Simultaneous inference for linear mixed model parameters with an application to small area estimation

Open access financiado por Universite de Geneve (article funding)European Regional Development Fund[Abstract]: Over the past decades, linear mixed models have attracted considerable attention in various fields of applied statistics. They are popular whenever clustered, hierarchical or longitudinal data are investigated. Nonetheless, statistical tools for valid simultaneous inference for mixed parameters are rare. This is surprising because one often faces inferential problems beyond the pointwise examination of fixed or mixed parameters. For example, there is an interest in a comparative analysis of cluster-level parameters or subject-specific estimates in studies with repeated measurements. We discuss methods for simultaneous inference assuming a linear mixed model. Specifically, we develop simultaneous prediction intervals as well as multiple testing procedures for mixed parameters. They are useful for joint considerations or comparisons of cluster-level parameters. We employ a consistent bootstrap approximation of the distribution of max-type statistic to construct our tools. The numerical performance of the developed methodology is studied in simulation experiments and illustrated in a data example on household incomes in small areas.Swiss National Science Foundation; 200021-192345,Swiss National Science Foundation; P2GEP2_195898Xunta de Galicia; ED431C 2020/14Ministerio de Ciencia e Innovación; PID2020-113578RB-I00Galician Innovation Agency/ Ministerio de Economía, empleo e industria; COV20/00604Xunta de Galicia; ED431G2019/0

Simultaneous Inference for Empirical Best Predictors With a Poverty Study in Small Areas

[Abstract]: Today, generalized linear mixed models (GLMM) are broadly used in many fields. However, the development of tools for performing simultaneous inference has been largely neglected in this domain. A framework for joint inference is indispensable to carry out statistically valid multiple comparisons of parameters of interest between all or several clusters. We therefore develop simultaneous confidence intervals and multiple testing procedures for empirical best predictors under GLMM. In addition, we implement our methodology to study widely employed examples of mixed models, that is, the unit-level binomial, the area-level Poisson-gamma and the area-level Poisson-lognormal mixed models. The asymptotic results are accompanied by extensive simulations. A case study on predicting poverty rates illustrates applicability and advantages of our simultaneous inference tools.The authors gratefully acknowledge the support from the Swiss National Science Foundation for the project 200021-192345. In addition, they acknowledge the support from the MINECO grants MTM2017-82724-R and MTM2014-52876-R, the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2016-015 and Centro Singular de Investigación de Galicia ED431G/01), all of them through the ERDF. The computations were performed at the University of Geneva on the Baobab cluster.Xunta de Galicia; ED431C-2016-015Xunta de Galicia; ED431G/0

Simultaneous Inference for Empirical Best Predictors with a Poverty Study in Small Areas

Today, generalized linear mixed models are broadly used in many fields. However, the development of tools for performing simultaneous inference has been largely neglected in this domain. A framework for joint inference is indispensable to carry out statistically valid multiple comparisons of parameters of interest between all or several clusters. We therefore develop simultaneous confidence intervals and multiple testing procedures for empirical best predictors under generalized linear mixed models. In addition, we implement our methodology to study widely employed examples of mixed models, that is, the unit-level binomial, the area-level Poisson-gamma and the area-level Poisson-lognormal mixed models. The asymptotic results are accompanied by extensive simulations. A case study on predicting poverty rates illustrates applicability and advantages of our simultaneous inference tools.Comment: 46 pages, 20 figures; simulations and data analysis expanded, additional remarks adde

Simple bootstrap for linear mixed effects under model misspecification

[Abstract]: Linear mixed effects are considered excellent predictors of cluster-level parameters in various domains. However, previous research has demonstrated that their performance is affected by departures from model assumptions. Given the common occurrence of these departures in empirical studies, there is a need for inferential methods that are robust to misspecifications while remaining accessible and appealing to practitioners. Statistical tools have been developed for cluster-wise and simultaneous inference for mixed effects under distributional misspecifications, employing a user-friendly semiparametric random effect bootstrap. The merits and limitations of this approach are discussed in the general context of model misspecification. Theoretical analysis demonstrates the asymptotic consistency of the methods under general regularity conditions. Simulations show that the proposed intervals are robust to departures from modelling assumptions, including asymmetry and long tails in the distributions of errors and random effects, outperforming competitors in terms of empirical coverage probability. Finally, the methodology is applied to construct confidence intervals for household income across counties in the Spanish region of Galicia.The authors gratefully acknowledge support from the Swiss National Science Foundation, projects 200021-192345 and P2GEP2-195898, as well as from the Instituto Galego de Estatística who provided us with the data set. In addition, this research has been supported by MICINN grant PID2020-113578RB-I00, and by Xunta de Galicia (Grupos de Referencia Competitiva ED431C 2020/14), GAIN (Galician Innovation Agency) and the Regional Ministry of Economy, Employment and Industry grant COV20/00604 and Centro de investigación del Sistema universitario de Galicia ED431G 2019/01, all of them through ERDF. The computations were performed at the University of Geneva using Baobab and Yggdrasil HPC Service and using the computational facilities of the Advanced Computing Research Centre, University of Bristol.Xunta de Galicia; ED431C 2020/14Xunta de Galicia; ED431G 2019/01Switzerland. Swiss National Science Foundation; 200021-192345Switzerland. Swiss National Science Foundation; P2GEP2-195898Xunta de Galicia; COV20/0060

Empirical best prediction under area-level Poisson mixed models

[Abstract] The paper studies the applicability of area-level Poisson mixed models to estimate small area counting indicators. Among the available procedures for fitting generalized linear models, the method of moments (MM) and the penalised quasi-likelihood (PQL) method are employed. The empirical best predictor (EBP) of the area mean is derived using MM and compared with plug-in alternatives using MM and PQL. The plug-in estimator using PQL is computationally faster and provides competitive performance with respect to EBP that involves high complex integrals. An approximation to the mean squared error (MSE) of the EBP is given and three MSE estimators are proposed. The first two MSE estimators are plug-in estimators without and with bias correction to the second order and the third one is based on parametric bootstrap. Several simulation experiments are carried out for analysing the behaviour of the EBP and for comparing the estimators of the MSE of the EBP. A good choice in practice is the bootstrap alternative since it performs similarly to the analytical versions and is computationally faster. The developed methodology and software are applied to data from the 2008 Spanish living condition survey. The target of the application is the estimation of poverty rates at province level.Xunta de Galicia; CN2012/130Ministerio de Ciencia e Innovación; MTM2013-41383-PMinisterio de Ciencia e Innovación; MTM2014-52876-RMinisterio de Ciencia e Innovación; MTM2011-22392Ministerio de Ciencia e Innovación; MTM2008-03010Ministerio de Ciencia e Innovación; MTM2012-37077-C02-0

Small area prediction of proportions and counts under a spatial Poisson mixed model

[Abstract]: This paper introduces an area-level Poisson mixed model with SAR(1) spatially correlated random effects. Small area predictors of proportions and counts are derived from the new model and the corresponding mean squared errors are estimated by parametric bootstrap. The behaviour of the introduced predictors is empirically investigated by running model-based simulation experiments. An application to real data from the Spanish living conditions survey of Galicia (Spain) is given. The target is the estimation of domain proportions of women under the poverty line.Supported by the Instituto Galego de Estatística, by MICINN Grants PID2020-113578RB-I00 and PGC2018-096840-B-I00, by the Generalitat Valenciana Grant PROMETEO/2021/063 and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C 2020/14), and by GAIN (Galician Innovation Agency) and the Regional Ministry of Economy, Employment and Industry Grant COV20/00604 and Centro de Investigación del Sistema Universitario de Galicia ED431G 2019/01, all of them through the ERDF.Generalitat Valenciana; PROMETEO/2021/063Xunta de Galicia; ED431C/2020/14Xunta de Galicia; COV20/00604Xunta de Galicia; ED431G/2019/0

Poverty Mapping Under Area-Level Random Regression Coefficient Poisson Models

[Abstract]: Under an area-level random regression coefficient Poisson model, this article derives small area predictors of counts and proportions and introduces bootstrap estimators of the mean squared errors (MSEs). The maximum likelihood estimators of the model parameters and the mode predictors of the random effects are calculated by a Laplace approximation algorithm. Simulation experiments are implemented to investigate the behavior of the fitting algorithm, the predictors, and the MSE estimators with and without bias correction. The new statistical methodology is applied to data from the Spanish Living Conditions Survey. The target is to estimate the proportions of women and men under the poverty line by province.This work was supported by the Ministry of Science and Innovation and the State Research Agency of the Spanish Government through the European Regional Development Fund (PID2022-136878NB-I00, PID2020-113578RB-I00 and PRE2021-100857 to Naomi Diz-Rosales funded by MCIN/AEI/10.13039/501100011033); by the Conselleria d’Innovació, Universitats, Ciéncia i Societat Digital of the Generalitat Valenciana (Prometeo/2021/063); by the Consellería de Cultura, Educación, Formación Profesional e Universidades of the Xunta de Galicia through the European Regional Development Fund (Competitive Reference Groups ED431C/2020/14, COV20/00604, and ED431G/2019/01); and by Centro de Investigación en Tecnologías de la Información y las Comunicaciones (CITIC) that is supported by Xunta de Galicia, collaboration agreement between the Consellería de Cultura, Educación, Formación Profesional e Universidades and the Galician universities for the reinforcement of the research centers of the Sistema Universitario de Galicia (CIGUS).Xunta de Galicia; ED431C/2020/14Xunta de Galicia; COV20/00604Xunta de Galicia; ED431G/2019/0

Mapping the Poverty Proportion in Small Areas under Random Regression Coefficient Poisson Models

Cursos e Congresos, C-155[Abstract] In a complex socio-economic context, policy makers need highly disaggregated poverty indicators. In this work, we develop a methodology in small area estimation to derive predictors of poverty proportions under a random regression coefficient Poisson model, introducing bootstrap estimators of mean squared errors. Maximum likelihood estimators of model parameters and random effects mode predictors are calculated using a Laplace approximation algorithm. Simulation experiments are conducted to investigate the behaviour of the fitting algorithm, the predictors and the mean squared error estimator. The new statistical methodology is applied to data from the Spanish survey of living conditions to map poverty proportions by province and sex, developing a tool to support policy decision makingXunta de Galicia; ED431C-2020/14This research is part of the grant PID2020-113578RB-I00, funded by MCIN/AEI/10.13039/501100011033/. It has also been supported by the Spanish grant PID2022-136878NB-I00, the Valencian grant Prometeo/2021/063, by the Xunta de Galicia (Competitive Reference ED431C-2020/14) and by CITIC that is supported by Xunta de Galicia, collaboration agreement between the Consellería de Cultura, Educación, Formación Profesional e Universidades and the Galician universities for the reinforcement of the research centres of the Sistema Universitario de Galicia (CIGUS). The first author was also sponsoredby the Spanish Grant for Predoctoral Research Trainees RD 103/2019 being this work part of grant PRE2021-100857, funded by MCIN/AEI/10.13039/501100011033/ and ESF