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My colleague wants to analyze some data after transforming the response variable by raising it to the power of $\frac18$ (that is, $y^{0.125}$).

I'm uncomfortable with this, but struggling to articulate why. I can't think of any mechanistic rationale for this transformation. Nor have I ever seen it before, and I worry that perhaps it inflates Type I error rates or something -- but I have nothing to support these concerns!

Additionally, my colleague finds that these transformed models outperform untransformed models in an AIC comparison. Does this, in itself, justify its use?

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    $\begingroup$ Just fyi, $y^{1/8}$ looks a lot like $\log(y)$ for many ranges of $y$. The log transformation is often justified in many cases (but also often used in unjustified cases as well). $\endgroup$ – Cliff AB Nov 23 '17 at 6:47
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    $\begingroup$ This is a related discussion $\endgroup$ – user603 Nov 23 '17 at 10:31
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    $\begingroup$ You cannot meaningfully compare AICs between models with transformed dependent variables. (Transforming the independent variable is fine.) $\endgroup$ – S. Kolassa - Reinstate Monica Nov 23 '17 at 11:06
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    $\begingroup$ @CliffAB Is right. The main practical difference between small positive powers and the logarithm is that you can take powers of zero. When there are a few zeros in the data (perhaps because of imprecision in how the numbers were recorded), sometimes a small power (0.1 or even 0.01) works as a substitute for the logarithm. (Better yet: use the Box-Cox transformation $y=(x^p-1)/p$ for small $p$.) Since very few natural laws involve a 1/8 power, though, and many involve exponential phenomena, using a log can sometimes provide better insight and interpretability than a small power. $\endgroup$ – whuber Nov 23 '17 at 14:13
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    $\begingroup$ This is a small riff on the idea that this transformation may be a substitute for logarithms if zeros occur. A logarithmic link for generalized linear models says that mean responses vary exponentially but doesn't assume that all their values are positive. So it tolerates some zeros in the data. Roughly the implication is that they should or would be positive if they could: e.g. reported zeros (zero specimens in sample, zero concentrations according to the machine) sometimes mean not detected. Despite its wonderful name Box-Cox seems oversold whenever there is a natural link in GLMs. $\endgroup$ – Nick Cox Nov 23 '17 at 18:21
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It is common practice to apply power transformations(Tukey, Box-Cox) with arbitrary values on the response. From that perspective, I see no particular concern regarding your value of 1/8 - if that transformation gives you good residuals, go for it.

Of course, any transformation changes the functional relationship that you fit, and it may be that 1/8 doesn't make sense mechanistically, but that would not be a concern to me when the purpose is not to extrapolate or fit parameters of a physical law, but to get a proper p-value on the sign of the effect (I'd argue that's the normal use case in a regression). For that purpose, your only concern is that the function fits to the data in the domain of your predictor values (wrt mean and residual variation), and that is easy to check.

If you are unsure about the best value for the power transformation, and want to compare between different options, you should not directly compare AIC / likelihood values because the power transformation changes the scale of the response. Fortunately, it turns out that it's relatively straightforward to calculate a correction for the transformation, such that different transformations can be compared via their (corrected) likelihood (see, e.g. here).

In R, this is implemented in MASS::boxcox - this is a convenient way to choose the right value for the power.

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