How does Bayesian Ordinal Regression differ from Bayesian Logistic Regression?

Question

How does Bayesian Ordinal Regression differ from Bayesian Logistic Regression? In particular, are there any good links on how Bayesian Ordinal Regression works, and how to implement it?

Answer 1

This question is identical to yours, except for the additional inquiry about Bayesian implementation. The answer provides a link to some course notes on the topic. As a brief summary, logistic regression assumes a binary response variable, and is typically modeled as

$$P(Y_i = 1) = g(x_i'\beta)$$ where $g(\cdot): \mathbb R \rightarrow (0,1)$ is called a link function. Strictly speaking, logistic regression always uses a logistic link function $g(t) = 1/(1+\exp(-x))$, but other link functions are available.

Ordinal regression is used when for ordinal response variables, i.e. when $Y_i$ takes values in the set $\{1, 2, \cdots J\}$ where the order of the categories is meaningful. Ordinal regression models this as, $$P(Y_i \leq j) = g(\theta_j + x_i'\beta)$$ with the assumption $$-\infty \equiv \theta_0 < \theta_1 < \cdots < \theta_{J-1} < \theta_J \equiv \infty.$$ Ordinal logistic regression again uses a logistic link function, inducing a "proportional log odds assumption". Peter McCullagh's original paper on this topic is pretty readable.

To apply these methods in a Bayesian setting, we need to specify priors on the parameters $\beta_1, \cdots \beta_p$ and $\theta_1, \cdots \theta_{J-1}$. A quick google search will turn up helpful results, such as STAN and BUGS resources for fitting these models. This paper uses WinBUGS to fit a Bayesian Ordinal Regression model on an Oral Health study. They assign vague normal priors on each regression coefficient $$\beta_k \stackrel{iid}{\sim} N(0, 10^6), \ k=1, \cdots p$$ and truncated vague normal priors on the latent "threshold" parameters. $$\theta_j \sim N(0, 10^6)I(\theta_{j-1}, \theta_{j+1}), \ j=1, \cdots J-1$$ where $I(a,b)$ is a truncation function on the interval $(a, b)$.