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Does the least squares method for regression assume that the regressors and response are continuous? In particular, I would like to fit a regression line for ranks. Can I use the least squares method to estimate the regression parameters?

What alternatives do I have if these assumptions are violated? Any references to regression of rank/ordinal/interval variables is appreciated.

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For ordinal target variables you should resort to ordinal regression. If you are not familiar with it at all I suggest to follow DevD reference to wiki.

If you are looking into readily available implementations for ordinal regression, I recommend to take a look at https://github.com/ck37/coral-ordinal and the 2 references therein with excellent exposition of ordinal probabilistic predictions and inference.

Cao, W., Mirjalili, V., & Raschka, S. (2019). Rank-consistent ordinal regression for neural networks. arXiv preprint arXiv:1901.07884, 6.

Shi X., Cao W., & Raschka S. (2021). Deep Neural Networks for Rank-Consistent Ordinal Regression Based On Conditional Probabilities. arXiv preprint arXiv:211108851

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