A comparison of asymptotically equivalent test statistics for regression transformation

SUMMARY A simulation comparison is made of six asymptotically equivalent test statistics for transformation of the response in regression; the tests typically have observed sizes up to twice the nominal size. A likelihood ratio statistic, a score statistic, and a Wald statistic behave very similarly, both at the null hypothesis and in terms of power. Values of all the statistics are highly positively correlated. The greater part of any differences between the statistics is concentrated in their standard deviations. To test hypotheses about parametric families of transformation, Box & Cox (1964) use a likelihood ratio test. More recently several other asymptotically equivalent tests have been suggested. In the present paper we present empirical comparisons of six tests in terms of size, power and diagnostic efficacy. The interesting relevance of these results to small-sample comparisons of score-, Wald- and likelihood ratio-tests is briefly discussed. The notation and background of the paper follow those of Box & Cox (1964). A vector of n response values Y is linearly related, with an initial constant term, to p — 1 explanatory variables whose values, and an initial column of ones, are contained in an n xp matrix X. The Box-Cox family of transformations, indexed by a scalar parameter A, is employed to produce the transformed response variable Y^'. The aim is to choose A, by maximum likelihood, so that Y^' follows the linear model y(A) = X/3 + e, (1-1) where e is a vector of errors which are mutually independent with normal distributions of zero mean and constant variance a2, and /3 is a vector of coefficients. The estimate A satisfies the equation