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From |
"Nick Cox" <n.j.cox@durham.ac.uk> |

To |
<statalist@hsphsun2.harvard.edu> |

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 and comments from Statalist members. ============================= 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 * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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