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Marginal permutation invariant covariance matrices with applications to linear models. (English) Zbl 1099.62074

Summary: The goal of the present paper is to perform a comprehensive study of the covariance structures in balanced linear models containing random factors which are invariant with respect to marginal permutations of the random factors. We shall focus on model formulation and interpretation rather than the estimation of parameters. It is proven that permutation invariance implies a specific structure for the covariance matrices. Useful results are obtained for the spectra of permutation invariant covariance matrices.
In particular, the reparameterization of random effects, i.e., imposing certain constraints, will be considered. There are many possibilities to choose reparameterization constraints in a linear model, however not every reparameterization keeps permutation invariance. The question is if there are natural restrictions on the random effects in a given model, i.e., such reparameterizations which are defined by the covariance structure of the corresponding factor. Examining relationships between the reparameterization conditions applied to the random factors of the models and the spectrum of the corresponding covariance matrices when permutation invariance is assumed, restrictions on the spectrum of the covariance matrix are obtained which lead to “sum-to-zero” reparameterization of the corresponding factor.

MSC:

62J10 Analysis of variance and covariance (ANOVA)
62F30 Parametric inference under constraints
62H99 Multivariate analysis
62F99 Parametric inference
15A99 Basic linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Andersson, S., Invariant normal models, Ann. statist., 3, 132-154, (1975) · Zbl 0373.62029
[2] Andersson, S.; Madsen, J., Symmetry and lattice conditional independence in a multivariate normal distribution (in multivariate analysis), Ann. statist., 26, 525-572, (1998) · Zbl 0943.62047
[3] Arnold, S., Applications of the theory of products of problems to certain patterned covariance matrices, Ann. statist., 1, 682-699, (1973) · Zbl 0274.62038
[4] Dawid, A.P., Symmetry models and hypotheses for structural data layouts, J. R. stat. soc. ser. B, 50, 1-34, (1988) · Zbl 0654.62056
[5] Jennrich, R.I.; Schluchter, M.D., Unbalanced repeated measures models with structured covariance matrices, Biometrics, 42, 805-820, (1986) · Zbl 0625.62052
[6] Jensen, S.T., Covariance hypotheses which are linear in both the covariance and the inverse covariance, Ann. statist., 16, 302-322, (1988) · Zbl 0653.62042
[7] McLean, R.A.; Sanders, W.L.; Stroup, W.W., A unified approach to mixed linear models, Amer. statist., 45, 54-64, (1991)
[8] Nahtman, T., Reparameterization of second-order interaction effects through the covariance matrix, Acta commentat. univ. Tartu. math., 6, 51-58, (2002) · Zbl 1021.62052
[9] Nahtman, T.; Möls, T., Characterization of fixed effects as special cases of generalized random factors, Tatra mt. math. publ., 26, 169-174, (2003) · Zbl 1154.62355
[10] T. Nahtman, Permutation invariance and reparameterizations in linear models, Dissertationes, Mathematicae Universitatis Tartuensis, 35, Tartu University Press, Tartu, 2004. · Zbl 1088.62088
[11] Olkin, I.; Press, S.J., Testing and estimation for a circular stationary model, Ann. math. statist., 40, 1358-1373, (1969) · Zbl 0186.51801
[12] Olkin, I., Testing and estimation for structures which are circularly symmetric in blocks, (), 183-195
[13] Olkin, I.; Viana, M., Correlation analysis of extreme observations from a multivariate normal distribution, J. amer. statist. assoc., 90, 1373-1379, (1995) · Zbl 0868.62050
[14] Perlman, M.D., Group symmetry covariance models, Statist. sci., 2, 421-425, (1987)
[15] Searle, S.R.; Henderson, H.V., Dispersion matrices for variance components models, J. amer. statist. assoc., 74, 465-470, (1979) · Zbl 0419.62061
[16] Speed, T.P., ANOVA models with random effects: an approach via symmetry, essays in time series and allied processes, J. appl. probab., 23A, 355-368, (1986)
[17] Speed, T.P.; Bailey, R.A., Factorial dispersion models, Internat. statist. rev., 55, 251-277, (1987) · Zbl 0652.62066
[18] Tjur, T., Analysis of variance and design of experiments, Scand. J. statist., 18, 273-322, (1991) · Zbl 0798.62078
[19] Viana, M.; Olkin, I., Symmetrically dependent models arising in visual assessment data, Biometrics, 56, 1188-1191, (2000) · Zbl 1060.62675
[20] Voss, D.T., Resolving the mixed models controversy, Amer. statist., 53, 352-356, (1999)
[21] Votaw, D.F., Testing compound symmetry in a normal multivariate distribution, Ann. math. statist., 19, 447-473, (1948) · Zbl 0033.07903
[22] Wilks, S.S., Sample criteria for testing equality of means, equality of variances and equality of covariances in a normal multivariate distribution, Ann. math. statist., 17, 257-281, (1946) · Zbl 0063.08259
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