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On the Asymptotic Distributions of Two Statistics for Two-Level Covariance Structure Models within the Class of Elliptical Distributions

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan  and
Peter M. Bentler
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Dr. Ke-Hai Yuan, Department of Psychology, University of Notre Dame, Notre Dame, IN 46556 USA. Email: kyuan@nd.edu
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Abstract

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes.

Keywords

Asymptoticselliptical distributionlikelihood ratio statisticrescaled statistic
Type
Theory and Methods
Information
Psychometrika , Volume 69 , Issue 3 , September 2004 , pp. 437 - 457
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper.

This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.

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