Inference Based on Ranks for the Multiple-Design Multivariate Linear Model

Abstract

Abstract Multivariate models more general than the standard multivariate linear model have received considerable attention in both the statistical and econometric literature; see Srivastava (1966, 1967, 1968) and Kleinbaum (1973). The multiple-design multivariate (MDM) linear model generalizes the standard multivariate linear model in the sense that a different design matrix is used for each response (dependent) variable. Important applications of the MDM linear model include changing covariates in repeated measurement designs and p-variate regression systems (p > 1) with different regressors for each of the p response variables. The latter situation has been described as “seemingly unrelated regression equations” in the econometric literature (Zellner 1962). In this article an approach to statistical inference is presented under the MDM linear model based on multivariate rank and aligned rank statistics. In Section 1 estimation and hypothesis testing for the MDM linear model are reviewed under parametric assumptions about the underlying multivariate distribution, such as a multivariate normal distribution or a continuous multivariate distribution that has a finite positive-definite variance—covariance matrix. In Sections 2 and 3 an approach to statistical inference based on ranks for the MDM linear model is presented. In contrast to the parametric assumptions, this approach requires only that the continuous multivariate distribution has a finite Fisher information matrix, which is less restrictive than a finite variance—covariance matrix. Thus, unlike the parametric analysis, heavy-tailed distributions without finite moments pose no problems in the rank approach. In particular, a multivariate linear rank statistic is constructed, and its asymptotic multivariate normality under a simple null hypothesis and a sequence of contiguous alternatives is established. These results lead to an asymptotic test of a simple null hypothesis and an expression for its asymptotic power. An R estimate of the parameter vector is derived via aligned rank statistics, and its asymptotic multivariate normality is established. A Wald-type statistic for an asymptotic test of a general linear hypothesis is constructed, and an expression for its asymptotic power based on a sequence of contiguous alternatives is developed. In Section 4 an example is presented in which the parametric and rank approaches are contrasted. The example consists of a pharmaceutical study with a four-way crossover design. To illustrate the nonrobustness of the parametric approach, the data set is reexamined after the introduction of random errors from multivariate normal and multivariate Cauchy distributions.