Ordered Logit

$$ \begin{aligned} & R_i \sim Categorical( \begin{bmatrix} \Pr(R = 1) \\ \Pr(R = 2) \\ \Pr(R = 3) \\ \Pr(R = 4) \end{bmatrix} )\\ & Pr(R_i = 4) = Pr(R_i \le 4) - Pr(R_i \le 3) \\ & Pr(R_i = 3) = Pr(R_i \le 3) - Pr(R_i \le 2) \\ & Pr(R_i = 2) = Pr(R_i \le 2) - Pr(R_i \le 1) \\ & Pr(R_i = 1) = Pr(R_i \le 1) \\ \\ & logit[ Pr(R_i \le 4) ] = logit(1) = \infty \\ & logit[ Pr(R_i \le 3) ] = log \frac{Pr(R_i \le 3)}{1 - Pr(R_i \le 3)} = \kappa_3 - \phi_i \\ & logit[ Pr(R_i \le 2) ] = log \frac{Pr(R_i \le 2)}{1 - Pr(R_i \le 2)} = \kappa_2 - \phi_i \\ & logit[ Pr(R_i \le 1) ] = logit[ Pr(R_i = 1) ] = log \frac{Pr(R_i = 1)}{1 - Pr(R_i = 1)} = \kappa_1 - \phi_i \\ \\ &\text{ or succintly as } \\ & logit( P_c ) = \kappa_c - \phi_i ~, \\ & ~~~ \text{where } P_c = Pr(R_i \le c) \text{, cutpoint } c = 1, 2, 3 \end{aligned} $$