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Tests of Homoscedasticity, Normality, and Missing Completely at Random for Incomplete Multivariate Data

Published online by Cambridge University Press:  01 January 2025

Mortaza Jamshidian  and
Siavash Jalal
Mortaza Jamshidian*
Affiliation:
California State University, Fullerton
Siavash Jalal
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Mortaza Jamshidian, Department of Mathematics, California State University, Fullerton, CA 92834, USA. E-mail: [email protected]
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Abstract

Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis. In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR). These tests of MCAR require large sample sizes n and/or large group sample sizes ni, and they usually fail when applied to nonnormal data. Hawkins (Technometrics 23:105–110, 1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when ni are small. This paper proposes a modification of this test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete. Moreover, it is shown that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and nonnormal data. It is explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality. Simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates. Also, a power study of the proposed tests indicates good power. The proposed methods use appropriate missing data imputations to impute missing data. Methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods. Examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.

Keywords

covariance structuresk-sample testmissing datamultiple imputationnonparametric teststructural equationstest of homogeneity of covariances
Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 4 , December 2010 , pp. 649 - 674
Copyright
Copyright © 2010 The Psychometric Society

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Footnotes

This research has been supported in part by the National Science Foundation Grant DMS-0437258 and the National Institute on Drug Abuse Grant 5P01DA001070-36. Siavash Jalal’s work was partly conducted while he was a graduate student at California State University, Fullerton. We would like to thank the Associate Editor, anonymous referees, and Ke-Hai Yuan for providing valuable comments that resulted in a much improved version of this paper.

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