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

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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 n i , 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 n i 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.

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References

  • Anderson, T.W., & Darling, D.A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49, 765–769.

    Article  Google Scholar 

  • Bentler, P.M., Kim, K.H., & Yuan, K.H. (2004). Testing homogeneity of covariances with infrequent missing data patterns. Unpublished Manuscript.

  • David, F.N. (1939). On Neyman’s “smooth” test for goodness of fit: I. Distribution of the criterion Ψ2 when the hypothesis tested is true. Biometrika, 31, 191–199.

    Google Scholar 

  • Fisher, R.A. (1932). Statistical methods for research workers (4th ed.). London: Oliver & Boyd.

    Google Scholar 

  • Hawkins, D.M. (1981). A new test for multivariate normality and homoscedasticity. Technometrics, 23, 105–110.

    Article  Google Scholar 

  • Hou, C.D. (2005). A simple approximation for the distribution of the weighted combination of non-independent or independent probabilities. Statistics and Probability Letters, 73, 179–187.

    Article  Google Scholar 

  • Jamshidian, M., & Bentler, P.M. (1999). ML estimation of mean and covariance structures with missing data using complete data routines. Journal of Educational and Behavioral Statistics, 24, 21–41.

    Google Scholar 

  • Jamshidian, M., & Schott, J. (2007). Testing equality of covariance matrices when data are incomplete. Computational Statistics and Data Analysis, 51, 4227–4239.

    Article  Google Scholar 

  • Kim, K.H., & Bentler, P.M. (2002). Tests of homogeneity of means and covariance matrices for multivariate incomplete data. Psychometrika, 67, 609–624.

    Article  Google Scholar 

  • Kruskal, W., & Wallis, W. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47, 583–621.

    Article  Google Scholar 

  • Ledwina, T. (1994). Data-driven version of Neyman’s smooth test of fit. Journal of the American Statistical Association, 89, 1000–1005.

    Article  Google Scholar 

  • Little, R.J.A. (1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association, 83, 1198–1202.

    Article  Google Scholar 

  • Little, R.J.A., & Rubin, D.B. (1987). Statistical analysis with missing data (1st edn.). New York: Wiley.

    Google Scholar 

  • Little, R.J.A., & Rubin, D.B. (2002). Statistical analysis with missing data (2nd edn.). New York: Wiley.

    Google Scholar 

  • Marhuenda, Y., Morales, D., & Pardo, M.C. (2005). A comparison of uniformity tests. Statistics, 39, 315–328.

    Article  Google Scholar 

  • Neyman, J. (1937). Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift, 20, 150–199.

    Google Scholar 

  • Rayner, J.C.W., & Best, D.J. (1990). Smooth tests of goodness of fit: An overview. International Statistical Review, 58, 9–17.

    Article  Google Scholar 

  • Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581–592.

    Article  Google Scholar 

  • Rubin, D.B. (1987). Multiple imputation for nonresponse in surveys. New York: Wiley.

    Book  Google Scholar 

  • Scholz, F.W., & Stephens, M.A. (1987). K-sample Anderson-Darling tests. Journal of the American Statistical Association, 82, 918–924.

    Article  Google Scholar 

  • Srivastava, M.S. (2002). Methods of multivariate statistics. New York: Wiley.

    Google Scholar 

  • Srivastava, M.S., & Dolatabadi, M. (2009). Multiple imputation and other resampling scheme for imputing missing observations. Journal of Multivariate Analysis, 100, 1919–1937.

    Article  Google Scholar 

  • Thas, O., & Ottoy, J.-P. (2004). An extension of the Anderson-Darling k-sample test to arbitrary sample space partition sizes. Journal of Statistical Computation and Simulation, 74, 651–665.

    Article  Google Scholar 

  • Yuan, K.H. (2009). Normal distribution pseudo ML for missing data: With applications to mean and covariance structure analysis. Journal of Multivariate Analysis, 100, 1900–1918.

    Article  Google Scholar 

  • Yuan, K.H., Bentler, P.M., & Zhang, W. (2005). The effect of skewness and kurtosis on mean and covariance structure analysis: The univariate case and its multivariate implication. Sociological Methods & Research, 34, 240–258.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, California State University, Fullerton, CA, 92834, USA

    Mortaza Jamshidian & Siavash Jalal

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  1. Mortaza Jamshidian
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  2. Siavash Jalal
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Correspondence to Mortaza Jamshidian.

Additional information

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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Jamshidian, M., Jalal, S. Tests of Homoscedasticity, Normality, and Missing Completely at Random for Incomplete Multivariate Data. Psychometrika 75, 649–674 (2010). https://doi.org/10.1007/s11336-010-9175-3

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  • DOI: https://doi.org/10.1007/s11336-010-9175-3

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