Correction for Bias Introduced by a Transformation of Variables

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. 1. Introduction. The problem of "normalizing" transformations has two different facets: one is concerned with the identity of transforming functions suitable for variables following a distribution with a particular shape or properties and the other with the nature of the statistics capable of serving as unbiased estimates in cases where a given transforming function appears to be successful. The literature on the first problem is rich (see, for example, [1], [2] and [8]). This paper is concerned with the second problem. Our purpose is to deduce minimum variance unbiased estimates of the effects of experimental treatments expressed in the original units. The solution is obtained for a broad category of transforming functions. The estimates of treatment effects expressed in the original units are customarily obtained by the inverse transformation of the estimates in transformed units. As is well known [1], [6], [7], this traditional estimate is biased. Occasionally, this bias is important. Further, the bias gains importance when a number of similar estimates of the same effect, obtained from independent sets of observations , are averaged in order to estimate the average effect. The random errors of the particular effects tend to average out but, in general, not the bias.