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Given that normality of the response variable is not an underlying assumption of linear regression and that for linear regression the assumption is only of the normality of residuals, what is the motivation for applying Box-Cox - or indeed any transformations (e.g. log) to make the response variable 'more normal'?

Is it:

  1. By making the response variable 'more normal', often the residuals will be more normal - as seems to be the motivation here https://data.library.virginia.edu/interpreting-log-transformations-in-a-linear-model/ for the log transform

  2. So that we can apply hypothesis testing / inferencing techniques that assume normality of the response variable, as this would seem to imply - https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Box-Cox_Transformation_for_Simple_Linear_Regression.pdf

  3. Some other factors?

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    $\begingroup$ Getting relationships more nearly linear and variability more nearly equal and distributions more nearly symmetric are indeed bigger deals than bringing any distribution closer to normal. The main deal with transformation is often just making it easier to see what is going on by allowing better focus on the primary question(s), More generally, the tags that you have included lead to many other threads. This isn't a new question. $\endgroup$
    – Nick Cox
    Aug 18 '21 at 8:57
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    $\begingroup$ Strictly, it's normality of errors, not of residuals, that is the ideal condition for plain or vanilla linear regression. $\endgroup$
    – Nick Cox
    Aug 18 '21 at 8:58
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    $\begingroup$ As far as I can see, the text linked in item 2 does (correctly) not state that this is about allowing inference that assumes normality of the response. $\endgroup$ Aug 18 '21 at 9:45
  • $\begingroup$ See stats.stackexchange.com/questions/310003/…. Often, the main contribution from the Box-Cox transformation is to stabilize the variance! $\endgroup$ Aug 18 '21 at 23:58

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