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##' Bayes factors or posterior samples for correlations.
##'
##' The Bayes factor provided by \code{ttestBF} tests the null hypothesis that
##' the true linear correlation \eqn{\rho}{rho} between two samples (\eqn{y}{y} and \eqn{x}{x})
##' of size \eqn{n}{n} from normal populations is equal to 0. The Bayes factor is based on Jeffreys (1961)
##' test for linear correlation. Noninformative priors are assumed for the population means and
##' variances of the two population; a shifted, scaled beta(1/rscale,1/rscale) prior distribution
##' is assumed for \eqn{\rho}{rho} (note that \code{rscale} is called \eqn{\kappa}{kappa} by
##' Ly et al. 2015; we call it \code{rscale} for consistency with other BayesFactor functions).
##'
##' For the \code{rscale} argument, several named values are recognized:
##' "medium.narrow", "medium", "wide", and "ultrawide". These correspond
##' to \eqn{r} scale values of \eqn{1/\sqrt(27)}{1/sqrt(27)}, \eqn{1/3}{1/3},
##' \eqn{1/\sqrt(3)}{1/sqrt(3)} and 1, respectively.
##'
##' The Bayes factor is computed via several different methods.
##' @title Function for Bayesian analysis of correlations
##' @param y first continuous variable
##' @param x second continuous variable
##' @param rscale prior scale. A number of preset values can be given as
##' strings; see Details.
##' @param nullInterval optional vector of length 2 containing
##' lower and upper bounds of an interval hypothesis to test, in correlation units
##' @param posterior if \code{TRUE}, return samples from the posterior instead
##' of Bayes factor
##' @param callback callback function for third-party interfaces
##' @param ... further arguments to be passed to or from methods.
##' @return If \code{posterior} is \code{FALSE}, an object of class
##' \code{BFBayesFactor} containing the computed model comparisons is
##' returned. If \code{nullInterval} is defined, then two Bayes factors will
##' be computed: The Bayes factor for the interval against the null hypothesis
##' that the probability is 0, and the corresponding Bayes factor for
##' the complement of the interval.
##'
##' If \code{posterior} is \code{TRUE}, an object of class \code{BFmcmc},
##' containing MCMC samples from the posterior is returned.
##' @export
##' @keywords htest
##' @author Richard D. Morey (\email{richarddmorey@@gmail.com})
##' @examples
##' bf = correlationBF(y = iris$Sepal.Length, x = iris$Sepal.Width)
##' bf
##' ## Sample from the corresponding posterior distribution
##' samples = correlationBF(y = iris$Sepal.Length, x = iris$Sepal.Width,
##' posterior = TRUE, iterations = 10000)
##' plot(samples[,"rho"])
##' @seealso \code{\link{cor.test}}
##' @references Ly, A., Verhagen, A. J. & Wagenmakers, E.-J. (2015).
##' Harold Jeffreys's Default Bayes Factor Hypothesis Tests: Explanation, Extension, and Application in Psychology.
##' Journal of Mathematical Psychology, Available online 28 August 2015, https://dx.doi.org/10.1016/j.jmp.2015.06.004.
##'
##' Jeffreys, H. (1961). Theory of probability, 3rd edn. Oxford, UK: Oxford University Press.
correlationBF <- function(y, x, rscale = "medium", nullInterval = NULL, posterior=FALSE, callback = function(...) as.integer(0), ...)
{
if(!is.null(nullInterval)){
if(any(nullInterval< -1) | any(nullInterval>1)) stop("nullInterval endpoints must be in [-1,1].")
nullInterval = range(nullInterval)
}
rscale = rpriorValues("correlation",,rscale)
if( length(y) != length(x) ) stop("Length of y and x must be the same.")
if(!is.null(nullInterval))
if(any(abs(nullInterval)>1))
stop("Invalid interval hypothesis; endpoints must be in [-1,1].")
if( length(y)<3 ) stop("N must be >2.")
n = length(x) - sum(is.na(y) | is.na(x))
if(n < 3) stop("Need at least 3 complete observations.")
hypNames = makeCorrHypothesisNames(rscale, nullInterval)
mod1 = BFcorrelation(type = "Jeffreys-beta*",
identifier = list(formula = "rho =/= 0", nullInterval = nullInterval),
prior=list(rscale=rscale, nullInterval = nullInterval),
shortName = hypNames$shortName,
longName = hypNames$longName
)
data = data.frame(y = y, x = x)
checkCallback(callback,as.integer(0))
if(posterior)
return(posterior(mod1, data = data, callback = callback, ...))
bf1 = compare(numerator = mod1, data = data)
if(!is.null(nullInterval)){
mod2 = mod1
attr(mod2@identifier$nullInterval, "complement") = TRUE
attr(mod2@prior$nullInterval, "complement") = TRUE
hypNames = makeCorrHypothesisNames(rscale, mod2@identifier$nullInterval)
mod2@shortName = hypNames$shortName
mod2@longName = hypNames$longName
bf2 = compare(numerator = mod2, data = data)
checkCallback(callback,as.integer(1000))
return(c(bf1, bf2))
}else{
checkCallback(callback,as.integer(1000))
return(c(bf1))
}
}
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