1. finishing up the bicycles and commute times and sids ...frederic/13/f16/day10.pdf1. finishing up...

Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.

1.FinishingupthebicyclesandcommutetimesandSIDSandBacktoSleepexamples.2.Comparing2means,breastfeedingandintelligenceexample.3.Paireddataandstudyingwithmusicexample.4.Simulationapproachwithpaireddataandbaseballexample.Readch7.

NOLECTURETHUNOV3!ReviewforthemidtermwillbeinclassNov1.RecallthereisalsonolectureorofficehourTueNov8.BringaPENCILandCALCULATORandanybooksornotesyouwanttothemidtermandfinal.HW3isdueTueNov1.4.CE.10,5.3.28,6.1.17,and6.3.14.In5.3.28d,usethetheory-basedformula.Youdonotneedtouseanapplet.Themidtermwillbeonch1-7.http://www.stat.ucla.edu/~frederic/13/F16.

1

BicyclingtoWork• WecannotgeneralizebeyondGrovesandhistwobikes.

• Alimitationisthatthisstudyisnotdouble-blind• Theresearcherandthesubject(whichhappenedtobethesamepersonhere)werenotblindtowhichtreatmentwasbeingused.

• Dr.Grovesknewwhichbikehewasriding,andthismighthaveaffectedhisstateofmindorhischoiceswhileriding.How?

• SIDS.Davies(1985)foundthatinHongKong,wherethecustomwasforchildrentosleepontheirbacks,theratesofSIDSwereverylow.

• 1992:BacktoSleepbeganintheUnitedStates.

BreastfeedingandIntelligenceExample6.3

BreastfeedingandIntelligence

• A1999studyin Pediatricsexaminedifchildrenwhowerebreastfedduringinfancydifferedfrombottle-fed.

• 323childrenrecruitedatbirthin1980-81fromfourWesternMichiganhospitals.

• Researchersdeemedtheparticipantsrepresentativeofthecommunityinsocialclass,maternaleducation,age,maritalstatus,andsexofinfant.

• Childrenwerefollowed-upatage4andassessedusingtheGeneral CognitiveIndex(GCI)• Ameasureofthechild’sintellectualfunctioning

• Researcherssurveyedparentsandrecordedifthechildhadbeenbreastfedduringinfancy.

BreastfeedingandIntelligence

• Explanatoryandresponsevariables.• Explanatoryvariable:Whetherthebabywasbreastfed.(Categorical)

• Responsevariable: Baby’sGCIatage4.(Quantitative)

• Isthisanexperimentoranobservationalstudy?• Cancause-and-effectconclusionsbedrawninthisstudy?

BreastfeedingandIntelligence

• Nullhypothesis: ThereisnorelationshipbetweenbreastfeedingduringinfancyandGCIatage4.

• Alternativehypothesis: ThereisarelationshipbetweenbreastfeedingduringinfancyandGCIatage4.

BreastfeedingandIntelligence

• µbreastfed =AverageGCIatage4forbreastfedchildren• µnot =AverageGCIatage4forchildrennotbreastfed

• H0: µbreastfed =µnot• Ha: µbreastfed ≠µnot

BreastfeedingandIntelligenceGroup Samplesize, n Samplemean Sample SDBreastfed 237 105.3 14.5NotBF 85 100.9 14.0

BreastfeedingandIntelligence

Thedifferenceinmeanswas4.4.• IfbreastfeedingisnotrelatedtoGCIatage4:

• Isitpossible adifferencethislargecouldhappenbychancealone?Yes

• Isitplausible(believable,fairlylikely)adifferencethislargecouldhappenbychancealone?• Wecaninvestigatethiswithsimulations.• Alternatively,wecanusetheory-basedmethods.

T-statistic• Tousetheory-basedmethodsinthemultiplemeansapplet,thet-statisticisused.

• Itissimplythenumberofstandarddeviationsourstatisticisaboveorbelowthemeanunderthenullhypothesis.

• 𝑡 = #$%$&#$&'()*+,$)-#&.-/1%23-45 = 6̅8(6̅9(:

;89

<8=;9

9

<9

�

• Here,t= ?:@.B(?::.C

(8E.F9

9GH =� 8E.I9

JF )

= 2.46.

• p-value~1.4or1.5%.[2*(1-pnorm(2.46))],orusept.

BreastfeedingandIntelligence

Meaningofthep-value:• IfbreastfeedingwerenotrelatedtoGCIatage4,thentheprobabilityofobservingadifferenceof4.4ormoreor-4.4orlessjustbychanceisabout1.4%.

• A95%CIcanalsobeobtainedusingthet-

distribution.TheSEis (?O.@9

PBQ+

� ?O.:9

S@) =1.79.

SothemarginoferrorismultiplierxSE.

BreastfeedingandIntelligence

• TheSEis (?O.@9

PBQ+

� ?O.:9

S@) =1.79.Themarginof

errorismultiplierxSE.• Themultipliershouldtechnicallybeobtainedusingthetdistribution,butforlargesamplesizesyougetalmostthesamemultiplierwithtandnormal.Use1.96fora95%CItoget4.40+/- 1.96x1.79=4.40+/- 3.51=(0.89,7.91).

• Thebookuses2insteadof1.96,andtheappletuses1.9756fromthet-distribution.Justuse1.96forthisclass.

BreastfeedingandIntelligence

• Wehavestrongevidenceagainstthenullhypothesisandcanconcludetheassociationbetweenbreastfeedingandintelligence hereisstatisticallysignificant.

• BreastfedbabieshavestatisticallysignificantlyhigheraverageGCIscoresatage4.

• Wecanseethisinboththesmallp-value(0.015)andtheconfidenceintervalthatsaysthemeanGCIforbreastfedbabiesis0.89to7.91pointshigherthanthatfornon-breastfedbabies.

BreastfeedingandIntelligence

• Towhatlargerpopulation(s)wouldyoubecomfortablegeneralizingtheseresults?• TheparticipantswereallchildrenborninWesternMichigan.

• Thislimitsthepopulationtowhomwecangeneralizetheseresults.

BreastfeedingandIntelligence

• CanyouconcludethatbreastfeedingimprovesaverageGCIatage4?• No.Thestudywasnotarandomizedexperiment.• Wecannotconcludeacause-and-effectrelationship.

• TheremightbealternativeexplanationsforthesignificantdifferenceinaverageGCIvalues.

• Whatmightsomeconfoundingfactorsbe?

BreastfeedingandIntelligence

• CanyouconcludethatbreastfeedingimprovesaverageGCIatage4?• No.Thestudywasnotarandomizedexperiment.• Wecannotconcludeacause-and-effectrelationship.

• TheremightbealternativeexplanationsforthesignificantdifferenceinaverageGCIvalues.• Maybebettereducatedmothersaremorelikelytobreastfeedtheirchildren

• Maybemothersthatbreastfeedspendmoretimewiththeirchildrenandinteractwiththemmore.

• Somemotherswhodonotbreastfeedarelesshealthyortheirbabieshaveweakerappetitesandthismightslowdowndevelopmentingeneral.

BreastfeedingandIntelligence

• Couldyoudesignastudythatallowsdrawingacause-and-effectconclusion?• Wewouldhavetorunanexperimentusingrandomassignmenttodeterminewhichmothersbreastfeedandwhichwouldnot.(Itwouldbeimpossibletodouble-blind.)

• Randomassignmentroughlybalancesoutallothervariables.

• Isitfeasible/ethicaltoconductsuchastudy?

StrengthofEvidence• Wealreadyknow:

• Assamplesizeincreases,thestrengthofevidenceincreases.

• Justaswithproportions,asthesamplemeansmovefartherapart,thestrengthofevidenceincreases.

MoreStrengthofEvidence• Ifthemeansarethesamedistanceapart,butthestandarddeviationschange,thenthestrengthofevidencechangestoo.

• Whichgivesstrongerevidenceagainstthenull?

MoreStrengthofEvidence• Ifthemeansarethesamedistanceapart,butthestandarddeviationschange,thenthestrengthofevidencechangestoo.

• Whichgivesstrongerevidenceagainstthenull?

• Smaller SDs lead to stronger evidence against the null.

EffectsonWidthofConfidenceIntervals

• Justasbefore:• Assamplesizeincreases,confidenceintervalwidthstendtodecrease.

• Asconfidencelevelincreases,confidenceintervalwidthsincrease.

• Thedifferenceinmeanswillnotaffectthewidth(marginoferror)butwillaffectthecenteroftheCI.

• Aswesawwithasinglemean,astheSDsofthesamplesincrease,thewidthoftheconfidenceintervalwillincrease.

PairedData.Chapter7

Introduction• Thepaireddatasetsinthischapterhaveonepairofquantitativeresponsevaluesforeachobs.unit.

• Thisallowsforacomparisonwheretheotherpossibleconfoundersareassimilaraspossiblebetweenthetwogroups.

• Paireddatastudiesremoveindividualvariabilitybylookingatthedifferencescoreforeachsubject.

• Reducingvariabilityindataimprovesinferences:• Narrowerconfidenceintervals.• Smallerp-valueswhenthenullhypothesisisfalse.• Lessinfluencefromconfoundingfactors.

3.Paireddataandstudyingwithmusicexample.Example7.1

StudyingwithMusic• Manystudentsstudywhilelisteningtomusic.• Doesithurttheirabilitytofocus?• In“CheckingItOut:Doesmusicinterferewithstudying?”StanfordProfCliffordNass claimsthehumanbrainlistenstosonglyricswiththesamepartthatdoeswordprocessing.

• Instrumentalmusicis,forthemostpart,processedontheothersideofthebrain,andNassclaimsthatlisteningtoinstrumentalmusichasvirtuallynointerferenceonreadingtext.

StudyingwithMusicConsidertheexperimentaldesigns:ExperimentA — Randomassignmentto2groups• 27studentswererandomlyassignedto1of2groups:

• Onegrouplistenstomusicwithlyrics.• Onegrouplistenstomusicwithoutlyrics.

• Studentsplayamemorizationgamewhilelisteningtotheparticularmusicthattheywereassigned.

StudyingwithMusicExperimentB— Paireddesignusingrepeatedmeasures• Allstudentsplaythememorizationgametwice:

• Oncewhilelisteningtomusicwithlyrics• Oncewhilelisteningtomusicwithoutlyrics.

ExperimentC— Paireddesignusingmatching• Sometimesrepeatingsomethingisimpossible(liketestingasurgicalprocedure)butwecanstillpair.• Testeachstudentonmemorization.• Matchstudentsupwithsimilarscoresandrandomly:

• Haveoneplaythegamewhilelisteningtomusicwithlyricsandtheotherwhilelisteningtomusicwithoutlyrics.

StudyingwithMusicWewillfocusontherepeatedmeasurestypeofpairing.• Whatifeveryonecouldrememberexactly2morewordswhentheylistenedtoasongwithoutlyrics?

• UsingExperimentA,therecouldbealotofoverlapbetweenthetwosetsofscoresanditwouldbedifficulttodetectadifference,asshownhere.

Without Lyrics

With Lyrics

StudyingwithMusic• Variabilityinpeople’smemorizationabilitiesmaymakeitdifficulttoseedifferencesbetweenthesongsinExperimentA.

• Thepaireddesignfocusesonthedifference inthenumberofwordsmemorized,insteadofthenumberofwordsmemorized.

• Bylookingatthisdifference,thevariabilityingeneralmemorizationabilityistakenaway.

StudyingwithMusic• InExperimentB,therewouldbenovariabilityatallinourhypotheticalexample.

• Whilethereissubstantialvariabilityinthenumberofwordsmemorizedbetweenstudents,therewouldbenovariabilityinthedifferenceinthenumberofwordsmemorized.Allvalueswouldbeexactly2.

• Hencewewouldhaveextremelystrongevidenceofadifferenceinabilitytomemorizewordsbetweenthetwotypesofmusic.

PairingandRandomAssignment

• Pairingoftenincreasespower,andmakesiteasiertodetectstatisticalsignificance.

• Canwemakecause-and-effectconclusionsinpaireddesign?

• Shouldwestillhaverandomassignment?

PairingandRandomAssignment

Inourmemorizingwithorwithoutlyricsexample:• Ifweseesignificantimprovementinperformance,isitattributabletothetypeofsong?

• Whataboutexperience?Couldthathavemadethedifference?

• Whatisabetterdesign?• Randomlyassigneachpersontowhichsongtheyhearfirst:withlyricsfirst,orwithout.

• Thiscancelsoutan“experience”effect

ParingandObservationalStudies

Youcanoftendomatchedpairsinobservationalstudies,whenyouknowthepotentialconfounderaheadoftime.Ifyouarestudyingwhethertheportacaval shuntdecreasestheriskofheartattack,youcouldmatcheachpatientgettingtheshuntwithapatientofsimilarhealthnotgettingtheshunt.Ifyouarestudyingwhetherlefthandedness causesdeath,andyouwanttoaccountforageinthepopulation,youcouldmatcheachleftiewitharightie ofthesameage,andcomparetheiragesatdeath.

4.Simulation-BasedApproachforAnalyzingPairedData,androundingfirstbaseexample.Section7.2

RoundingFirstBaseExample7.2

RoundingFirstBase• Imagineyou’vehitalinedriveandaretryingtoreachsecondbase.

• Doesthepaththatyoutaketoroundfirstbasemakemuchofadifference?• Narrowangle• Wideangle

Narrow

Wide

RoundingFirstBase

• Woodward(1970)investigatedthesebaserunningstrategies.

• Hetimed22differentrunnersfromaspot35feetpasthometoaspot15feetbeforesecond.

• Eachrunnerusedeachstrategy(paireddesign),witharestinbetween.

• Heusedrandomassignmenttodecidewhichpatheachrunnershoulddofirst.

• Thispaireddesigncontrolsfortherunner-to-runnervariability.

FirstBase• Whataretheobservationalunitsinthisstudy?

• Therunners(22total)• Whatvariablesarerecorded?Whataretheirtypesandroles?• Explanatoryvariable:baserunningmethod:wideornarrowangle(categorical)

• Responsevariable:timefromhomeplatetosecondbase(quantitative)

• Isthisanobservationalstudyoranexperiment?• Randomizedexperiment.

Theresults

TheStatistics

• Thereisalotofoverlapinthedistributionsandsubstantialvariability.

• Itisdifficulttodetectadifferencebetweenthemethodswhentheseissomuchvariation.

•

Mean SDNarrow 5.534 0.260Wide 5.459 0.273

RoundingFirstBase

• However,thesedataareclearlypaired.• Thepairedresponsevariableistimedifferenceinrunningbetweenthetwomethodsandwecanusethisinanalyzingthedata.

TheDifferencesinTimes

TheDifferencesinTimes

• Meandifferenceis�̅�d=0.075seconds• StandarddeviationofthedifferencesisSDd =0.0883sec.

• Thisstandarddeviationof0.0883issmallerthantheoriginalstandarddeviationsoftherunningtimes,whichwere0.260and0.273.