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I have fish data from 2 sites and am looking to see if there is a difference in some hormone concentrations (continuous) between the sites. However, it turned out the size of the fish differed between sites so I also want to include this in my model, leaving me with one categorical (2 levels) and one continuous independent variable.

So far I have made linear models and used a multiple linear regression but now I'm wondering if that is the right thing to do when one of the independent variables is categorical?

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Yes, multiple linear regression is appropriate here. It would work even if there were only categorical predictors, because an ANOVA is just a linear regression.

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