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I am making my first thorough exploration of transformations.

My primary goal is to improve the normality of the residuals following a mixed effects model fit. For some of my response variables, the residuals (and original data) show a good approximation of a normal distribution. For others, there is substantial skew. I have tried several standard transformations (log, sqrt, squared, etc.) and for most response variables one of these works well.

I know there are arguments for and against response variable transformations, but at least some things I have read indicate that this empirical approach is acceptable.

Assuming that is indeed acceptable, my question is whether it is also acceptable to choose an arbitrary power transformation. One set of typical transformations include raising the response variable to some integer power: -3, -2, -1, 1, 2 etc. (cube root, through reciprocal, no transform, and squared). In one of my response variables, none of these provides a good approximation of normality. Instead, I found that raising the response variable to a fractional power (0.25) worked quite well. Theoretically, one could also find that raising to the power 1.5 works better than no transformation or the squared transformation.

I have not been able to find any discussion of whether this is a good or bad idea.

So, to summarize: assuming that empirical selection of a transformation to improve normality is acceptable, should the power transformations explored be limited to integer powers or can any fractional power be considered?

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    $\begingroup$ Search this site for «Box-Cox transformation» $\endgroup$ – kjetil b halvorsen Dec 10 '17 at 12:37
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    $\begingroup$ Suppose I run Box-Cox which indicates say a power of 0.49 but I then choose a power of 0.5 (square root) instead. I made a choice based on simplicity and ease of reporting and interpretation as well as what a method indicates. Your question is complicated by your emphasis on normal residuals. In principle and in practice it is more important to get the systematic structure of the model about right. So, depending on emphasis on getting closer to normality, getting closer to equal scatter, and getting closer to linearity, different transformations could each seem best. $\endgroup$ – Nick Cox Dec 10 '17 at 14:41
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    $\begingroup$ I would not want any toolkit of transformations to omit square root or cube root. It's also very surprising that you mention logarithms but omit the detail that logarithms correspond to power 0. Experiences vary (a lot), but mine is that the logarithm is by far the most useful transform. Nor is it ruled out by zeros in the data: use a logarithmic link function. Yet another point is that choosing the same transformation in similar situations often is preferable to choosing different transformations ad hoc. $\endgroup$ – Nick Cox Dec 10 '17 at 14:46
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    $\begingroup$ Thanks @NickCox and others. Summarizing... 1) be careful that outliers do not drive an inappropriate choice of transformation, 2) balancing normality, equal scatter, and linearity are all important, 3) Box-Cox is an important addition I need to educate myself about given its goal to optimize transformation to achieve normality (but not scatter and linearity), and 3) my specific case of a mixed effects model complicates matters, although there is a (I believe new) r package that tackles the complexity: boxcoxmix. $\endgroup$ – Russell Wyeth Dec 11 '17 at 20:19
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    $\begingroup$ Does this answer your question? Intuition behind Box-Cox transform $\endgroup$ – kjetil b halvorsen Oct 2 at 16:12