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I am thinking about making a regression on these data. As a dependent variable, I want to use percent of people in a county voting for Clinton in 1992. Is this appropriate?

I do not know if I should use linear or logistic regression. Can I even use logistic regression if my dependent variable is not categorical?

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Logistic regression can use frequency weighting to model proportions as a number of yes/no votes. You should be able to get the county voting numbers and/or population from census information and merge them together. Even with weights, it's important to remember that this is still an ecological study.

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