Some Remarks on the Equalities of Predictors in Linear Mixed Models

Melike Yiğit; Nesrin Güler; Melek Eriş Büyükkaya

doi:10.36753/mathenot.892258

EN

Some Remarks on the Equalities of Predictors in Linear Mixed Models

Abstract

Consider a transformed linear mixed model (TLMM) obtained pre-multiplying a linear mixed model (LMM) M : y = Zα + Rγ + e by a given matrix. This work concerns the problem of the equalities of linear predictors under the considered two LMMs under general assumptions. We characterize the equalities between the best linear unbiased predictors (BLUPs) under the LMM and its TLMM by using various rank formulas of block matrices and elementary matrix operations.

Keywords

BLUP, equalities, linear mixed model, random vectors, transformed model

References

[1] Alalouf, I. S., Styan, G. P. H.: Characterizations of estimability in the general linear model. Ann. Stat. 7, 194-200 (1979).
[2] Arendacká, B., Puntanen, S.: Further remarks on the connection between fixed linear model and mixed linear model. Stat. Papers. 56 (4), 1235-1247 (2015).
[3] Baksalary, J. K., Kala, S.: Linear transformations preserving best linear unbiased estimators in a general Gauss-Markoff model. Ann. Stat. 9, 913-916 (1981).
[4] Dong, B., Guo, W., Tian, Y.: On relations between BLUEs under two transformed linear models. J. Multivariate Anal. 131, 279-292 (2014).
[5] Drygas, H.: Sufficiency and completeness in the general Gauss-Markov model. Sankhya ̄, Ser A. 45, 88-98 (1983).
[6] Goldberger, A. S.: Best linear unbiased prediction in the generalized linear regression model. J. Amer. Statist. Assoc. 57, 369-375 (1962).
[7] Güler, N.: On relations between BLUPs under two transformed linear random-effects models. Commun. Statist. Simulation and Computation. (2020). https://doi.org/10.1080/03610918.2020.1757709
[8] Harville, D.: Extension of the Gauss-Markov theorem to include the estimation of random effects. Ann. Stat. 4, 384-395 (1976).
[9] Haslett, S. J., Puntanen, S.: Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat. Papers. 51 (2), 465-475 (2010).
[10] Haslett, S. J., Puntanen, S.: On the equality of the BLUPs under two linear mixed models. Metrika. 74, 381-395 (2011).

[11] Isotalo, J., Puntanen, S.: Linear prediction sufficiency for new observations in the general Gauss–Markov model.
Commun. Statist. Theory and Methods. 35, 1011-1023 (2006).
[12] Liu, X., Wang, Q. W.: Equality of the BLUPs under the mixed linear model when random components and errors are correlated. J. Multivariate Anal. 116, 297-309 (2013).
[13] Marsaglia, G., Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra. 2, 269-292 (1974).
[14] Penrose, R.: Generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
[15] Puntanen, S., Styan, G. P. H. , Isotalo, J.: Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, Heidelberg (2011).
[16] Rao, C. R.: Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J. Multivariate Anal. 3, 276-292 (1973).
[17] Sun, Y., Jiang B., Jiang, H.: Computations of predictors/estimators under a linear random–effects model with parameter restrictions. Commun. Statist. Theory and Methods. 48 (14), 3482-3497 (2019).
[18] Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices. Southeast Asian Bull. Math. 25, 745-755 (2002).
[19] Tian, Y.: On equalities for BLUEs under misspecified Gauss-Markov models. Acta Mathematica Sinica. Eng. Ser. 25 (11), 1907-1920 (2009).
[20] Tian, Y.: A new derivation of BLUPs under random-effects model. Metrika. 78, 905-918 (2015).
[21] Tian, Y.: A matrix handling of predictions under a general linear random-effects model with new observations. Electron. J. Linear Algebra. 29, 30-45 (2015).
[22] Tian, Y.: Transformation approaches of linear random-effects models. Stat. Methods Appl. 26 (4), 583-608 (2017).
[23] Tian, Y., Cheng, S.: The maximal and minimal ranks of A-BXC with applications. New York J. Math. 9, 345-362 (2003).
[24] Tian, Y., Puntanen, S.: On the equivalence of estimations under a general linear model and its transformed models. Linear Algebra Appl. 430, 2622-2641 (2009).
[25] Tian, Y., Jiang, B.: An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices. Linear Multilinear Algebra. 64 (12), 2351-2367 (2016).
[26] Tian, Y., Jiang B.: Matrix rank/inertia formulas for least-squares solutions with statistical applications. Spec. Matrices. 4 (1), 130-140 (2016).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Melike Yiğit ^*
0000-0002-9205-7842
Türkiye

Nesrin Güler
0000-0003-3233-5377
Türkiye

Melek Eriş Büyükkaya
0000-0002-6207-5687
Türkiye

Publication Date

December 31, 2021

Submission Date

March 6, 2021

Acceptance Date

November 25, 2021

Published in Issue

Year 2021 Volume: 9 Number: 4

DOI

https://doi.org/10.36753/mathenot.892258

IZ

https://izlik.org/JA29GN79WT

APA

Yiğit, M., Güler, N., & Eriş Büyükkaya, M. (2021). Some Remarks on the Equalities of Predictors in Linear Mixed Models. Mathematical Sciences and Applications E-Notes, 9(4), 185-193. https://doi.org/10.36753/mathenot.892258

AMA

1.Yiğit M, Güler N, Eriş Büyükkaya M. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. 2021;9(4):185-193. doi:10.36753/mathenot.892258

Chicago

Yiğit, Melike, Nesrin Güler, and Melek Eriş Büyükkaya. 2021. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes 9 (4): 185-93. https://doi.org/10.36753/mathenot.892258.

EndNote

Yiğit M, Güler N, Eriş Büyükkaya M (December 1, 2021) Some Remarks on the Equalities of Predictors in Linear Mixed Models. Mathematical Sciences and Applications E-Notes 9 4 185–193.

IEEE

[1]M. Yiğit, N. Güler, and M. Eriş Büyükkaya, “Some Remarks on the Equalities of Predictors in Linear Mixed Models”, Math. Sci. Appl. E-Notes, vol. 9, no. 4, pp. 185–193, Dec. 2021, doi: 10.36753/mathenot.892258.

ISNAD

Yiğit, Melike - Güler, Nesrin - Eriş Büyükkaya, Melek. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes 9/4 (December 1, 2021): 185-193. https://doi.org/10.36753/mathenot.892258.

JAMA

1.Yiğit M, Güler N, Eriş Büyükkaya M. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. 2021;9:185–193.

MLA

Yiğit, Melike, et al. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 4, Dec. 2021, pp. 185-93, doi:10.36753/mathenot.892258.

Vancouver

1.Melike Yiğit, Nesrin Güler, Melek Eriş Büyükkaya. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. 2021 Dec. 1;9(4):185-93. doi:10.36753/mathenot.892258