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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.