I am using different regression models that can predict an ordered categorical variable from a metric variable. For example, I want to regress Happiness (in 1-5 ratings) on Money (a metric variable) using ordered probit regression:
$Happiness \sim log(Dollars)$
But I came across a fundamental question: How can I compare the accuracy of different models in predicting ordered categorical data?
If I do an ordered probit regression, cross-validating model prediction accuracy with 80% data for training and 20% for validation, I can compare the probability distribution of real data with what probit provided, but in this way, the order is ignored, and for example if it predicts every rating $Y$ with the opposite rating $(Y+4)\%5+1$ (exactly opposite), the result doesn't change!
So, what should I do? I found some visual correlation measures, but nothing quantitative.
Figure 1. Fictitious report for a probit regression, based on this logit regression