I recently reread some statistics books and noted something weird: They all discuss the assumptions of linear regression and mention the need for a normal distributed dependent variable. In the next step, there is always a discussion of variables which are non-normal and potential "cures" like sqrt, log, inverse etc.

But the books are never going to discuss how to interpret the transformed variable - which strikes me as odd, since a sqrt function changes the interpretation from "the GDP of a state increases by 1 when the social-freedom index increases by 1" to "the square root of GDP of a state increases by 1 when the social-freedom index increases by 1", which is quite a difference.

Can somebody point me to some good explanation about this?

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    $\begingroup$ Note that the assumption, when it's made, is of normally distributed errors; the dependent variable certainly doesn't need to be normally distributed. $\endgroup$ – Scortchi - Reinstate Monica Apr 7 '14 at 9:03
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    $\begingroup$ What books are you reading? Good books on regression and transformation 1) Do not get the assumptions of regression wrong and 2) Do discuss the effects of transformation on interpretation. $\endgroup$ – Peter Flom - Reinstate Monica Apr 7 '14 at 9:50
  • $\begingroup$ I am, at the moment, not certain whether I just mssremebered the books or the books are indeed stating that the dependent variable should be normal distributed. As a side question: can the dependend variable be non-normal and the errors not? The books in question are Andy Fields Introduction into statistics using R and Tabachnick & Fidell. $\endgroup$ – Christian Sauer Apr 7 '14 at 10:07
  • $\begingroup$ Of course it can: think of estimating the difference in means between two groups as a special case of regression; if the difference is large compared to the error variance the dependent variable will be bimodal. $\endgroup$ – Scortchi - Reinstate Monica Apr 7 '14 at 11:22
  • $\begingroup$ Some books definitely get this one wrong, and both those books you mention do contain errors (Field seems to have quite a few), but I doubt they'd get such a basic detail wrong. $\endgroup$ – Glen_b -Reinstate Monica Apr 7 '14 at 11:52

You have partly answered your own question: It changes the interpretation in just the way you have mentioned. But if you want to see the interpretation on the original variable, one way is to graph the original variable (Y) against the independent variable X. If you have multiple X, you can make a graph for each, or you can look at Y at different combinations of values for the different X's.

  • $\begingroup$ Thank you for the idea - I guess I sought an "official" explanation ;) $\endgroup$ – Christian Sauer Apr 7 '14 at 10:51

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