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Both Box-Cox and Yeo-Johnson transform non-normal distribution into a normal distribution. However, Box-Cox requires all samples to be positive, while Yeo-Johnson has no restrictions.

To me, it seems that Yeo-Johnson is superior to Box-Cox. Is there any reason why I shouldn't always blindly use Yeo-Johnson over Box-cox ? (ex: back-transform, interpretability, computation efficiency...)

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Interpretability is a major issue.

The power parameter is different for positive and negative values; and the transformation therefore has a different interpretation for positive and negative values.

When you have both, the transformation over the whole range seems somewhat arbitrary and it is very tricky to explain - especially to a lay audience.

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I have found transformations like the cube root

$$ \text{sign}(y)\ |y|^{1/3}$$ and the so-called neglog

$$\text{sign}(y)\ \ln(1 + |y|)$$ and the inverse hyperbolic sine

$$\text{asinh}(y)$$

helpful for responses that can be negative, zero, or positive and that are skewed and/or have very long tails.

Their advantages include

  1. pulling in the tails relative to zero

  2. working smoothly around 0

  3. preserving the sign of the response (which usually has substantive or scientific meaning: consider profit/loss or increase/decrease)

  4. having derivatives that are easy to work with

  5. having exact or approximate limiting behaviour as usually desired, in particular that for $y \gg 0$ the second is close to $\ln y$.

In these cases there is no question of choosing a different power for positive and negative values, and any analyst should want to see a very strong justification for that.

The practical test of a transformation is often just that it aids visualization. It's not essential that coefficients from regressions are easy to interpret. If the cube root of $y$ changes linearly with predictors, that is what you have found. The coefficients don't need chit-chat; they are the gradients in that space (which you should be using for visualization).

Despite deeply impressive work on estimating a transformation, it is not often that I choose a transformation formally. Logarithms, reciprocals and square roots are often obvious candidates on a variety of grounds. Try out candidates and see if they help.

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  • $\begingroup$ One reason for using those "obvious candidates" apart from "often easier to interpret" is that the more you pre-process your data by optimising a data dependent objective function, the more danger there is that later results will be biased and over-optimistic (underestimating uncertainty, as standard measures do not take into account the uncertainty in pre-processing). $\endgroup$ Aug 18 at 10:23
  • $\begingroup$ Indeed. I am a little contrarian on that. Looking at the data to see what analysis makes sense is to me a bigger plus than the minus you're referring to. The usual argument seems to be that B's analysis trying many things and then choosing a particular transformation is more problematic than A's analysis which just chose that transformation directly because prior experience made it seem obvious. I find that hard to swallow Also, being open about what you considered and didn't use sounds good, but are you supposed to document all that you tried that you realised was plain silly? $\endgroup$
    – Nick Cox
    Aug 18 at 10:40
  • $\begingroup$ In my view this is very subtle and not black and white. There are pros and cons of looking at the data and making decisions based on that, and I agree that in many cases the pros outweigh the cons. My point was that less bias is to be expected if one only decides between say three options that are quite different from each other (no transformation, logs, roots for example), rather than picking one from a continuous set by optimisation, as in Box-Cox, Yeo-Johnson. It's a compromise though - there will be bias, and there's some possibility to adapt to the data. $\endgroup$ Aug 18 at 11:01
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    $\begingroup$ I think I agree. Ironically, or otherwise, the original paper by Box and Cox doesn't encourage or endorse letting a program choose your transformation for you (not that at the time code was easily available for that in any case). The two examples end up as choosing variously logarithm and reciprocal as transformations , as close to the powers given by a calculation and (vital!) as natural for each problem. $\endgroup$
    – Nick Cox
    Aug 18 at 11:06

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