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Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix

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  • Dayanand Naik
  • Shantha Rao

Abstract

In this article we consider a set of t repeated measurements on p variables (or characteristics) on each of the n individuals. Thus, data on each individual is a p 2 t matrix. The n individuals themselves may be divided and randomly assigned to g groups. Analysis of these data using a MANOVA model, assuming that the data on an individual has a covariance matrix which is a Kronecker product of two positive definite matrices, is considered. The well-known Satterthwaite type approximation to the distribution of a quadratic form in normal variables is extended to the distribution of a multivariate quadratic form in multivariate normal variables. The multivariate tests using this approximation are developed for testing the usual hypotheses. Results are illustrated on a data set. A method for analysing unbalanced data is also discussed.

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  • Dayanand Naik & Shantha Rao, 2001. "Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(1), pages 91-105.
    • Handle: RePEc:taf:japsta:v:28:y:2001:i:1:p:91-105
    DOI: 10.1080/02664760120011626
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    References listed on IDEAS

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    1. Huynh Huynh & Leonard S. Feldt, 1976. "Estimation of the Box Correction for Degrees of Freedom from Sample Data in Randomized Block and Split-Plot Designs," Journal of Educational and Behavioral Statistics, , vol. 1(1), pages 69-82, March.
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    3. Khuri, A. I. & Mathew, T. & Nel, D. G., 1994. "A Test to Determine Closeness of Multivariate Satterthwaite's Approximation," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 201-209, October.
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    1. Lu, Nelson & Zimmerman, Dale L., 2005. "The likelihood ratio test for a separable covariance matrix," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 449-457, July.
    2. Glanz, Hunter & Carvalho, Luis, 2018. "An expectation–maximization algorithm for the matrix normal distribution with an application in remote sensing," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 31-48.
    3. Kohli, Priya & Garcia, Tanya P. & Pourahmadi, Mohsen, 2016. "Modeling the Cholesky factors of covariance matrices of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 87-100.
    4. Filipiak, Katarzyna & Klein, Daniel, 2017. "Estimation of parameters under a generalized growth curve model," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 73-86.
    5. Anuradha Roy & Ricardo Leiva, 2008. "Testing of a Structures Covariance Matrix for Three-Level Repeated Measures Data," Working Papers 0037, College of Business, University of Texas at San Antonio.
    6. Lee, Keunbaik & Lee, Chang-Hoon & Kwak, Min-Sun & Jang, Eun Jin, 2021. "Analysis of multivariate longitudinal data using ARMA Cholesky and hypersphere decompositions," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    7. Mirosław Krzyśko & Wojciech Łukaszonek & Waldemar Wołyński, 2018. "Canonical Correlation Analysis In The Case Of Multivariate Repeated Measures Data," Statistics in Transition New Series, Polish Statistical Association, vol. 19(1), pages 75-85, March.
    8. Kim, Chulmin & Zimmerman, Dale L., 2012. "Unconstrained models for the covariance structure of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 104-118.
    9. Manceur, A.M. & Dutilleul, P., 2013. "Unbiased modified likelihood ratio tests for simple and double separability of a variance–covariance structure," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 631-636.
    10. Feng, Sanying & Lian, Heng & Xue, Liugen, 2016. "A new nested Cholesky decomposition and estimation for the covariance matrix of bivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 98-109.
    11. Geoffrey Colin L. Peterson & Dong Li & Brian J. Reich & Donald Brenner, 2017. "Spatial prediction of crystalline defects observed in molecular dynamic simulations of plastic damage," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(10), pages 1761-1784, July.
    12. Wojciech Łukaszonek, 2017. "A Multidimensional And Dynamised Classification Of Polish Provinces Based On Selected Features Of Higher Education In 2002–2013," Statistics in Transition New Series, Polish Statistical Association, vol. 18(2), pages 271-290, June.
    13. Anestis Touloumis & Simon Tavaré & John C. Marioni, 2015. "Testing the mean matrix in high-dimensional transposable data," Biometrics, The International Biometric Society, vol. 71(1), pages 157-166, March.
    14. Lingzhe Guo & Reza Modarres, 2020. "Testing the equality of matrix distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 289-307, June.
    15. Filipiak, Katarzyna & Klein, Daniel & Roy, Anuradha, 2016. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 105-124.
    16. Driezen, Kassandra & Adriaensen, Frank & Rondinini, Carlo & Doncaster, C. Patrick & Matthysen, Erik, 2007. "Evaluating least-cost model predictions with empirical dispersal data: A case-study using radiotracking data of hedgehogs (Erinaceus europaeus)," Ecological Modelling, Elsevier, vol. 209(2), pages 314-322.
    17. Sean L Simpson & Lloyd J Edwards & Martin A Styner & Keith E Muller, 2014. "Kronecker Product Linear Exponent AR(1) Correlation Structures for Multivariate Repeated Measures," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-10, February.
    18. Hao, Chengcheng & Liang, Yuli & Mathew, Thomas, 2016. "Testing variance parameters in models with a Kronecker product covariance structure," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 182-189.
    19. Katarzyna Filipiak & Daniel Klein & Anuradha Roy, 2015. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Working Papers 0148mss, College of Business, University of Texas at San Antonio.
    20. Martin Ohlson & Zhanna Andrushchenko & Dietrich Rosen, 2011. "Explicit estimators under m-dependence for a multivariate normal distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(1), pages 29-42, February.
    21. Viroli, Cinzia, 2012. "On matrix-variate regression analysis," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 296-309.

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