# Generalization of cumulative probability models for ordinal Y

There are many models in existence for ordinal $Y$, for example the proportional odds ordinal logistic model, the continuation ratio model, and the cumulative probit model. The first and third of these are in a class of models specifying Prob$(Y \leq j | X) = F(\alpha_{j} + X\beta)$ where $F$ is a non-decreasing function (e.g., logistic or normal CDF). This can be generalized by having a family of functions (e.g., a 3-parameter family) and estimating the parameters of this family in addition to $\alpha, \beta$. Or perhaps there is a way to do this non-parametrically in $F$. Has anyone seen any methods developed along these lines? Among other things, this would be a competitor of quantile regression.

• My basic understanding of these ordinal regression methods is that they extensions of the GLM (usually a link function is applied to the LHS). $F$ is a cumulative link. The advantage of explicitly stating $F$ is that the parameters are interpretable. But if $F$ is allowed to be anything, then perhaps the objective is more for prediction than inference. Is that correct? How could one articulate the constraints relative to a general additive multinomial model? – AdamO Jan 31 '18 at 21:26
• I think that's mainly correct. You could still do inference though if you are willing to talk about effect ratios that are not quite as specific as odds or hazard ratios. – Frank Harrell Feb 1 '18 at 11:55