---
title: "pR2D2ord prior for Ordinal Regression"
author: "Eric Yanchenko"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{pR2D2ord prior for Ordinal Regression}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

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.

```{r}
    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.

```{r}
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.

```{r}
out <- R2D2ordinal::ord_r2d2(x=X, y=Y, K=K, hyper=hyper, warmup=500, iter=1500, verbose=FALSE)
fit <- rstan::extract(out)
```

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

```{r}
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()
```

