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A linear regression would give me a possitive estimate of the slope. For a one unit increase in x, I would get a so and so increase in y.

I can also split the data for the y variable between high and low, dichotomizing it. And calculate the estimated increase in log odds for a one unit increase in x.

Is there even a point in doing so?

It seems like the question can be answered using linear regression.

I don't see the point in using logistic regression unless the output is necessarily binary (gender, political affiliation etc)

Even if we are interested in a output of,say,having high income vs low income, you can just have income as a continuous spectrum and use ols to get the answer.

Is there something I'm missing?