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Imagine that I have one variable from a 7-point Likert item, and my predictor is a categorical variable with three levels. Imagine that my sample size is 100. Can I run a linear regression lm(ordinal_var ~ categorical_pred), or should I use something like a cumulative ordinal analysis?

What if my sample size is 10,000?

What if my predictor is continuous?

What if I use the mean of several Likert items as the dependent variable?

I am asking about sample size because I heard that if you have enough sample size, the results will be robust regardless of whether you use linear regression or a more appropriate analysis.

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Regardless of the sample size, nobody prohibits you from using linear regression. People sometimes use linear regression in situations where the dependent variable is binary, counts, or ordinal. In each case, there are better alternatives to linear regression, that are designed for handling such data (e.g. logistic regression, Poisson regression, ordinal regression, IRT models, etc). As George Box said, "all models are wrong, but some are useful".

If using linear regression for Likert data you would be implicitly making an assumption that the spacing between the categories is the same, so "Strongly disagree" is as far as to "disagree" as "agree", etc. This may or may not be a problem, depending on the nature of your research. With linear regression also nothing stops it from making predictions that are invalid (that "agrees" outside of the scale, etc). So using it possibly leads to poorer results.

On another hand, in some cases, models like linear regression may be desirable, because they may be easier to interpret, scale better to large amounts of data, and for other reasons. It's about picking the right tool for the job, where the definition of "right" depends on your needs. In some cases "right" means "precise", in others it means "easy to use". For some jobs, you simply do not need anything more than a hammer.

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