c(3.5,3.5,2.5,0.5) , mgp=c(2.25,0.7,0) )
  
  # Select thinned steps in chain for plotting of posterior predictive curves:
  chainLength = NROW( mcmcChain )
  nCurvesToPlot = 30
  stepIdxVec = seq( 1 , chainLength , floor(chainLength/nCurvesToPlot) )
  xRange = range( y )
  xLim = c( xRange[1]-0.1*(xRange[2]-xRange[1]) , 
            xRange[2]+0.1*(xRange[2]-xRange[1]) )
  xVec = seq( xLim[1] , xLim[2] , length=200 )
  maxY = max( dt( 0 , df=max(nu[stepIdxVec]) ) / min(sigma[stepIdxVec]) )
  # Plot data y and smattering of posterior predictive curves:
  stepIdx = 1
  plot( xVec , dt( (xVec-mu[stepIdxVec[stepIdx]])/sigma[stepIdxVec[stepIdx]] , 
                   df=nu[stepIdxVec[stepIdx]] )/sigma[stepIdxVec[stepIdx]] , 
        ylim=c(0,maxY) , cex.lab=1.75 ,
        type="l" , col="skyblue" , lwd=1 , xlab="y" , ylab="p(y)" , 
        main="Data w. Post. Pred." )
  for ( stepIdx in 2:length(stepIdxVec) ) {
    lines(xVec, dt( (xVec-mu[stepIdxVec[stepIdx]])/sigma[stepIdxVec[stepIdx]] , 
                    df=nu[stepIdxVec[stepIdx]] )/sigma[stepIdxVec[stepIdx]] , 
          type="l" , col="skyblue" , lwd=1 )
  }
  histBinWd = median(sigma)/2
  histCenter = mean(mu)
  histBreaks = sort( c( seq( histCenter-histBinWd/2 , min(xVec)-histBinWd/2 ,
                             -histBinWd ),
                        seq( histCenter+histBinWd/2 , max(xVec)+histBinWd/2 ,
                             histBinWd ) , xLim ) )
  histInfo = hist( y , plot=FALSE , breaks=histBreaks )
  yPlotVec = histInfo$density 
  yPlotVec[ yPlotVec==0.0 ] = NA
  xPlotVec = histInfo$mids
  xPlotVec[ yPlotVec==0.0 ] = NA
  points( xPlotVec , yPlotVec , type="h" , lwd=3 , col="red" )
  text( max(xVec) , maxY , bquote(N==.(length(y))) , adj=c(1.1,1.1) )
  
  # Plot posterior distribution of parameter nu:
  paramInfo = plotPost( log10(nu) , col="skyblue" , # breaks=30 ,
                       showCurve=showCurve ,
                       xlab=bquote("log10("*nu*")") , cex.lab = 1.75 , showMode=TRUE ,
                       main="Normality" ) #  (<0.7 suggests kurtosis)
  summaryInfo = rbind( summaryInfo , paramInfo )
  rownames(summaryInfo)[NROW(summaryInfo)] = "log10(nu)"
  
  # Plot posterior distribution of parameters mu1, mu2, and their difference:
  xlim = range( c( mu , compValm ) )
  paramInfo = plotPost( mu ,  xlim=xlim , cex.lab = 1.75 , ROPE=ROPEm ,
                       showCurve=showCurve , compVal = compValm ,
                       xlab=bquote(mu) , main=paste("Mean") , 
                       col="skyblue" )
  summaryInfo = rbind( summaryInfo , paramInfo )
  rownames(summaryInfo)[NROW(summaryInfo)] = "mu"
  
  # Plot posterior distribution of sigma:
  xlim=range( sigma )
  paramInfo = plotPost( sigma ,  xlim=xlim , cex.lab = 1.75 , ROPE=ROPEsd ,
                       showCurve=showCurve , compVal=compValsd ,
                       xlab=bquote(sigma) , main=paste("Std. Dev.") , 
                       col="skyblue" , showMode=TRUE )
  summaryInfo = rbind( summaryInfo , paramInfo )
  rownames(summaryInfo)[NROW(summaryInfo)] = "sigma"
  
  # Plot of estimated effect size:
  effectSize = ( mu - compValm ) / sigma
  paramInfo = plotPost( effectSize , compVal=0 ,  ROPE=ROPEeff ,
                       showCurve=showCurve ,
                       xlab=bquote( (mu-.(compValm)) / sigma ),
                       showMode=TRUE , cex.lab=1.75 , main="Effect Size" , col="skyblue" )
  summaryInfo = rbind( summaryInfo , paramInfo )
  rownames(summaryInfo)[NROW(summaryInfo)] = "effSz"
  
  return( summaryInfo )

  } # end of function BEST1Gplot

#==============================================================================



# MODIFIED 2015 DEC 02.
# John K. Kruschke  
# johnkruschke@gmail.com
# http://www.indiana.edu/~kruschke/BEST/
#
# This program is believed to be free of errors, but it comes with no guarantee! 
# The user bears all responsibility for interpreting the results. 
# Please check the webpage above for updates or corrections.
#
### ***************************************************************
### ******** SEE FILE BESTexample.R FOR INSTRUCTIONS **************
### ***************************************************************

# Load various essential functions:
source("DBDA2E-utilities.R") 

BESTmcmc = function( y1 , y2 , 
                     priorOnly=FALSE , showMCMC=FALSE ,
                     numSavedSteps=20000 , thinSteps=1 ,
                     mu1PriorMean = mean(c(y1,y2)) ,
                     mu1PriorSD = sd(c(y1,y2))*5 ,
                     mu2PriorMean = mean(c(y1,y2)) ,
                     mu2PriorSD = sd(c(y1,y2))*5 ,
                     sigma1PriorMode = sd(c(y1,y2)) ,
                     sigma1PriorSD = sd(c(y1,y2))*5 ,
                     sigma2PriorMode = sd(c(y1,y2)) ,
                     sigma2PriorSD = sd(c(y1,y2))*5 ,
                     nuPriorMean = 30 ,
                     nuPriorSD = 30 ,
                     runjagsMethod=runjagsMethodDefault , 
                     nChains=nChainsDefault ) { 
  # This function generates an MCMC sample from the posterior distribution.
  # Description of arguments:
  # showMCMC is a flag for displaying diagnostic graphs of the chains.
  #    If F (the default), no chain graphs are displayed. If T, they are.
  
  #------------------------------------------------------------------------------
  # THE DATA.
  # Load the data:
  y = c( y1 , y2 ) # combine data into one vector
  x = c( rep(1,length(y1)) , rep(2,length(y2)) ) # create group membership code
  Ntotal = length(y)
  # Specify the data and prior constants in a list, for later shipment to JAGS:
  if ( priorOnly ) {
    dataList = list(
      #  y = y ,
      x = x ,
      Ntotal = Ntotal ,
      mu1PriorMean = mu1PriorMean ,
      mu1PriorSD = mu1PriorSD ,
      mu2PriorMean = mu2PriorMean ,
      mu2PriorSD = mu2PriorSD ,
      Sh1 = gammaShRaFromModeSD( mode=sigma1PriorMode , 
                                 sd=sigma1PriorSD )$shape ,
      Ra1 = gammaShRaFromModeSD( mode=sigma1PriorMode , 
                                 sd=sigma1PriorSD )$rate ,
      Sh2 = gammaShRaFromModeSD( mode=sigma2PriorMode , 
                                 sd=sigma2PriorSD )$shape ,
      Ra2 = gammaShRaFromModeSD( mode=sigma2PriorMode , 
                                 sd=sigma2PriorSD )$rate ,
      ShNu = gammaShRaFromMeanSD( mean=nuPriorMean , sd=nuPriorSD )$shape ,
      RaNu = gammaShRaFromMeanSD( mean=nuPriorMean , sd=nuPriorSD )$rate
    )
  } else {
    dataList = list(
      y = y ,
      x = x ,
      Ntotal = Ntotal ,
      mu1PriorMean = mu1PriorMean ,
      mu1PriorSD = mu1PriorSD ,
      mu2PriorMean = mu2PriorMean ,
      mu2PriorSD = mu2PriorSD ,
      Sh1 = gammaShRaFromModeSD( mode=sigma1PriorMode , 
                                 sd=sigma1PriorSD )$shape ,
      Ra1 = gammaShRaFromModeSD( mode=sigma1PriorMode , 
                                 sd=sigma1PriorSD )$rate ,
      Sh2 = gammaShRaFromModeSD( mode=sigma2PriorMode , 
                                 sd=sigma2PriorSD )$shape ,
      Ra2 = gammaShRaFromModeSD( mode=sigma2PriorMode , 
                                 sd=sigma2PriorSD )$rate ,
      ShNu = gammaShRaFromMeanSD( mean=nuPriorMean , sd=nuPriorSD )$shape ,
      RaNu = gammaShRaFromMeanSD( mean=nuPriorMean , sd=nuPriorSD )$rate
    )
  }
  
  #----------------------------------------------------------------------------
  # THE MODEL.
  modelString = "
  model {
    for ( i in 1:Ntotal ) {
      y[i] ~ dt( mu[x[i]] , 1/sigma[x[i]]^2 , nu )
    }
    mu[1] ~ dnorm( mu1PriorMean , 1/mu1PriorSD^2 )  # prior for mu[1]
    sigma[1] ~ dgamma( Sh1 , Ra1 )     # prior for sigma[1]
    mu[2] ~ dnorm( mu2PriorMean , 1/mu2PriorSD^2 )  # prior for mu[2]
    sigma[2] ~ dgamma( Sh2 , Ra2 )     # prior for sigma[2]
    nu ~ dgamma( ShNu , RaNu ) # prior for nu
  }
  " # close quote for modelString
  # Write out modelString to a text file
  writeLines( modelString , con="BESTmodel.txt" )
  
  #------------------------------------------------------------------------------
  # INTIALIZE THE CHAINS.
  # Initial values of MCMC chains based on data:
  mu = c( mean(y1) , mean(y2) )
  sigma = c( sd(y1) , sd(y2) )
  # Regarding initial values in next line: (1) sigma will tend to be too big if 
  # the data have outliers, and (2) nu starts at 5 as a moderate value. These
  # initial values keep the burn-in period moderate.
  initsList = list( mu = mu , sigma = sigma , nu = 5 )
  
  #------------------------------------------------------------------------------
  # RUN THE CHAINS
  
  parameters = c( "mu" , "sigma" , "nu" )     # The parameters to be monitored
  adaptSteps = 500               # Number of steps to "tune" the samplers
  burnInSteps = 1000
  
  runJagsOut <- run.jags( method=runjagsMethod , model="BESTmodel.txt" , monitor=parameters , data=dataList , inits=initsList , n.chains=nChains , adapt=adaptSteps , burnin=burnInSteps , sample=ceiling(numSavedSteps/nChains) , thin=thinSteps , summarise=FALSE , plots=FALSE ) codaSamples = as.mcmc.list( runJagsOut ) # resulting codaSamples object has these indices: # codaSamples[[ chainIdx ]][ stepIdx , paramIdx ] ## Original version, using rjags: # nChains = 3 # nIter = ceiling( ( numSavedSteps * thinSteps ) / nChains ) # # Create, initialize, and adapt the model: # jagsModel = jags.model( "BESTmodel.txt" , data=dataList , inits=initsList , # n.chains=nChains , n.adapt=adaptSteps ) # # Burn-in: # cat( "Burning in the MCMC chain...\n" ) # update( jagsModel , n.iter=burnInSteps ) # # The saved MCMC chain: # cat( "Sampling final MCMC chain...\n" ) # codaSamples = coda.samples( jagsModel , variable.names=parameters , # n.iter=nIter , thin=thinSteps ) #------------------------------------------------------------------------------ # EXAMINE THE RESULTS if ( showMCMC ) { for ( paramName in varnames(codaSamples) ) { diagMCMC( codaSamples , parName=paramName ) } } # Convert coda-object codaSamples to matrix object for easier handling. # But note that this concatenates the different chains into one long chain. # Result is mcmcChain[ stepIdx , paramIdx ] mcmcChain = as.matrix( codaSamples ) return( mcmcChain ) } # end function BESTmcmc #============================================================================== BESTsummary = function( y1 , y2 , mcmcChain ) { #source("HDIofMCMC.R") # in DBDA2E-utilities.R mcmcSummary = function( paramSampleVec , compVal=NULL ) { meanParam = mean( paramSampleVec ) medianParam = median( paramSampleVec ) dres = density( paramSampleVec ) modeParam = dres$x[which.max(dres$y)] hdiLim = HDIofMCMC( paramSampleVec ) if ( !is.null(compVal) ) { pcgtCompVal = ( 100 * sum( paramSampleVec > compVal ) 
                      / length( paramSampleVec ) )
    } else {
      pcgtCompVal=NA
    }
    return( c( meanParam , medianParam , modeParam , hdiLim , pcgtCompVal ) )
  }
  # Define matrix for storing summary info:
  summaryInfo = matrix( 0 , nrow=9 , ncol=6 , dimnames=list(
    PARAMETER=c( "mu1" , "mu2" , "muDiff" , "sigma1" , "sigma2" , "sigmaDiff" ,
                 "nu" , "nuLog10" , "effSz" ),
    SUMMARY.INFO=c( "mean" , "median" , "mode" , "HDIlow" , "HDIhigh" ,
                    "pcgtZero" ) 
  ) )
  summaryInfo[ "mu1" , ] = mcmcSummary( mcmcChain[,"mu[1]"] )
  summaryInfo[ "mu2" , ] = mcmcSummary( mcmcChain[,"mu[2]"] )
  summaryInfo[ "muDiff" , ] = mcmcSummary( mcmcChain[,"mu[1]"]
                                           - mcmcChain[,"mu[2]"] , 
                                           compVal=0 )
  summaryInfo[ "sigma1" , ] = mcmcSummary( mcmcChain[,"sigma[1]"] )
  summaryInfo[ "sigma2" , ] = mcmcSummary( mcmcChain[,"sigma[2]"] )
  summaryInfo[ "sigmaDiff" , ] = mcmcSummary( mcmcChain[,"sigma[1]"]
                                              - mcmcChain[,"sigma[2]"] , 
                                              compVal=0 )
  summaryInfo[ "nu" , ] = mcmcSummary( mcmcChain[,"nu"] )
  summaryInfo[ "nuLog10" , ] = mcmcSummary( log10(mcmcChain[,"nu"]) )
  
  N1 = length(y1)
  N2 = length(y2)
  effSzChain = ( ( mcmcChain[,"mu[1]"] - mcmcChain[,"mu[2]"] ) 
                 / sqrt( ( mcmcChain[,"sigma[1]"]^2 
                           + mcmcChain[,"sigma[2]"]^2 ) / 2 ) ) 
  summaryInfo[ "effSz" , ] = mcmcSummary( effSzChain , compVal=0 )
  # Or, use sample-size weighted version:
  # effSz = ( mu1 - mu2 ) / sqrt( ( sigma1^2 *(N1-1) + sigma2^2 *(N2-1) ) 
  #                               / (N1+N2-2) )
  # Be sure also to change plot label in BESTplot function, below.
  return( summaryInfo )
}

#==============================================================================

BESTplot = function( y1 , y2 , mcmcChain , ROPEm=NULL , ROPEsd=NULL , 
                     ROPEeff=NULL , showCurve=FALSE , pairsPlot=FALSE ) {
  # This function plots the posterior distribution (and data).
  # Description of arguments:
  # y1 and y2 are the data vectors.
  # mcmcChain is a list of the type returned by function BTT.
  # ROPEm is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the difference of means.
  # ROPEsd is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the difference of standard deviations.
  # ROPEeff is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the effect size.
  # showCurve is TRUE or FALSE and indicates whether the posterior should
  #   be displayed as a histogram (by default) or by an approximate curve.
  # pairsPlot is TRUE or FALSE and indicates whether scatterplots of pairs
  #   of parameters should be displayed.
  mu1 = mcmcChain[,"mu[1]"]
  mu2 = mcmcChain[,"mu[2]"]
  sigma1 = mcmcChain[,"sigma[1]"]
  sigma2 = mcmcChain[,"sigma[2]"]
  nu = mcmcChain[,"nu"]
  if ( pairsPlot ) {
    # Plot the parameters pairwise, to see correlations:
    openGraph(width=7,height=7)
    nPtToPlot = 1000
    plotIdx = floor(seq(1,length(mu1),by=length(mu1)/nPtToPlot))
    panel.cor = function(x, y, digits=2, prefix="", cex.cor, ...) {
      usr = par("usr"); on.exit(par(usr))
      par(usr = c(0, 1, 0, 1))
      r = (cor(x, y))
      txt = format(c(r, 0.123456789), digits=digits)[1]
      txt = paste(prefix, txt, sep="")
      if(missing(cex.cor)) cex.cor <- 0.8/strwidth(txt)
      text(0.5, 0.5, txt, cex=1.25 ) # was cex=cex.cor*r
    }
    pairs( cbind( mu1 , mu2 , sigma1 , sigma2 , log10(nu) )[plotIdx,] ,
           labels=c( expression(mu[1]) , expression(mu[2]) , 
                     expression(sigma[1]) , expression(sigma[2]) , 
                     expression(log10(nu)) ) , 
           lower.panel=panel.cor , col="skyblue" )
  }
  # source("plotPost.R") # in DBDA2E-utilities.R
  # Set up window and layout:
  openGraph(width=6.0,height=8.0)
  layout( matrix( c(4,5,7,8,3,1,2,6,9,10) , nrow=5, byrow=FALSE ) )
  par( mar=c(3.5,3.5,2.5,0.5) , mgp=c(2.25,0.7,0) )
  
  # Select thinned steps in chain for plotting of posterior predictive curves:
  chainLength = NROW( mcmcChain )
  nCurvesToPlot = 30
  stepIdxVec = seq( 1 , chainLength , floor(chainLength/nCurvesToPlot) )
  xRange = range( c(y1,y2) )
  if ( isTRUE( all.equal( xRange[2] , xRange[1] ) ) ) {
    meanSigma = mean( c(sigma1,sigma2) )
    xRange = xRange + c( -meanSigma , meanSigma )
  }
  xLim = c( xRange[1]-0.1*(xRange[2]-xRange[1]) , 
            xRange[2]+0.1*(xRange[2]-xRange[1]) )
  xVec = seq( xLim[1] , xLim[2] , length=200 )
  maxY = max( dt( 0 , df=max(nu[stepIdxVec]) ) /
                min(c(sigma1[stepIdxVec],sigma2[stepIdxVec])) )
  # Plot data y1 and smattering of posterior predictive curves:
  stepIdx = 1
  plot( xVec , dt( (xVec-mu1[stepIdxVec[stepIdx]])/sigma1[stepIdxVec[stepIdx]] , 
                   df=nu[stepIdxVec[stepIdx]] )/sigma1[stepIdxVec[stepIdx]] , 
        ylim=c(0,maxY) , cex.lab=1.75 ,
        type="l" , col="skyblue" , lwd=1 , xlab="y" , ylab="p(y)" , 
        main="Data Group 1 w. Post. Pred." )
  for ( stepIdx in 2:length(stepIdxVec) ) {
    lines(xVec, dt( (xVec-mu1[stepIdxVec[stepIdx]])/sigma1[stepIdxVec[stepIdx]] , 
                    df=nu[stepIdxVec[stepIdx]] )/sigma1[stepIdxVec[stepIdx]] , 
          type="l" , col="skyblue" , lwd=1 )
  }
  histBinWd = median(sigma1)/2
  histCenter = mean(mu1)
  histBreaks = sort( c( seq( histCenter-histBinWd/2 , min(xVec)-histBinWd/2 ,
                             -histBinWd ),
                        seq( histCenter+histBinWd/2 , max(xVec)+histBinWd/2 ,
                             histBinWd ) , xLim ) )
  histInfo = hist( y1 , plot=FALSE , breaks=histBreaks )
  yPlotVec = histInfo$density 
  yPlotVec[ yPlotVec==0.0 ] = NA
  xPlotVec = histInfo$mids
  xPlotVec[ yPlotVec==0.0 ] = NA
  points( xPlotVec , yPlotVec , type="h" , lwd=3 , col="red" )
  text( max(xVec) , maxY , bquote(N[1]==.(length(y1))) , adj=c(1.1,1.1) )
  # Plot data y2 and smattering of posterior predictive curves:
  stepIdx = 1
  plot( xVec , dt( (xVec-mu2[stepIdxVec[stepIdx]])/sigma2[stepIdxVec[stepIdx]] , 
                   df=nu[stepIdxVec[stepIdx]] )/sigma2[stepIdxVec[stepIdx]] , 
        ylim=c(0,maxY) , cex.lab=1.75 , 
        type="l" , col="skyblue" , lwd=1 , xlab="y" , ylab="p(y)" , 
        main="Data Group 2 w. Post. Pred." )
  for ( stepIdx in 2:length(stepIdxVec) ) {
    lines(xVec, dt( (xVec-mu2[stepIdxVec[stepIdx]])/sigma2[stepIdxVec[stepIdx]] , 
                    df=nu[stepIdxVec[stepIdx]] )/sigma2[stepIdxVec[stepIdx]] , 
          type="l" , col="skyblue" , lwd=1 )
  }
  histBinWd = median(sigma2)/2
  histCenter = mean(mu2)
  histBreaks = sort( c( seq( histCenter-histBinWd/2 , min(xVec)-histBinWd/2 ,
                             -histBinWd ),
                        seq( histCenter+histBinWd/2 , max(xVec)+histBinWd/2 ,
                             histBinWd ) , xLim ) )
  histInfo = hist( y2 , plot=FALSE , breaks=histBreaks )
  yPlotVec = histInfo$density 
  yPlotVec[ yPlotVec==0.0 ] = NA
  xPlotVec = histInfo$mids
  xPlotVec[ yPlotVec==0.0 ] = NA
  points( xPlotVec , yPlotVec , type="h" , lwd=3 , col="red" )
  text( max(xVec) , maxY , bquote(N[2]==.(length(y2))) , adj=c(1.1,1.1) )
  
  # Plot posterior distribution of parameter nu:
  histInfo = plotPost( log10(nu) , col="skyblue" , # breaks=30 ,
                       showCurve=showCurve ,
                       xlab=bquote("log10("*nu*")") , cex.lab = 1.75 , 
                       cenTend=c("mode","median","mean")[1] ,
                       main="Normality" ) #  (<0.7 suggests kurtosis)
  
  # Plot posterior distribution of parameters mu1, mu2, and their difference:
  xlim = range( c( mu1 , mu2 ) )
  histInfo = plotPost( mu1 ,  xlim=xlim , cex.lab = 1.75 ,
                       showCurve=showCurve ,
                       xlab=bquote(mu[1]) , main=paste("Group",1,"Mean") , 
                       col="skyblue" )
  histInfo = plotPost( mu2 ,  xlim=xlim , cex.lab = 1.75 ,
                       showCurve=showCurve ,
                       xlab=bquote(mu[2]) , main=paste("Group",2,"Mean") , 
                       col="skyblue" )
  histInfo = plotPost( mu1-mu2 , compVal=0 ,  showCurve=showCurve ,
                       xlab=bquote(mu[1] - mu[2]) , cex.lab = 1.75 , ROPE=ROPEm ,
                       main="Difference of Means" , col="skyblue" )
  
  # Plot posterior distribution of param's sigma1, sigma2, and their difference:
  xlim=range( c( sigma1 , sigma2 ) )
  histInfo = plotPost( sigma1 ,  xlim=xlim , cex.lab = 1.75 ,
                       showCurve=showCurve ,
                       xlab=bquote(sigma[1]) , main=paste("Group",1,"Std. Dev.") , 
                       col="skyblue" , cenTend=c("mode","median","mean")[1] )
  histInfo = plotPost( sigma2 ,  xlim=xlim , cex.lab = 1.75 ,
                       showCurve=showCurve ,
                       xlab=bquote(sigma[2]) , main=paste("Group",2,"Std. Dev.") , 
                       col="skyblue" , cenTend=c("mode","median","mean")[1] )
  histInfo = plotPost( sigma1-sigma2 , 
                       compVal=0 ,  showCurve=showCurve ,
                       xlab=bquote(sigma[1] - sigma[2]) , cex.lab = 1.75 , 
                       ROPE=ROPEsd ,
                       main="Difference of Std. Dev.s" , col="skyblue" , 
                       cenTend=c("mode","median","mean")[1] )
  
  # Plot of estimated effect size. Effect size is d-sub-a from 
  # Macmillan & Creelman, 1991; Simpson & Fitter, 1973; Swets, 1986a, 1986b.
  effectSize = ( mu1 - mu2 ) / sqrt( ( sigma1^2 + sigma2^2 ) / 2 )
  histInfo = plotPost( effectSize , compVal=0 ,  ROPE=ROPEeff ,
                       showCurve=showCurve ,
                       xlab=bquote( (mu[1]-mu[2])
                                    /sqrt((sigma[1]^2 +sigma[2]^2 )/2 ) ),
                       cenTend=c("mode","median","mean")[1] , cex.lab=1.0 , main="Effect Size" , 
                       col="skyblue" )
  # Or use sample-size weighted version:
  # Hedges 1981; Wetzels, Raaijmakers, Jakab & Wagenmakers 2009.
  # N1 = length(y1)
  # N2 = length(y2)
  # effectSize = ( mu1 - mu2 ) / sqrt( ( sigma1^2 *(N1-1) + sigma2^2 *(N2-1) )
  #                                    / (N1+N2-2) )
  # Be sure also to change BESTsummary function, above.
  # histInfo = plotPost( effectSize , compVal=0 ,  ROPE=ROPEeff ,
  #          showCurve=showCurve ,
  #          xlab=bquote( (mu[1]-mu[2])
  #          /sqrt((sigma[1]^2 *(N[1]-1)+sigma[2]^2 *(N[2]-1))/(N[1]+N[2]-2)) ),
  #          cenTend=c("mode","median","mean")[1] , cex.lab=1.0 , main="Effect Size" , col="skyblue" )
  return( BESTsummary( y1 , y2 , mcmcChain ) )
} # end of function BESTplot

#==============================================================================

BESTpower = function( mcmcChain , N1 , N2 , ROPEm , ROPEsd , ROPEeff ,
                      maxHDIWm , maxHDIWsd , maxHDIWeff , nRep=200 ,
                      mcmcLength=10000 , saveName="BESTpower.Rdata" ,
                      showFirstNrep=0 ) {
  # This function estimates power.
  # Description of arguments:
  # mcmcChain is a matrix of the type returned by function BESTmcmc.
  # N1 and N2 are sample sizes for the two groups.
  # ROPEm is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the difference of means.
  # ROPEsd is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the difference of standard deviations.
  # ROPEeff is a two element vector, such as c(-1,1), specifying the limit
  #   of the ROPE on the effect size.
  # maxHDIWm is the maximum desired width of the 95% HDI on the difference 
  #   of means.
  # maxHDIWsd is the maximum desired width of the 95% HDI on the difference 
  #   of standard deviations.
  # maxHDIWeff is the maximum desired width of the 95% HDI on the effect size.
  # nRep is the number of simulated experiments used to estimate the power.
  #source("HDIofMCMC.R") # in DBDA2E-utilities.R
  #source("HDIofICDF.R") # in DBDA2E-utilities.R
  chainLength = NROW( mcmcChain )
  # Select thinned steps in chain for posterior predictions:
  stepIdxVec = seq( 1 , chainLength , floor(chainLength/nRep) )
  goalTally = list( HDIm.gt.ROPE = c(0) ,
                    HDIm.lt.ROPE = c(0) ,
                    HDIm.in.ROPE = c(0) ,
                    HDIm.wd.max = c(0) ,
                    HDIsd.gt.ROPE = c(0) ,
                    HDIsd.lt.ROPE = c(0) ,
                    HDIsd.in.ROPE = c(0) ,
                    HDIsd.wd.max = c(0) ,
                    HDIeff.gt.ROPE = c(0) ,
                    HDIeff.lt.ROPE = c(0) ,
                    HDIeff.in.ROPE = c(0) ,
                    HDIeff.wd.max = c(0) )
  power     = list( HDIm.gt.ROPE = c(0,0,0) ,
                    HDIm.lt.ROPE = c(0,0,0) ,
                    HDIm.in.ROPE = c(0,0,0) ,
                    HDIm.wd.max = c(0,0,0) ,
                    HDIsd.gt.ROPE = c(0,0,0) ,
                    HDIsd.lt.ROPE = c(0,0,0) ,
                    HDIsd.in.ROPE = c(0,0,0) ,
                    HDIsd.wd.max = c(0,0,0) ,
                    HDIeff.gt.ROPE = c(0,0,0) ,
                    HDIeff.lt.ROPE = c(0,0,0) ,
                    HDIeff.in.ROPE = c(0,0,0) ,
                    HDIeff.wd.max = c(0,0,0) )
  nSim = 0
  for ( stepIdx in stepIdxVec ) {
    nSim = nSim + 1
    cat( "\n:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::\n" )
    cat( paste( "Power computation: Simulated Experiment" , nSim , "of" , 
                length(stepIdxVec) , ":\n\n" ) )
    # Get parameter values for this simulation:
    mu1Val = mcmcChain[stepIdx,"mu[1]"]
    mu2Val = mcmcChain[stepIdx,"mu[2]"]
    sigma1Val = mcmcChain[stepIdx,"sigma[1]"]
    sigma2Val = mcmcChain[stepIdx,"sigma[2]"]
    nuVal = mcmcChain[stepIdx,"nu"]
    # Generate simulated data:
    y1 = rt( N1 , df=nuVal ) * sigma1Val + mu1Val    
    y2 = rt( N2 , df=nuVal ) * sigma2Val + mu2Val    
    # Get posterior for simulated data:
    simChain = BESTmcmc( y1, y2, numSavedSteps=mcmcLength, thinSteps=1, 
                         showMCMC=FALSE )
    if (nSim <= showFirstNrep ) { BESTplot( y1, y2, simChain, ROPEm=ROPEm, ROPEsd=ROPEsd, ROPEeff=ROPEeff ) } # Assess which goals were achieved and tally them: HDIm = HDIofMCMC( simChain[,"mu[1]"] - simChain[,"mu[2]"] ) if ( HDIm[1] > ROPEm[2] ) { 
      goalTally$HDIm.gt.ROPE = goalTally$HDIm.gt.ROPE + 1 }
    if ( HDIm[2] < ROPEm[1] ) { goalTally$HDIm.lt.ROPE = goalTally$HDIm.lt.ROPE + 1 } if ( HDIm[1] > ROPEm[1] &  HDIm[2] < ROPEm[2] ) { 
      goalTally$HDIm.in.ROPE = goalTally$HDIm.in.ROPE + 1 } 
    if ( HDIm[2] - HDIm[1] < maxHDIWm ) { goalTally$HDIm.wd.max = goalTally$HDIm.wd.max + 1 } HDIsd = HDIofMCMC( simChain[,"sigma[1]"] - simChain[,"sigma[2]"] ) if ( HDIsd[1] > ROPEsd[2] ) { 
      goalTally$HDIsd.gt.ROPE = goalTally$HDIsd.gt.ROPE + 1 }
    if ( HDIsd[2] < ROPEsd[1] ) { goalTally$HDIsd.lt.ROPE = goalTally$HDIsd.lt.ROPE + 1 } if ( HDIsd[1] > ROPEsd[1] &  HDIsd[2] < ROPEsd[2] ) { 
      goalTally$HDIsd.in.ROPE = goalTally$HDIsd.in.ROPE + 1 } 
    if ( HDIsd[2] - HDIsd[1] < maxHDIWsd ) { goalTally$HDIsd.wd.max = goalTally$HDIsd.wd.max + 1 } HDIeff = HDIofMCMC( ( simChain[,"mu[1]"] - simChain[,"mu[2]"] ) / sqrt( ( simChain[,"sigma[1]"]^2 + simChain[,"sigma[2]"]^2 ) / 2 ) ) if ( HDIeff[1] > ROPEeff[2] ) { 
      goalTally$HDIeff.gt.ROPE = goalTally$HDIeff.gt.ROPE + 1 }
    if ( HDIeff[2] < ROPEeff[1] ) { goalTally$HDIeff.lt.ROPE = goalTally$HDIeff.lt.ROPE + 1 } if ( HDIeff[1] > ROPEeff[1] &  HDIeff[2] < ROPEeff[2] ) { 
      goalTally$HDIeff.in.ROPE = goalTally$HDIeff.in.ROPE + 1 } 
    if ( HDIeff[2] - HDIeff[1] < maxHDIWeff ) { goalTally$HDIeff.wd.max = goalTally$HDIeff.wd.max + 1 } for ( i in 1:length(goalTally) ) { s1 = 1 + goalTally[[i]] s2 = 1 + ( nSim - goalTally[[i]] ) power[[i]][1] = s1/(s1+s2) power[[i]][2:3] = HDIofICDF( qbeta , shape1=s1 , shape2=s2 ) } cat( "\nAfter ", nSim , " Simulated Experiments, Power Is: \n( mean, HDI_low, HDI_high )\n" ) for ( i in 1:length(power) ) { cat( names(power[i]) , round(power[[i]],3) , "\n" ) } save( nSim , power , file=saveName ) } return( power ) } # end of function BESTpower #============================================================================== makeData = function( mu1 , sd1 , mu2 , sd2 , nPerGrp , pcntOut=0 , sdOutMult=2.0 , rnd.seed=NULL ) { # Auxilliary function for generating random values from a # mixture of normal distibutions. if(!is.null(rnd.seed)){set.seed(rnd.seed)} # Set seed for random values. nOut = ceiling(nPerGrp*pcntOut/100) # Number of outliers. nIn = nPerGrp - nOut # Number from main distribution. if ( pcntOut > 100 | pcntOut < 0 ) { stop("pcntOut must be between 0 and 100.") } if ( pcntOut > 0 & pcntOut < 1 ) { warning("pcntOut is specified as percentage 0-100, not proportion 0-1.") } if ( pcntOut > 50 ) {
    warning("pcntOut indicates more than 50% outliers; did you intend this?")
  }
  if ( nOut < 2 & pcntOut > 0 ) {
    stop("Combination of nPerGrp and pcntOut yields too few outliers.")
  }
  if ( nIn < 2 ) { stop("Too few non-outliers.") } sdN = function( x ) { sqrt( mean( (x-mean(x))^2 ) ) } # genDat = function( mu , sigma , Nin , Nout , sigmaOutMult=sdOutMult ) { y = rnorm(n=Nin) # Random values in main distribution. y = ((y-mean(y))/sdN(y))*sigma + mu # Translate to exactly realize desired. yOut = NULL if ( Nout > 0 ) {
      yOut = rnorm(n=Nout)                 # Random values for outliers
      yOut=((yOut-mean(yOut))/sdN(yOut))*(sigma*sdOutMult)+mu # Realize exactly.
    }
    y = c(y,yOut)                      # Concatenate main with outliers.
    return(y)
  }
  y1 = genDat( mu=mu1 , sigma=sd1 , Nin=nIn , Nout=nOut )
  y2 = genDat( mu=mu2 , sigma=sd2 , Nin=nIn , Nout=nOut )
  #
  # Set up window and layout:
  openGraph(width=7,height=7) # Plot the data.
  layout(matrix(1:2,nrow=2))
  histInfo = hist( y1 , main="Simulated Data" , col="pink2" , border="white" , 
                   xlim=range(c(y1,y2)) , breaks=30 , prob=TRUE )
  text( max(c(y1,y2)) , max(histInfo$density) , 
        bquote(N==.(nPerGrp)) , adj=c(1,1) )
  xlow=min(histInfo$breaks)
  xhi=max(histInfo$breaks)
  xcomb=seq(xlow,xhi,length=1001)
  lines( xcomb , dnorm(xcomb,mean=mu1,sd=sd1)*nIn/(nIn+nOut) +
           dnorm(xcomb,mean=mu1,sd=sd1*sdOutMult)*nOut/(nIn+nOut) , lwd=3 )
  lines( xcomb , dnorm(xcomb,mean=mu1,sd=sd1)*nIn/(nIn+nOut) , 
         lty="dashed" , col="green", lwd=3)
  lines( xcomb , dnorm(xcomb,mean=mu1,sd=sd1*sdOutMult)*nOut/(nIn+nOut) , 
         lty="dashed" , col="red", lwd=3)
  histInfo = hist( y2 , main="" , col="pink2" , border="white" , 
                   xlim=range(c(y1,y2)) , breaks=30 , prob=TRUE )
  text( max(c(y1,y2)) , max(histInfo$density) , 
        bquote(N==.(nPerGrp)) , adj=c(1,1) )
  xlow=min(histInfo$breaks)
  xhi=max(histInfo$breaks)
  xcomb=seq(xlow,xhi,length=1001)
  lines( xcomb , dnorm(xcomb,mean=mu2,sd=sd2)*nIn/(nIn+nOut) +
           dnorm(xcomb,mean=mu2,sd=sd2*sdOutMult)*nOut/(nIn+nOut) , lwd=3)
  lines( xcomb , dnorm(xcomb,mean=mu2,sd=sd2)*nIn/(nIn+nOut) , 
         lty="dashed" , col="green", lwd=3)
  lines( xcomb , dnorm(xcomb,mean=mu2,sd=sd2*sdOutMult)*nOut/(nIn+nOut) , 
         lty="dashed" , col="red", lwd=3)
  #
  return( list( y1=y1 , y2=y2 ) )
}

#==============================================================================


# Version of May 26, 2012. Re-checked on 2015 May 08.
# John K. Kruschke  
# johnkruschke@gmail.com
# http://www.indiana.edu/~kruschke/BEST/
#
# This program is believed to be free of errors, but it comes with no guarantee! 
# The user bears all responsibility for interpreting the results. 
# Please check the webpage above for updates or corrections.
#
### ***************************************************************
### ******** SEE FILE BESTexample.R FOR INSTRUCTIONS **************
### ***************************************************************

# OPTIONAL: Clear R's memory and graphics:
rm(list=ls())  # Careful! This clears all of R's memory!
graphics.off() # This closes all of R's graphics windows.

# Get the functions loaded into R's working memory:
source("BEST.R")

#-------------------------------------------------------------------------------
# RETROSPECTIVE POWER ANALYSIS.
# !! This section assumes you have already run BESTexample.R !!
# Re-load the saved data and MCMC chain from the previously conducted 
# Bayesian analysis. This re-loads the variables y1, y2, mcmcChain, etc.
load( "BESTexampleMCMC.Rdata" )
power = BESTpower( mcmcChain , N1=length(y1) , N2=length(y2) , 
                  ROPEm=c(-0.1,0.1) , ROPEsd=c(-0.1,0.1) , ROPEeff=c(-0.1,0.1) , 
                  maxHDIWm=2.0 , maxHDIWsd=2.0 , maxHDIWeff=0.2 , nRep=1000 ,
                  mcmcLength=10000 , saveName = "BESTexampleRetroPower.Rdata" ) 

#-------------------------------------------------------------------------------
# PROSPECTIVE POWER ANALYSIS, using fictitious strong data.
# Generate large fictitious data set that expresses hypothesis:
prospectData = makeData( mu1=108, sd1=17, mu2=100, sd2=15, nPerGrp=1000, 
                         pcntOut=10, sdOutMult=2.0, rnd.seed=NULL )
y1pro = prospectData$y1 # Merely renames simulated data for convenience below.
y2pro = prospectData$y2 # Merely renames simulated data for convenience below.
# Generate Bayesian posterior distribution from fictitious data:
# (uses fewer than usual MCMC steps because it only needs nRep credible 
# parameter combinations, not a high-resolution representation)
mcmcChainPro = BESTmcmc( y1pro , y2pro , numSavedSteps=2000 )  
postInfoPro = BESTplot( y1pro , y2pro , mcmcChainPro , pairsPlot=TRUE )  
save( y1pro, y2pro, mcmcChainPro, postInfoPro, 
      file="BESTexampleProPowerMCMC.Rdata" )
# Now compute the prospective power for planned sample sizes:
N1plan = N2plan = 50 # specify planned sample size
powerPro = BESTpower( mcmcChainPro , N1=N1plan , N2=N2plan , showFirstNrep=5 ,
                   ROPEm=c(-1.5,1.5) , ROPEsd=c(-0.0,0.0) , ROPEeff=c(-0.0,0.0) , 
                   maxHDIWm=15.0 , maxHDIWsd=10.0 , maxHDIWeff=1.0 , nRep=1000 ,
                   mcmcLength=10000 , saveName = "BESTexampleProPower.Rdata" ) 

#-------------------------------------------------------------------------------
Go to top
Various plots
BeanplotsDensity and Rug plotDirac function - Interdisciplinarity versus OverspecialisationGoogle TrendsJASP prior & posterior plot
#https://cran.r-project.org/web/packages/beanplot/beanplot.pdf
par( mfrow = c( 1, 4 ) )
with(dataexp2, beanplot(v00, ylim = c(0,10), col="lightgray", main = "v00", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp2, beanplot(v01, ylim = c(0,10), col="lightgray", main = "v01", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp2, beanplot(v10, ylim = c(0,10), col="darkgray", main = "v10", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp2, beanplot(v11, ylim = c(0,10), col="darkgray", main = "v11", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))


#with ylim and titles
with(dataexp2, beanplot(v00, ylab = "VAS Brightness rating", xlab = "beanplot showing distribution and mean", 
ylim = c(0,10), col="darkgray", main = "v00", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, 
overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))




install.packages ("beanplot")
library("beanplot")
par( mfrow = c( 1, 2 ) )
with(dataexp2, beanplot(v00, col="darkgray", main = "v00", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp2, beanplot(v01, col="darkgray", main = "v01", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
par( mfrow = c( 1, 2 ) )
with(dataexp2, beanplot(v10, col="darkgray", main = "v10", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp2, beanplot(v11, col="darkgray", main = "v11", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))

stripchart(variable ~ factor, vertical=TRUE, method="stack", 
  ylab="variable", data=StackedData)
  
#two sided
with(dataexp1, beanplot(v00, v10, col="darkgray", main = "v00 vs. v10", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "both", jitter = NULL, beanlinewd = 2))

#verticalpar( mfrow = c( 2, 1 ) )
install.packages ("beanplot")
library("beanplot")
par( mfrow = c( 2, 1 ) ) # layout matrix

with(dataexp1, beanplot(v00, v10, main = "v00 vs. v10", kernel = "gaussian", cut = 3, col = list("black", c("grey", "white")), xlab="VAS",
cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = T, side = "both", jitter = NULL, beanlinewd = 2))

with(dataexp1, beanplot(v01, v11, main = "v01 vs. v11", kernel = "gaussian", cut = 3, col = list("black", c("grey", "white")), xlab="VAS",
cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = T, side = "both", jitter = NULL, beanlinewd = 2))


#exp1
dataexp1 <- 
  read.table("http://www.irrational-decisions.com/phd-thesis/dataexp1.csv",
   header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
   
par( mfrow = c( 1, 2) )
with(dataexp1, beanplot(v00, ylim = c(0,7), col="lightgray", main = "v00", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp1, beanplot(v10, ylim = c(0,7), col="lightgray", main = "v01", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))

par( mfrow = c( 1, 2) )
with(dataexp1, beanplot(v01, ylim = c(3,10), col="darkgray", main = "v10", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))
with(dataexp1, beanplot(v11, ylim = c(3,10), col="darkgray", main = "v11", kernel = "gaussian", cut = 3, cutmin = -Inf, cutmax = Inf, overallline = "mean", horizontal = FALSE, side = "no", jitter = NULL, beanlinewd = 2))



#histogram and rug plot

plot(density(dataexp2$v00))
rug(dataexp2$v00, col=2, lwd=3.5)

##############

library(MASS)
plot(density(dataexp2$v00, bw=5)) 
rug(dataexp2$v00+ rnorm(length(dataexp2$v00), sd=5), col=2, lwd=3.5)


x = c(5)
y   = c(9)
plot(x,y,type="h",xlim=c(0,10),ylim=c(0,10),lwd=2,col="blue",ylab="knowledge", xlab="diversity")


install.packages("gtrendsR")
library(gtrendsR)
gtrends(c("bayesian inference", "p-value"), time = "all") # since 2004

res <- gtrends(c("Bayesian inference", "Markov Chain Monte Carlo", "ANOVA"), geo = c("US", "GB", "DE", "CA",))

res <- gtrends("Markov Chain Monte Carlo", geo = c("US", "GB", "DE"))

plot(res)

#get vars

length(res)
str(res)
res$interest_over_time
ff<-res$interest_over_time

scatterplot(ff$hits~ff$date| geo, reg.line=FALSE, smooth=FALSE, spread=FALSE, boxplots=FALSE, span=0.5, ellipse=FALSE, levels=c(.5, .9), by.groups=TRUE, data=ff)

res2 <- gtrends(c("p-value", "bayes factor"))


library(plotrix)

.likelihoodShiftedT <- function(par, data) {
    
    -sum(log(dt((data - par[1])/par[2], par[3])/par[2]))
    
}

.dposteriorShiftedT <- function(x, parameters, oneSided) { # function that returns the posterior density if (oneSided == FALSE) { dt((x - parameters[1])/parameters[2], parameters[3])/parameters[2] } else if (oneSided == "right") { ifelse(x >= 0, (dt((x - parameters[1])/parameters[2], parameters[3])/parameters[2])/pt((0 - 
            parameters[1])/parameters[2], parameters[3], lower.tail = FALSE), 
            0)
        
    } else if (oneSided == "left") {
        
        ifelse(x <= 0, (dt((x - parameters[1])/parameters[2], parameters[3])/parameters[2])/pt((0 - 
            parameters[1])/parameters[2], parameters[3], lower.tail = TRUE), 
            0)
        
    }
    
}

.dprior <- function(x, r, oneSided = oneSided) {
    
    # function that returns the prior density
    
    if (oneSided == "right") {
        
        y <- ifelse(x < 0, 0, 2/(pi * r * (1 + (x/r)^2)))
        return(y)
    }
    
    if (oneSided == "left") {
        
        y <- ifelse(x > 0, 0, 2/(pi * r * (1 + (x/r)^2)))
        return(y)
    } else {
        
        return(1/(pi * r * (1 + (x/r)^2)))
    }
}

.plotPosterior.ttest <- function(x = NULL, y = NULL, paired = FALSE, oneSided = FALSE, 
    BF, BFH1H0 = TRUE, iterations = 10000, rscale = "medium", lwd = 2, 
    cexPoints = 1.5, cexAxis = 1.2, cexYlab = 1.5, cexXlab = 1.5, cexTextBF = 1.4, 
    cexCI = 1.1, cexLegend = 1.2, lwdAxis = 1.2, addInformation = TRUE, 
    dontPlotData = FALSE) {
    
    if (addInformation) {
        
        par(mar = c(5.6, 5, 7, 4) + 0.1, las = 1)
        
    } else {
        
        par(mar = c(5.6, 5, 4, 4) + 0.1, las = 1)
    }
    
    if (dontPlotData) {
        
        plot(1, type = "n", xlim = 0:1, ylim = 0:1, bty = "n", axes = FALSE, 
            xlab = "", ylab = "")
        
        axis(1, at = 0:1, labels = FALSE, cex.axis = cexAxis, lwd = lwdAxis, 
            xlab = "")
        axis(2, at = 0:1, labels = FALSE, cex.axis = cexAxis, lwd = lwdAxis, 
            ylab = "")
        
        mtext(text = "Density", side = 2, las = 0, cex = cexYlab, line = 3.25)
        mtext(expression(paste("Effect size", ~delta)), side = 1, cex = cexXlab, 
            line = 2.5)
        
        return()
    }
    
    if (rscale == "medium") {
        r <- sqrt(2)/2
    }
    if (rscale == "wide") {
        r <- 1
    }
    if (rscale == "ultrawide") {
        r <- sqrt(2)
    }
    if (mode(rscale) == "numeric") {
        r <- rscale
    }
    
    if (oneSided == FALSE) {
        nullInterval <- NULL
    }
    if (oneSided == "right") {
        nullInterval <- c(0, Inf)
    }
    if (oneSided == "left") {
        nullInterval <- c(-Inf, 0)
    }
    
    # sample from delta posterior
    samples <- BayesFactor::ttestBF(x = x, y = y, paired = paired, posterior = TRUE, 
        iterations = iterations, rscale = r)
    
    delta <- samples[, "delta"]
    
    # fit shifted t distribution
    if (is.null(y)) {
        
        deltaHat <- mean(x)/sd(x)
        N <- length(x)
        df <- N - 1
        sigmaStart <- 1/N
        
    } else if (paired) {
        
        deltaHat <- mean(x - y)/sd(x - y)
        N <- length(x)
        df <- N - 1
        sigmaStart <- 1/N
        
    } else if (!is.null(y) && !paired) {
        
        N1 <- length(x)
        N2 <- length(y)
        sdPooled <- sqrt(((N1 - 1) * var(x) + (N2 - 1) * var(y))/(N1 + 
            N2))
        deltaHat <- (mean(x) - mean(y))/sdPooled
        df <- N1 + N2 - 2
        sigmaStart <- sqrt(N1 * N2/(N1 + N2))
    }
    
    if (sigmaStart < 0.01) 
        sigmaStart <- 0.01
    
    parameters <- try(silent = TRUE, expr = optim(par = c(deltaHat, sigmaStart, 
        df), fn = .likelihoodShiftedT, data = delta, method = "BFGS")$par)
    
    if (class(parameters) == "try-error") {
        
        parameters <- try(silent = TRUE, expr = optim(par = c(deltaHat, 
            sigmaStart, df), fn = .likelihoodShiftedT, data = delta, method = "Nelder-Mead")$par)
    }
    
    if (BFH1H0) {
        
        BF10 <- BF
        BF01 <- 1/BF10
        
    } else {
        
        BF01 <- BF
        BF10 <- 1/BF01
    }
    
    
    # set limits plot
    xlim <- vector("numeric", 2)
    
    if (oneSided == FALSE) {
        
        xlim[1] <- min(-2, quantile(delta, probs = 0.01)[[1]])
        xlim[2] <- max(2, quantile(delta, probs = 0.99)[[1]])
        
        if (length(x) < 10) {
            
            if (addInformation) {
                
                stretch <- 1.52
            } else {
                
                stretch <- 1.4
            }
            
        } else {
            
            stretch <- 1.2 } } if (oneSided == "right") { # if (length(delta[delta >= 0]) < 10) return('Plotting is not
        # possible: To few posterior samples in tested interval')
        
        xlim[1] <- min(-2, quantile(delta[delta >= 0], probs = 0.01)[[1]])
        xlim[2] <- max(2, quantile(delta[delta >= 0], probs = 0.99)[[1]])
        
        if (any(is.na(xlim))) {
            
            xlim[1] <- min(-2, .qShiftedT(0.01, parameters, oneSided = "right"))
            xlim[2] <- max(2, .qShiftedT(0.99, parameters, oneSided = "right"))
            
        }
        
        stretch <- 1.32
    }
    
    if (oneSided == "left") {
        
        # if (length(delta[delta <= 0]) < 10) return('Plotting is not
        # possible: To few posterior samples in tested interval')
        
        xlim[1] <- min(-2, quantile(delta[delta <= 0], probs = 0.01)[[1]])
        xlim[2] <- max(2, quantile(delta[delta <= 0], probs = 0.99)[[1]])
        
        if (any(is.na(xlim))) {
            
            xlim[1] <- min(-2, .qShiftedT(0.01, parameters, oneSided = "left"))
            xlim[2] <- max(2, .qShiftedT(0.99, parameters, oneSided = "left"))
            
        }
        
        stretch <- 1.32
    }
    
    xticks <- pretty(xlim)
    
    ylim <- vector("numeric", 2)
    
    ylim[1] <- 0
    dmax <- optimize(function(x) .dposteriorShiftedT(x, parameters = parameters, 
        oneSided = oneSided), interval = range(xticks), maximum = TRUE)$objective
    ylim[2] <- max(stretch * .dprior(0, r, oneSided = oneSided), stretch * 
        dmax)  # get maximum density
    
    # calculate position of 'nice' tick marks and create labels
    yticks <- pretty(ylim)
    xlabels <- formatC(xticks, 1, format = "f")
    ylabels <- formatC(yticks, 1, format = "f")
    
    # compute 95% credible interval & median:
    if (oneSided == FALSE) {
        
        CIlow <- quantile(delta, probs = 0.025)[[1]]
        CIhigh <- quantile(delta, probs = 0.975)[[1]]
        medianPosterior <- median(delta)
        
        if (any(is.na(c(CIlow, CIhigh, medianPosterior)))) {
            
            CIlow <- .qShiftedT(0.025, parameters, oneSided = FALSE)
            CIhigh <- .qShiftedT(0.975, parameters, oneSided = FALSE)
            medianPosterior <- .qShiftedT(0.5, parameters, oneSided = FALSE)
        }
    }
    
    if (oneSided == "right") {
        
        CIlow <- quantile(delta[delta >= 0], probs = 0.025)[[1]]
        CIhigh <- quantile(delta[delta >= 0], probs = 0.975)[[1]]
        medianPosterior <- median(delta[delta >= 0])
        
        if (any(is.na(c(CIlow, CIhigh, medianPosterior)))) {
            
            CIlow <- .qShiftedT(0.025, parameters, oneSided = "right")
            CIhigh <- .qShiftedT(0.975, parameters, oneSided = "right")
            medianPosterior <- .qShiftedT(0.5, parameters, oneSided = "right")
        }
    }
    
    if (oneSided == "left") {
        
        CIlow <- quantile(delta[delta <= 0], probs = 0.025)[[1]]
        CIhigh <- quantile(delta[delta <= 0], probs = 0.975)[[1]]
        medianPosterior <- median(delta[delta <= 0])
        
        if (any(is.na(c(CIlow, CIhigh, medianPosterior)))) {
            
            CIlow <- .qShiftedT(0.025, parameters, oneSided = "left")
            CIhigh <- .qShiftedT(0.975, parameters, oneSided = "left")
            medianPosterior <- .qShiftedT(0.5, parameters, oneSided = "left")
        }
        
    }
    
    posteriorLine <- .dposteriorShiftedT(x = seq(min(xticks), max(xticks), 
        length.out = 1000), parameters = parameters, oneSided = oneSided)
    
    xlim <- c(min(CIlow, range(xticks)[1]), max(range(xticks)[2], CIhigh)) plot(1, 1, xlim = xlim, ylim = range(yticks), ylab = "", xlab = "", type = "n", axes = FALSE) lines(seq(min(xticks), max(xticks), length.out = 1000), posteriorLine, lwd = lwd) lines(seq(min(xticks), max(xticks), length.out = 1000), .dprior(seq(min(xticks), max(xticks), length.out = 1000), r = r, oneSided = oneSided), lwd = lwd, lty = 3) axis(1, at = xticks, labels = xlabels, cex.axis = cexAxis, lwd = lwdAxis) axis(2, at = yticks, labels = ylabels, , cex.axis = cexAxis, lwd = lwdAxis) if (nchar(ylabels[length(ylabels)]) > 4) {
        
        mtext(text = "Density", side = 2, las = 0, cex = cexYlab, line = 4)
        
    } else if (nchar(ylabels[length(ylabels)]) == 4) {
        
        mtext(text = "Density", side = 2, las = 0, cex = cexYlab, line = 3.25)
        
    } else if (nchar(ylabels[length(ylabels)]) < 4) {
        
        mtext(text = "Density", side = 2, las = 0, cex = cexYlab, line = 2.85)
        
    }
    
    mtext(expression(paste("Effect size", ~delta)), side = 1, cex = cexXlab, 
        line = 2.5)
    
    points(0, .dprior(0, r, oneSided = oneSided), col = "black", pch = 21, 
        bg = "grey", cex = cexPoints)
    
    if (oneSided == FALSE) {
        
        heightPosteriorAtZero <- .dposteriorShiftedT(0, parameters = parameters, 
            oneSided = oneSided)
        
    } else if (oneSided == "right") {
        
        posteriorLineLargerZero <- posteriorLine[posteriorLine > 0]
        heightPosteriorAtZero <- posteriorLineLargerZero[1]
        
    } else if (oneSided == "left") {
        
        posteriorLineLargerZero <- posteriorLine[posteriorLine > 0]
        heightPosteriorAtZero <- posteriorLineLargerZero[length(posteriorLineLargerZero)]
    }
    
    points(0, heightPosteriorAtZero, col = "black", pch = 21, bg = "grey", 
        cex = cexPoints)
    
    ### 95% credible interval
    
    # enable plotting in margin
    par(xpd = TRUE)
    
    yCI <- grconvertY(dmax, "user", "ndc") + 0.04
    yCI <- grconvertY(yCI, "ndc", "user")
    
    arrows(CIlow, yCI, CIhigh, yCI, angle = 90, code = 3, length = 0.1, 
        lwd = lwd)
    
    medianText <- formatC(medianPosterior, digits = 3, format = "f")
    
    if (addInformation) {
        
        # display BF10 value
        offsetTopPart <- 0.06
        
        yy <- grconvertY(0.75 + offsetTopPart, "ndc", "user")
        yy2 <- grconvertY(0.806 + offsetTopPart, "ndc", "user")
        
        xx <- min(xticks) if (BF10 >= 1000000 | BF01 >= 1000000) {
            
            BF10t <- formatC(BF10, 3, format = "e")
            BF01t <- formatC(BF01, 3, format = "e")
        }
        
        if (BF10 < 1000000 & BF01 < 1000000) {
            
            BF10t <- formatC(BF10, 3, format = "f")
            BF01t <- formatC(BF01, 3, format = "f")
        }
        
        if (oneSided == FALSE) {
            
            text(xx, yy2, bquote(BF[10] == .(BF10t)), cex = cexTextBF, 
                pos = 4)
            text(xx, yy, bquote(BF[0][1] == .(BF01t)), cex = cexTextBF, 
                pos = 4)
        }
        
        if (oneSided == "right") {
            
            text(xx, yy2, bquote(BF["+"][0] == .(BF10t)), cex = cexTextBF, 
                pos = 4)
            text(xx, yy, bquote(BF[0]["+"] == .(BF01t)), cex = cexTextBF, 
                pos = 4)
        }
        
        if (oneSided == "left") {
            
            text(xx, yy2, bquote(BF["-"][0] == .(BF10t)), cex = cexTextBF, 
                pos = 4)
            text(xx, yy, bquote(BF[0]["-"] == .(BF01t)), cex = cexTextBF, 
                pos = 4)
        }
        
        yy <- grconvertY(0.756 + offsetTopPart, "ndc", "user")
        yy2 <- grconvertY(0.812 + offsetTopPart, "ndc", "user")
        
        CIText <- paste("95% CI: [", bquote(.(formatC(CIlow, 3, format = "f"))), 
            ", ", bquote(.(formatC(CIhigh, 3, format = "f"))), "]", sep = "")
        medianLegendText <- paste("median =", medianText)
        
        text(max(xticks), yy2, medianLegendText, cex = 1.1, pos = 2)
        text(max(xticks), yy, CIText, cex = 1.1, pos = 2)
        
        # probability wheel
        if (max(nchar(BF10t), nchar(BF01t)) <= 4) {
            xx <- grconvertX(0.44, "ndc", "user")
        }
        
        if (max(nchar(BF10t), nchar(BF01t)) == 5) {
            xx <- grconvertX(0.44 + 0.001 * 5, "ndc", "user")
        }
        
        if (max(nchar(BF10t), nchar(BF01t)) == 6) {
            xx <- grconvertX(0.44 + 0.001 * 6, "ndc", "user")
        }
        
        if (max(nchar(BF10t), nchar(BF01t)) == 7) {
            xx <- grconvertX(0.44 + 0.002 * max(nchar(BF10t), nchar(BF01t)), 
                "ndc", "user")
        }
        
        if (max(nchar(BF10t), nchar(BF01t)) == 8) {
            xx <- grconvertX(0.44 + 0.003 * max(nchar(BF10t), nchar(BF01t)), "ndc", "user") } if (max(nchar(BF10t), nchar(BF01t)) > 8) {
            xx <- grconvertX(0.44 + 0.005 * max(nchar(BF10t), nchar(BF01t)), 
                "ndc", "user")
        }
        
        yy <- grconvertY(0.788 + offsetTopPart, "ndc", "user")
        
        # make sure that colored area is centered
        radius <- 0.06 * diff(range(xticks))
        A <- radius^2 * pi
        alpha <- 2/(BF01 + 1) * A/radius^2
        startpos <- pi/2 - alpha/2
        
        # draw probability wheel
        plotrix::floating.pie(xx, yy, c(BF10, 1), radius = radius, col = c("darkred", 
            "white"), lwd = 2, startpos = startpos)
        
        yy <- grconvertY(0.865 + offsetTopPart, "ndc", "user")
        yy2 <- grconvertY(0.708 + offsetTopPart, "ndc", "user")
        
        if (oneSided == FALSE) {
            
            text(xx, yy, "data|H1", cex = cexCI)
            text(xx, yy2, "data|H0", cex = cexCI)
        }
        
        if (oneSided == "right") {
            
            text(xx, yy, "data|H+", cex = cexCI)
            text(xx, yy2, "data|H0", cex = cexCI)
        }
        
        if (oneSided == "left") {
            
            text(xx, yy, "data|H-", cex = cexCI)
            text(xx, yy2, "data|H0", cex = cexCI)
        }
        
        # add legend
        CIText <- paste("95% CI: [", bquote(.(formatC(CIlow, 3, format = "f"))), 
            " ; ", bquote(.(formatC(CIhigh, 3, format = "f"))), "]", sep = "")
        
        medianLegendText <- paste("median =", medianText)
    }
    
    mostPosterior <- mean(delta > mean(range(xticks)))
    
    if (mostPosterior >= 0.5) {
        
        legendPosition <- min(xticks)
        legend(legendPosition, max(yticks), legend = c("Posterior", "Prior"), 
            lty = c(1, 3), bty = "n", lwd = c(lwd, lwd), cex = cexLegend, 
            xjust = 0, yjust = 1, x.intersp = 0.6, seg.len = 1.2)
    } else {
        
        legendPosition <- max(xticks)
        legend(legendPosition, max(yticks), legend = c("Posterior", "Prior"), 
            lty = c(1, 3), bty = "n", lwd = c(lwd, lwd), cex = cexLegend, 
            xjust = 1, yjust = 1, x.intersp = 0.6, seg.len = 1.2)
    }
}

### generate data ###

set.seed(1)
x <- rnorm(30, 0.15)

### calculate Bayes factor ###

library(BayesFactor)
BF <- extractBF(ttestBF(x, rscale = "medium"), onlybf = TRUE)

### plot ###

.plotPosterior.ttest(x = x, rscale = "medium", BF = BF)
Simulations
#monte carlo t test simulation


#install.packages("MonteCarlo")
library(MonteCarlo)



# Define function that generates data and applies the method of interest

ttest<-function(n,loc,scale){
  
  # generate sample:
    sample<-rnorm(n, loc, scale)
  
  # calculate test statistic:
    stat<-sqrt(n)*mean(sample)/sd(sample)
  
  # get test decision:
    decision<-abs(stat)>1.96
  
  # return result:
    return(list("decision"=decision))
}

# define parameter grid:

  n_grid<-c(50,100,250,500)
  loc_grid<-seq(0,1,0.2)
  scale_grid<-c(1,2)

# collect parameter grids in list:
  param_list=list("n"=n_grid, "loc"=loc_grid, "scale"=scale_grid)

# run simulation:

  MC_result<-MonteCarlo(func=ttest, nrep=1000, param_list=param_list)

  summary(MC_result)
R to WordPress
#install.packages("RWordPress", repos="http://www.omegahat.org/R", build=TRUE)
library(RWordPress)
options(WordPressLogin=c(user="password"),
        WordPressURL="http://your_wp_installation.org/xmlrpc.php")
#[code lang='r']
#...
#[/code]
knit_hooks$set(output=function(x, options) paste("\\[code\\]\n", x, "\\[/code\\]\n", sep=""))
knit_hooks$set(source=function(x, options) paste("\\[code lang='r'\\]\n", x, "\\[/code\\]\n", sep=""))

knit2wp <- function(file) {
    require(XML)
    content <- readLines(file)
    content <- htmlTreeParse(content, trim=FALSE)

    ## WP will add the h1 header later based on the title, so delete here
    content$children$html$children$body$children$h1 <- NULL
    content <- paste(capture.output(print(content$children$html$children$body,
                                          indent=FALSE, tagSeparator="")),
                     collapse="\n")
    content <- gsub("<?.body>", "", content)         # remove body tag
    
    ## enclose code snippets in SyntaxHighlighter format
    content <- gsub("<?pre><code class=\"r\">", "\\[code lang='r'\\]\\\n",
                    content)
    content <- gsub("<?pre><code class=\"no-highlight\">", "\\[code\\]\\\n",
                    content)
    content <- gsub("<?pre><code>", "\\[code\\]\\\n", content)
    content <- gsub("<?/code></pre>", "\\[/code\\]\\\n", content)
    return(content)
}

newPost(content=list(description=knit2wp('rerWorkflow.html'),
                     title='Workflow: Post R markdown to WordPress',
                     categories=c('R')),
        publish=FALSE)

postID <- 99                    # post id returned by newPost()
editPost(postID,
         content=list(description=knit2wp('rerWorkflow.html'),
                      title='Workflow: Post R markdown to WordPress',
                      categories=c('R')),
         publish=FALSE)

 

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Exploring the diagnosticity of the p-valueP-value distribution and power curves for an independent two-tailed t-testBIC approximation for ANOVA designsWhen does a significant p-value indicate a true effect?




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