I have been performing a regression using ad dependent variable an ordinal variable. I later realized I had to use an ordinal regression instead. Nevertheless, I wondered what could be the consequences of that mistake for my analysis. Anyone can explain me that?
A short explanation: MLR uses dummy coding to run a series of binary logistic regressions for outcomes, while OLR uses proportional odds to model outcome as being in a category or any previous category. You conserve information in OLR when outcome is ordered.
Additional reading:
- https://www.theanalysisfactor.com/logistic-regression-models-for-multinomial-and-ordinal-variables/
- http://www.karlin.mff.cuni.cz/~antoch/robust10/PREDNASKY/CTVRTEK_DOPOLEDNE/zezula.pdf
- https://stats.stackexchange.com/questions/155737/what-is-the-difference-between-multinomial-and-ordinal-logistic-regression
Professor Long in his 1997 book (Regression Models for Categorical and Limited Dependent Variables) says:
"Multinomial logit model (MNLM) can be thought of as simultaneous estimating binary logits for all possible comparisons among the outcome categories". On the very same page he adds "if a dependent variable is ordinal and a model for nominal variables is used, there is a loss of efficiency since information is being ignored".
There are many ordinal logistic regression (OLR) types. The most famous among them is indeed the proportional odds model (POM) which also travels with the name cumulative logit model. See:
- Ananth, C. V., & Kleinbaum, D. G. (1997). Regression models for ordinal responses: A review of methods and applications. International Journal of Epidemiology. https://doi.org/10.1093/ije/26.6.1323]
- Greenland, S. (1994). Alternative models for ordinal logistic regression. Statistics in Medicine. https://doi.org/10.1002/sim.4780131607
Each of those models, POM included, have assumptions and restrictions.
Besides the mentioned loss in efficiency, I think you are loosing an easy interpretation of your model. The ordinal model parameters allow you to calculate the predicted height of your outcome value given your covariates. In the Multinomial model the coefficients allow you to derive the probability of being in an (unorderd) class of your dependent variable. This is much more difficult to interpret if you want to recover notion about the ordinal structure of your dependent variable.
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