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Let's assume we made a survey where participants had to evaluate us on a scale from 1 to 5. We want to analyse how different variables (let's say gender or age for instance) have an effect on the answers.

As we have discrete data, my first idea was to use a $\chi^2$-test for independence, however that would not reflect the "order" of the different answers (the test does not care about "3 is the midpoint between 2 and 4" and other similar relationships).

For that matter, I thought about fitting a linear regression model. Apart from that advantage, it is also easier to include multiple variables at the same time. However, I am worried that the discrete nature of the output makes regression unsuitable.

Finally, I considered working on our problem as a multinomial classification one, but we are back at the "ordering" issue.

What is the best way to attack this problem?

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This is a classic situation calling for a form of ordinal regression. Ordinal regression specifically takes into account the ordering among response categories that you wish to preserve.

There are several ways to implement ordinal regression, depending on how the 5 response levels differ from each other and how those differences are related to differences in the values of the predictor variables. This site's explanation of ordered-logit models provides a good introduction, describes what a linear regression would mean in this type of situation (sometimes a valid choice), and includes many further references.

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  • $\begingroup$ Yes! Ordinal regression is exactly what I was looking for! $\endgroup$ – David Jun 6 '19 at 13:35

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