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I've looked into other questions that dealt with what regression models to use when the dependent variable is continuous between 0 and 1 (some people have mentioned fractional logit, GLM, and even OLS), but not something that is the same question as I have.

I have a dependent variable that can only take four possible values between 0 and 1, namely 0, 1/3, 2/3, and 1. Which regression models would work well with this kind of dependent variable?

Thank you in advance.

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That looks like a discrete, ordered variable, which makes ordinal regression (e.g., ordinal logistic regression) appropriate for it. If you don't want to make the proportional odds assumption, you can also treat the variable as a nominal variable and run multinomial regression (e.g., multinomial logistic regression) on it. Ordinal regression would be the most interpretable model, though.

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