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Year 2021, Volume: 50 Issue: 4, 1185 - 1211, 06.08.2021
https://doi.org/10.15672/hujms.773667

Abstract

References

  • [1] A. Akharif and M. Hallin, Efficient detection of random coefficients in autoregressive models, Ann. Statist. 31 (2), 675-704, 2003.
  • [2] A. Azzalini and A. Capitanio, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, J. R. Stat. Soc. Ser. B. Stat. Methodol. 65 (2), 367-389, 2003.
  • [3] D.M. Bates, M. Machler, B. Bolker and S. Walker, Fitting linear mixed effects models using lme4, J. Stat. Softw. 67 (1), 1-48, 2015.
  • [4] N. Bennala, M. Hallin and D. Paindaveine, Pseudo-Gaussian and rank-based optimal tests for random individual effects in large n small t panels, J. Econometrics 170 (1), 50-67, 2012.
  • [5] C. Delphine, M. Hallin and D. Paindaveine, On the estimation of cross-information quantities in rank-based inference, in Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in Honor of Professor Jana Jurečkovà, Institute of Mathematical Statistics, 7, 35-45, 2010.
  • [6] E. Demidenko, Mixed Models: Theory and Applications with R, John Wiley and Sons, 2013.
  • [7] R. Drikvandi and S. Noorian, Testing random effects in linear mixed-effects models with serially correlated errors, Biom. J. 61 (4), 802-812, 2019.
  • [8] R. Drikvandi, A. Khodadadi and G. Verbeke, Testing variance components in balanced linear growth curve models, J. Appl. Stat. 39 (3), 563-572, 2012.
  • [9] J.J. Droesbeke and J. Fine, Inférence non paramétrique, Les statistiques de rangs, Éditions de l’Université de Bruxelles, 1996.
  • [10] M. Fihri, A. Akharif, A. Mellouk and M. Hallin, Efficient pseudo-Gaussian and rankbased detection of random regression coefficients, J. Nonparametr. Stat. 32 (2), 367- 402, 2020.
  • [11] J.L. Foulley, C. Delmas and C. Robert-Granié, Méthodes du maximum de vraisemblance en modèle linéaire mixte, J Soc Stat Paris 143 (1-2), 5-52, 2002.
  • [12] J. Hájek and Z. Šidák, Theory of Rank Tests, Academic Press, New York, 1967.
  • [13] J. Hájek, Local asymptotic minimax and admissibility in estimation, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics, Berkeley, CA: University of California Press, 1972.
  • [14] M. Hallin and B.J.M. Werker, Semiparametric efficiency, distribution-freeness and invariance, Bernoulli 9 (1), 137-165, 2003.
  • [15] J.P. Kreiss, On adaptive estimation in stationary ARMA processes, Ann. Statist. 15 (1), 112-133, 1987.
  • [16] N.M. Laird and J.H. Ware, Random-effects models for longitudinal data, Biometrics 38 (4), 963-974, 1982.
  • [17] L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986.
  • [18] L.M. Le Cam and G.L. Yang, Asymptotics in Statistics: Some Basic Concepts, 2 edn, Springer-Verlag, New York, 2000.
  • [19] A. Lmakri, A. Akharif and A. Mellouk, Optimal detection of bilinear dependence in short panels of regression data, Rev. Colomb. Estad. 43 (2), 143-171, 2020.
  • [20] C.H. Morrell, Likelihood ratio testing of variance components in the linear mixedeffects model using restricted maximum likelihood, Biometrics 54 (4), 1560-1568, 1998.
  • [21] G.E. Noether, On a theorem by wald and wolfowitz, Ann. Math. Statist. 20 (3), 455- 458, 1949.
  • [22] J. Pinheiro and D. Bates, Mixed-Effects Models in S and S-PLUS, Springer Science and Business Media, 2006.
  • [23] R.F. Potthoff and S.N. Roy, A generalized multivariate analysis of variance model useful especially for growth curve problems, Biometrika 51 (3-4), 313-326, 1964.
  • [24] K. Rao, R. Drikvandi and B. Saville, Permutation and Bayesian tests for testing random effects in linear mixed-effects models, Stat. Med. 38 (25), 5034-5047, 2019.
  • [25] M. Regis, A. Brini, N. Nooraee, R. Haakma and E.R. van den Heuvel, The linear mixed model: model formulation, identifiability and estimation, Comm. Statist. Simulation Comput. Doi:10.1080/03610918.2019.1694153, 2019.
  • [26] A.R. Swensen, The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend, J. Multivariate Anal. 16 (1), 54-70, 1985.
  • [27] G. Verbeke, Linear Mixed Models for Longitudinal Data, Linear Mixed Models in Practice, Springer, New York, 63-153, 1997.

Optimal tests for random effects in linear mixed models

Year 2021, Volume: 50 Issue: 4, 1185 - 1211, 06.08.2021
https://doi.org/10.15672/hujms.773667

Abstract

In the past decade, mixed-effects modeling has received a great deal of attention in the applied and theoretical statistical literature. They are very flexible tools in analyzing repeated measures, panel data, cross-sectional data, and hierarchical data. However, the complex nature of these models has motivated researchers to study different aspects of this problem. One of which is to test the significance of random effects used to model unobserved heterogeneity in the population. The method of likelihood ratio test based on the normality assumption of the error term and random effects has been proposed. However, this assumption does not necessarily hold in practice. In this paper, we propose an optimal test based on the so-called uniform local asymptotic normality to detect the possible presence of random effects in linear mixed models. We show that the proposed test statistic is, consistent, locally asymptotically optimal even for a model that does not require the traditional assumption of normality and is comparable to the classical L.ratio-test when the standard assumptions are met. Finally, simulation studies and real data analysis are also conducted to empirically examine the performance of this procedure.

References

  • [1] A. Akharif and M. Hallin, Efficient detection of random coefficients in autoregressive models, Ann. Statist. 31 (2), 675-704, 2003.
  • [2] A. Azzalini and A. Capitanio, Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, J. R. Stat. Soc. Ser. B. Stat. Methodol. 65 (2), 367-389, 2003.
  • [3] D.M. Bates, M. Machler, B. Bolker and S. Walker, Fitting linear mixed effects models using lme4, J. Stat. Softw. 67 (1), 1-48, 2015.
  • [4] N. Bennala, M. Hallin and D. Paindaveine, Pseudo-Gaussian and rank-based optimal tests for random individual effects in large n small t panels, J. Econometrics 170 (1), 50-67, 2012.
  • [5] C. Delphine, M. Hallin and D. Paindaveine, On the estimation of cross-information quantities in rank-based inference, in Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in Honor of Professor Jana Jurečkovà, Institute of Mathematical Statistics, 7, 35-45, 2010.
  • [6] E. Demidenko, Mixed Models: Theory and Applications with R, John Wiley and Sons, 2013.
  • [7] R. Drikvandi and S. Noorian, Testing random effects in linear mixed-effects models with serially correlated errors, Biom. J. 61 (4), 802-812, 2019.
  • [8] R. Drikvandi, A. Khodadadi and G. Verbeke, Testing variance components in balanced linear growth curve models, J. Appl. Stat. 39 (3), 563-572, 2012.
  • [9] J.J. Droesbeke and J. Fine, Inférence non paramétrique, Les statistiques de rangs, Éditions de l’Université de Bruxelles, 1996.
  • [10] M. Fihri, A. Akharif, A. Mellouk and M. Hallin, Efficient pseudo-Gaussian and rankbased detection of random regression coefficients, J. Nonparametr. Stat. 32 (2), 367- 402, 2020.
  • [11] J.L. Foulley, C. Delmas and C. Robert-Granié, Méthodes du maximum de vraisemblance en modèle linéaire mixte, J Soc Stat Paris 143 (1-2), 5-52, 2002.
  • [12] J. Hájek and Z. Šidák, Theory of Rank Tests, Academic Press, New York, 1967.
  • [13] J. Hájek, Local asymptotic minimax and admissibility in estimation, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics, Berkeley, CA: University of California Press, 1972.
  • [14] M. Hallin and B.J.M. Werker, Semiparametric efficiency, distribution-freeness and invariance, Bernoulli 9 (1), 137-165, 2003.
  • [15] J.P. Kreiss, On adaptive estimation in stationary ARMA processes, Ann. Statist. 15 (1), 112-133, 1987.
  • [16] N.M. Laird and J.H. Ware, Random-effects models for longitudinal data, Biometrics 38 (4), 963-974, 1982.
  • [17] L. Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986.
  • [18] L.M. Le Cam and G.L. Yang, Asymptotics in Statistics: Some Basic Concepts, 2 edn, Springer-Verlag, New York, 2000.
  • [19] A. Lmakri, A. Akharif and A. Mellouk, Optimal detection of bilinear dependence in short panels of regression data, Rev. Colomb. Estad. 43 (2), 143-171, 2020.
  • [20] C.H. Morrell, Likelihood ratio testing of variance components in the linear mixedeffects model using restricted maximum likelihood, Biometrics 54 (4), 1560-1568, 1998.
  • [21] G.E. Noether, On a theorem by wald and wolfowitz, Ann. Math. Statist. 20 (3), 455- 458, 1949.
  • [22] J. Pinheiro and D. Bates, Mixed-Effects Models in S and S-PLUS, Springer Science and Business Media, 2006.
  • [23] R.F. Potthoff and S.N. Roy, A generalized multivariate analysis of variance model useful especially for growth curve problems, Biometrika 51 (3-4), 313-326, 1964.
  • [24] K. Rao, R. Drikvandi and B. Saville, Permutation and Bayesian tests for testing random effects in linear mixed-effects models, Stat. Med. 38 (25), 5034-5047, 2019.
  • [25] M. Regis, A. Brini, N. Nooraee, R. Haakma and E.R. van den Heuvel, The linear mixed model: model formulation, identifiability and estimation, Comm. Statist. Simulation Comput. Doi:10.1080/03610918.2019.1694153, 2019.
  • [26] A.R. Swensen, The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend, J. Multivariate Anal. 16 (1), 54-70, 1985.
  • [27] G. Verbeke, Linear Mixed Models for Longitudinal Data, Linear Mixed Models in Practice, Springer, New York, 63-153, 1997.
There are 27 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Yassine Ou Larbi 0000-0002-5390-3137

Rachid El Halimi This is me 0000-0002-3846-8827

Abdelhadi Akharif This is me 0000-0002-6279-9503

Amal Mellouk This is me 0000-0002-8931-9709

Publication Date August 6, 2021
Published in Issue Year 2021 Volume: 50 Issue: 4

Cite

APA Ou Larbi, Y., Halimi, R. E., Akharif, A., Mellouk, A. (2021). Optimal tests for random effects in linear mixed models. Hacettepe Journal of Mathematics and Statistics, 50(4), 1185-1211. https://doi.org/10.15672/hujms.773667
AMA Ou Larbi Y, Halimi RE, Akharif A, Mellouk A. Optimal tests for random effects in linear mixed models. Hacettepe Journal of Mathematics and Statistics. August 2021;50(4):1185-1211. doi:10.15672/hujms.773667
Chicago Ou Larbi, Yassine, Rachid El Halimi, Abdelhadi Akharif, and Amal Mellouk. “Optimal Tests for Random Effects in Linear Mixed Models”. Hacettepe Journal of Mathematics and Statistics 50, no. 4 (August 2021): 1185-1211. https://doi.org/10.15672/hujms.773667.
EndNote Ou Larbi Y, Halimi RE, Akharif A, Mellouk A (August 1, 2021) Optimal tests for random effects in linear mixed models. Hacettepe Journal of Mathematics and Statistics 50 4 1185–1211.
IEEE Y. Ou Larbi, R. E. Halimi, A. Akharif, and A. Mellouk, “Optimal tests for random effects in linear mixed models”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 4, pp. 1185–1211, 2021, doi: 10.15672/hujms.773667.
ISNAD Ou Larbi, Yassine et al. “Optimal Tests for Random Effects in Linear Mixed Models”. Hacettepe Journal of Mathematics and Statistics 50/4 (August 2021), 1185-1211. https://doi.org/10.15672/hujms.773667.
JAMA Ou Larbi Y, Halimi RE, Akharif A, Mellouk A. Optimal tests for random effects in linear mixed models. Hacettepe Journal of Mathematics and Statistics. 2021;50:1185–1211.
MLA Ou Larbi, Yassine et al. “Optimal Tests for Random Effects in Linear Mixed Models”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 4, 2021, pp. 1185-11, doi:10.15672/hujms.773667.
Vancouver Ou Larbi Y, Halimi RE, Akharif A, Mellouk A. Optimal tests for random effects in linear mixed models. Hacettepe Journal of Mathematics and Statistics. 2021;50(4):1185-211.