The score statistic for regression transformation

SUMMARY The score statistic for regression transformation using a parametric family is developedfurther; a simple formula for the approximate variance matrix of score statistics, basedon asymptotic theory, is applied to yield the score statistic T s for regression transforma-tion. This is compared to Atkinson's T D statistic for the same problem, and yields animprovement to its standardization. Simulations indicate that the significance levels ofthe proposed score statistic T s are very close to their nominal values. Some key words: Power transformation; Regression; Score statistic; Simulation; Tests for normality. 1. INTRODUCTION The work of Box & Cox (1964) has been the basis of many investigations into theuse of transformations in regression analysis, and the current state of the topic is wellsummarized in the monographs of Cook & Weisberg (1982) and Atkinson (1985). Theprincipal aim of the original work was to transform a response or dependent variable sothat it would satisfy the normality and other standard assumptions of regression analysis.The authors used a parametric family of transformations to achieve this, and placedemphasis on estimating its parameters. In subsequent work, Andrews (1971) was moreconcerned with testing transformation parameters, while Atkinson (1973) introducedanother statistic for this purpose from consideration of both Andrews's statistic and scoreor C(a) tests. The present paper develops a standardized score statistic in the case of ascalar transformation parameter; in effect, it is found to provide the standardization ofAtkinson's statistic, an aspect which Atkinson (1973) believed to be theoretically intract-able, and avoided by using an approximate regression argument.