Stage 1: Cumulative Logit Regression
The cumulative logit regression model predicts probabilities for an ordinal outcome \(Y=j\) for \(j=1, \dots , J\) levels, where for demonstration \(J=3\) and the reference outcome level is \(Y=3\). Let \(\pi_j = \text{Pr}\left[ Y \le j \right]\), then with continuous predictor \(X\) and linear parameters \(\{ \alpha_j,\beta_j \}\) the cumulative logit regression model is: