A weighted multivariate sign test for cluster-correlated data

We consider the multivariate location problem with cluster-correlated data. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Under weak assumptions, the test statistic is asymptotically distributed as a chi-squared random variable as the number of clusters goes to infinity. The asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality. Optimal weights maximizing Pitman asymptotic efficiency are provided. These weights depend on the cluster sizes and on the intracluster correlation. Several approaches for estimating these weights are presented. Using Pitman asymptotic efficiency, we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster. A multivariate weighted t-test is also introduced. The finite-sample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive. A real data example illustrates the practical application of the methodology. Copyright 2007, Oxford University Press.

[1]  Sharon Lohr,et al.  Robust Estimation of Multivariate Covariance Components , 2005, Biometrics.

[2]  R. Glynn,et al.  Incorporation of Clustering Effects for the Wilcoxon Rank Sum Test: A Large‐Sample Approach , 2003, Biometrics.

[3]  Somnath Datta,et al.  Rank-Sum Tests for Clustered Data , 2005 .

[4]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[5]  B. Leroux,et al.  Analysis of clustered data: A combined estimating equations approach , 2002 .

[6]  Hannu Oja,et al.  Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices , 2006 .

[7]  Hannu Oja,et al.  A weighted spatial median for clustered data , 2006, Stat. Methods Appl..

[8]  L. Dümbgen.  On Tyler's M-Functional of Scatter in High Dimension , 1998 .

[9]  B. Rosner,et al.  Use of the Mann-Whitney U-test for clustered data. , 1999, Statistics in medicine.

[10]  Hannu Oja,et al.  On the multivariate spatial median for clustered data , 2007 .

[11]  Denis Larocque,et al.  An affine‐invariant multivariate sign test for cluster correlated data , 2002 .

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  Hannu Oja,et al.  Multivariate Nonparametric Tests , 2004 .

[14]  Somnath Datta,et al.  Marginal Analyses of Clustered Data When Cluster Size Is Informative , 2003, Biometrics.

[15]  Pranab Kumar Sen,et al.  Within‐cluster resampling , 2001 .

[16]  R. L. Ebel,et al.  Estimation of the reliability of ratings , 1951 .

[17]  Ronald H. Randles,et al.  A Simpler, Affine-Invariant, Multivariate, Distribution-Free Sign Test , 2000 .

[18]  J. Ware,et al.  Applied Longitudinal Analysis , 2004 .