Robust estimation of fixed effect parameters and variances of linear mixed models: the minimum density power divergence approach
Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the d...
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| Veröffentlicht in: | Advances in statistical analysis : AStA : a journal of the German Statistical Society Jg. 108; H. 1; S. 127 - 157 |
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| Abstract | Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the density power divergence (DPD) family, which measures the discrepancy between two probability density functions, has been successfully used to build robust estimators with high stability associated with minimal loss in efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent but non-identically distributed observations for LMMs according to the variance components model. We prove that the theoretical properties hold, including consistency and asymptotic normality of the estimators. The influence function and sensitivity measures are computed to explore the robustness properties. As a data-based choice of the MDPDE tuning parameter
α
is very important, we propose two candidates as “optimal” choices, where optimality is in the sense of choosing the strongest downweighting that is necessary for the particular data set. We conduct a simulation study comparing the proposed MDPDE, for different values of
α
, with S-estimators, M-estimators and the classical maximum likelihood estimator, considering different levels of contamination. Finally, we illustrate the performance of our proposal on a real-data example. |
|---|---|
| AbstractList | Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the density power divergence (DPD) family, which measures the discrepancy between two probability density functions, has been successfully used to build robust estimators with high stability associated with minimal loss in efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent but non-identically distributed observations for LMMs according to the variance components model. We prove that the theoretical properties hold, including consistency and asymptotic normality of the estimators. The influence function and sensitivity measures are computed to explore the robustness properties. As a data-based choice of the MDPDE tuning parameter α is very important, we propose two candidates as “optimal” choices, where optimality is in the sense of choosing the strongest downweighting that is necessary for the particular data set. We conduct a simulation study comparing the proposed MDPDE, for different values of α, with S-estimators, M-estimators and the classical maximum likelihood estimator, considering different levels of contamination. Finally, we illustrate the performance of our proposal on a real-data example. Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the density power divergence (DPD) family, which measures the discrepancy between two probability density functions, has been successfully used to build robust estimators with high stability associated with minimal loss in efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent but non-identically distributed observations for LMMs according to the variance components model. We prove that the theoretical properties hold, including consistency and asymptotic normality of the estimators. The influence function and sensitivity measures are computed to explore the robustness properties. As a data-based choice of the MDPDE tuning parameter $$\alpha$$ α is very important, we propose two candidates as “optimal” choices, where optimality is in the sense of choosing the strongest downweighting that is necessary for the particular data set. We conduct a simulation study comparing the proposed MDPDE, for different values of $$\alpha$$ α , with S-estimators, M-estimators and the classical maximum likelihood estimator, considering different levels of contamination. Finally, we illustrate the performance of our proposal on a real-data example. Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the density power divergence (DPD) family, which measures the discrepancy between two probability density functions, has been successfully used to build robust estimators with high stability associated with minimal loss in efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent but non-identically distributed observations for LMMs according to the variance components model. We prove that the theoretical properties hold, including consistency and asymptotic normality of the estimators. The influence function and sensitivity measures are computed to explore the robustness properties. As a data-based choice of the MDPDE tuning parameter α is very important, we propose two candidates as “optimal” choices, where optimality is in the sense of choosing the strongest downweighting that is necessary for the particular data set. We conduct a simulation study comparing the proposed MDPDE, for different values of α , with S-estimators, M-estimators and the classical maximum likelihood estimator, considering different levels of contamination. Finally, we illustrate the performance of our proposal on a real-data example. |
| Author | Saraceno, Giovanni Basu, Ayanendranath Ghosh, Abhik Agostinelli, Claudio |
| Author_xml | – sequence: 1 givenname: Giovanni orcidid: 0000-0002-1753-2367 surname: Saraceno fullname: Saraceno, Giovanni email: gsaracen@buffalo.edu organization: Department of Mathematics, University of Trento, Department of Biostatistics, University at Buffalo – sequence: 2 givenname: Abhik surname: Ghosh fullname: Ghosh, Abhik organization: Interdisciplinary Statistical Research Unit, Indian Statistical Institute – sequence: 3 givenname: Ayanendranath surname: Basu fullname: Basu, Ayanendranath organization: Interdisciplinary Statistical Research Unit, Indian Statistical Institute – sequence: 4 givenname: Claudio surname: Agostinelli fullname: Agostinelli, Claudio organization: Department of Mathematics, University of Trento |
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| References | McCulloch, C.E., Searle, S.R.: Generalized, Linear, and Mixed Models. John Wiley & Sons, Wiley Series in Probability and Statistics (2001) HugginsRMOn the robust analysis of variance components models for pedigree dataAust. J. Stat.19933514357123968310.1111/j.1467-842X.1993.tb01311.x WelshAHRichardsonAM13 approaches to the robust estimation of mixed modelsHandbook Stat.19971534338410.1016/S0169-7161(97)15015-5 RichardsonAMWelshAHRobust restricted maximum likelihood in mixed linear modelsBiometrics19955141429143910.2307/2533273 HampelFRContributions to the Theory of Robust Estimation1968BerkeleyUniversity of California PinheiroJCLiuCWuYNEfficient algorithms for robust estimation in linear mixed-effects models using the multivariate t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} distributionJ. Comput. Graph. Stat.2001102249276193970010.1198/10618600152628059 ChristensenRMixed models and variance componentsPlane Answers to Complex Questions: The Theory of Linear Models2011New York, NYSpringer29133110.1007/978-1-4419-9816-3_12 YauKKWKukAYCRobust estimation in generalized linear mixed modelsJ. Royal Stat. Soc.2002641101117188184710.1111/1467-9868.00327 LangeKLLittleRJATaylorJMGRobust statistical modeling using the t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} distributionJ. Am. Stat. Assoc.1989844088818961134486 Koller, M.: Robust estimation of linear mixed models. PhD thesis, ETH Zürich (2013) HampelFRThe influence curve and its role in robust estimationJ. Am. Stat. Assoc.19746934638339336265710.1080/01621459.1974.10482962 Pinheiro, J., Bates, D., R Core Team: Nlme: Linear and Nonlinear Mixed Effects Models. (2022). R package version 3.1-157. https://CRAN.R-project.org/package=nlme HugginsRMStaudteRGVariance components models for dependent cell populationsJ. Am. Stat. Assoc.1994894251929 BasuAHarrisIRHjortNJonesMCRobust and efficient estimation by minimizing a density power divergenceBiometrika1998853549559166587310.1093/biomet/85.3.549 CastillaEGhoshAMartinNPardoLRobust semiparametric inference for polytomous logistic regression with complex survey designAdv. Data Anal. Classificat.2021153701734430698010.1007/s11634-020-00430-7 KollerKrobustlmm: An R package for robust estimation of linear mixed-effects modelsJ. Stat. Softw.20167561242016fsts.book.....K10.18637/jss.v075.i06 StahelWAWelshAApproaches to robust estimation in the simplest variance components modelJ. Stat. Plan. Infer.1994572295319144024110.1016/S0378-3758(96)00050-X PotthoffRFRoySNA generalized multivariate analysis of variance model useful especially for growth curve problemsBiometrika1964513/431332618106210.2307/2334137 Agostinelli, C., Yohai, V.J.: robustvarComp: Robust Estimation for Variance Component Models. (2019). R package version 0.1-6 GhoshABasuARobust and efficient estimation in the parametric proportional hazards model under random censoringStat. Med.2019382752835299403240410.1002/sim.837731660630 AgostinelliCYohaiVJComposite robust estimators for linear mixed modelsJ. Am. Stat. Assoc.20161115161764177436017341:CAS:528:DC%2BC2sXlsVKmug%3D%3D10.1080/01621459.2015.1115358 GhoshABasuARobust estimation in generalized linear models: The density power divergence approachTEST201625269290349351910.1007/s11749-015-0445-3 CoptSVictoria-FeserMPHigh breakdown inference in the mixed linear modelJ. Am. Stat. Assoc.20061012923001:CAS:528:DC%2BD2sXms1Oqsg%3D%3D10.1198/016214505000000772 GhoshABasuARobust estimation for independent non-homogeneous observations using density power divergence with applications to linear regressionElectr. J. Stat.20137242024563117102 RichardsonAMBounded influence estimation in the mixed linear modelJ. Am. Stat. Assoc.199792437154161143610410.1080/01621459.1997.10473612 R Core Team: R: A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2019). R Foundation for Statistical Computing SinhaSKRobust analysis of generalized linear mixed modelsJ. Am. Stat. Assoc.200499466451460206283010.1198/016214504000000340 BasuAParkCShioyaHStatistical Inference: The Minimum Distance Approach2011Chapman and HallCRC Press10.1201/b10956 CastillaEGhoshAMartinNPardoLNew robust statistical procedures for the polytomous logistic regression modelsBiometrics201874412821291390814610.1111/biom.1289029772052 GhoshARobust inference under the beta regression model with application to health care studiesStat. Methods Med. Res.2019283871888392289610.1177/096228021773814229179655 HugginsRMA robust approach to the analysis of repeated measuresBiometrics1993493715720124348710.2307/2532192 A Ghosh (473_CR11) 2016; 25 A Ghosh (473_CR10) 2013; 7 S Copt (473_CR8) 2006; 101 AH Welsh (473_CR30) 1997; 15 E Castilla (473_CR6) 2021; 15 WA Stahel (473_CR29) 1994; 57 SK Sinha (473_CR28) 2004; 99 R Christensen (473_CR7) 2011 473_CR25 RM Huggins (473_CR16) 1993; 49 A Ghosh (473_CR9) 2019; 28 RM Huggins (473_CR17) 1994; 89 RF Potthoff (473_CR24) 1964; 51 KKW Yau (473_CR31) 2002; 64 RM Huggins (473_CR15) 1993; 35 AM Richardson (473_CR27) 1995; 51 A Basu (473_CR3) 1998; 85 E Castilla (473_CR5) 2018; 74 A Ghosh (473_CR12) 2019; 38 473_CR21 473_CR1 A Basu (473_CR4) 2011 FR Hampel (473_CR13) 1968 473_CR22 473_CR18 K Koller (473_CR19) 2016; 75 JC Pinheiro (473_CR23) 2001; 10 FR Hampel (473_CR14) 1974; 69 C Agostinelli (473_CR2) 2016; 111 AM Richardson (473_CR26) 1997; 92 KL Lange (473_CR20) 1989; 84 |
| References_xml | – reference: StahelWAWelshAApproaches to robust estimation in the simplest variance components modelJ. Stat. Plan. Infer.1994572295319144024110.1016/S0378-3758(96)00050-X – reference: KollerKrobustlmm: An R package for robust estimation of linear mixed-effects modelsJ. Stat. Softw.20167561242016fsts.book.....K10.18637/jss.v075.i06 – reference: Pinheiro, J., Bates, D., R Core Team: Nlme: Linear and Nonlinear Mixed Effects Models. (2022). R package version 3.1-157. https://CRAN.R-project.org/package=nlme – reference: BasuAParkCShioyaHStatistical Inference: The Minimum Distance Approach2011Chapman and HallCRC Press10.1201/b10956 – reference: CastillaEGhoshAMartinNPardoLNew robust statistical procedures for the polytomous logistic regression modelsBiometrics201874412821291390814610.1111/biom.1289029772052 – reference: PotthoffRFRoySNA generalized multivariate analysis of variance model useful especially for growth curve problemsBiometrika1964513/431332618106210.2307/2334137 – reference: WelshAHRichardsonAM13 approaches to the robust estimation of mixed modelsHandbook Stat.19971534338410.1016/S0169-7161(97)15015-5 – reference: R Core Team: R: A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2019). R Foundation for Statistical Computing – reference: AgostinelliCYohaiVJComposite robust estimators for linear mixed modelsJ. Am. Stat. Assoc.20161115161764177436017341:CAS:528:DC%2BC2sXlsVKmug%3D%3D10.1080/01621459.2015.1115358 – reference: CoptSVictoria-FeserMPHigh breakdown inference in the mixed linear modelJ. Am. Stat. Assoc.20061012923001:CAS:528:DC%2BD2sXms1Oqsg%3D%3D10.1198/016214505000000772 – reference: BasuAHarrisIRHjortNJonesMCRobust and efficient estimation by minimizing a density power divergenceBiometrika1998853549559166587310.1093/biomet/85.3.549 – reference: RichardsonAMWelshAHRobust restricted maximum likelihood in mixed linear modelsBiometrics19955141429143910.2307/2533273 – reference: YauKKWKukAYCRobust estimation in generalized linear mixed modelsJ. Royal Stat. Soc.2002641101117188184710.1111/1467-9868.00327 – reference: Agostinelli, C., Yohai, V.J.: robustvarComp: Robust Estimation for Variance Component Models. (2019). R package version 0.1-6 – reference: GhoshARobust inference under the beta regression model with application to health care studiesStat. Methods Med. Res.2019283871888392289610.1177/096228021773814229179655 – reference: PinheiroJCLiuCWuYNEfficient algorithms for robust estimation in linear mixed-effects models using the multivariate t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} distributionJ. Comput. Graph. Stat.2001102249276193970010.1198/10618600152628059 – reference: CastillaEGhoshAMartinNPardoLRobust semiparametric inference for polytomous logistic regression with complex survey designAdv. Data Anal. Classificat.2021153701734430698010.1007/s11634-020-00430-7 – reference: GhoshABasuARobust estimation in generalized linear models: The density power divergence approachTEST201625269290349351910.1007/s11749-015-0445-3 – reference: HugginsRMOn the robust analysis of variance components models for pedigree dataAust. J. Stat.19933514357123968310.1111/j.1467-842X.1993.tb01311.x – reference: SinhaSKRobust analysis of generalized linear mixed modelsJ. Am. Stat. Assoc.200499466451460206283010.1198/016214504000000340 – reference: ChristensenRMixed models and variance componentsPlane Answers to Complex Questions: The Theory of Linear Models2011New York, NYSpringer29133110.1007/978-1-4419-9816-3_12 – reference: LangeKLLittleRJATaylorJMGRobust statistical modeling using the t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} distributionJ. Am. Stat. Assoc.1989844088818961134486 – reference: RichardsonAMBounded influence estimation in the mixed linear modelJ. Am. Stat. Assoc.199792437154161143610410.1080/01621459.1997.10473612 – reference: HampelFRContributions to the Theory of Robust Estimation1968BerkeleyUniversity of California – reference: GhoshABasuARobust estimation for independent non-homogeneous observations using density power divergence with applications to linear regressionElectr. J. Stat.20137242024563117102 – reference: HugginsRMA robust approach to the analysis of repeated measuresBiometrics1993493715720124348710.2307/2532192 – reference: McCulloch, C.E., Searle, S.R.: Generalized, Linear, and Mixed Models. John Wiley & Sons, Wiley Series in Probability and Statistics (2001) – reference: GhoshABasuARobust and efficient estimation in the parametric proportional hazards model under random censoringStat. Med.2019382752835299403240410.1002/sim.837731660630 – reference: HampelFRThe influence curve and its role in robust estimationJ. Am. Stat. 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| SubjectTerms | Asymptotic methods Datasets Divergence Econometrics Economics Finance Influence functions Insurance Management Mathematical models Mathematics and Statistics Maximum likelihood estimators Normality Optimization Original Paper Parameter estimation Probability density functions Probability Theory and Stochastic Processes Robustness Statistics Statistics for Business |
| Title | Robust estimation of fixed effect parameters and variances of linear mixed models: the minimum density power divergence approach |
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