# A Matrix-Variate Regression Model with Canonical States: An Application to Elderly Danish Twins

## DOI:

https://doi.org/10.6092/issn.1973-2201/5473## Keywords:

Linear Regression, Matrix-variate normal distribution, Maximum Likelihood, Structural equation modeling, Twin data## Abstract

In many situations we observe a set of variables in different states (e.g. times, replicates, locations) and the interest can be to regress the matrix-variate observed data on a set of covariates. We dene a novel matrix-variate regression model characterized by canonical components with the aim of analyzing the effect of covariates in describing the variability within and between the different states. Despite the seeming complexity, inference can be easily performed through maximum likelihood. We derive the inferential properties of the model estimators and a general approach for hypothesis testing. Finally, the proposed method is applied to data coming from the Longitudinal Study of Aging Danish Twins (LSADT), so to investigate the causes of variation in cognitive functioning.

## References

L. ANDERLUCCI, C. VIROLI (2015). Covariance Pattern Mixture Models for the Analysis of Multivariate Heterogeneous Longitudinal Data. The Annals of Applied Statistics, 9, pp. 777-800.

K. E. BASFORD, G. J. MCLACHLAN (1985). The Mixture Method of Clustering applied to three-way data. Journal of Classification, 2, pp. 109-125.

L. BRIEN, G. FITZMAURICE (2005). Regression models for the analysis of longitudinal Gaussian data from multiple sources. Statistics in Medicine, 24, pp. 1725-1744.

K. CHRISTENSEN, J. W. VAUPEL (2009). Longitudinal Study of Aging Danish Twins, 1995. ICPSR21041-v1. Ann Arbor, MI: Inter-university Consortium for Political and Social Research.

K. CHRISTENSEN, N. V HOLM, M. MCGUE, L. CORDER, J. W. VAUPEL (1999). A Danish population-based twin study on general health in the elderly. Journal of Aging and Health, 11, pp. 49-64.

J. M. DICKEY (1967). Matrix variate generalizations of the multivariate t distribution and the inverted multivariate t distribution. The Annals of Mathematical Statistics. 2, pp. 511-518.

E. DRIGALENKO (1998). How sib pairs reveal linkage. The American Journal of Human Genetics, 63, pp. 1242–1245.

P. DUTILLEUL (1999). The MLE algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation, 64, pp. 105-123.

R. C. ELSTON, S. BUXBAUM, K. B. JACOBS, J. M. OLSON (2000). Haseman and Elston revisited. Genetics Epidemiology, 19, pp. 1-17.

M. G. GENTON (2007). Separable approximations of space-time covariance matrices. Environmetrics, Special Issue for METMA3, 18, pp. 681-695.

A. K. GUPTA, D. K. NAGAR (2000). Matrix Variate Distributions. Chapman & Hall/CRC.

J. K. HASEMAN, R. C. ELSTON (1972). The investigation of linkage between a quantitative trait and a marker locus. Behavior Genetics, 2, pp. 3-19.

M. P. LAWTON, E. M. BRODY (1969). Assessment of older people: Self-maintaining and instrumental activities of daily living. Gerontologist, 9, pp. 179-186.

K. V. MARDIA, J. T. KENT, J, M. BIBBY (2003). Multivariate Analysis. Academic Press.

M. MCGUE, K. CHRISTENSEN (2007). Social activity and healthy aging: A study of aging Danish twins. Twin Research and Human Genetics, 10, pp. 255-265.

G. J. MCLACHLAN, D. PEEL (2000). Finite Mixture Models. Wiley, New York.

M. W. MITCHELL, M. G. GENTON, M. L. GUMPERTZ (2005). Testing for separability of space-time covariances. Envinronmetrics, 16, pp. 819-831.

M. W. MITCHELL, M. G. GENTON, M. L. GUMPERTZ (2006). A likelihood ratio test for separability of covariances. Journal of Multivariate Analysis, 97, pp. 1025-1043.

M. C. NEALE, S. M. BOKER, G. XIE, H. H. MAES (2006). Mx: Statistical Modeling, (Seventh Edition), Virginia Commonwealth University, Department of Psychiatry, Richmond, VA.

M. C. NEALE (2003). A finite mixture distribution model for data collected from twins. Twin Research, 6, pp. 235-239.

S. K. NG, G. J MCLACHLAN (2014). Mixture of random effects models for clustering multilevel growth trajectories. Computational Statistics & Data Analysis, 71, pp. 43-51.

C. VIROLI (2011). Finite mixtures of matrix normal distributions for classifying three-way data. Statistics and Computing, 21, pp. 511-522.

C. VIROLI (2012). On matrix-variate regression analysis. Journal of Multivariate Analysis, 111, pp. 296-309.

H. WANG, M. WEST (2009). Bayesian Analysis of Matrix Normal Graphical Models. Biometrika, 96, pp. 821-834.

F. A. WRIGHT (1997). The phenotypic difference discards sib-pair QTL linkage information. The American Journal of Human Genetics, 60, pp. 740-742.

Y. M. ZHANG, H. Y LU, L. L. YAO (2008). Multiple quantitative trait loci Haseman-Elston regression using all markers on the entire genome, Theoretical and Applied Genetics, 117, pp. 683-690.

H. ZHOU, L. LI (2014). Regularized matrix regression. Journal of the Royal Statistical Society - Series B, 76, pp. 463-483.

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*Statistica*,

*74*(4), 367–381. https://doi.org/10.6092/issn.1973-2201/5473

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