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First, the title of the question is so broad that it would need a book or two to answer. But in the text, you seem to be concerned mostly with collinearity and want to use correlations to weed out some variables. I'll just answer about that. Correlations are not a good tool for diagnosing collinearity. You can have problematic collinearity with very low ...


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Since you say : However, the relationship between $\textbf{y}$ and $\textbf{X}$ is not strictly linear over the entire domain, but could be better modeled as such within several subgroups $g$ (and coefficients are more meaningful if defined for each subgroup): it sounds to me very much like a mixed effects model (of which multilevel and hierarchical ...


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