Model-Based Recursive Partitioning with Adjustment for Measurement Error: Applied to the Cox’s Proportional Hazards and Weibull Model

259 Pages·2015·1.51 MB·English

by  Hanna Birke (auth.)

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Hanna Birke
Model-Based Recursive
Partitioning with
Adjustment for
Measurement Error
Applied to the Cox’s Proportional
Hazards and Weibull Model
Foreword by Prof. Dr. Thomas Augustin
Hanna Birke
Düsseldorf, Germany
BestMasters
ISBN 978-3-658-08504-9 ISBN 978-3-658-08505-6 (eBook)
DOI 10.1007/978-3-658-08505-6
Library of Congress Control Number: 2015930262
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Foreword
Inrecentyearsdata-analyticmethodsoriginatingfromtheareaofmachinelearning
havegainedalsostrongmomentuminStatistics. Statisticianshavecontributed
tothemethodologicalfoundationsofthesealgorithmicmethodsandsubstantially
extendedthem,andthusthesemethodshavebecomepartofstatisticalmethodol-
ogy. Aparticularlypopularexamplearerecursivepartitioningmethods,including
so-called classification and regression trees. Their basic rationale is to analyse
multivariate data by successively splitting the sample and thus partitioning it
into sub-groups that are homogeneous with respect to an outcome variable of
interest. Originally,thesemethods,whicharetypicallyverypowerfulinprediction
problems,havebeenunderstoodasmodel-free,providingafundamentalalternative
toregressionmodelling,contrastingtheexplanation-orientedregressionmodelling
culturewiththeprediction-orientedalgorithmicculture.
Thisseeminglyfundamentalantagonismisovercomebythemodel-basedrecursive
partitioning ("MOB") approach of Zeileis, Hothorn & Hornik (2008; Journal of
Computational and Graphical Statistics), where, in simplified terms, regression
modelsarespilt: Firstaregressionmodelrelatingthecorecovariatestotheresponse
variableisestimatedonthewholesample,andthenitisinvestigatedforparameter
instability, i.e. whether the individual score contributions suggest a substantial
structuralchangeintheparametervaluesovertheorderedvaluesofsomeadditional
variable. Ifso,thesampleissplit,andseparateregressionmodelsareestimated
forthesub-groups,whichthenaretakenasnewsamplesforaniterativerepetition
oftheproceduresuntilastoppingcriterionisfulfilled. Intheend,theprocedure
givestheuserapartitionwhosemembersarehomogeneouswithrespecttothecore
regression model, and thus parsimonious regression models of high explanatory
powerareobtained-if(!) nomeasurementerroroccurs.
However,substantialmeasurementerrorisquiteacommonproblem,forinstance,in
econometricsandbiometrics. Therefore,thedevelopmentofpowerfulmeasurement
correctionmethodshasbeenanintensivelystudiedareaintraditionalregression
models. Birkeisthefirstwhotransferredsomeofthesemethodstothemodel-based
VI Foreword
recursive partitioning context. Her basic technical insight is that for the MOB
methodology it is sufficient to consider arbitrary unbiased estimation functions,
insteadofthecommonlyusedlikelihood-basedscore-function. Birkethereforerelies
on measurement error corrected score functions in terms of Nakamura, and she
isindeedabletoextendthewholeMOBframeworktothesituationofclassical
additivecovariatemeasurementerror. Thisincludesasophisticatedimplementa-
tion for the Weibull and the Cox model that makes this computationally quite
demandingprocedurestillpracticallyapplicable. Awelldesignedsimulationstudy
corroboratesthehighpowerofthedevelopedmethodology.
By incorporating measurement error correction methods in a very elegant and
efficient way, Birke’s work boosts the practical application and relevance of the
model-basedrecursivepartitioningmethodology.
Munich,November2014 Prof. Dr. ThomasAugustin
Short Profile of the Working
Group
TheworkinggroupFoundationsofStatisticsandTheirApplications (head: Prof.
Dr. ThomasAugustin;www.statistik.lmu.de/institut/ag/agmg/index.html)
is one of eight working groups/chairs at the Department of Statistics at LMU
Munich.
Theresearchvisionofthegroupistocontributetothedevelopmentofapowerful
comprehensive statistical methodology where the often limited data quality is
explicitlyreflectedinstatisticalmodellingandinference. Incontrasttotraditional
methods,whichindispensablyrequirestrong,oftenempiricallynottenableassump-
tionsondataformation,thenovelmethodsdeliberatelyaimatdrawingreliable
conclusions from complex data, leading to deeper insights into social, economic
andbiologicalstructuresandprocesses.
Thereforeonemainfocusofresearchisonreliableinferencebyset-valuedmethods
(impreciseprobabilities,partialidentification)and(generalized)measurementerror
modelling. Furtherresearchinterestsincludeotherissuesinthefoundationsofsta-
tisticalinference,statisticalaspectsofempiricalsocialresearchandpsychometrics,
survivalanalysis,graphicalmodelsanddifferentmachinelearningmethods.
Acknowledgment
Thismaster’sthesiswaswrittenunderthedirectionofProf. Dr. ThomasAugustin
attheDepartmentofStatisticsoftheLudwig-Maximilians-UniversityMunich.
Inparticular,IwouldliketoexpressmygratitudetoProf. Dr. ThomasAugustin
for the interesting and relevant topic of the thesis, for the good cooperation as
wellasforhishelpfulandvaluablesuggestions. Iappreciatehissupportatany
time. Inaddition,IwishtothankTeresaforproofreadingmythesisandforher
constructivecomments.
I would like to thank my ’stät girls’ Ina, Kathrin, Sonja and Teresa as well as
Markus. Iamgratefultohavefoundsuchamazingfriendswhohaveaccompanied
methroughoutmystudies.
Furthermore,IthankJan,Michael,mysisterLenaandespeciallymyparentsfor
theirconstantsupportandencouragement.
Düsseldorf,November2014 HannaBirke
Contents
1. Introduction 1
2. Theoretical Background 3
2.1. CoxModelandMeasurementError . . . . . . . . . . . . . . . . . . 3
2.1.1. MeasurementErrorModel . . . . . . . . . . . . . . . . . . . 5
2.1.2. AdjustmentforMeasurementErrorinCoxModel . . . . . . 6
2.1.3. Corrected Log-Likelihood and Score Function in Cox and
WeibullModel. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2. Model-BasedRecursivePartitioning . . . . . . . . . . . . . . . . . . 21
2.2.1. IntroductiontoClassificationandRegressionTrees . . . . . 22
2.2.2. Model-BasedRecursivePartitioning. . . . . . . . . . . . . . 24
3. Implementation 35
3.1. Model-BasedRecursivePartitioninginR . . . . . . . . . . . . . . . 35
3.2. FittingaMOBWeibullModelwithAdjustmentforMeasurement
Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4. Simulation Study 55
4.1. CheckthefitFunction . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.1. CheckthefitFunction-onlyW . . . . . . . . . . . . . . . 57
4.1.2. CheckthefitFunction-WandF . . . . . . . . . . . . . . 66
4.2. StructuralChanges . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.1. StructuralChanges-single. . . . . . . . . . . . . . . . . . . 72
4.2.2. StructuralChanges-multi . . . . . . . . . . . . . . . . . . . 80
4.3. OneGlobalModelFitvs. MOB . . . . . . . . . . . . . . . . . . . . 85
4.4. ConclusionoftheSimulationStudy . . . . . . . . . . . . . . . . . . 92
5. Conclusion 95
Appendix 101