Practical Regression and Anova using R - The Comprehensive R

213 Pages·2009·0.99 MB·English

Practical Regression and Anova using R
JulianJ.Faraway
July2002
1
Copyright (cid:0)c 1999,2000,2002JulianJ.Faraway
Permissiontoreproduceindividualcopiesofthisbookforpersonaluseisgranted. Multiplecopiesmay
be createdfor nonprofitacademicpurposes —a nominalcharge to cover theexpenseof reproductionmay
bemade. Reproductionforprofitisprohibitedwithoutpermission.
Preface
There are many books on regression and analysis of variance. These books expect different levels of pre-
paredness and place different emphases on the material. This book is not introductory. It presumes some
knowledgeofbasicstatisticaltheoryandpractice. Studentsareexpectedtoknowtheessentialsofstatistical
inferencelikeestimation,hypothesistestingandconfidenceintervals. Abasicknowledgeofdataanalysisis
presumed. Somelinearalgebraandcalculusisalsorequired.
The emphasis of this text is on the practice of regression and analysis of variance. The objective is to
learn what methods are available and moreimportantly, when they shouldbe applied. Many examples are
presented to clarify the use of the techniques and to demonstrate what conclusions can be made. There
is relatively less emphasis on mathematical theory, partly because some prior knowledge is assumed and
partly because the issues are better tackled elsewhere. Theory is important because it guides the approach
we take. I take a wider view of statistical theory. It is not just the formal theorems. Qualitative statistical
conceptsarejustasimportantinStatisticsbecausetheseenableustoactuallydoitratherthanjusttalkabout
it. Thesequalitative principlesarehardertolearnbecausetheyaredifficulttostatepreciselybuttheyguide
thesuccessfulexperiencedStatistician.
Dataanalysiscannotbelearntwithoutactuallydoingit. Thismeansusingastatisticalcomputingpack-
age. Thereisawidechoiceofsuchpackages. Theyaredesignedfordifferentaudiencesandhavedifferent
strengths and weaknesses. I have chosen to use R (ref. Ihaka and Gentleman (1996)). Why do I use R ?
Theareseveralreasons.
1. Versatility. R is a also a programming language, so I am not limited by the procedures that are
preprogrammedbyapackage. ItisrelativelyeasytoprogramnewmethodsinR .
2. Interactivity. Data analysis is inherently interactive. Some older statistical packages were designed
when computing was more expensive and batch processing of computations was the norm. Despite
improvementsinhardware,theoldbatchprocessingparadigmlivesonintheiruse. R doesonething
atatime,allowingustomakechangesonthebasisofwhatweseeduringtheanalysis.
3. R is based on S from which the commercial package S-plus is derived. R itself is open-source
software and may be freely redistributed. Linux, Macintosh, Windows and other UNIX versions
are maintained and can be obtained from the R-project at www.r-project.org. R is mostly
compatible with S-plus meaning that S-plus could easily be used for the examples given in this
book.
4. Popularity. SAS is the most common statistics package in general but R or S is most popular with
researchers in Statistics. A look at common Statistical journals confirms this popularity. R is also
popularforquantitative applicationsinFinance.
The greatest disadvantage of R is that it is not so easy to learn. Some investment of effort is required
beforeproductivitygainswillberealized. ThisbookisnotanintroductiontoR.Thereisashortintroduction
2
3
in the Appendix but readers are referred to the R-project web site at www.r-project.orgwhere you
can find introductory documentation and information about books on R . I have intentionally included in
thetextallthecommandsusedtoproducetheoutputseeninthisbook. Thismeansthatyoucanreproduce
these analyses and experimentwith changes and variations before fully understandingR . The reader may
choosetostartworkingthroughthistextbeforelearningR andpickitupasyougo.
The web site for this book is at www.stat.lsa.umich.edu/˜faraway/bookwhere data de-
scribedinthisbookappears. Updateswillappeartherealso.
ThankstothebuildersofR withoutwhomthisbookwouldnothavebeenpossible.
Contents
1 Introduction 8
1.1 Beforeyoustart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.2 DataCollection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 InitialDataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 WhentouseRegressionAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Estimation 16
2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 LinearModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 MatrixRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Estimatingb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Leastsquaresestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Examplesofcalculatingbˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Whyisbˆ agoodestimate? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Gauss-Markov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 MeanandVarianceofbˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.10 Estimatings 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.11 GoodnessofFit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.12 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Inference 26
3.1 Hypothesisteststocomparemodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 SomeExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Testofallpredictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.2 Testingjustonepredictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.3 Testingapairofpredictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.4 Testingasubspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 ConcernsaboutHypothesisTesting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 ConfidenceIntervalsforb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Confidenceintervalsforpredictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 Whatcangowrong? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9.1 Sourceandqualityofthedata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4
CONTENTS 5
3.9.2 Errorcomponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.9.3 StructuralComponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.10 InterpretingParameterEstimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 ErrorsinPredictors 55
5 GeneralizedLeastSquares 59
5.1 Thegeneralcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 WeightedLeastSquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 IterativelyReweightedLeastSquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6 TestingforLackofFit 65
6.1 s 2 known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 s 2 unknown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7 Diagnostics 72
7.1 ResidualsandLeverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2 StudentizedResiduals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.3 Anoutliertest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4 InfluentialObservations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.5 ResidualPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.6 Non-ConstantVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.7 Non-Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.8 AssessingNormality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.9 Half-normalplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.10 CorrelatedErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8 Transformation 95
8.1 Transformingtheresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2 Transformingthepredictors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.2.1 BrokenStickRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.2.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.3 RegressionSplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.4 ModernMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9 ScaleChanges,PrincipalComponentsandCollinearity 106
9.1 ChangesofScale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.2 PrincipalComponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.3 PartialLeastSquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.4 Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.5 RidgeRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
10 VariableSelection 124
10.1 HierarchicalModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
10.2 StepwiseProcedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.2.1 ForwardSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.2.2 StepwiseRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.3 Criterion-basedprocedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
CONTENTS 6
10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11 StatisticalStrategyandModelUncertainty 134
11.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
12 ChicagoInsuranceRedlining-acompleteexample 138
13 RobustandResistantRegression 150
14 MissingData 156
15 AnalysisofCovariance 160
15.1 Atwo-levelexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
15.2 Codingqualitative predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
15.3 AThree-levelexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
16 ANOVA 168
16.1 One-WayAnova. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
16.1.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
16.1.2 Estimationandtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
16.1.3 Anexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
16.1.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
16.1.5 MultipleComparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
16.1.6 Contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
16.1.7 Scheffe´’stheoremformultiplecomparisons . . . . . . . . . . . . . . . . . . . . . . 177
16.1.8 Testingforhomogeneityofvariance . . . . . . . . . . . . . . . . . . . . . . . . . . 179
16.2 Two-WayAnova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
16.2.1 Oneobservationpercell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.2.2 Morethanoneobservationpercell . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.2.3 Interpretingtheinteractioneffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.2.4 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
16.3 Blockingdesigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
16.3.1 RandomizedBlockdesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
16.3.2 RelativeadvantageofRCBDoverCRD . . . . . . . . . . . . . . . . . . . . . . . . 190
16.4 LatinSquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
16.5 BalancedIncompleteBlockdesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
16.6 Factorialexperiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A RecommendedBooks 204
A.1 BooksonR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.2 BooksonRegressionandAnova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
B R functionsanddata 205
CONTENTS 7
C QuickintroductiontoR 207
C.1 Readingthedatain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
C.2 NumericalSummaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
C.3 GraphicalSummaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.4 Selectingsubsetsofthedata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
C.5 LearningmoreaboutR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Chapter 1
Introduction
1.1 Before you start
Statistics starts with a problem, continues with the collection of data, proceeds with the data analysis and
finishes with conclusions. It is a common mistake of inexperienced Statisticians to plunge into a complex
analysiswithoutpayingattentiontowhattheobjectivesareorevenwhetherthedataareappropriateforthe
proposedanalysis. Lookbeforeyouleap!
1.1.1 Formulation
The formulationof a problem is often more essential than its solutionwhich may be merelya
matterofmathematicalorexperimentalskill. AlbertEinstein
Toformulatetheproblemcorrectly, youmust
1. Understand the physical background. Statisticians often work in collaboration with others and need
tounderstandsomethingaboutthesubjectarea. Regardthisasanopportunitytolearnsomethingnew
ratherthanachore.
2. Understandtheobjective. Again,oftenyouwillbeworkingwithacollaboratorwhomaynotbeclear
about what the objectives are. Beware of “fishing expeditions” - if you look hard enough, you’ll
almostalwaysfindsomethingbutthatsomethingmayjustbeacoincidence.
3. Make sure you know what the client wants. Sometimes Statisticians perform an analysis far more
complicated than the client really needed. You may find that simpledescriptive statisticsare all that
areneeded.
4. Put the problem into statistical terms. This is a challenging step and where irreparable errors are
sometimesmade. Oncetheproblemistranslatedintothelanguageof Statistics,thesolutionisoften
routine. Difficulties with this step explain why Artificial Intelligence techniques have yet to make
muchimpactinapplicationtoStatistics. Definingtheproblemishardtoprogram.
That a statistical method can read in and process the data is not enough. The results may be totally
meaningless.
8
1.1. BEFOREYOUSTART 9
1.1.2 DataCollection
It’simportanttounderstandhowthedatawascollected.
Are the data observational or experimental? Are the data a sample of convenience or were they
(cid:0)
obtained via a designed sample survey. How the data were collected has a crucial impact on what
conclusionscanbemade.
Istherenon-response? Thedatayoudon’tseemaybejustasimportantasthedatayoudosee.
(cid:0)
Aretheremissingvalues? Thisisacommonproblemthatistroublesomeandtimeconsumingtodeal
(cid:0)
with.
Howarethedatacoded? Inparticular, howarethequalitative variablesrepresented.
(cid:0)
Whatarethe unitsof measurement? Sometimesdatais collectedor representedwithfar moredigits
(cid:0)
thanarenecessary. Considerroundingifthiswillhelpwiththeinterpretationorstoragecosts.
Bewareofdataentryerrors. Thisproblemisalltoocommon—almostacertaintyinanyrealdataset
(cid:0)
ofatleastmoderatesize. Performsomedatasanitychecks.
1.1.3 InitialDataAnalysis
Thisisacriticalstepthatshouldalwaysbeperformed. Itlookssimplebutitisvital.
Numericalsummaries-means,sds,five-numbersummaries,correlations.
(cid:0)
Graphicalsummaries
(cid:0)
– Onevariable-Boxplots,histogramsetc.
– Twovariables-scatterplots.
– Manyvariables-interactive graphics.
Lookforoutliers,data-entryerrorsandskewedorunusualdistributions. Arethedatadistributed asyou
expect?
Getting data into a form suitable for analysis by cleaning out mistakes and aberrations is often time
consuming. Itoftentakesmoretimethanthedataanalysisitself. Inthiscourse,allthedatawillbereadyto
analyzebutyoushouldrealizethatinpracticethisisrarelythecase.
Let’s look at an example. The National Institute of Diabetes and Digestive and Kidney Diseases
conducted a study on 768 adult female Pima Indians living near Phoenix. The following variables were
recorded: Number of times pregnant, Plasma glucose concentration a 2 hours in an oral glucose tolerance
test,Diastolicbloodpressure(mmHg),Tricepsskinfoldthickness(mm),2-Hourseruminsulin(muU/ml),
Bodymassindex (weightin kg/(heightinm2)),Diabetespedigreefunction,Age(years)andatestwhether
thepatientshowssignsofdiabetes(coded0ifnegative,1ifpositive). ThedatamaybeobtainedfromUCI
Repositoryofmachinelearningdatabasesathttp://www.ics.uci.edu/˜mlearn/MLRepository.html.
Ofcourse,beforedoinganythingelse,oneshouldfindoutwhatthepurposeofthestudywasandmore
abouthowthedatawascollected. Butlet’sskipaheadtoalookatthedata: