On the Comparative Anatomy of Transformations

Abstract

The attention of statisticians has usually been focussed on single transformations, rather than on families of transformations. With a growing appreciation of the advantages of examining the behavior of data or approximations over whole families of transformations (Moore and Tukey [2], Anscombe and Tukey [1]), there arises a need for rationally planned charts for representing families of transformations. The contributions which (i) the topology of the family and (ii) a definition of the strength of a transformation can make to charting are studied in general and applied to the charting of the simple family of transformations. This family is defined to include all transformations of the form $$y \text{is replaced by} z = (y + c)^p$$ and all their limits. It thus includes $z = \log (y + c), z = e^{my}$ and the special case \begin{equation*}z = \begin{cases}0, & y = y_\min,\\1, & \text{otherwise},\end{cases}\end{equation*} where $y_\min$ is the least value of $y$ either (i) present in the data or (ii) possible, as well as all linear transformations of these transformations. Experience having shown that transformations with $p \leqq 1$ are much more frequently useful than any others, the charts developed, presented, and exemplified here are restricted to the part of the simple family--its central region--for which $p \leqq 1$. Separate charts are presented for two cases which should cover most cases which arise in practice: (1) Where, as with counted data and small counts, the least reasonable value for $y + c = 0$, and this value is likely to occur; (2) Where $y + c$ is always safely $>0$, and the range of $y$ is through not many powers of 10.