Ordered logistic model

Description

The ordinal model is a model of ordered discrete data.

Definition

For discrete ordered outcome \(y\) and predictors \(x\), the model is:

\[ \begin{align} y_i \sim \text{orderedLogistic}(\eta_i, c) \\ \eta_i = \alpha + \beta \cdot x_i \end{align} \]

Parameters needing priors

• \(\alpha\) (intercept)
• \(\beta\) (predictor weights)
• \(c\) (cutpoints)

Prior for \(\alpha\)

Prior for \(\beta\)

Prior for \(c\)

Induced Dirichlet prior

Betancourt (2019) suggests using an induced Dirichlet prior on the cutpoint \(c\).

Stan code:

// Copyright 2019 Michael Betancourt (BSD-3)
functions {
  real induced_dirichlet_lpdf(vector c, vector alpha, real phi) {
    int K = num_elements(c) + 1;
    vector[K - 1] sigma = inv_logit(phi - c);
    vector[K] p;
    matrix[K, K] J = rep_matrix(0, K, K);

    // Induced ordinal probabilities
    p[1] = 1 - sigma[1];
    for (k in 2:(K - 1))
      p[k] = sigma[k - 1] - sigma[k];
    p[K] = sigma[K - 1];

    // Baseline column of Jacobian
    for (k in 1:K) J[k, 1] = 1;

    // Diagonal entries of Jacobian
    for (k in 2:K) {
      real rho = sigma[k - 1] * (1 - sigma[k - 1]);
      J[k, k] = - rho;
      J[k - 1, k] = rho;
    }

    return   dirichlet_lpdf(p | alpha)
           + log_determinant(J);
  }
}

See also

• Stan Function Reference
• Stan Users Guide

References

Betancourt, Michael. 2019. “Ordinal Regression.” https://betanalpha.github.io/assets/case_studies/ordinal_regression.html.