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I have a dataset where I am trying to enforce normality on positively-skewed variables. I've found that consecutive log transformations help in achieving normality but am wondering if there is any statistical or theoretical violation that is breached when consecutive log transformations are performed.

Thanks in advance for any information or advice offered!

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    $\begingroup$ Since it's only an abstract question, you give us no basis for a statistical answer. Mathematically there is (of course) nothing wrong with it provided the log-log transformation is defined. Thus, the original values all must be greater than 1. Statistically, this will often (but not always!) be a poor idea, but that depends on the purpose, context, and assumptions behind the transformation. That's why I asked "why." $\endgroup$
    – whuber
    Jul 1 '13 at 19:37
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    $\begingroup$ Your model only requires the residuals to be approximately normal. It does not require the response variable itself to be normal. An, in fact, that is only for a GLM for a normal distribution. For other GLMs, such as Poisson or Binomial regression, you definitely do not want the residuals to be Normal! $\endgroup$
    – whuber
    Jul 1 '13 at 19:45
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    $\begingroup$ Right, I understand that... so in this case, can one perform multiple, consecutive log transformations to acquire a normal distribution of residuals? $\endgroup$
    – Thraupidae
    Jul 1 '13 at 19:47
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    $\begingroup$ Thraupidae Where possible you should also consider the meaning of the transformed random variable. For example, in some cases times are frequently more skewed than a lognormal, even conditionally, but the reciprocal of time is a speed or rate, which has a natural interpretation and so would be a good candidate to at least consider, rather than loglog. It is the case that loglog transformations (or variants as noted by @NickCox) do sometimes occur, but in many circumstances there are more interpretable alternatives to look at. $\endgroup$
    – Glen_b
    Jul 2 '13 at 1:34
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    $\begingroup$ The other thing is to consider modelling the distribution itself. For example, GLMs include quite skew possibilities like the Inverse Gaussian, whose log is still right skew; with GLMs you can model the mean of the original variable. [Yet another possibility, where a transformation is the natural scale on which to think about a variable, would be to consider a simple transformation as well as a GLM -- such as a Gamma model fitted to log-data, for example.] $\endgroup$
    – Glen_b
    Jul 2 '13 at 1:39
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I think it helps here to be concrete, not abstract.

Think about the support of such transformations. ln $x$ requires that $x > 0$, ln ln $x$ requires that $x > 1$, ln ln ln $x$ that $x > e = \exp(1)$, and so forth. Can you think of variables that naturally satisfy such limits? Also, if one limit is natural, the others won't be. So, while you have a mathematical family here, it is not statistically very useful.

Also, even if by accident or otherwise your data satisfy these limits, these are very severe transformations, even ln repeated twice. They may be the ultimate outlier tamers for large positive values, but they also may have the opposite effect of creating outliers by separating arbitrarily small values just above the lower limits.

If you want something stronger than the logarithm for transforming highly skewed positive values, the (negative) reciprocal is the best candidate. It has the added virtues that its units of measurement are easy to think about and its interpretation is often easy (e.g. times and rates are reciprocals of each other). (A negative sign is optional here if preserving order is important.)

A quite different and much more positive point is that functions such as the loglog $-$ln($-$ln($x$)) and complementary loglog ln($−$ln$ (1−x))$ can be very useful for $0 < x < 1$ and appear as link functions for generalized linear models.

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  • $\begingroup$ Cheers! This brings up a lot of great points. Interpretability was something I was concerned about and what you've written here regarding that makes a lot of sense. Thanks! $\endgroup$
    – Thraupidae
    Jul 2 '13 at 14:31

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