Author: Eric Yanchenko

# pR2D2ord prior for Ordinal Regression

#### Eric Yanchenko

This package implements the \(\mathsf{pR2D2ord}\) prior from Pseudo-R2D2 for high-dimensional ordinal regression by Eric Yanchenko. In this vignette, we explain how to use the main functions in this package. Please make sure that ‘rstan’ is installed before running this package

First, we simulate some ordinal regression data.

        n = 100 # Number of data points
        p = 5   # Number of covariates
        K = 3   # Number of response categories

        # Regression coefficients
        beta <- c(1, -1, 2, -2, 0)

        # Generate covariates
        X <- matrix(rnorm(n*p), nrow=n, ncol=p)

        # Generate latent term
        eta = X%*%beta
        z <- rnorm(n, eta, 1)

        # Set cut-off values
        tau <- c(-Inf, 0, 2/3, Inf)

        # Find response values
        Y <- findInterval(z, tau)

Next, we find the optimal hyperparameters for the generalized Inverse Gaussian distribution in order to induce \(R^2_M\sim\mathsf{Beta}(a,b)\) where \(R^2_M\) is McFadden’s coefficient-of-determination.

    a = 1
    b = 10
    # In practice, this function should be run.
    # hyper <- R2D2ordinal::find_param(a, b, n, K, nreps = 5, no_cores=10)
    hyper = c(0.045, 1.05, 1.05)

Given the hyper-parameter values, we can fit the model in Stan. Note that since we found the optimal hyper-parameters above, we don’t need this function to re-compute them. Otherwise, we would set a=1,b=10 and leave hyper=NULL in the following function call.

    out <- R2D2ordinal::ord_r2d2(x=X, y=Y, K=K, hyper=hyper, warmup=500, iter=1500, verbose=FALSE)
    #> Warning: There were 4 divergent transitions after warmup. See
    #> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
    #> to find out why this is a problem and how to eliminate them.
    #> Warning: Examine the pairs() plot to diagnose sampling problems
    fit <- rstan::extract(out)

Given the posterior samples, we can plot the posterior distribution of \(\beta_1\), \(\beta_2\) and \(W\).

    df = dplyr::tibble(W = fit$W,
                beta1 = fit$beta[,1],
                beta2 = fit$beta[,2])

    ggplot2::ggplot(df, ggplot2::aes(x=beta1))+
      ggplot2::geom_histogram(bins=25)+
      ggplot2::geom_vline(xintercept = beta[1], color="red")+
      ggplot2::xlab("beta1")+
      ggplot2::ylab("Count")+
      ggplot2::theme_bw()

    ggplot2::ggplot(df, ggplot2::aes(x=beta2))+
      ggplot2::geom_histogram(bins=25)+
      ggplot2::geom_vline(xintercept = beta[2], color="red")+
      ggplot2::xlab("beta2")+
      ggplot2::ylab("Count")+
      ggplot2::theme_bw()

    ggplot2::ggplot(df, ggplot2::aes(x=W))+
      ggplot2::geom_histogram(bins=25)+
      ggplot2::xlab("W")+
      ggplot2::ylab("Count")+
      ggplot2::theme_bw()