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If you want to regress y on x, where multiple y's are observed at each x, is it ever better to instead take the mean at each x, and the use those means for the regression? Does it depend on the distributional assumptions?

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    $\begingroup$ Taking the means of the x's is probably not the smartest thing to do if you have high skew. I would choose to regress y on x instead of y on mean(x). This will allow you to determine outliers during your residual analysis. $\endgroup$ – Christopher Aden Dec 25 '10 at 21:54
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When you use the means you are removing much of the variation of the y's around their averages. This would be incorrect if you are assessing the relationships between the individual y's and the x's. In particular it will cause you to overestimate the correlation between the y's and the x's and give you too much confidence in the estimated regression coefficients. If your purpose is to assess the relationship between the average y and the x's, that might be ok. (One subtlety occurs when the numbers of y's involved in the averages vary, because then you should be using weighted least squares methods rather than ordinary least squares.) Just beware of the ecological fallacy, which is the mistake of using the aggregated (i.e., averaged) y's in place of their individual values.

These considerations are independent of any distributional assumptions.

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