On matrix-variate regression analysis

Three-way data arise in different application domains when multiple responses are measured at different time points or locations. A new regression model for analyzing three-way data is proposed. By assuming the matrix normal distribution for the error term, we will show that the proposed model represents the natural generalization of multiple and multivariate regression analysis. Inferential properties of the model estimators are derived. The model fit is illustrated on a real application.

[1]  Age K. Smilde,et al.  Multiway covariates regression models , 1999 .

[2]  James M. Dickey,et al.  Matricvariate Generalizations of the Multivariate $t$ Distribution and the Inverted Multivariate $t$ Distribution , 1967 .

[3]  D. Theobald,et al.  Empirical Bayes hierarchical models for regularizing maximum likelihood estimation in the matrix Gaussian Procrustes problem , 2006, Proceedings of the National Academy of Sciences.

[4]  Dietrich von Rosen,et al.  The growth curve model: a review , 1991 .

[5]  D. Zimmerman,et al.  The likelihood ratio test for a separable covariance matrix , 2005 .

[6]  Dayanand N. Naik,et al.  Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix , 2001 .

[7]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[8]  Stephen P. Gurden,et al.  A comparison of multiway regression and scaling methods , 2001 .

[9]  Cinzia Viroli,et al.  Model based clustering for three-way data structures , 2011 .

[10]  R. Bro.  Multiway calibration. Multilinear PLS , 1996 .

[11]  P. Krishnaiah,et al.  16 Likelihood ratio tests for mean vectors and covariance matrices , 1980 .

[12]  S. Wold,et al.  Multi‐way principal components‐and PLS‐analysis , 1987 .

[13]  Garrett M Fitzmaurice,et al.  Regression models for the analysis of longitudinal Gaussian data from multiple sources , 2005, Statistics in medicine.

[14]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[15]  M. Genton,et al.  A likelihood ratio test for separability of covariances , 2006 .

[16]  Age K. Smilde,et al.  Multiway multiblock component and covariates regression models , 2000 .

[17]  Cinzia Viroli,et al.  Finite mixtures of matrix normal distributions for classifying three-way data , 2011, Stat. Comput..

[18]  Marc G. Genton,et al.  Testing for separability of space–time covariances , 2005 .

[19]  Gregory C. Reinsel,et al.  Estimation and Prediction in a Multivariate Random Effects Generalized Linear Model , 1984 .

[20]  R. Potthoff,et al.  A generalized multivariate analysis of variance model useful especially for growth curve problems , 1964 .

[21]  Bayesian Analysis of a Growth Curve Model with a General Autoregressive Covariance Structure , 2000 .

[22]  S. Lundbye-Christensen,et al.  A multivariate growth curve model for pregnancy. , 1991, Biometrics.

[23]  John J. McArdle.  Growth Curve Analysis , 2001 .

[24]  G. Reinsel,et al.  Multivariate Repeated-Measurement or Growth Curve Models with Multivariate Random-Effects Covariance Structure , 1982 .

[25]  B P Korin,et al.  On the distribution of a statistic used for testing a covariance matrix. , 1968, Biometrika.

[26]  Larry Norton,et al.  Growth Curve Analysis , 2003 .

[27]  A. Rukhin.  Matrix Variate Distributions , 1999, The Multivariate Normal Distribution.

[28]  Sophia Rabe-Hesketh,et al.  Generalized latent variable models: multilevel, longitudinal, and structural equation models , 2004 .

[29]  Risto Lehtonen,et al.  Multilevel Statistical Models , 2005 .

[30]  P. Dutilleul.  The mle algorithm for the matrix normal distribution , 1999 .

[31]  Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up , 2008 .

[32]  K. Fang,et al.  Generalized Multivariate Analysis , 1990 .

[33]  M. West,et al.  Bayesian analysis of matrix normal graphical models. , 2009, Biometrika.

[34]  Truc T. Nguyen.  A Note on Matrix Variate Normal Distribution , 1997 .

[35]  N. Rao Chaganty,et al.  Analysis of multivariate longitudinal data using quasi-least squares , 2002 .

[36]  J. Hooper,et al.  Simultaneous Equations and Canonical Correlation Theory , 1959 .

[37]  A. Dawid.  Some matrix-variate distribution theory: Notational considerations and a Bayesian application , 1981 .