Simulating Data with SAS

362 Pages·2013·7.851 MB·English

by  Rick Wicklin

The correct bibliographic citation for this manual is as follows: Wicklin, Rick. 2013. Simulating Data with SAS®.
Cary, NC: SAS Institute Inc.
Simulating Data with SAS®
Copyright © 2013, SAS Institute Inc., Cary, NC, USA
ISBN 978-1-61290-622-5 (electronic book)
ISBN 978-1-61290-332-3
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Contents

Acknowledgments v

I Essentials of Simulating Data 1
Chapter 1. Introduction to Simulation ................................................................................................... 3
Chapter 2. Simulating Data from Common Univariate Distributions ................................................. 11
Chapter 3. Preliminary and Background Information ......................................................................... 29

II Basic Simulation Techniques 49
Chapter 4. Simulating Data to Estimate Sampling Distributions ......................................................... 51
Chapter 5. Using Simulation to Evaluate Statistical Techniques ......................................................... 73
Chapter 6. Strategies for Efficient and Effective Simulation ............................................................... 93

III Advanced Simulation Techniques 107
Chapter 7. Advanced Simulation of Univariate Data ......................................................................... 109
Chapter 8. Simulating Data from Basic Multivariate Distributions ................................................... 129
Chapter 9. Advanced Simulation of Multivariate Data ...................................................................... 153
Chapter 10. Building Correlation and Covariance Matrices ................................................................. 175

IV Applications of Simulation in Statistical Modeling 195
Chapter 11. Simulating Data for Basic Regression Models ................................................................. 197
Chapter 12. Simulating Data for Advanced Regression Models .......................................................... 225
Chapter 13. Simulating Data from Times Series Models ..................................................................... 251
Chapter 14. Simulating Data from Spatial Models ............................................................................... 263
Chapter 15. Resampling and Bootstrap Methods ................................................................................. 281
Chapter 16. Moment Matching and the Moment-Ratio Diagram ......................................................... 297

V Appendix 323
Appendix A. A SAS/IML Primer .......................................................................................................... 325

Index 339

Acknowledgments
IwouldliketothankRobertRodriguezandPhilGibbsforpointingouttheneedforabookabout
simulatingdatainSAS.“Simulation”isavasttopic,andearlydiscussionswiththemhelpedmeto
whittledownthepossibletopics. BobandMauraStokesprovidedmanyopportunitiesformeto
developthismaterialbyinvitingmetopresentpapersandworkshopsatconferences. Mysupervisors
atSASfullysupportedmeasIpreparedforandparticipatedintheseconferences.
IthankthemanySASuserswhoencouragedmetowriteabookthatemphasizesthepracticalside
of simulation. Discussions with SAS users helped me to determine what topics are of practical
importancetostatisticiansandanalystsinbusinessandindustry.
I thank my colleagues at SAS from whom I have learned many statistical and programming
techniques. Special thanks to Randy Tobias, who always provides sound advice and statistical
wisdomformynaivequestions. ThanksalsotoTimArnoldandWarrenKuhfeldfortheir‘saslatex’
documentation system that automatically produced all tables and graphs in this book from the
programsthatappearinthetext.
I thank my editor, John West, and the other employees at SAS Press for their work producing
andpromotingthebook. Ithanktworeviewers, ClementStoneandBobPearson, whoprovided
insightfulcommentsaboutthebook’scontentandorganization.
Thankstoseveralcolleaguesandfriendswhoreadandcommentedonearlydraftsofthisbook. This
includesthefollowingindividuals: RobAgnelli,JasonBrinkley,TonyaChapman,SteveDenham,
BruceElsheimer,BetsyEnstrom,PhilGibbs,EmilyLada,PushpalMukhopadhyay,BillRaynor,
RobertRodriguez,JimSeabolt,UdoSglavo,YingSo,JillTao,RandyTobias,IanWakeling,Donna
Watts,andMinZhu.
Finally,Iwouldliketothankmywife,Nancy,forherconstantsupport,andmyparentsforinstilling
inmealoveoflearning.

Part I
Essentials of Simulating Data

Chapter 1
Introduction to Simulation
Contents
1.1 OverviewofSimulationofData . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 TheGoalofThisBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 WhoShouldReadThisBook? . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 TheSAS/IMLLanguage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 ComparingtheDATAStepandSAS/IMLLanguage . . . . . . . . . . . . . 6
1.6 OverviewofThisBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 ObtainingtheProgramsUsedinThisBook . . . . . . . . . . . . . . . . . . 8
1.8 SpecializedSimulationToolsinSASSoftware . . . . . . . . . . . . . . . . 8
1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Overview of Simulation of Data
There are many kinds of simulation. Climate scientists use simulation to model the interactions
between the earth’s atmosphere, oceans, and land. Astrophysicists use simulation to model the
evolutionofgalaxies. Biologistsusesimulationtomodelthespreadofepidemicsandtheeffectsof
vaccinationprograms. Engineersusesimulationtostudythesafetyandfuelefficiencyofautomobile
andairplanedesigns. Inthesesimulationsofphysicalsystems,scientistsmodelrealityandusea
computertostudythemodelundervariousconditions.
Statisticiansalsobuildmodels. Forexample,asimplemodelofhumanheightmightassumethat
heightisnormallydistributedinthepopulation. Thisisausefulmodel,butitturnsoutthathuman
heightsarenotactuallynormallydistributed(Schilling,Watkins,andWatkins2002). Evenifyou
restrict the data to a single gender, there are more very tall and very short people than would be
expectedfromanormaldistributionofheights.
Ifasetofdataisonlyapproximatelynormal,whatdoesthatmeanforstatisticalteststhatassume
normality? Ifyoucomputeat testtocomparethemeansoftwogroups—atestthatassumesthatthe
twounderlyingpopulationsarenormallydistributed—howsensitiveisyourconclusiontotheactual
populationdistribution? Ifthepopulationsareslightlynonnormal,doesthatinvalidatethet test? Or
aretheresultsfairlyrobusttodeviationsfromnormality?
Onewaytoanswerthesequestionsistosimulatedatafromnonnormalpopulations. Ifyouconstruct
adistributionalmodel,thenyoucangeneraterandomsamplesfromthemodelandexaminehowthe
t testperformsonthesimulateddata. Simulationgivesyoucompletecontroloverthecharacteristics
ofthepopulationmodelfromwhichthe(simulated)dataaredrawn.
Simulatingdataisalsousefulforcomparingtwodifferentstatisticaltechniques. PerhapsTechniqueA
performs better on skewed data than Technique B. Perhaps Technique B is more robust to the
4 Chapter1: IntroductiontoSimulation
presenceofoutliers. Toapracticingstatistician,thiskindofinformationisquitevaluable. AsGentle
(2009,p. xi)says,“Learningtosimulatedatawithgivencharacteristicsmeansthatoneunderstands
thosecharacteristics. Applyingstatisticalmethodstosimulateddata...helpsusbettertounderstand
thosemethodsandtheprinciplesunderlyingthem.”
This book is about simulating data in SAS software. This book demonstrates how to generate
observationsfrompopulationsthathavespecifiedstatisticalcharacteristics. Inthisbook,thephrases
“simulating data,” “generating a random sample,” and “sampling from a distribution” are used
interchangeably.
A large portion of this book is about learning how to construct statistical models (distributions)
that have certain statistical properties. Skewed distributions, fat-tailed distributions, bimodal
distributions—theseareafewexamplesofmodelsthatyoucanconstructbyusingthetechniques
inthisbook. Thisbookalsopresentstechniquesforgeneratingdatafromcorrelatedmultivariate
distributions. EachtechniqueisaccompaniedbyaSASprogram.
Althoughthisbookusesstatistics,itsaudienceisnotlimitedtostatisticians. Itisahow-tobookfor
statisticalprogrammerswhouseSASsoftwareandwhowanttosimulatedataefficiently.
Inshort,thisbookdescribeshowtowriteSASprogramsthatsimulatedatawithawiderangeof
characteristics,anddescribeshowtousethatdatatounderstandtheperformanceandapplicabilityof
statisticaltechniques.
1.2 The Goal of This Book
Thegoalofthisbookistoprovidetips,techniques,andexamplesforefficientlysimulatingdatain
SASsoftware.
Datasimulationisafundamentaltechniqueinstatisticalprogrammingandresearch. Toevaluate
statisticalmethods,youoftenneedtocreatedatawithknownproperties,bothrandomandnonrandom.
Thisbookcontainsmorethanonehundredannotatedprogramsthatsimulatedatawithspecified
characteristics. Youcanusesimulateddatatoestimatetheprobabilityofanevent,toestimatethe
samplingdistributionofastatistic,toestimatethecoverageprobabilitiesofconfidenceintervals,and
toevaluatetherobustnessofastatisticaltest.
Someprogramsarepresentedintwoforms,firstbyusingtheDATAstepandthenagainbyusingthe
SAS/IMLlanguage. Bypresentingthesamealgorithmintwodifferentways,thenoviceSAS/IML
programmercanlearnhowtowritesimulationsintheSAS/IMLlanguage. Laterchaptersthatdiscuss
multivariatesimulationusetheSAS/IMLlanguageheavily. Ifyouareseriousaboutsimulation,you
shouldinvestthetimetolearnhowtousetheSAS/IMLlanguageefficiently.
Although this book covers many standard examples of data distributions, there are many other
examples that are not covered. However, many techniques that are described in this book are
generallyapplicable. Forexample,Section7.5describeshowtogeneraterandomsamplesfroma
mixtureofnormaldistributions. Thesametechniquecanbeusedtosimulatedatafromamixtureof
otherdistributions.
Thebookalsoincludesmorethan100exercises. Manyexercisesextendtheresultsofasectionto
otherdistributionsortorelatedproblems. Theexercisesprovidepracticalprogrammingproblems
thatencourageyoutomasterthematerialbeforemovingontothenextsection. Mostexerciseswill