Ordered Probit Regression
Description
Given an ordered variable (e.g. a Likert scale item) with \(K+1\) responses, the ordered probit models the relative probability of each response as \[\begin{gather}p(\text{response}=k \mid \{ \theta_k \}) = f(\theta_k) - f(\theta_{k-1})\end{gather}\] where \(f\) is a cumulative distribution function.
We assume that item responses are drawn from a latent distribution. This distribution is then divided into \(K+1\) areas that represent the probability of obtaining a particular response by \(K\) thresholds.
Fitting the model involves estimating the threshold values that govern the probability of each response.
Example: if we have \(6\) possible responses on a survey question where the order matters (“Highly likely” = 6 > “Highly unlikely” = 1), then the relative probability of choosing “Highly likely” is \[\begin{gather} p(\text{response}=6 \mid \{ \theta_6 \}) = f(\theta_6) - f(\theta_{5}) \\ = f(\infty) - f(\theta_{5}) = 1 - f(\theta_5) \end{gather}\]
\(f\) is usually a cumulative standard normal distribution due to its nice interprative properties with \(\mu=0\) and \(\sigma=1\).
Interpretation
- Estimated are the threshold values that describe the probabiltiies of observing a particular ordered response
- Assuming a standard normal distribution for the responses \(\tilde{Y}\), beta coefficients correspond to changes in the mean of the underlying response distribution by a one unit increase in the predictor or regressor.
- So if \(X_1\) corresponds to a categorical variable, \(\beta_1=0.5\) corresponds to half of a standard deviation amount of change in the mean of the response variable \(\tilde{Y}\), relative to the reference class (e.g. maladaptive versus competent)