Regarding how to transform leptokurtic distribution to normality

Question

I followed Nick Cox's advice from here (How to transform leptokurtic distribution to normality?) and used "sign(.) * abs(.)^(1/3)" as suggested. It worked for me and I was wondering if anyone has a published reference for this formula? or if you could point me in the right direction?

Answer 1

@whuber's comment is to the point and a good succinct summary. But some more can be added to flesh out a slender argument.

The transformation mentioned is cube root $x^{1/3}$, written in pseudo- or generic code: the problem tackled is that powering in software is rarely programmed to catch odd integer roots (i.e. roots $1/3$, $1/5, \dots$), even though they are perfectly well defined for negative arguments (e.g. $-2$ is the cube root of $-8$). Thus to take cube roots of negative numbers, we negate them beforehand and restore the original sign afterwards.

The merits of cube root with roughly symmetric but long-, heavy- or fat-tailed distributions are most clear when values are roughly symmetric about zero. Sometimes this happens any way (curvatures from digital elevation models are a case in point), or it can be achieved by working with deviations, most usefully from the median or some other resistant estimator of location or level of the distribution, deviations that are thus negative, zero or positive.

Cube root then satisfies these properties.

1. Sign is preserved: negative, zero and positive values remain so after transformation.

2. Furthermore, the transformation is monotonic.

3. Tails tend to be pulled in, so kurtosis (and often skewness) may be much reduced. There is, however, no guarantee that distributions will be closer to normality, let alone even approximately normal.

4. The transformation is smooth around zero and indeed steepest at zero with slope 1. On many displays of transformed data, unless there are exact zeros, positive and negative values are thus pulled apart, which can echo any qualitative distinction between them (e.g. firms making a profit and firms making a loss, convex slopes and concave slopes).

The transformation is perhaps the simplest transformation with these properties that should be familiar to those who profited from pre-university mathematics. $\text{sign}(x) \ln(|x| + 1)$ is likely to seem too unfamiliar or too contrived and $\text{arsinh}(x)$ is likely to seem too esoteric, if it is remembered at all. However, anecdotal evidence is that cube root is also too infrequently used to remain anywhere so familiar as say quadratic, logarithmic or exponential functions, so it may well need to be explained at least briefly to almost all audiences.

Cube roots are naturally classical, as the cube root of a volume is a length. More statistical uses of the transformation go back at least to the 1950s, but its applicability to negative, zero and positive values alike has only rarely been pointed out, perhaps because it appears too obvious mathematically to deserve any emphasis.

These arguments and some others (e.g. the relationship between cube roots and gamma distributions) are sketched in

Cox, N.J. 2011. Cube roots. Stata Journal 11: 149-154. http://www.stata-journal.com/article.html?article=st0223

which includes some further references.