stat 13, intro. to statistical methods for the life and ...frederic/13/sum17/day06.pdf · stat 13,...
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Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.
1. Collect hw2. 2. More problems with studies, coverage, adherer bias and clofibrate example. 3. More about confounding factors. 4. Confounding and lefties example. 5. Comparing two proportions using numerical and visual summaries,
good or bad year example. 6. Comparing 2 proportions with CIs + testing using simulation, dolphin example.7. Comparing 2 props. with theory-based testing, smoking and gender example. 8. Five number summary, IQR, and geysers. 9. Comparing two means with simulations and bicycling to work example.
Read ch5 and 6. The midterm will be on ch 1-6. http://www.stat.ucla.edu/~frederic/13/sum17 .Bring a PENCIL and CALCULATOR and any books or notes you want to the midterm and final.
HW3 4.CE.10, 5.3.28, 6.1.17, and 6.3.14. 4.CE.10 starts out "Studies have shown that children in the U.S.
who have been spanked have a significantly lower IQ score on average...."5.3.28 starts out "Recall the data from the Physicians' Health Study: Of the
11,034 physicians who took the placebo ...."6.1.17 starts out "The graph below displays the distribution of word lengths ...."
6.3.14 starts out "In an article titled 'Unilateral Nostril Breathing Influences Lateralized Cognitive Performance' that appeared ...."
1
1.HandinHW2!2.Moreproblemswithstudies,coverage,adhererbiasandClofibrate.Surveysareobservational.• Coverageisacommonissue.Coverageistheextenttowhich
thepeopleyousampledfromrepresenttheoverallpopulation.Asurveyatafancyresearchhospitalinawealthyneighborhoodmayyieldpatiencewithhigherincomes,highereducation,etc.
• Non-responsebiasisanothercommonproblem.Poorcoveragemeansthepeoplegettingthesurveydonotrepresentthegeneralpopulation.Non-responsebiasmeansthatoutofthepeopleyougavethesurveyto,thepeopleactuallyfillingitoutandsubmittingitaredifferentfromthepeoplewhodidnot.
• Sameexactissuesinwebsurveys.
Moreproblemswithstudies,andClofibrate example.Non-responsebiasissimilartoadhererbias,inexperiments.Adrugcalledclofibrate wastestedon3,892middle-agedmenwithhearttrouble.Itwassupposedtopreventheartattacks.1,103assignedatrandomtotakeclofibrate,2,789toplacebo(lactose)group.Subjectswerefollowedfor5years.Isthisanexperimentoranobservationalstudy?
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
Moreproblemswithstudies,andClofibrate example.Non-responsebiasissimilartoadhererbias,inexperiments.Adrugcalledclofibrate wastestedon3,892middle-agedmenwithhearttrouble.Itwassupposedtopreventheartattacks.1,103assignedatrandomtotakeclofibrate,2,789toplacebo(lactose)group.Subjectswerefollowedfor5years.Isthisanexperimentoranobservationalstudy?
Itisanexperiment.DoesClofibrate work?Clofibrate patientswhodiedduringfollowup
adherers 15%non-adherers 25%total 20%
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Thosewhotookclofibrate didmuchbetterthanthosewhodidn'tkeeptakingclofibrate.Doesthismeanclofibrate works?
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Thosewhoadheredtoplaceboalsodidmuchbetterthanthosewhostoppedadhering.
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Allinalltherewaslittledifferencebetweenthetwogroups.
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Adherersdidbetterthannon-adherers,notbecauseofclofibrate,butbecausetheywerehealthieringeneral.Why?
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Adherersdidbetterthannon-adherers,notbecauseofclofibrate,butbecausetheywerehealthieringeneral.Why?• adherersarethetypetoengageinhealthierbehavior.• sickpatientsarelesslikelytoadhere.
3.Moreaboutconfoundingfactors.• Byaconfoundingfactor,wemeananalternativeexplanation
thatcouldexplaintheapparentrelationshipbetweenthetwovariables,eveniftheyarenotcausallyrelated.Typicallythisisdonebyfindinganotherdifferencebetweenthetreatmentandcontrolgroup.Forinstance,differentstudieshaveexaminedsmokersandnon-smokersandhavefoundthatsmokershavehigherratesoflivercancer.Oneexplanationwouldbethatsmokingcauseslivercancer.Butisthereanyother,alternativeexplanation?
• Onealternativewouldbethatthesmokerstendtodrinkmorealcohol,anditisthealcohol,notthesmoking,thatcauseslivercancer.
3.Moreaboutconfoundingfactors.• Anotherplausibleexplanationisthatthesmokersareprobably
olderonaveragethanthenon-smokers,andolderpeoplearemoreatriskforallsortsofcancerthanyoungerpeople.
• Anothermightbethatsmokersengageinotherunhealthyactivitiesmorethannon-smokers.
• Notethatifonesaidthat“smokingmakesyouwanttodrinkalcoholwhichcauseslivercancer,”thatwouldnotbeavalidconfoundingfactor,sinceinthatexplanation,smokingeffectiveiscausallyrelatedtolivercancerrisk.
3.Moreaboutconfoundingfactors.• Aconfoundingfactormustbeplausiblylinkedtoboththe
explanatoryandresponsevariables.Soforinstancesaying“perhapsahigherproportionofthesmokersaremen”wouldnotbeaveryconvincingconfoundingfactor,unlessyouhavesomereasontothinkgenderisstronglylinkedtolivercancer.
• Anotherexample:left-handednessandageatdeath.PsychologistsDianeHalpernandStanleyCoren lookedat1,000deathrecordsofthosewhodiedinSouthernCaliforniainthelate1980s andearly1990sandcontactedrelativestoseeifthedeceasedwererighthanded orlefthanded.Theyfoundthattheaverageagesatdeathofthelefthanded was66,andfortherighthanded itwas75.Theirresultswerepublishedinprestigiousscientificjournals,NatureandtheNewEnglandJournalofMedicine.
Moreaboutconfoundingfactors.Allsortsofcausalconclusionsweremadeabouthowthisshowsthatthestressofbeinglefthanded inourrighthanded worldleadstoprematuredeath.
Moreaboutconfoundingfactors.• Isthisanobservationalstudyoranexperiment?
Moreaboutconfoundingfactors.• Isthisanobservationalstudyoranexperiment?Itisanobservationalstudy.• Arethereplausibleconfoundingfactorsyoucanthinkof?
Moreaboutconfoundingfactors.• Aconfoundingfactoristheageofthetwopopulationsin
general.Leftiesinthe1980swereonaverageyoungerthanrighties.Manyoldleftieswereconvertedtorightiesatinfancy,intheearly20thcentury,butthispracticehassubsided.Thusinthe1980sand1990s,therewererelativelyfewoldleftiesbutmanyyoungleftiesintheoverallpopulation.Thisaloneexplainsthediscrepancy.
Unit2.ComparingTwoGroups
• InUnit1,welearnedthebasicprocessofstatisticalinferenceusingtestsandconfidenceintervals.Wedidallthisbyfocusingonasingleproportion.
• InUnit2,wewilltaketheseideasandextendthemtocomparingtwogroups.Wewillcomparetwoproportions,twoindependentmeans,andpaireddata.
5.Comparingtwoproportionsusingnumericalandvisualsummaries,andthegoodorbadyearexample.
Section5.1
Example5.1:PositiveandNegativePerceptions
• Considerthesetwoquestions:– Areyouhavingagoodyear?– Areyouhavingabadyear?
• Dopeopleanswereachquestioninsuchawaythatwouldindicatethesameanswer?(e.g.YesforthefirstoneandNoforthesecond.)
PositiveandNegativePerceptions
• Researchersquestioned30students(randomlygivingthemoneofthetwoquestions).
• Theythenrecordedifapositiveornegativeresponsewasgiven.
• Theywantedtoseeifthewordingofthequestioninfluencedtheanswers.
Positiveandnegativeperceptions
• Observationalunits– The30students
• Variables– Questionwording(goodyearorbadyear)– Perceptionoftheiryear(positiveornegative)
• Whichistheexplanatoryvariableandwhichistheresponse variable?
• Isthisanobservationalstudyorexperiment?
Individual TypeofQuestion
Response Individual TypeofQuestion
Response
1 GoodYear Positive 16 GoodYear Positive2 GoodYear Negative 17 BadYear Positive3 BadYear Positive 18 GoodYear Positive4 GoodYear Positive 19 GoodYear Positive5 GoodYear Negative 20 GoodYear Positive6 BadYear Positive 21 BadYear Negative7 GoodYear Positive 22 GoodYear Positive8 GoodYear Positive 23 BadYear Negative9 GoodYear Positive 24 GoodYear Positive10 BadYear Negative 25 BadYear Negative11 GoodYear Negative 26 GoodYear Positive12 BadYear Negative 27 BadYear Negative13 GoodYear Positive 28 GoodYear Positive14 BadYear Negative 29 BadYear Positive15 GoodYear Positive 30 BadYear Negative
RawDatainaSpreadsheet
Two-WayTables
• Atwo-waytableorganizesdata– Summarizestwo categoricalvariables– Alsocalledcontingencytable
• Arestudentsmorelikelytogiveapositiveresponseiftheyweregiventhegoodyearquestion?
GoodYear BadYear TotalPositiveresponse 15 4 19Negativeresponse 3 8 11Total 18 12 30
Two-WayTables
• Conditionalproportionswillhelpusbetterdetermineifthereisanassociationbetweenthequestionaskedandthetypeofresponse.
• Wecanseethatthesubjectswiththepositivequestionweremorelikely torespondpositively.
GoodYear BadYear TotalPositiveresponse 15/18 ≈0.83 4/12≈0.33 19Negativeresponse 3 8 11Total 18 12 30
SegmentedBarGraphs
• Wecanalsousesegmentedbargraphstoseethisassociation betweenthe"goodyear"questionandapositiveresponse.
Statistic
GoodYear BadYear TotalPositiveresponse 15(83%) 4(33%) 19Negativeresponse 3 8 11Total 18 12 30
� Thestatisticwewillmainlyusetosummarizethistableisthedifferenceinproportionsofpositiveresponsesis0.83− 0.33=0.50.
AnotherStatistic
GoodYear BadYear TotalPositiveresponse 15(83%) 4(33%) 19Negativeresponse 3 8 11Total 18 12 30
� Anotherstatisticthatisoftenused,calledrelativerisk,istheratiooftheproportions:0.83/0.33=2.5.
� Wecansaythatthosewhoweregiventhegoodyearquestionwere2.5timesaslikelytogiveapositiveresponse.
NoAssociation
• Fordatatoshownoassociation,theproportionsofpositiveresponsesshouldbethesameforthosegettingeachquestiontype.
• Sincetheoverallpositiveresponsewas19/30(63%),ifthereisnoassociationweshouldhave63%ofthe18thatgotthegoodyearquestionwithapositiveresponse(11.4)and63%ofthe12thatgotthebadyearquestionshouldgiveapositiveresponse(7.6).Thefollowingtableistheclosestpossible.
Good Year Bad Year TotalPositiveresponse 11(61%) 8(67%) 19Negativeresponse 7 4 11Total 18 12 30
6. Comparing two proportions with CIs and testing using simulation, dolphin example.
Section5.2
SwimmingwithDolphins
Example5.2
SwimmingwithDolphinsIsswimmingwithdolphinstherapeuticforpatientssufferingfromclinicaldepression?
• ResearchersAntonioli andReveley (2005),inBritishMedicalJournal,recruited30subjectsaged18-65withaclinicaldiagnosisofmildtomoderatedepression
• Discontinuedantidepressantsandpsychotherapy4weekspriortoandthroughouttheexperiment
• 30subjectswenttoanislandnearHonduraswheretheywererandomlyassignedtotwotreatmentgroups
SwimmingwithDolphins• Bothgroupsengagedinonehourofswimmingandsnorkeling
eachday• Onegroupswaminthepresenceofdolphinsandtheother
groupdidnot• Participantsinbothgroupshadidenticalconditionsexceptfor
thedolphins• Aftertwoweeks,eachsubjects’levelofdepressionwas
evaluated,asithadbeenatthebeginningofthestudy• Theresponsevariableiswhetherornotthesubjectachieved
substantialreductionindepression
SwimmingwithDolphins
Nullhypothesis:Dolphinsdonothelp.– Swimmingwithdolphinsisnotassociatedwithsubstantialimprovementindepression
Alternativehypothesis:Dolphinshelp.– Swimmingwithdolphinsincreases theprobabilityofsubstantialimprovementindepressionsymptoms
SwimmingwithDolphins• Theparameteristhe(long-run)differencebetweenthe
probabilityofimprovingwhenreceivingdolphintherapyandtheprob.ofimprovingwiththecontrol(𝜋dolphins - 𝜋control)
• Sowecanwriteourhypothesesas:H0:𝜋dolphins - 𝜋control=0.Ha:𝜋dolphins- 𝜋control>0.or
H0: 𝜋dolphins= 𝜋control
Ha:𝜋dolphins> 𝜋control
(Note:wearenotsayingourparametersequalanycertainnumber.)
SwimmingwithDolphins
Results:
Dolphingroup
Controlgroup
Total
Improved 10(66.7%) 3(20%) 13
DidNot Improve 5 12 17Total 15 15 30
Thedifferenceinproportionsofimproversis:𝒑$𝒅 − 𝒑$𝒄 =0.667– 0.20=0.467.
SwimmingwithDolphins
• Therearetwopossibleexplanationsforanobserveddifferenceof0.467.– Atendencytobemorelikelytoimprovewithdolphins(alternativehypothesis)
– The13subjectsweregoingtoshowimprovementwithorwithoutdolphinsandrandomchanceassignedmoreimproverstothedolphins(nullhypothesis)
SwimmingwithDolphins
• Ifthenullhypothesisistrue(noassociationbetweendolphintherapyandimprovement)wewouldhave13improversand17non-improversregardlessofthegrouptowhichtheywereassigned.
• Hencetheassignmentdoesn’tmatterandwecanjustrandomlyassignthesubjects’resultstothetwogroupstoseewhatwouldhappenunderatruenullhypothesis.
SwimmingwithDolphins• Wecansimulatethiswithcards– 13cardstorepresenttheimprovers– 17cardsrepresentthenon-improvers
• Shufflethecards– put15inonepile(dolphintherapy)– put15inanother(controlgroup)
SwimmingwithDolphins• ComputetheproportionofimproversintheDolphinTherapygroup
• ComputetheproportionofimproversintheControlgroup
• Thedifferenceinthesetwoproportionsiswhatcouldjustaswellhavehappenedundertheassumptionthereisnoassociationbetweenswimmingwithdolphinsandsubstantialimprovementindepression.
20.0%Improvers66.7%Improvers
DolphinTherapyControlNon-
improver
Improver
Improver
Improver
Improver
Improver
Improver
Improver
ImproverImprover
Improver
Improver
Improver
ImproverNon-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
40.0%Improvers 46.7%Improvers0.400– 0.467=-0.067
DifferenceinSimulatedProportions
33.3%Improvers53.3%Improvers 46.7%Improvers40.0%Improvers
Non-improver
Improver
Non-improver
Improver
Improver
Non-improver
Improver
Improver
ImproverNon-improver
Non-improver
Non-improver
Non-improver
ImproverNon-improver
Non-improver
Improver
Improver
Non-improver
Non-improver
Non-improver
Improver
ImproverImprover
Improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
0.533– 0.333=0.200
DifferenceinSimulatedProportions
DolphinTherapy Control
53.3%Improvers 33.3%Improvers40.0%Improvers46.7%Improvers
Non-improver
Improver
Non-improver
Improver
Improver
Non-improver
Improver
Improver
ImproverNon-improver
Non-improver
Non-improver
Non-improver
ImproverNon-improver
Non-improver
Improver
Improver
Non-improver
Non-improver
Non-improver
Improver
ImproverImprover
Improver
Non-improver
Non-improver
Non-improver
Non-improver
Non-improver
0.467– 0.400=0.067
DifferenceinSimulatedProportions
DolphinTherapyControl
MoreSimulations-0.067-0.333 -0.200
0.067 0.2000.333
0.467-0.200
-0.200
-0.200-0.067 -0.067
-0.067
-0.067 -0.067-0.067
0.067
0.067
0.067
0.067
0.067
0.0670.200
0.200
0.200
0.3330.333Onlyonesimulatedstatisticsoutof30wasas
largeorlargerthanourobserveddifferenceinproportionsof0.467,henceourp-valueforthisnulldistributionis1/30≈0.03.
DifferenceinSimulatedProportions
SwimmingwithDolphins• Wedid1000repetitionstodevelopanulldistribution.
SwimmingwithDolphins
• 13outof1000resultshadadifferenceof0.467orhigher(p-value=0.013).
• 0.467is(.*+,-((../0
≈ 2.52 SDabovezero.� Usingeitherthep-valueorstandardizedstatistic,wehavestrongevidenceagainstthenullandcanconcludethattheimprovementduetoswimmingwithdolphinswasstatisticallysignificant.
SwimmingwithDolphins
� A95%confidenceintervalforthedifferenceintheprobabilityusingthestandarddeviationfromthenulldistributionis0.467+ 2(0.185)=0.467+ 0.370or(0.097to0.837)
• Weare95%confidentthatwhenallowedtoswimwithdolphins,theprobabilityofimprovingisbetween0.097and0.837higherthanwhennodolphinsarepresent.
• Howdoesthisintervalbackupourconclusionfromthetestofsignificance?
SwimmingwithDolphins
• Canwesaythatthepresenceofdolphinscaused thisimprovement?– Sincethiswasarandomizedexperiment,andassumingeverythingwasidenticalbetweenthegroups,wehavestrongevidencethatdolphinswerethecause
• Canwegeneralizetoalargerpopulation?– Maybemildtomoderatelydepressed18-65yearoldpatientswillingtovolunteerforthisstudy
– Wehavenoevidencethatrandomselectionwasusedtofindthe30subjects."Outpatients,recruitedthroughannouncementsontheinternet,radio,newspapers,andhospitals."
7.Comparingtwoproportions:Theory-BasedApproach,andsmokingandgenderexample.
Section5.3
Introduction
• Justaswithasingleproportion,wecanoftenpredictresultsofasimulationusingatheory-basedapproach.
• Thetheory-basedapproachalsogivesasimplerwaytogenerateaconfidenceintervals.
• ThemainnewmathematicalfacttouseistheSEforthedifferencebetweentwoproportionsis
�̂�(1 − �̂�) .9:+ .
9<
� .
Parents’SmokingStatusandtheirBabies’Gender
Example5.3
SmokingandGender
• Howdoesparents’behavioraffectthegenderoftheirchildren?
• Fukudaetal.(2002)foundthefollowinginJapan.– Outof565birthswherebothparentssmokedmorethan
apackaday,255wereboys.Thisis45.1%boys.– Outof3602birthswherebothparentsdidnotsmoke,
1975wereboys.This54.8%boys.– Intotal,outof4170births,2230wereboys,whichis
53.5%.• Otherstudieshaveshownareducedmaletofemale
birthratiowherehighconcentrationsofotherenvironmentalchemicalsarepresent(e.g.industrialpollution,pesticides)
SmokingandGender• A segmentedbargraphand2-waytable• Let’scomparetheproportionstoseeifthedifferenceis
statisticallysignificantly.
BothSmoked Neither Smoked
Boy 255(45.1%) 1,975(54.8%)
Girl 310 1,627
Total 565 3,602
SmokingandGender
NullHypothesis:• Thereisnoassociationbetween
smokingstatusofparentsandsexofchild.
• Theprobabilityofhavingaboyisthesameforparentswhosmokeanddon’tsmoke.
• 𝜋smoking - 𝜋nonsmoking =0
SmokingandGender
AlternativeHypothesis:• Thereisanassociationbetweensmokingstatusofparentsandsexofchild.
• Theprobabilityofhavingaboyisnotthesameforparentswhosmokeanddon’tsmoke
• 𝜋smoking - 𝜋nonsmoking ≠0
SmokingandGender
• Whataretheobservationalunitsinthestudy?• Whatarethevariablesinthisstudy?• Whichvariableshouldbeconsideredtheexplanatoryvariableandwhichtheresponsevariable?
• Whatistheparameterofinterest?• Canyoudrawcause-and-effectconclusionsforthisstudy?
SmokingandGender
Usingthe3SStrategytoassesthestrength1.Statistic:• Theproportionofboysborntononsmokersminustheproportionofboysborntosmokersis0.548– 0.451=0.097.
SmokingandGender
2.Simulate:• Manyrepetitionsofshufflingthe2230boysand1937girlstothe565smokingand3602nonsmokingparents
• Calculatethedifferenceinproportionsofboysbetweenthegroupsforeachrepetition.
• Shufflingsimulatesthenullhypothesisofnoassociation
SmokingandGender3.Strengthofevidence:• Nothingasextremeas
ourobservedstatistic(≥0.097or≤−0.097)occurredin5000repetitions,
• HowmanySDsis0.097abovethemean?Z=0.097/0.023=4.22usingsimulations.Whataboutusingthetheory-basedapproach?
SmokingandGender
• Noticethenulldistributioniscenteredatzeroandisbell-shaped.
• Thiscanbeapproximatedbythenormaldistribution.
Formulas
• Thetheory-basedapproachyieldsz=4.30.
𝑧 =�̂�. − �̂�@
�̂�(1 − �̂�) 1𝑛.+ 1𝑛@
�
• Here𝑧 = .0*/-.*0.
.0B0(.-.0B0) :CDE<F
:GDG
�=4.30.
• p-valueis2*(1-pnorm(4.30))=0.00171%.
SmokingandGender
• Fukudaetal.(2002)foundthefollowinginJapan.– Outof3602birthswherebothparentsdidnotsmoke,
1975wereboys.This54.8%boys.– Outof565birthswherebothparentssmokedmorethan
apackaday,255wereboys.Thisis45.1%boys.– Intotal,outof4170births,2230wereboys,whichis
53.5%boys.
Formulas• Howdowefindthemarginoferrorforthedifferencein
proportions?
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 ⨯�̂�.(1 − �̂�.)
𝑛.+�̂�@(1 − �̂�@)
𝑛@
�
• Themultiplierisdependentupontheconfidencelevel.– 1.645for90%confidence– 1.96for95%confidence– 2.576for99%confidence
• Wecanwritetheconfidenceintervalintheform:– statistic± marginoferror.
SmokingandGender• Ourstatisticistheobservedsampledifferenceinproportions,
0.097.
• Pluggingin1.96 ⨯ RS:(.-RS:)9:
+ RS<(.-RS<)9<
� =0.044,
weget0.097± 0.044asour95%CI.• Wecouldalsowritethisintervalas(0.053,0.141).• Weare95%confidentthattheprobabilityofaboybaby
whereneitherfamilysmokesminustheprobabilityofaboybabywherebothparentssmokeisbetween0.053and0.141.
Aclarificationontheformulas• Themarginoferrorforthedifferenceinproportionsis
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 ⨯ SE,whereSE = RS:(.-RS:)9:
+ RS<(.-RS<)9<
�
Intesting,thenullhypothesisisnodifferencebetweenthetwogroups,soweusedtheSE
�̂�(1 − �̂�)𝑛.
+�̂�(1 − �̂�)
𝑛@
�
where�̂� istheproportioninbothgroupscombined.Butin
CIs,weusetheformula RS:(.-RS:)9:
+ RS<(.-RS<)9<
� becausewe
arenotassuming�̂�. =�̂�@withCIs.
SmokingandGender
• Howwouldtheintervalchangeiftheconfidencelevelwas99%?
• TheSE= RS:(.-RS:)9:
+ RS<(.-RS<)9<
� =.0224.
• Previously,fora95%CI,itwas0.097± 1.96x.0224=0.097± 0.044.
• Fora99%CI,itis0.097± 2.576x.0224=0.097± 0.058.
SmokingandGender• Writtenasthestatistic± marginoferror,the99%CIforthedifferencebetweenthetwoproportionsis
0.097± 0.058.• Marginoferror– 0.058forthe99%confidenceinterval– 0.044forthe95%confidenceinterval
SmokingandGender
• Howwouldthe95%confidenceintervalchangeifwewereestimating
𝜋smoker – 𝜋nonsmoker
insteadof𝜋nonsmoker – 𝜋smoker?
SmokingandGender
• (−0.141,−0.053)or−0.097± 0.044insteadof
• (0.053,0.141)or 0.097± 0.044.
• Thenegativesignsindicatetheprobabilityofaboyborntosmokingparentsislowerthanthatfornonsmokingparents.
SmokingandGender
ValidityConditionsofTheory-Based• Sameaswithasingleproportion.• Shouldhaveatleast10observationsineachofthecellsofthe2x2table.
SmokingParents Non-smokingParents
Total
Male 255 1975 2230Female 310 1627 1937Total 565 3602 4167
SmokingandGender• Thestrongsignificantresultinthisstudyyieldedquiteabitofpresswhenitcameout.
• Soonotherstudiescameoutwhichfoundnorelationshipbetweensmokingandgender(Parazinni etal.2004,Obel etal.2003).
• James(2004)arguedthatconfoundingvariableslike socialfactors,diet,environmentalexposureorstresswerethereasonfortheassociationbetweensmokingandgenderofthebaby.Theseareallconfoundedsinceitwasanobservationalstudy.Differentstudiescouldeasilyhavehaddifferentlevelsoftheseconfoundingfactors.
8.Fivenumbersummary,IQR,andgeysers.
6.1:ComparingTwoGroups:QuantitativeResponse6.2:ComparingTwoMeans:Simulation-BasedApproach6.3:ComparingTwoMeans:Theory-BasedApproach
Section 6.1ExploringQuantitativeData
Quantitativevs.CategoricalVariables
• Categorical– Valuesforwhicharithmeticdoesnotmakesense.– Gender,ethnicity,eyecolor…
• Quantitative– Youcanaddorsubtractthevalues,etc.– Age,height,weight,distance,time…
GraphsforaSingleVariable
Categorical
Quantitative
BarGraph DotPlot
ComparingTwoGroupsGraphically
Categorical
Quantitative
NotationCheck
Statistics� �̅� Samplemean� �̂� Sampleproportion.
Parameters� 𝜇 Populationmean� 𝜋 Population
proportionorprobability.
Statisticssummarizeasampleandparameterssummarizeapopulation
Quartiles
• Suppose25%oftheobservationsliebelowacertainvaluex.Thenxiscalledthelowerquartile(or25th percentile).
• Similarly,if25%oftheobservationsaregreaterthanx,thenxiscalledtheupperquartile (or75thpercentile).
• Thelowerquartilecanbecalculatedbyfindingthemedian,andthendeterminingthemedianofthevaluesbelowtheoverallmedian.Similarlytheupperquartileismedian{xi :xi> overallmedian}.
IQRandFive-NumberSummary• Thedifferencebetweenthequartilesiscalledtheinter-
quartilerange (IQR),anothermeasureofvariabilityalongwithstandarddeviation.
• Thefive-numbersummary forthedistributionofaquantitativevariableconsistsoftheminimum,lowerquartile,median,upperquartile,andmaximum.
• TechnicallytheIQRisnottheinterval(25thpercentile,75thpercentile),butthedifference75th percentile– 25th .
• Differentsoftwareusedifferentconventions,butwewillusetheconventionthat,ifthereisarangeofpossiblequantiles,youtakethemiddleofthatrange.
• Forexample,supposedataare1,3,7,7,8,9,12,14.• M=7.5,25th percentile=5,75th percentile=10.5.IQR=5.5.
IQRandFive-NumberSummary• Formediansandquartiles,wewillusetheconvention,if
thereisarangeofpossibilities,takethemiddleoftherange.• InR,thisistype=2.type=1meanstaketheminimum.• x=c(1,3,7,7,8,9,12,14)• quantile(x,.25,type=2)##5.5• IQR(x,type=2)##5.5• IQR(x,type=1)##6.Canyouseewhy?
• Forexample,supposedataare1,3,7,7,8,9,12,14.• M=7.5,25th percentile=5,75th percentile=10.5.IQR=5.5.
GeyserEruptions
Example6.1
OldFaithfulInter-EruptionTimes
• Howdothefive-numbersummaryandIQRdifferforinter-eruptiontimesbetween1978and2003?
OldFaithfulInter-EruptionTimes
• 1978IQR=81– 58=23• 2003IQR=98– 87=11
Boxplots
MinQlower MedQupper Max
Boxplots(Outliers)• Adatavaluethatismorethan1.5× IQRabovetheupperquartileorbelowthelowerquartileisconsideredanoutlier.
• Whentheseoccur,thewhiskersonaboxplotextendouttothefarthestvaluenotconsideredanoutlierandoutliersarerepresentedbyadotoranasterisk.
CancerPamphletReadingLevels
• Shortetal.(1995)comparedreadinglevelsofcancerpatientsandreadabilitylevelsofcancerpamphlets.Whatisthe:– Medianreadinglevel?– Meanreadinglevel?
• Arethedataskewedonewayortheother?
• Skewedabittotheright• Meantotherightofmedian
ComparingTwoMeans:Simulation-BasedApproachand
bicyclingtoworkexample.Section 6.2
Comparisonwithproportions.
• Wewillbecomparingmeans,muchthesamewaywecomparedtwoproportionsusingrandomizationtechniques.
• Thedifferencehereisthattheresponsevariableisquantitative(theexplanatoryvariableisstillbinarythough).Soifcardsareusedtodevelopanulldistribution,numbersgoonthecardsinsteadofwords.
BicyclingtoWorkExample6.2
BicyclingtoWork• Doesbicycleweightaffectcommutetime?• BritishMedicalJournal(2010)presentedtheresultsofa
randomizedexperimentdonebyJeremyGroves,who wantedtoknowifbicycleweightaffectedhiscommutetowork.
• For56days(JanuarytoJuly)Grovestossedacointodecideifhewouldbikethe27milestoworkonhiscarbonframebike(20.9lbs)orsteelframebicycle(29.75lbs).
• Herecordedthecommutetimeforeachtrip.
BicyclingtoWork
• Whataretheobservationalunits?– Eachtriptoworkonthe56differentdays.
• Whataretheexplanatoryandresponsevariables?– ExplanatoryiswhichbikeGrovesrode(categorical–binary)
– Responsevariableishiscommutetime(quantitative)
BicyclingtoWork
• Nullhypothesis: Commutetimeisnotaffectedbywhichbikeisused.
• Alternativehypothesis: Commutetimeisaffectedbywhichbikeisused.
BicyclingtoWork• Inchapter5weusedthedifferenceinproportions of
“successes”betweenthetwogroups.• Nowwewillcomparethedifferenceinaverages between
thetwogroups.• Theparametersofinterestare:– µcarbon =Longtermaveragecommutetimewithcarbonframedbike
– µsteel =Longtermaveragecommutetimewithsteelframedbike.
BicyclingtoWork
• µisthepopulationmean.Itisaparameter.• Usingthesymbolsµcarbon andµsteel,wecanrestatethehypotheses.
• H0: µcarbon =µsteel• Ha: µcarbon ≠µsteel .
BicyclingtoWork
Remember:• Thehypothesesareaboutthelongterm associationbetweencommutetimeandbikeused,notjusthis56trips.
• Hypothesesarealwaysaboutpopulationsorprocesses,notthesampledata.
BicyclingtoWork
Samplesize Samplemean SampleSD
Carbonframe 26 108.34min 6.25min
Steelframe 30 107.81min 4.89 min
BicyclingtoWork
• Thesampleaverageandvariabilityforcommutetimewashigherforthecarbonframebike
• Doesthisindicateatendency?• Orcouldahigheraveragejustcomefromtherandomassignment?PerhapsthecarbonframebikewasrandomlyassignedtodayswheretrafficwasheavierorweathersloweddownDr.Grovesonhiswaytowork?
BicyclingtoWork
• Isitpossible togetadifferenceof0.53minutesifcommutetimeisn’taffectedbythebikeused?
• ThesametypeofquestionwasaskedinChapter5forcategoricalresponsevariables.
• Thesameanswer.Yesit’spossible,howlikelythough?
BicyclingtoWork
• The3SStrategyStatistic:
• Chooseastatistic:• Theobserveddifferenceinaveragecommutetimes�̅�carbon – �̅�steel =108.34- 107.81
=0.53minutes
BicyclingtoWork
Simulation:• Wecanimaginesimulatingthisstudywithindexcards.–Writeall56timeson56cards.
• Shuffleall56cardsandrandomlyredistributeintotwostacks:– Onewith26cards(representingthetimesforthecarbon-framebike)
– Another30cards(representingthetimesforthesteel-framebike)
BicyclingtoWork
Simulation(continued):• Shufflingassumesthenullhypothesisofnoassociationbetweencommutetimeandbike
• Aftershufflingwecalculatethedifferenceintheaveragetimesbetweenthetwostacksofcards.
• Repeatthismanytimestodevelopanulldistribution• Let’sseewhatthislookslike
mean=107.87mean=108.34
CarbonFrameSteelFrame114116 123
113
113
118 109106111
119
mean=108.27
108.27– 107.87=0.40
103103 112
102
110
102 107100101
104
103105 106 102111
106108 106105 107
mean=107.81
116116 118
113
113
113 105104110
109
111111 105
102
106
109 10898103
110
112102 106 102101
105
ShuffledDifferencesinMeans
mean=108.37mean=108.27
CarbonFrameSteelFrame114116 123
113
113
118 109106111
119
mean=107.69
107.69– 108.37=-0.68
103103 112
102
110
102 107100101
104
103105 106 102111
106108 106105 107
mean=107.87
116116 118
113
113
113 105104110
109
111111 105
102
106
109 10898103
110
112102 106 102101
105
ShuffledDifferencesinMeans
mean=108.13mean=107.69
CarbonFrameSteelFrame114116 123
113
113
118 109106111
119
mean=107.97
107.97– 108.13=-0.16
103103 112
102
110
102 107100101
104
103105 106 102111
106108 106105 107
mean=108.37
116116 118
113
113
113 105104110
109
111111 105
102
106
109 10898103
110
112102 106 102101
105
ShuffledDifferencesinMeans
MoreSimulations-2.11-1.20 -1.21
-1.93 -1.53-1.11
0.71-0.52
1.79
0.022.53 1.90
-0.98
0.81 0.551.89
-0.31
-2.50
0.38
-1.51
0.22
1.500.13
0.44
1.46
-0.64-1.10Nineteenofour30simulatedstatisticswereas
ormoreextremethanourobserveddifferenceinmeansof0.53,henceourestimatedp-valueforthisnulldistributionis19/30=0.63.
ShuffledDifferencesinMeans
BicyclingtoWork• Using1000simulations,weobtainap-valueof72%.• Whatdoesthisp-valuemean?• Ifmeancommutetimesforthebikesarethesamein
thelongrun,andwerepeatedrandomassignmentofthelighterbiketo26daysandtheheavierto30days,adifferenceasextremeas0.53minutesormorewouldoccurinabout72%oftherepetitions.
• Therefore,wedonothavestrongevidencethatthecommutetimesforthetwobikeswilldifferinthelongrun.ThedifferenceobservedbyDr.Grovesisnotstatisticallysignificant.
BicyclingtoWork
• HaveweproventhatthebikeGroveschoosesisnotassociatedwithcommutetime?(Canweconcludethenull?)– No,alargep-valueisnot“strongevidencethatthenullhypothesisistrue.”
– Itsuggeststhatthenullhypothesisisplausible– Therecouldbeasmalllong-termdifference.Buttherealsocouldbenodifference.
BicyclingtoWork
• Imaginewewanttogeneratea95%confidenceintervalforthelong-rundifferenceinaveragecommutingtime.– Sampledifferenceinmeans± 1.96⨯SEforthedifferencebetweenthetwomeans
• Fromsimulations,theSE=standarddeviationofthedifferences=1.47.
• 0.53± 1.96(1.47)=0.53± 2.88• -2.35to3.41.• Whatdoesthismean?
BicyclingtoWork
• Weare95%confidentthatthetruelongtermdifference(carbon– steel)inaveragecommutingtimesisbetween-2.41and3.47minutes.Thecarbonframedbikeisbetween2.41minutesfasterand3.47minutesslowerthanthesteelframedbike.
• Doesitmakesensethattheintervalcontains0,basedonourp-value?
BicyclingtoWork
Scopeofconclusions• Canwegeneralizeourconclusiontoalargerpopulation?
• TwoKeyquestions:–Wasthesamplerandomlyobtainedandrepresentativeoftheoverallpopulationofinterest?
–Wasthisanexperiment?Weretheobservationalunitsrandomlyassignedtotreatments?
BicyclingtoWork
• Wasthesamplerepresentativeofanoverallpopulation?
• WhataboutthepopulationofalldaysDr.Grovesmightbiketowork?– No,Grovescommutedonconsecutivedaysinthisstudyanddidnotincludeallseasons.
• Wasthisanexperiment?Weretheobservationalunitsrandomlyassignedtotreatments?– Yes,heflippedacoinforthebike.–Wecanprobablydrawcause-and-effectconclusionshere.
BicyclingtoWork
• WecannotgeneralizebeyondGrovesandhistwobikes.
• Alimitationisthatthisstudyisnotdouble-blind– Theresearcherandthesubject(whichhappenedtobethesamepersonhere)werenotblindtowhichtreatmentwasbeingused.
– Dr.Grovesknewwhichbikehewasriding,andthismighthaveaffectedhisstateofmindorhischoiceswhileriding.