Predict ordered logit probabilities for out of of sample outcome values

I have a discrete grid of possible outcomes and I have a sample including some, but not all of these outcomes, along with values of explanatory values

I have fit an ordered logit model to the sample using mnrfit, but what I would like to do is now generate predicted probabilities over all outcomes in the grid. The mnrval function does exactly what I want, but only for the observed categories. I would like to extend this even to unobserved categories. Is this possible?

Answer 1

[Saarthak Gupta]

Hi Mahmood,

I understand you want to give the cumulative probabilities of response categories that are not available in the training data.

An ordered logit model is a multinomial model for ordinal responses with the logit link function.

This model captures the log cumulative odds of a response under the assumption that the effects of predictor variables are the same for all categories on the logarithmic scale. It essentially means that for all categories cj:

Note: Log cumulative odds of a response is the logarithm of the ratio of the probability that the response belongs to a category with a value less than or equal to category j, P(y ≤ cj), and the probability that a response belongs to a category with a value greater than category j, P(y > cj).

From the above formula, we can derive the cumulative probability of a response cj:

You may observe that the cumulative probability depends on the value of alpha (intercept) estimated for that response.

For responses not available in the training data, we cannot estimate the value for the intercept, and hence cannot give the cumulative (or categorical) probability for that response.

However, you can give a lower bound on the cumulative probability of a response ci. For ci > cj (cj being the predecessor of ci in the ordering of response categories):

Refer to the following MATLAB documentation for further reference:

• Multinomial Models for Ordinal Responses: https://www.mathworks.com/help/stats/multinomial-models-for-ordinal-responses.html
• fitmnr: https://www.mathworks.com/help/stats/fitmnr.html
• MultinomialRegression: https://www.mathworks.com/help/stats/multinomialregression.html