Table Of ContentSpringerBriefs in Statistics
JSS Research Series in Statistics
Shonosuke Sugasawa · Tatsuya Kubokawa
Mixed-Effects Models
and Small Area
Estimation
SpringerBriefs in Statistics
JSS Research Series in Statistics
Editors-in-Chief
NaotoKunitomo,TheInstituteofMathematicalStatistics,Tachikawa,Tokyo,Japan
AkimichiTakemura,TheCenterforDataScienceEducationandResearch,Shiga
University,Hikone,Shiga,Japan
SeriesEditors
GenshiroKitagawa,MeijiInstituteforAdvancedStudyofMathematicalSciences,
Nakano-ku,Tokyo,Japan
ShigeyukiMatsui,GraduateSchoolofMedicine,NagoyaUniversity,Nagoya,
Aichi,Japan
ManabuIwasaki,SchoolofDataScience,YokohamaCityUniversity,Yokohama,
Kanagawa,Japan
YasuhiroOmori,GraduateSchoolofEconomics,TheUniversityofTokyo,
Bunkyo-ku,Tokyo,Japan
MasafumiAkahira,InstituteofMathematics,UniversityofTsukuba,Tsukuba,
Ibaraki,Japan
MasanobuTaniguchi,SchoolofFundamentalScienceandEngineering,Waseda
University,Shinjuku-ku,Tokyo,Japan
HiroeTsubaki,TheInstituteofStatisticalMathematics,Tachikawa,Tokyo,Japan
SatoshiHattori,FacultyofMedicine,OsakaUniversity,Suita,Osaka,Japan
KosukeOya,SchoolofEconomics,OsakaUniversity,Toyonaka,Osaka,Japan
TaijiSuzuki,SchoolofEngineering,UniversityofTokyo,Tokyo,Japan
The current research of statistics in Japan has expanded in several directions in
linewithrecenttrendsinacademicactivitiesintheareaofstatisticsandstatistical
sciencesovertheglobe.ThecoreoftheseresearchactivitiesinstatisticsinJapanhas
beentheJapanStatisticalSociety(JSS).Thissociety,theoldestandlargestacademic
organizationforstatisticsinJapan,wasfoundedin1931byahandfulofpioneerstatis-
ticiansandeconomistsandnowhasahistoryofabout80years.Manydistinguished
scholarshavebeenmembers,includingtheinfluentialstatisticianHirotuguAkaike,
whowasapastpresidentofJSS,andthenotablemathematicianKiyosiItô,whowas
anearliermemberoftheInstituteofStatisticalMathematics(ISM),whichhasbeen
acloselyrelatedorganizationsincetheestablishmentofISM.Thesocietyhastwo
academicjournals:theJournaloftheJapanStatisticalSociety(EnglishSeries)and
theJournaloftheJapanStatisticalSociety(JapaneseSeries).ThemembershipofJSS
consistsofresearchers,teachers,andprofessionalstatisticiansinmanydifferentfields
includingmathematics,statistics,engineering,medicalsciences,governmentstatis-
tics,economics,business,psychology,education,andmanyothernatural,biological,
and social sciences. The JSS Series of Statistics aims to publish recent results of
currentresearchactivities intheareas ofstatisticsand statisticalsciences inJapan
thatotherwisewouldnotbeavailableinEnglish;theyarecomplementarytothetwo
JSSacademicjournals,bothEnglishandJapanese.Becausethescopeofaresearch
paperinacademicjournalsinevitablyhasbecomenarrowlyfocusedandcondensed
in recent years, this series is intended to fill the gap between academic research
activitiesandtheformofasingleacademicpaper.Theserieswillbeofgreatinterest
toawideaudienceofresearchers,teachers,professionalstatisticians,andgraduate
studentsinmanycountrieswhoareinterestedinstatisticsandstatisticalsciences,in
statisticaltheory,andinvariousareasofstatisticalapplications.
·
Shonosuke Sugasawa Tatsuya Kubokawa
Mixed-Effects Models
and Small Area Estimation
ShonosukeSugasawa TatsuyaKubokawa
CenterforSpatialInformationScience FacultyofEconomics
UniversityofTokyo UniversityofTokyo
Kashiwa-shi,Chiba,Japan Tokyo,Japan
ISSN 2191-544X ISSN 2191-5458 (electronic)
SpringerBriefsinStatistics
ISSN 2364-0057 ISSN 2364-0065 (electronic)
JSSResearchSeriesinStatistics
ISBN 978-981-19-9485-2 ISBN 978-981-19-9486-9 (eBook)
https://doi.org/10.1007/978-981-19-9486-9
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2023
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Singapore
Preface
Thisbookprovidesaself-containedintroductionofmixed-effectsmodelsandsmall
area estimation techniques. In particular, it focuses on both introducing classical
theory and reviewing the latest methods. It first introduces basic issues of mixed-
effects models, such as parameter estimation, random effects prediction, variable
selection,andasymptotictheory.Standardmixed-effectsmodelsusedinsmallarea
estimation, known as Fay–Herriot model and nested error regression model, are
then introduced. Both frequentist and Bayesian approaches are given to compute
predictors of small area parameters of interest. For measuring uncertainty of the
predictors,several methods tocalculate meansquared errorsandconfidence inter-
vals are discussed. Various advanced approaches using mixed-effects models are
introduced,coveringfromfrequentisttoBayesianapproaches.Thisbookishelpful
forresearchersandgraduatestudentsinvariousfieldsrequiringdataanalysisskills
aswellasinmathematicalstatistics.
The authors would like to thank Professor Masafumi Akahira for giving us the
opportunityofpublishingthisbook.Theworkofthefirstauthorwassupportedin
partbyGrant-in-AidforScientificResearch(21H00699)fromtheJapanSocietyfor
thePromotion ofScience (JSPI).Theworkofthesecond author wassupported in
partbyGrant-in-AidforScientificResearch(18K11188)fromtheJSPI.
Tokyo,Japan ShonosukeSugasawa
September2022 TatsuyaKubokawa
v
Contents
1 Introduction ................................................... 1
References ..................................................... 2
2 GeneralMixed-EffectsModelsandBLUP ........................ 5
2.1 Mixed-EffectsModelsandExamples .......................... 5
2.2 BestLinearUnbiasedPredictors .............................. 9
2.3 REMLandGeneralEstimatingEquations ...................... 12
2.4 AsymptoticProperties ....................................... 13
2.5 ProofsoftheAsymptoticResults ............................. 18
References ..................................................... 21
3 MeasuringUncertaintyofPredictors ............................. 23
3.1 EBLUPandtheMeanSquaredError .......................... 23
3.2 ApproximationoftheMSE .................................. 24
3.3 EvaluationoftheMSEUnderNormality ....................... 29
3.4 EstimationoftheMSE ...................................... 31
3.5 ConfidenceIntervals ........................................ 33
References ..................................................... 35
4 BasicMixed-EffectsModelsforSmallAreaEstimation ............ 37
4.1 BasicArea-LevelModel ..................................... 37
4.1.1 Fay–HerriotModel ................................... 37
4.1.2 AsymptoticPropertiesofEBLUP ...................... 39
4.2 BasicUnit-LevelModels .................................... 49
4.2.1 NestedErrorRegressionModel ........................ 49
4.2.2 AsymptoticPropertiesofEBLUP ...................... 51
References ..................................................... 56
5 HypothesisTestsandVariableSelection .......................... 57
5.1 Test Procedures for a Linear Hypothesis on Regression
Coefficients ................................................ 57
5.2 InformationCriteriaforVariableorModelSelection ............. 61
References ..................................................... 66
vii
viii Contents
6 AdvancedTheoryofBasicSmallAreaModels .................... 67
6.1 AdjustedLikelihoodMethods ................................ 67
6.1.1 StrictlyPositiveEstimateofRandomEffectVariance ...... 67
6.1.2 AdjustedLikelihoodforEmpiricalBayesConfidence
Intervals ............................................ 69
6.1.3 AdjustedLikelihoodforSolvingMultipleSmallArea
EstimationProblems ................................. 72
6.2 ObservedBestPrediction .................................... 72
6.3 RobustMethods ............................................ 75
6.3.1 Unit-LevelModels ................................... 75
6.3.2 Area-LevelModels ................................... 77
References ..................................................... 80
7 SmallAreaModelsforNon-normalResponseVariables ............ 83
7.1 GeneralizedLinearMixedModels ............................ 83
7.2 NaturalExponentialFamilieswithConjugatePriors ............. 85
7.3 UnmatchedSamplingandLinkingModels ..................... 86
7.4 ModelswithDataTransformation ............................. 89
7.4.1 Area-LevelModelsforPositiveValues .................. 89
7.4.2 Area-LevelModelsforProportions ..................... 90
7.4.3 Unit-LevelModelsandEstimatingFinitePopulation
Parameters .......................................... 91
7.5 ModelswithSkewedDistributions ............................ 95
References ..................................................... 98
8 ExtensionsofBasicSmallAreaModels ........................... 99
8.1 FlexibleModelingofRandomEffects ......................... 99
8.1.1 UncertaintyofthePresenceofRandomEffects ........... 99
8.1.2 Modeling Random Effects via Global–Local
ShrinkagePriors ..................................... 104
8.2 MeasurementErrorsinCovariates ............................ 107
8.2.1 MeasurementErrorsintheFay–HerriotModel ........... 107
8.2.2 MeasurementErrorsintheNestedErrorRegression
Model .............................................. 110
8.3 NonparametricandSemiparametricModeling .................. 111
8.4 ModelingHeteroscedasticVariance ........................... 113
8.4.1 ShrinkageEstimationofSamplingVariances ............. 113
8.4.2 HeteroscedasticVarianceinNestedErrorRegression
Models ............................................. 116
References ..................................................... 120
Chapter 1
Introduction
Theterm‘smallarea’or‘smalldomain’referstoasmallgeographicalregionsuch
asacounty,municipalityorstate,orasmalldemographicgroupsuchasaspecific
age–sex–race group. In the estimation of a characteristic of such a small group,
the direct estimate based on only on the data from the small group is likely to be
unreliable, because only the small number of observations are available from the
small group. The problem of small area estimation is how to produce a reliable
estimateforthecharacteristicofthesmallgroup,andthesmallareaestimationhas
beenactivelyandextensivelystudiedfromboththeoreticalandpracticalaspectsdue
to an increasing demand for reliable small area estimates from public and private
sectors. The articles by Ghosh and Rao (1994) and Pfeffermann (2013) give good
reviews and motivations, and the comprehensive book by Rao and Molina (2015)
coversallthemaindevelopmentsinsmallareaestimation.Morerecentreviewonthe
useofmixedmodelsinsmallareaestimationisgiveninSugasawaandKubokawa
(2020). Also see Demidenko (2004) for general mixed models and Pratesi (2016)
foranalysisofpovertydatabysmallareaestimation.Inthispaper,wedescribethe
detailsofclassicalmethodsandgiveareviewofrecentdevelopments,whichwillbe
helpfulforreaderswhoareinterestedinthistopic.
Toimprovetheaccuracyofdirectsurveyestimates,wemakeuseoftherelevant
supplementaryinformationsuchasdatafromotherrelatedareasandcovariatedata
fromothersources.Thelinearmixedmodels(LMM)enableusto‘borrowstrength’
fromtherelevantsupplementarydata,andtheresultingmodel-basedestimatorsor
thebestlinearunbiasedpredictors(BLUP)providereliableestimatesforthesmall
areacharacteristics.TheBLUPshrinksthedirectestimatesinsmallareastowarda
stable quantity constructed by pooling all the data, thereby BLUP is characterized
by the effects of pooling and shrinkage of the data. These two features of BLUP
mainlycomefromthestructureoflinearmixedmodelsdescribedas(observation)
= (common parameters) + (random effects) + (error terms), namely, the shrinkage
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2023 1
S.SugasawaandT.Kubokawa,Mixed-EffectsModelsandSmallAreaEstimation,
JSSResearchSeriesinStatistics,https://doi.org/10.1007/978-981-19-9486-9_1
2 1 Introduction
effectarisesfromtherandomeffects,andthepoolingeffectisduetothesetupofthe
common parameters. While BLUP was originally proposed by Henderson (1950),
empiricalversionofBLUP(EBLUP)isrelatedtotheclassicalshrinkageestimator
studiedbyStein(1956),whoestablishedanalyticallythatEBLUPimprovesonthe
samplemeanswhenthenumberofsmallareasislargerthanorequaltothree.This
factshowsnotonlythatEBLUPhasalargerprecisionthanthesamplemean,butalso
thatasimilarconceptcameoutatthesametimebyHenderson(1950)forpractical
useandStein(1956)fortheoreticalinterest.Basedonthesehistoricalbackgrounds,
therehavebeenalotofmethodsproposedsofar.
Asaformerpartofthisbook,wefirstintroducedetailsoftheoryoflinearmixed
modelsandBLUP(orEBLUP)inChap.2.Sincemeasuringthevariabilityorriskof
EBLUPisanimportanttaskinsmallareaestimation,wefocusonmeansquarederrors
andpredictionintervalsinChap.3,anddescribeseveralmethodsbasedonasymptotic
calculations and simulation-based methods such as jackknife and bootstrap. Our
argument tries to keep generality without assuming normality assumptions for the
distributionasmuchaspossible.Wethenintroducetwobasicsmallareamodels,the
Fay–Herriotmodel(FayandHerriot1979)andthenestederrorregression(Battese
etal.1988)inChap.4.InChap.5,weprovidebasictechniquesofhypothesistesting
andvariableselectioninlinearmixedmodels,whichcanimmediatelybeappliedto
thebasicsmallareamodels.
Asalatterpartofthisbook,wefocusmoreontechniquesofsmallareaestimation
basedonmixed-effectsmodels,withsomeexamplesinplaces.First,inChap.6,we
explainadvancedtheoryofbasicsmallareamodelstohandlepracticalproblems.We
mainlyfocusonthreetechniques,adjustedlikelihoodmethodsforestimatingrandom
effectsvariance,observedbestpredictionforrandomeffectsandrobustprediction
andfittingofthesmallareamodels.InChap.7,weintroduceseveraltechniquesto
handlenon-normalresponsevariablesinsmallareaestimation,includinggeneralized
linearmixedmodels,modelswithdatatransformation,andmodelswithnon-normal
distributions.Finally,wereviewseveralextensionsofthebasicsmallareamodelsin
Chap.8.Thetopicstreatedthereareflexiblemodelingofrandomeffects,measure-
menterrormodels,nonparametricandsemiparametricmodels,andheteroscedastic
variancemodels.
References
BatteseG,HarterR,FullerW(1988)Anerror-componentsmodelforpredictionofcountycrop
areasusingsurveyandsatellitedata.JAmStatAssoc83:28–36
DemidenkoE(2004)Mixedmodels:theoryandapplications.Wiley
FayR,HerriotR(1979)Estimatorsofincomeforsmallareaplaces:AnapplicationofJames-Stein
procedurestocensus.JAmStatiAssoc74:341–353
GhoshM,RaoJ(1994)Smallareaestimation:anappraisal.StatSci9:55–76
HendersonC(1950)Estimationofgeneticparameters.AnnMathStat21:309–310
PfeffermannD(2013)Newimportantdevelopmentsinsmallareaestimation.StatSci28:40–68
PratesiM(ed)(2016)Analysisofpovertydatabysmallareaestimation.Wiley
