Groupe d’études et de recherche en analyse des décisions

G-2008-63

Nonparametric Analysis of Clustered Multivariate Data

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There has been a wide interest to extend univariate and multivariate nonparametric procedures to clustered and hierarchical data. Traditionally, parametric mixed models have been used to account for the correlation structures among the dependent observational units. In this work we outline how multivariate nonparametric procedures for one-sample and several samples problems can be extended to cluster-dependent data problems. Mixed models notation involving design matrices for fixed and random effects is used throughout. For a suitable chosen score function, the asymptotic variance formulas are derived and limiting distributions under the null hypothesis and under a sequence of contiguous alternatives, as well as the limiting distribution for the corresponding estimates, are given. The approach based on a general score function also shows how maximum likelihood estimates or M-estimates, for example, behave with clustered data. Small sample procedures based on sign change and permutation principles are discussed. Further development of nonparametric methods for cluster correlated data would benefit from the notation already familiar to statisticians working under normality assumptions.

, 20 pages