Solved – Is the linear probability model generalisable to ordered logit/probit regressions

Question

I have a set of data where the dependent variable is an ordered response with 7 levels and I've fitted an ordered logit model to the data, and now I want to conduct some robustness checks on the estimates.

Is the Linear Probability Model for binary responses generalisable for ordinal data with multiple categories? I.e. If treat my dependent variable as numeric and use least squares will the interpretation of the coefficients be the same as in the binary case?

I'm aware that there are other methods for robustness checks for ordinal models, but I haven't been able to find any documentation on the topic and am curious.

Answer 1

That is not a robustness check because the ordinary linear model is guaranteed not to fit. It will yield probabilities estimates outside $[0,1]$. A better approach to checking the assumptions of an ordinal regression model are:

1. First, relax the assumptions allowing for nonlinear effects using regression splines
2. Then, check the equal slopes (parallelism) assumption.

For the logistic ordinal model (proportional odds model) the equal slopes (proportional odds) assumption can be checked in several ways, including:

1. Fit a series of binary logistic models for different cutoffs of $Y$ and plot the regression coefficients vs. cutoff and check for constancy
2. Construct partial residual plots for all cutoffs of $Y$. One often has to collapse infrequent $Y$ categories to carry this out.
3. Plot the logit of the empirical cumulative distribution of $Y$ stratified by very important predictors and check for parallelism.