R/jzs_partcorSD.R

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

jzs_partcorSD <- 
  function(V1,V2,control,
           SDmethod=c("fit.st","dnorm","splinefun","logspline"),
           alternative=c("two.sided","less","greater"),
           n.iter=10000,n.burnin=500,
           standardize=TRUE){
    
    runif(1) # defines .Random.seed
    
    if(standardize==TRUE){
      M <- (V1-mean(V1))/sd(V1)
      Y <- (V2-mean(V2))/sd(V2)
      X <- (control-mean(control))/sd(control)
    } else {
      M <- V1
      Y <- V2
      X <- control      
    }
    
    n <- length(V1)
    
    #==========================================================
    # load JAGS models
    #==========================================================
    
    
    jagsmodelpartialcorrelation <- 
      
      "####### Cauchy-prior on beta and tau' #######
model
    
{
    
    for (i in 1:n)
    
{
    mu[i] <- intercept + theta[1]*x[i,1] + theta[2]*x[i,2]
    y[i]   ~ dnorm(mu[i],phi)
    
}
    
    # uninformative prior on intercept alpha, 
    # 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)
    
    # Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS.
        # New Jersey: John Wiley & Sons, Inc. p. 167
    # calculation of the inverse matrix of V
    inverse.V <- inverse(V)
    # calculation of the elements of prior precision matrix
    for(i in 1:2)
{ 
    for (j in 1:2)
{
    prior.T[i,j] <- inverse.V[i,j] * phi/g
}
}
    # multivariate prior for the beta vector
    theta[1:2] ~ dmnorm( mu.theta, prior.T )
    for(i in 1:2) { mu.theta[i] <- 0 }
    
}
    
    # 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)
    # Also note: JAGS does not want [,] structure
    #-----------------------------------------------------------------------------
    "
    
    jags.model.file2 <- tempfile(fileext=".txt")
    write(jagsmodelpartialcorrelation,jags.model.file2)
    
    #==========================================================
    # BF FOR PARTIAL CORRELATION (MY|X)
    #==========================================================
    
    x <- cbind(X,M)
    y <- Y
    
    V <- solve(t(x)%*%x) #NB I switched to the notation from Ntzoufras, p. 167
    
    jags.data   <- list("n", "x", "y", "V")
    jags.params <- c("theta")
    jags.inits  <-  list(
      list(theta = c(0.0,0.3)),  #chain 1 starting value
      list(theta = c(0.3, 0.0)), #chain 2 starting value
      list(theta = c(-.15,.15))) #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.file2)
    
    beta <- jagssamples$BUGSoutput$sims.list$theta[,2]
        
    #------------------------------------------------------------------
    
    if(SDmethod[1]=="fit.st"){
      
      mydt <- function(x, m, s, df) dt((x-m)/s, df)/s
      
      bar <- try({
        fit.t <- fit.st(beta)
        nu    <- as.numeric(fit.t$par.ests[1]) #degrees of freedom
        mu    <- as.numeric(fit.t$par.ests[2]) 
        sigma <- abs(as.numeric(fit.t$par.ests[3])) # This is a hack -- with high n occasionally
        # sigma switches sign. 
      })
      
      if(!("try-error"%in%class(bar))){
        
        # BAYES FACTOR BETA
        BF <- 1/(mydt(0,mu,sigma,nu)/dcauchy(0))
        
      } else {
        
        warning("fit.st did not converge, alternative optimization method was used.","\n")
        
        mydt2 <- function(pars){
          
          m <- pars[1]
          s <- abs(pars[2])  # no negative standard deviation
          df <- abs(pars[3]) # no negative degrees of freedom
          
          -2*sum(dt((beta-m)/s, df,log=TRUE)-log(s))
        }
        
        res <- optim(c(mean(beta),sd(beta),20),mydt2)$par
        
        m <- res[1]
        s <- res[2]
        df <- res[3]
        
        # ALTERNATIVE BAYES FACTOR PARTIAL CORRELATION
        BF <- 1/(mydt2(0,m,s,df)/dcauchy(0))
      }
      
      #-------------------------
      
    } else if(SDmethod[1]=="dnorm"){
      
      BF <- 1/(dnorm(0,mean(beta),sd(beta))/dcauchy(0)) 
      
      #-------------------------
      
    } else if(SDmethod[1]=="splinefun"){
      f <- splinefun(density(beta))
      BF <- 1/(f(0)/dcauchy(0))
      
      #-------------------------
      
    } else if (SDmethod[1]=="logspline"){
      fit.posterior <- logspline(beta)
      posterior.pp  <- dlogspline(0, fit.posterior) # this gives the pdf at point b2 = 0
      prior.pp      <- dcauchy(0)                   # height of prior at b2 = 0
      BF           <- prior.pp/posterior.pp
        
    }
    
    #-------------------------------------------------------
    
    # one-sided test?
    
    # save BF for one-tailed test
    # BF21 = 2*{proportion posterior samples of beta < 0}
    
    propposterior_less <- sum(beta<0)/length(beta)
    propposterior_greater <- sum(beta>0)/length(beta)
    
    # 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(beta)
    }
    
    if(propposterior_greater==0){
      propposterior_greater <- 1/length(beta)
    }
    
    BF21_less <- 2*propposterior_less
    BF21_greater <- 2*propposterior_greater
    
    if(alternative[1]=="less"){
      # BF10 = p(D|b~cauchy(0,1))/p(D|b=0)
      BF10 <- BF
      
      # BF21 = p(D|b~cauchy-(0,1))/p(D|b~cauchy(0,1))
      # BF21 = 2*{proportion posterior samples of beta < 0}
      BF21 <- BF21_less
      
      BF <- BF10*BF21
      
    } else if(alternative[1]=="greater"){
      # BF10 = p(D|b~cauchy(0,1))/p(D|b=0)
      BF10 <- BF
      
      # BF21 = p(D|b~cauchy+(0,1))/p(D|b~cauchy(0,1))
      # BF21 = 2*{proportion posterior samples of beta > 0}
      BF21 <- BF21_greater
      
      BF <- BF10*BF21
      
    }
    
    #---------------------------------------------------
    
    # convert BFs to posterior probability
    # prob cannot be exactly 1 or 0
    prob_b <- BF/(BF+1)
    
    if(prob_b == 1){
      prob_b <- prob_b - .Machine$double.eps
    }
    if(prob_b == 0){
      prob_b <- prob_b + .Machine$double.eps
    }
    
    #====================================================
    
    res <- list(PartCoef=mean(beta),
                BayesFactor=BF,
                PosteriorProbability=prob_b,
                beta_samples=beta,
                jagssamples=jagssamples)
    
    class(res) <- c("jzs_med","list")
    class(res$jagssamples) <- "rjags"
    class(res$beta_samples) <- "CI"
    
    return(res)
  }