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Biometrika Advance Access originally published online on May 10, 2007
Biometrika 2007 94(2):267-283; doi:10.1093/biomet/asm026
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Copyright © 2007 Biometrika Trust

Articles

A weighted multivariate sign test for cluster-correlated data

Denis Larocque

Department of Quantitative Methods, HEC Montréal, 3000 chemin de la, Côte-Sainte-Catherine, Montréal, Québec, Canada, H3 T 2A7, Canada

Jaakko Nevalainen

Department of Mathematics, Statistics and Philosophy, 33014 University of Tampere, Finland

Hannu Oja

Tampere School of Public Health, 33014 University of Tampere, Finland

denis.larocque{at}hec.ca

jaakko.nevalainen{at}uta.fi

Hannu.Oja{at}uta.fi

Received for publication 1 October 2005. Revision received 1 August 2006.
  Abstract

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.

Key Words: Affine-invariance • Clustered observations • Intraclass correlation • Multivariate location problem • One-way random effect • Spatial sign test


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