R/jzs_cor.R

In BayesMed: Default Bayesian Hypothesis Tests for Correlation, Partial Correlation, and Mediation

jzs_cor <-
  function(V1,V2,
           alternative=c("two.sided","less","greater"),
           n.iter=10000,n.burnin=500,standardize=TRUE){
    
    runif(1) # defines .Random.seed
    
    # standardize variables
    if(standardize==TRUE){
      X <- (V1-mean(V1))/sd(V1)
      Y <- (V2-mean(V2))/sd(V2)
    }else {
      M <- V1
      Y <- V2
    }
    
    r <- cor(X,Y)
    n <- length(X)
    
    # main function to analytically calculate the BF for correlation
    # see Wetzels, R. & Wagenmakers, E.-J. (2012). A default Bayesian hypothesis test for correlations and partial correlations. Psychonomic Bulletin & Review.
    jzs_corbf <- function(r,n){
      int <- function(r,n,g){
        a <- .5*((n-2)*log(1+g)-(n-1)*log(1+g*(1-r^2)))
        exp(a)*dinvgamma(g,shape=.5,scale=n/2)
      }
      bf10 <- integrate(int,lower=0,upper=Inf,r=r,n=n)$value
      return(bf10)
    }
    
    BF <- jzs_corbf(r,n)
    
    ### the next part is needed to impose an order restriction
    ### for the order restrictions we need to estimate the posterior samples
    
    #==========================================================
    # load JAGS models
    #==========================================================
    
    jagsmodelcorrelation <- 
      
      "####### Cauchy-prior on single regression coefficient #######
    model
    
{
    
    for (i in 1:n)
    
{
    mu[i] <- intercept + alpha*x[i]
    y[i]   ~ dnorm(mu[i],phi)
    
}
    
    # uninformative prior on the intercept intercept, 
    # Jeffreys' prior on precision phi
    intercept ~ dnorm(0,.0001)
    phi   ~ dgamma(.0001,.0001)
    #phi   ~ dgamma(0.0000001,0.0000001) #JAGS accepts even this
    #phi   ~ dgamma(0.01,0.01)           #WinBUGS wants this
    
    # inverse-gamma prior on g:
    g       <- 1/invg 
    a.gamma <- 1/2
    b.gamma <- n/2    
    invg     ~ dgamma(a.gamma,b.gamma)
    
    
    # g-prior on beta:
    vari <- (g/phi) * invSigma 
    prec <- 1/vari
    alpha    ~ dnorm(0, prec)
}
    
    # Explanation------------------------------------------------------------------ 
    # Prior on g:
    # We know that g ~ inverse_gamma(1/2, n/2), with 1/2 the shape
    # parameter and n/2 the scale parameter.
    # It follows that 1/g ~ gamma(1/2, 2/n).
    # However, BUGS/JAGS uses the *rate parameterization* 1/theta instead of the
    # scale parametrization theta. Hence we obtain, in de BUGS/JAGS rate notation:
    # 1/g ~ dgamma(1/2, n/2)
    #------------------------------------------------------------------------------
    "
    
    jags.model.file1 <- tempfile(fileext=".txt")
    write(jagsmodelcorrelation,jags.model.file1)
    
    #========================================================================
    # Estimate Posterior Distribution for the Correlation Coefficient Alpha
    #========================================================================
    
    x <- X
    y <- Y
    
    invSigma <- solve(t(x)%*%x)
    
    jags.data   <- list("n", "x", "y", "invSigma")
    jags.params <- c("alpha", "g")
    jags.inits  <-  list(
      list(alpha = 0.0), #chain 1 starting value
      list(alpha = -0.3), #chain 2 starting value
      list(alpha = 0.3)) #chain 3 starting value
    
    jagssamples <- jags(data=jags.data, inits=jags.inits, jags.params, 
                       n.chains=3, n.iter=n.iter, DIC=T,
                       n.burnin=n.burnin, n.thin=1, model.file=jags.model.file1)
    
    # estimate the posterior regression coefficient and scaling factor g
    alpha <- jagssamples$BUGSoutput$sims.list$alpha[,1]
    g  <- jagssamples$BUGSoutput$sims.list$g
    
    #-------------------------------------------------------
    
    # one-sided test?
    
    # save BF for one-tailed test
    # BF21 = 2*{proportion posterior samples of alpha < 0}
    
    propposterior_less <- sum(alpha<0)/length(alpha)
    propposterior_greater <- sum(alpha>0)/length(alpha)
    
    # posterior proportion cannot be zero, because this renders a BF of zero
    # none of the samples of the parameter follow the restriction
    # ergo: the posterior proportion is smaller than 1/length(parameter)
    
    if(propposterior_less==0){
      propposterior_less <- 1/length(alpha)
    }
    
    if(propposterior_greater==0){
      propposterior_greater <- 1/length(alpha)
    }
    
    BF21_less <- 2*propposterior_less
    BF21_greater <- 2*propposterior_greater
    
    
    if(alternative[1]=="less"){
      # BF10 = p(D|a~cauchy(0,1))/p(D|a=0)
      BF10 <- BF
      
      # BF21 = p(D|a~cauchy-(0,1))/p(D|a~cauchy(0,1))
      # BF21 = 2*{proportion posterior samples of alpha < 0}
      BF21 <- BF21_less
      
      BF <- BF10*BF21
      
    } else if(alternative[1]=="greater"){
      # BF10 = p(D|a~cauchy(0,1))/p(D|a=0)
      BF10 <- BF
      
      # BF21 = p(D|a~cauchy+(0,1))/p(D|a~cauchy(0,1))
      # BF21 = 2*{proportion posterior samples of alpha > 0}
      BF21 <- BF21_greater
      
      BF <- BF10*BF21
      
    }
    
    #--------------------------------------------------------
    
    # convert BFs to posterior probability
    # prob cannot be exactly 1 or 0
    prob_r <- BF/(BF+1)
    
    if(prob_r == 1){
      prob_r <- prob_r - .Machine$double.eps
    }
    if(prob_r == 0){
      prob_r <- prob_r + .Machine$double.eps
    }
    
    #==================================================
    
    # convert posterior samples for the regression coefficient x-y to correlation
    cor_coef <- alpha*(sd(x)/sd(y))
    
    #===================================================
    
    res <- list(Correlation=mean(cor_coef),
                BayesFactor=BF,
                PosteriorProbability=prob_r,
                alpha_samples=alpha,
                jagssamples=jagssamples)
    
    class(res) <- c("jzs_med","list")
    class(res$jagssamples) <- "rjags"
    class(res$alpha_samples) <- "CI"
    
    return(res)
  }