A method for determining the effect of imbalance

Abstract

In the analysis of variance, the efficiency of a statistical procedure depends on the values of some function, g, chosen appropriately. For example, g can be the variance of an estimator of a variance component, or the power of a test associated with a given model. Of particular interest in this concern is the behavior of g under a variety of experimental conditions. For unbalanced data, g depends in a complex way on the cell frequencies (design), and on a set of parameters denoted by θ. It is therefore quite difficult, in general, to determine how imbalance affects the value of g. The degree of imbalance of a data set can be quantified by the value of a measure, denoted by ø, as shown in Khuri ( Biometrical J. 29 (1987), 383–396). Studying the behavior of the function g under different patterns of imbalance would be greatly simplified if g were directly related to ø. Unfortunately, the dependence of g on the cell frequencies is not, in general, expressible explicitly in terms of ø. However, if the functional form of g can be adequately approximated by an empirical model given in terms of ø and θ, then it is possible to use ø to assess the effect of imbalance on g. The derivation of such a model can be conveniently achieved by means of response surface techniques. This is demonstrated using, as an example of g, the variance of an analysis of variance ( ANOVA) estimator. A key element in the development of this demonstration is an algorithm for generating cell frequencies having a specified degree of imbalance. A numerical example is given to illustrate the proposed methodology.