| | | |

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

#### Müge BORAZAN ÇELİKBIÇAK [1] , Serpil AKTAŞ ALTUNAY [2]

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.
MANOVA, Repeated measurement data, Multvariate Laplace Distribution, Em Algorithm, repeated measures MANOVA
• [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.
Primary Language en Statistics and Probability Statistics Orcid: 0000-0002-5796-9192Author: Müge BORAZAN ÇELİKBIÇAK (Primary Author)Institution: JANDARMA VE SAHİL GÜVENLİK AKADEMİSİ, JANDARMA VE SAHİL GÜVENLİK FAKÜLTESİ, GÜVENLİK BİLİMLERİ BÖLÜMÜCountry: Turkey Orcid: 0000-0003-3364-6388Author: Serpil AKTAŞ ALTUNAYInstitution: HACETTEPE ÜNİVERSİTESİ, FEN FAKÜLTESİ, İSTATİSTİK BÖLÜMÜCountry: Turkey Publication Date : August 6, 2020
 Bibtex @research article { hujms743041, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1533 - 1549}, doi = {10.15672/hujms.743041}, title = {Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution}, key = {cite}, author = {Borazan Çeli̇kbıçak, Müge and Aktaş, Serpil} } APA Borazan Çeli̇kbıçak, M , Aktaş, S . (2020). Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1533-1549 . DOI: 10.15672/hujms.743041 MLA Borazan Çeli̇kbıçak, M , Aktaş, S . "Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1533-1549 Chicago Borazan Çeli̇kbıçak, M , Aktaş, S . "Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1533-1549 RIS TY - JOUR T1 - Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution AU - Müge Borazan Çeli̇kbıçak , Serpil Aktaş Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.743041 DO - 10.15672/hujms.743041 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1533 EP - 1549 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.743041 UR - https://doi.org/10.15672/hujms.743041 Y2 - 2020 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution %A Müge Borazan Çeli̇kbıçak , Serpil Aktaş %T Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.743041 %U 10.15672/hujms.743041 ISNAD Borazan Çeli̇kbıçak, Müge , Aktaş, Serpil . "Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution". Hacettepe Journal of Mathematics and Statistics 49 / 4 (August 2020): 1533-1549 . https://doi.org/10.15672/hujms.743041 AMA Borazan Çeli̇kbıçak M , Aktaş S . Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1533-1549. Vancouver Borazan Çeli̇kbıçak M , Aktaş S . Parameter estimates for two-way repeated measurement MANOVA based on multivariate Laplace distribution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1533-1549.

Authors of the Article