Estimating Polychoric Correlation Matrix with Stan

Hi all,

I’m working on a Bayesian approach to “gernalizability analysis” with an ordinal data set. I used brms to fit cumulative models to then calculate “variance partitioning”.

Generalizability studies typically report a reliability statistic for the scale, this is often Cronbach’s alpha. But, my view is that the typical approach is specific to frequentist models of metric/interval data.

After some reading, I think calculating an “ordinal alpha” based on the polychoric correlation matrix would be the best approach.

There are some existing packages in R to calculate polychoric correlation matrices (e.g., polychor,, psych::polychoric) but these are restricted to maximum likelihood estimation.

I’m wondering if anyone is aware of a stan-based approach for estimating a polychoric correlation matrix. I’d like to use a Stan/Bayesian approach to fit with the rest of the analysis and provide distributions of the estimates.

I don’t have the programming skills to implement this, so I’m wondering if anyone has already attempted the estimation.

As an appendix, this is the implementation in the polychor package

polychor <-
    function (x, y, ML=FALSE, control=list(), std.err=FALSE, maxcor=.9999){
        f <- function(pars) {
            if (length(pars) == 1){
                rho <- pars
                if (abs(rho) > maxcor) rho <- sign(rho)*maxcor
                row.cuts <- rc
                col.cuts <- cc
            }
            else {
                rho <- pars[1]
                if (abs(rho) > maxcor) rho <- sign(rho)*maxcor
                row.cuts <- pars[2:r]
                col.cuts <- pars[(r+1):(r+c-1)]
                if (any(diff(row.cuts) < 0) || any(diff(col.cuts) < 0)) return(Inf)
            }
            P <- binBvn(rho, row.cuts, col.cuts)
            - sum(tab * log(P))
        }
        tab <- if (missing(y)) x else table(x, y)
        zerorows <- apply(tab, 1, function(x) all(x == 0))
        zerocols <- apply(tab, 2, function(x) all(x == 0))
        zr <- sum(zerorows)
        if (0 < zr) warning(paste(zr, " row", suffix <- if(zr == 1) "" else "s",
                                  " with zero marginal", suffix," removed", sep=""))
        zc <- sum(zerocols)
        if (0 < zc) warning(paste(zc, " column", suffix <- if(zc == 1) "" else "s",
                                  " with zero marginal", suffix, " removed", sep=""))
        tab <- tab[!zerorows, ,drop=FALSE]  
        tab <- tab[, !zerocols, drop=FALSE] 
        r <- nrow(tab)
        c <- ncol(tab)
        if (r < 2) {
            warning("the table has fewer than 2 rows")
            return(NA)
        }
        if (c < 2) {
            warning("the table has fewer than 2 columns")
            return(NA)
        }
        n <- sum(tab)
        rc <- qnorm(cumsum(rowSums(tab))/n)[-r]
        cc <- qnorm(cumsum(colSums(tab))/n)[-c]
        if (ML) {
            result <- optim(c(optimise(f, interval=c(-1, 1))$minimum, rc, cc), f,
                            control=control, hessian=std.err)
            if (result$par[1] > 1){
                result$par[1] <- maxcor
                warning(paste("inadmissible correlation set to", maxcor))
            }
            else if (result$par[1] < -1){
                result$par[1] <- -maxcor
                warning(paste("inadmissible correlation set to -", maxcor, sep=""))
            }
            if (std.err) {
                chisq <- 2*(result$value + sum(tab * log((tab + 1e-6)/n)))
                df <- length(tab) - r - c
                result <- list(type="polychoric",
                               rho=result$par[1],
                               row.cuts=result$par[2:r],
                               col.cuts=result$par[(r+1):(r+c-1)],
                               var=solve(result$hessian),
                               n=n,
                               chisq=chisq,
                               df=df,
                               ML=TRUE)
                class(result) <- "polycor"
                return(result)
            }
            else return(as.vector(result$par[1]))
        }
        else if (std.err){
            result <- optim(0, f, control=control, hessian=TRUE, method="BFGS")
            if (result$par > 1){
                result$par <- maxcor
                warning(paste("inadmissible correlation set to", maxcor))
            }
            else if (result$par < -1){
                result$par <- -maxcor
                warning(paste("inadmissible correlation set to -", maxcor, sep=""))
            }
            chisq <- 2*(result$value + sum(tab *log((tab + 1e-6)/n)))
            df <- length(tab) - r - c 
            result <- list(type="polychoric",
                           rho=result$par,
                           var=1/result$hessian,
                           n=n,
                           chisq=chisq,
                           df=df,
                           ML=FALSE)
            class(result) <- "polycor"
            return(result)
        }
        else optimise(f, interval=c(-maxcor, maxcor))$minimum
    }

Another thing is that the polychor function produces maximum likelihood estimates of the pairwise correlation between two variables, but the overall correlation matrix among K variables may not be positive definite.

To do the Bayesian analogue, you basically need the stuff at

and to estimate the correlation matrix, the latent z variables, and the cutpoints simultaneously. Hopefully, that is enough to get started.

Thanks, Ben!

The link provided and code for the tobit model are well above my understanding, but provide a good starting place!

I should have included in my initial post that the estimation of the polychromic correlation is the first step in estimating an ordinal alpha (reliability) coefficient.

I’m trying to take a Bayesian approach to what is presented and discussed in this paper that provides an R-based example workflow in the appendix.