For each \(j \text{th}\) outcome level up to \(J-1\), a cumulative logit is estimated with its own regression parameters \(\{ \hat{\alpha}_j, \hat{\beta}_j \}\). The formulation in Eq. \ref{eq:2} is known as non-proportional odds, where the log-odds \(\hat{\beta}_j\) differs between outcome levels. Simplification is possible in a proportional odds formulation where constant log-odds \(\beta_j=\beta\) is assumed among outcome levels \cite{mccullagh1989generalized}.
Let \(\hat{\pi}_{ij} = \hat{\text{Pr}} \left[ Y_{ij} \leq j|X_i \right]\), then in terms of the parameter estimates \(\{ \hat{\alpha}_j,\hat{\beta}_j \} \) the predicted cumulative probability for the \(j\text{th}\) outcome level at the \(i \text{th}\) observation is: