A Quick Method for Choosing a Transformation

The choice of non-linear transformations in the analysis of data can frequently be simplified by restricting the possible transformations to a particular family. Tukey has shown that the simple has many desirable properties from this point of view. This family can be represented as the set of solutions to a third order differential equation and the constant of this equation provides a convenient index of the family. This index may be approximated by substituting the given data into the corresponding difference equation. The resulting approximation can then be used for rough solutions or as a starting value for the iterative solution of the maximum likelihood equations given by Turner. Two examples are provided to demonstrate the procedure. In the analysis of data in the physical and engineering sciences it is not uncommon to consider the possibility of applying a transformation to one or more of the variables under study. The most obvious situation of this type occurs when the dependent variable is not linearly related to the independent variable and there is no reason to think that the underlying relation is a polynomial of a particular degree. Polynomials are, of course, both easy to handle and widely used. However, higher degree polynomials raise serious numerical problems and leave much to be desired from the point of view of interpretation. We take it as self-evident that almost any physical scientist would rather consider