Title: Informative transformation of responses that can be positive or negative
Authors: Anthony Atkinson - London School of Economics (United Kingdom) [presenting]
Marco Riani - University of Parma (Italy)
Abstract: The parametric family of power transformations to approximate normality analysed by Box and Cox can be applied only to positive data. This transformation has been generalized to allow for the inclusion of zero and negative response values, which arise for example in data on GNP growth and company profits and in the differences in measurements before and after treatment. The aim is to describe the use of constructed variables to provide an approximate score statistic for the transformation which avoids the numerical optimization required for estimation of the transformation parameter using maximum likelihood. The resulting statistic is based on aggregate properties of the data. Robust analysis of the data with the forward search provides a series of subsets of the data of increasing size, ordered by closeness to the fitted model for each subset size. The ``fan plot'' of the statistics for these subsets against subset size clearly indicates the effect of individual observations, especially outliers, on the estimated transformation parameter. The score test is extended to determine whether positive or negative observations require different transformations, leading to an informative extended fan plot. The methods will be illustrated with several examples, one from Darwin on cross- and self-fertilized plants. There will be some discussion of the distributions of the test statistics.