Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution

Year 2020, Volume: 49 Issue: 4, 1533 - 1549, 06.08.2020

Müge Borazan Çelikbıçak , Serpil Aktaş

https://doi.org/10.15672/hujms.743041

Abstract

Repeated measures data describe multiple measurements taken from the same experimental unit under the different treatment conditions. In particular, researches with repeated measures data in various fields such as health and behavioral sciences, education, and psychology have an important role in applied statistics. There are many methods used to analyze the results of research designs planned with these measurements. The most important difference between these methods is the assumptions on which the models are based. One of the most important assumptions needed by classical methods is the normality assumption. Many methods are valid under the assumption of normality. However, it is not always possible to hold this assumption in applications. For this reason, in the analysis of repeated measures data, different distributions are necessary that can provide flexibility beyond the normal distribution, especially in cases where the assumption of normality does not hold. In this study, it is proposed to use Multivariate Laplace distribution (MLD) which is an alternative distribution in cases where normality assumption does not hold by examining the multivariate variance analysis model (MANOVA). Under MLD assumption, the parameter estimates for the Two-way Repeated Measures MANOVA model are carried out with the maximum likelihood (ML) estimation and ML estimates are obtained via the EM Algorithm.

Keywords

MANOVA, Repeated measurement data, Multvariate Laplace Distribution, Em Algorithm, repeated measures MANOVA

References

• [1] T.W. Anderson and K.T. Fang, Theory and Applications of Elliptically Contoured and Related Distributions, Standford,CA, 1990.
• [2] T.W. Anderson and K.T. Fang, On The Theory of Multivariate Elliptically Contoured Distributions and Their Applications, Standford,CA, 1992.
• [3] O. Arslan, An Alternative Multivariate Skew Laplace distribution: properties and estimation, Statist. Probab. Lett., 51, 865−887, 2010.
• [4] O. Arslan, Maximum Likelihood Parameter Estimation for the Multivariate Skew- Slash Distribution, Statist. Probab. Lett.,79, 2158−2165, 2009.
• [5] O. Arslan and A.I. Genç, A Generalization of the Multivariate Slash Distribution, J. Statist. Plann. Inference, 139, 1164−1170, 2009.
• [6] M. Borazan Çelikbıçak, Eliptik Konturlu Dağılımlara Dayalı Çok Değişkenli Tekrarlı Ölçümlü Varyans Analizi, Hacettepe Üniversitesi, Fen Bilimleri Enstitüsü, Basılmamış Doktora Tezi Ankara, 2020.
• [7] Y.M. Bulut, Matris değişkenli Laplace dağılımı: Özellikleri ve parametre tahmini, İstatistikçiler Dergisi: İstatistik ve Aktüerya, 11, 32−41, 2018.
• [8] N. Çelik, Anova Modellerinde Çarpık Dağılımlar Kullanılarak Dayanıklı İstatistiksel Sonuç Çıkarım ve Uygulamaları, Ankara Üniversitesi, Doktora Tezi, 2012.
• [9] C.S. Davis , Statistical Methods for the Analysis of Repeated Measurements, 2003.
• [10] J.G. Dias and M. Wedel, An empirical comparison of EM, SEM and MCMC performance for problematic Gaussian mixture likelihoods, Stat. Comput., 14, 323−332, 2004.
• [11] F.Z. Doğru and O. Arslan, Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling, Comm. Statist. Theory Methods, 46, 10879−10896, 2017.
• [12] K.T. Fang, S.Kotz, K.W. Ng, Symmetric Multivariate and Related Distributions, 2018.
• [13] J. Fox, M. Friendly and S. Weisberg, Hypothesis tests for Multivariate Linear Model Using Car Package, The R Journal, 5, 39−52, 2013.
• [14] S. Friedrich, F. Konietschke and M. Pauly, Analysis of Multivariate Data and Repeated Measures Designs with the R Package MANOVA.RM, 2018.
• [15] M. Geraci and M.C. Borja, Notebook: The Laplace distribution, Significance, 15, 10−11, 2018.
• [16] A.K. Gupta and T. Varga, Elliptically Contoured Models in Statistics, Springer Science and Business Media Dordrecht, 1993.
• [17] D.J. Hand and C.C. Taylor, Multivariate Analysis of Variance and Repeated Measures a Practical Approach for Behaviorual Scientists, Chapman and Hall, London, 1987.
• [18] S. Kotz, T.J., Kozubowski,and K. Podgórski, The Laplace Distribution and Generalizations, 2001.
• [19] T.J. Kozubowski, K., Podgórski and I. Rychlik, , Multivariate Generalized Laplace distribution and Related Random Fields, J. Multivariate Anal., 113, 59−72, 2013.
• [20] M. Krzysko, T. Smialowski and W. Wolynski, Analysis of Multivariate Repeated Measures Data using a MANOVA model and Principal Components, Biometrical Letters, 51, 103−114, 2014.
• [21] V. Kumar, P. Mehta and G. Shukla, Multivariate Analysis of Repeated Measures Data, 6, 133−148, 2013.
• [22] J.K. Lindsey, Multivariate Elliptically Contoured Distributions for Repeated Measurements, Biometrics,55, 1277−1280, 1999.
• [23] J.K. Lindsey and P.J. Lindsey, Multivariate Distributions with Correlation Matrices for Nonlinear Repeated Measurements , Comput. Statist. Data Anal., 50, 720−732, 2006.
• [24] G. McLachlan and T. Krishnan, The EM Algorithm and Extensions(2nd ed), John Wiley and Sons, Inc., 2008.
• [25] D.K. Mcgraw and F. Wagner, Symmetric Distribution, IEEE Trans. Inform. Theory, IT., 14(1), 110−120, 1968.
• [26] D.F. Morrison, Multivariate Analysis of Variance, Encyclopedia of Biostatistics, 2005.
• [27] S. Nadarajah, The Kotz-Type Distribution with Applications, Statistics (Ber.),37 , 341−358, 2003.
• [28] R.G. O’Brien and M.K. Kaiser, MANOVA method for Analyzing Repeated Measures Designs, An Extensive Primer, Psychological bulletin, 97, 316−333, 1985.
• [29] K. Plungpongpun, Analysis of Multivariate Data Using Kotz Type Distribution, 2003.
• [30] H. Visk, On the Parameter Estimation of the Asymmetric Multivariate Laplace Distribution, Comm. Statist. Theory Methods, 38, 461−470, 2009.
• [31] F.G. Yavuz and O. Arslan, Linear mixed model with Laplace distribution (LLMM), Statist. Papers, 59, 271−289, 2018.