Parameter Estimation for Generalized Random Coefficient in the Linear Mixed Models
Keywords:
Linear mixed model, inference for linear model, conditional least squares, weighted conditional least squares, mean-squared errorsAbstract
The analysis of longitudinal data, comprising repeated measurements of the same individuals over time, requires models with a random effects because traditional linear regression is not suitable and makes the strong assumption that the measurements are independent, which is often unrealistic for longitudinal data. However, values repeatedly measured in the same individual are usually correlated, and ignoring the correlation between repeated measurements may lead to biased estimates as well as invalid P-values and confidence intervals. Therefore, careful consideration is needed to enable valid inference of covariate effects on longitudinal responses. In this regard, Residual Maximum Likelihood (REML) analysis is the most widely used method to estimate parameters. This method is based on the assumption that there is no correlation between the random effects and the error term (or residual effects). Hence, it is unclear if the failure to meet this assumption will affect the conclusions when conducting a real datasets study. In the present article, we propose Conditional Least Squares (CLS) and Weighted Conditional Least-Squares (WCLS) methods for estimating the model parameters, considering the presence of the correlation between the random effects and the error term. The choice of these methods is motivated by the fact that it is not necessary to specify the distributions of random effects and/or errors. These methods are illustrated via simulation studies that were performed with different sample sizes and different parameter values. In addition, we compared these methods with traditional estimators. This comparison is made by utilizing the ratio of their mean square error. Our results highlight that the weighted conditional least-squares estimator is efficient and attractive compared to the restricted maximum likelihood when random effects are permitted to be correlated with the error term. Also, real data analysis is conducted to confirm the advantages of the improved method.
References
Anderson RL. Bancroft TA. Statistical Theory in Research. New York: McGrawHill; 1952.
Bates DM. Pinheiro JC. Linear and nonlinear mixed-effects models. Proceeding of the 10th annual conference on applied statistics in agriculture; 1998. pp. 1-22.
Breslow NE. Clayton DG. Approximate inference in generalized linear mixed models. J Am Stat Assoc. 1993; 88(421): 9-25.
Cornillon PA. Matzner-Lober E. Rgression: thorie et applications. Springer; 2007.
Davezies L. Modles effets fixes, effets alatoires, modles mixtes ou multi-niveaux: proprits et mises en oeuvre des modlisations de l’htrognit dans le cas de donnes groupes; 2011.
Demidenko E. Mixed models: theory and applications with R. John Wiley and Sons; 2013.
Diggle P, Diggle PJ, Heagerty P, Liang KY, Zeger S. Analysis of longitudinal data. Oxford University Press; 2002.
Drikvandi R, Khodadadi A, Verbeke G. Testing variance components in balanced linear growth curve models. J Appl Stat. 2012; 39(3): 563-572.
Foulley JL, Delmas C, Robert-Grani C. Mthodes du maximum de vraisemblance en modle linaire mixte. Journal de la socit franaise de statistique. 2002; 143(1-2): 5-52.
Hartley HO, Rao JN. Maximum-likelihood estimation for the mixed analysis of variance model. Biometrika. 1967; 54(1-2): 93-108.
Harville DA. Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc. 1977; 72(358): 320-338.
Hwang SY, Basawa IV. Parameter estimation for generalized random coefficient autoregressive processes. J Stat Plan Infer. 1998; 68(2): 323-337.
Ibragimov IA, Hasminskii RZ. Statistical Estimation: Asymptotic Theory. New York: SpringerVerlag; 1981.
Jiang J. Linear and generalized linear mixed models and their applications. Springer Science and Business Media; 2007.
Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics. 1982; 38(4): 963-974.
Ou Larbi Y, Halimi RE, Akharif A, Mellouk A. Optimal tests for random effects in linear mixed models . Hacet J Math Stat. 2021; 50(4): 1185-1211.
Pinheiro J, Bates D. Mixed-effects models in S and S-PLUS. Springer Science and Business Media;
Regis M, Brini A, Nooraee N, Haakma R, van den Heuvel ER. The t linear mixed model: model formulation, identifiability and estimation. Commun Stat Simulat. 2019; 51(5): 2318-2342.
Swamy PAVB. Statistical inference in random coefficient regression models (Vol. 55). Springer Science and Business Media; 2012.
Shao J. Asymptotic distribution of the weighted least squares estimator. Ann I Stat Math. 1989; 41(2): 365-382.
Thompson Jr. WA. The problem of negative estimates of variance components. Ann Math Stat. 1962; 33(1): 273-289.
Verbeke G. Linear mixed models for longitudinal data. In Linear mixed models in practice. New York: Springer; 1997.
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