# st: -transint- updated on SSC

 From "Nick Cox" To Subject st: -transint- updated on SSC Date Wed, 23 Nov 2005 00:52:20 -0000

```Thanks to Kit Baum, a revised version of -transint-
is now available from SSC.

-transint- is just a Stata help file containing some material
I wrote on transformations. Thus if you install -transint-,
-help transint- or -whelp transint- will give you a view of
this material.

This arose out of various teaching and advising within my
Department. As I am a geographer preparing material for geographers,
the material should be considered in that light. I have not
found a treatment of transformations that meets my idea of what
I wanted to be available, so I wrote one. My students and
colleagues are not statisticians, any more than I am, but the
colleagues concerned are strong quantitatively-minded scientists.

Naturally the hope is that this material will be interesting or
useful to others. I would appreciate comments, including information
on errors or omissions.

The previous version was published in 1999 and is re-released as
transint6.hlp with some small fixes. The new version is much longer.
You will be able to read -transint- in versions of Stata before 8,
but much or all of the SMCL mark-up will be visible to you.

The stimulus for re-releasing this was discovering a vein of ideas on
how to transform variables that are both negative and positive, but
heavy-tailed. This extract from transint.hlp may give some of the
flavour. It also indicates a topic on which I welcome references

============================= extract

Transformations for variables that are both positive and negative

Most of the literature on transformations focuses on one or both of two
related situations: the variable concerned is strictly positive; or it is
zero or positive. If the first situation does not hold, some transformations
do not yield real number results (notably, logarithms and reciprocals); if
the second situation does not hold, then some other transformations do not
yield real number results or more generally do not appear useful (notably,
cube roots, square roots or squares).

However, in some situations response variables in particular can be both
positive and negative. This is common whenever the response is a balance,
change, difference or derivative. Although such variables are often skew, the
most awkward property that may invite transformation is heavy (long or fat)
tails, high kurtosis in one terminology.  Zero usually has a strong
substantive meaning, so that we wish to preserve the distinction between
negative, zero and positive values. (Note that Celsius or Fahrenheit
temperatures do not really qualify here, as their zero points are
statistically arbitrary, for all the importance of whether water melts or
freezes.)

In these circumstances, experience with right-skewed and strictly positive
variables might suggest looking for a transformation that behaves like ln x
when x is positive and like ln(-x) when x is negative.  This still leaves the
problem of what to do with zeros. In addition, it is clear from any sketch
that (in Stata terms)

cond(x <= 0, ln(-x), ln(x))

would be useless. One way forward is to use

ln(-x + 1)    if x <= 0,
ln(x + 1)     if x > 0.

This can also be written

sign(x) ln(|x| + 1)

where sign(x) is 1 if x > 0, 0 if x == 0 and -1 if x < 0.  This function
passes through the origin, behaves like x for small x, positive and negative,
and like sign(x) ln(abs(x)) for large |x|.  The gradient is steepest at 1 at
x = 0, so the transformation pulls in extreme values relative to those near
the origin.  It has recently been dubbed the neglog transformation (Whittaker
et al. 2005).  An earlier reference is John and Draper (1980).  In Stata
language, this could be

cond(x <= 0, ln(-x + 1), ln(x + 1))

or

sign(x) * ln(abs(x) + 1)

A suitable generalisation of powers other than 0 is

-[(-x + 1)^p - 1] / p    if x <= 0,
[(x + 1)^p - 1] / p    if x > 0.

Transformations that affect skewness as well as heavy tails in variables that
are both positive and negative were discussed by Yeo and Johnson (2000).

Another possibility in this terrain is to apply the inverse hyperbolic
function arsinh (also known as arg sinh and arcsinh). This is the inverse of
the sinh function, which in turn is defined as

sinh(x) = (exp(x) - exp(-x)) / 2.

The arsinh function can be computed in Stata as

ln(x + sqrt(x^2 + 1))

It too passes through the origin and is steepest at the origin.  For large
|x| it behaves like sign(x) ln(|2x|). So in practice neglog(x) and arsinh(x)
have loosely similar effects.

John, J.A. and N.R. Draper. 1980.  An alternative family of transformations.
Applied Statistics 29: 190-197.

Whittaker, J., J. Whitehead and M. Somers. 2005.  The neglog transformation
and quantile regression for the analysis of a large credit scoring
database.  Applied Statistics 54: 863-878.

Yeo, I. and R.A. Johnson. 2000.  A new family of power transformations to
improve normality or symmetry.  Biometrika 87: 954-959.

============================= end of extract

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```