Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13,...
-
Upload
phungkhanh -
Category
Documents
-
view
219 -
download
3
Transcript of Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13,...
![Page 1: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/1.jpg)
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
![Page 2: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/2.jpg)
1.HandinHW2!2.Moreproblemswithstudies,coverage,adhererbiasandClofibrate.Surveysareobservational.• Coverageisacommonissue.Coverageistheextenttowhich
thepeopleyousampledfromrepresenttheoverallpopulation.Asurveyatafancyresearchhospitalinawealthyneighborhoodmayyieldpatiencewithhigherincomes,highereducation,etc.
• Non-responsebiasisanothercommonproblem.Poorcoveragemeansthepeoplegettingthesurveydonotrepresentthegeneralpopulation.Non-responsebiasmeansthatoutofthepeopleyougavethesurveyto,thepeopleactuallyfillingitoutandsubmittingitaredifferentfromthepeoplewhodidnot.
• Sameexactissuesinwebsurveys.
![Page 3: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/3.jpg)
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%
![Page 4: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/4.jpg)
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%
![Page 5: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/5.jpg)
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Thosewhotookclofibrate didmuchbetterthanthosewhodidn'tkeeptakingclofibrate.Doesthismeanclofibrate works?
![Page 6: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/6.jpg)
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Thosewhoadheredtoplaceboalsodidmuchbetterthanthosewhostoppedadhering.
![Page 7: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/7.jpg)
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Allinalltherewaslittledifferencebetweenthetwogroups.
![Page 8: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/8.jpg)
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Adherersdidbetterthannon-adherers,notbecauseofclofibrate,butbecausetheywerehealthieringeneral.Why?
![Page 9: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/9.jpg)
Clofibrate patientswhodiedduringfollowupadherers 15%non-adherers 25%total 20%
--------------------------------------------------------------------------Placebo
adherers 15%nonadherers 28%total 21%
Adherersdidbetterthannon-adherers,notbecauseofclofibrate,butbecausetheywerehealthieringeneral.Why?• adherersarethetypetoengageinhealthierbehavior.• sickpatientsarelesslikelytoadhere.
![Page 10: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/10.jpg)
3.Moreaboutconfoundingfactors.• Byaconfoundingfactor,wemeananalternativeexplanation
thatcouldexplaintheapparentrelationshipbetweenthetwovariables,eveniftheyarenotcausallyrelated.Typicallythisisdonebyfindinganotherdifferencebetweenthetreatmentandcontrolgroup.Forinstance,differentstudieshaveexaminedsmokersandnon-smokersandhavefoundthatsmokershavehigherratesoflivercancer.Oneexplanationwouldbethatsmokingcauseslivercancer.Butisthereanyother,alternativeexplanation?
• Onealternativewouldbethatthesmokerstendtodrinkmorealcohol,anditisthealcohol,notthesmoking,thatcauseslivercancer.
![Page 11: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/11.jpg)
3.Moreaboutconfoundingfactors.• Anotherplausibleexplanationisthatthesmokersareprobably
olderonaveragethanthenon-smokers,andolderpeoplearemoreatriskforallsortsofcancerthanyoungerpeople.
• Anothermightbethatsmokersengageinotherunhealthyactivitiesmorethannon-smokers.
• Notethatifonesaidthat“smokingmakesyouwanttodrinkalcoholwhichcauseslivercancer,”thatwouldnotbeavalidconfoundingfactor,sinceinthatexplanation,smokingeffectiveiscausallyrelatedtolivercancerrisk.
![Page 12: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/12.jpg)
3.Moreaboutconfoundingfactors.• Aconfoundingfactormustbeplausiblylinkedtoboththe
explanatoryandresponsevariables.Soforinstancesaying“perhapsahigherproportionofthesmokersaremen”wouldnotbeaveryconvincingconfoundingfactor,unlessyouhavesomereasontothinkgenderisstronglylinkedtolivercancer.
• Anotherexample:left-handednessandageatdeath.PsychologistsDianeHalpernandStanleyCoren lookedat1,000deathrecordsofthosewhodiedinSouthernCaliforniainthelate1980s andearly1990sandcontactedrelativestoseeifthedeceasedwererighthanded orlefthanded.Theyfoundthattheaverageagesatdeathofthelefthanded was66,andfortherighthanded itwas75.Theirresultswerepublishedinprestigiousscientificjournals,NatureandtheNewEnglandJournalofMedicine.
![Page 13: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/13.jpg)
Moreaboutconfoundingfactors.Allsortsofcausalconclusionsweremadeabouthowthisshowsthatthestressofbeinglefthanded inourrighthanded worldleadstoprematuredeath.
![Page 14: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/14.jpg)
Moreaboutconfoundingfactors.• Isthisanobservationalstudyoranexperiment?
![Page 15: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/15.jpg)
Moreaboutconfoundingfactors.• Isthisanobservationalstudyoranexperiment?Itisanobservationalstudy.• Arethereplausibleconfoundingfactorsyoucanthinkof?
![Page 16: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/16.jpg)
Moreaboutconfoundingfactors.• Aconfoundingfactoristheageofthetwopopulationsin
general.Leftiesinthe1980swereonaverageyoungerthanrighties.Manyoldleftieswereconvertedtorightiesatinfancy,intheearly20thcentury,butthispracticehassubsided.Thusinthe1980sand1990s,therewererelativelyfewoldleftiesbutmanyyoungleftiesintheoverallpopulation.Thisaloneexplainsthediscrepancy.
![Page 17: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/17.jpg)
Unit2.ComparingTwoGroups
• InUnit1,welearnedthebasicprocessofstatisticalinferenceusingtestsandconfidenceintervals.Wedidallthisbyfocusingonasingleproportion.
• InUnit2,wewilltaketheseideasandextendthemtocomparingtwogroups.Wewillcomparetwoproportions,twoindependentmeans,andpaireddata.
![Page 18: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/18.jpg)
5.Comparingtwoproportionsusingnumericalandvisualsummaries,andthegoodorbadyearexample.
Section5.1
![Page 19: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/19.jpg)
Example5.1:PositiveandNegativePerceptions
• Considerthesetwoquestions:– Areyouhavingagoodyear?– Areyouhavingabadyear?
• Dopeopleanswereachquestioninsuchawaythatwouldindicatethesameanswer?(e.g.YesforthefirstoneandNoforthesecond.)
![Page 20: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/20.jpg)
PositiveandNegativePerceptions
• Researchersquestioned30students(randomlygivingthemoneofthetwoquestions).
• Theythenrecordedifapositiveornegativeresponsewasgiven.
• Theywantedtoseeifthewordingofthequestioninfluencedtheanswers.
![Page 21: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/21.jpg)
Positiveandnegativeperceptions
• Observationalunits– The30students
• Variables– Questionwording(goodyearorbadyear)– Perceptionoftheiryear(positiveornegative)
• Whichistheexplanatoryvariableandwhichistheresponse variable?
• Isthisanobservationalstudyorexperiment?
![Page 22: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/22.jpg)
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
![Page 23: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/23.jpg)
Two-WayTables
• Atwo-waytableorganizesdata– Summarizestwo categoricalvariables– Alsocalledcontingencytable
• Arestudentsmorelikelytogiveapositiveresponseiftheyweregiventhegoodyearquestion?
GoodYear BadYear TotalPositiveresponse 15 4 19Negativeresponse 3 8 11Total 18 12 30
![Page 24: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/24.jpg)
Two-WayTables
• Conditionalproportionswillhelpusbetterdetermineifthereisanassociationbetweenthequestionaskedandthetypeofresponse.
• Wecanseethatthesubjectswiththepositivequestionweremorelikely torespondpositively.
GoodYear BadYear TotalPositiveresponse 15/18 ≈0.83 4/12≈0.33 19Negativeresponse 3 8 11Total 18 12 30
![Page 25: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/25.jpg)
SegmentedBarGraphs
• Wecanalsousesegmentedbargraphstoseethisassociation betweenthe"goodyear"questionandapositiveresponse.
![Page 26: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/26.jpg)
Statistic
GoodYear BadYear TotalPositiveresponse 15(83%) 4(33%) 19Negativeresponse 3 8 11Total 18 12 30
� Thestatisticwewillmainlyusetosummarizethistableisthedifferenceinproportionsofpositiveresponsesis0.83− 0.33=0.50.
![Page 27: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/27.jpg)
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.
![Page 28: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/28.jpg)
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
![Page 29: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/29.jpg)
6. Comparing two proportions with CIs and testing using simulation, dolphin example.
Section5.2
![Page 30: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/30.jpg)
SwimmingwithDolphins
Example5.2
![Page 31: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/31.jpg)
SwimmingwithDolphinsIsswimmingwithdolphinstherapeuticforpatientssufferingfromclinicaldepression?
• ResearchersAntonioli andReveley (2005),inBritishMedicalJournal,recruited30subjectsaged18-65withaclinicaldiagnosisofmildtomoderatedepression
• Discontinuedantidepressantsandpsychotherapy4weekspriortoandthroughouttheexperiment
• 30subjectswenttoanislandnearHonduraswheretheywererandomlyassignedtotwotreatmentgroups
![Page 32: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/32.jpg)
SwimmingwithDolphins• Bothgroupsengagedinonehourofswimmingandsnorkeling
eachday• Onegroupswaminthepresenceofdolphinsandtheother
groupdidnot• Participantsinbothgroupshadidenticalconditionsexceptfor
thedolphins• Aftertwoweeks,eachsubjects’levelofdepressionwas
evaluated,asithadbeenatthebeginningofthestudy• Theresponsevariableiswhetherornotthesubjectachieved
substantialreductionindepression
![Page 33: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/33.jpg)
SwimmingwithDolphins
Nullhypothesis:Dolphinsdonothelp.– Swimmingwithdolphinsisnotassociatedwithsubstantialimprovementindepression
Alternativehypothesis:Dolphinshelp.– Swimmingwithdolphinsincreases theprobabilityofsubstantialimprovementindepressionsymptoms
![Page 34: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/34.jpg)
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.)
![Page 35: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/35.jpg)
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.
![Page 36: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/36.jpg)
SwimmingwithDolphins
• Therearetwopossibleexplanationsforanobserveddifferenceof0.467.– Atendencytobemorelikelytoimprovewithdolphins(alternativehypothesis)
– The13subjectsweregoingtoshowimprovementwithorwithoutdolphinsandrandomchanceassignedmoreimproverstothedolphins(nullhypothesis)
![Page 37: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/37.jpg)
SwimmingwithDolphins
• Ifthenullhypothesisistrue(noassociationbetweendolphintherapyandimprovement)wewouldhave13improversand17non-improversregardlessofthegrouptowhichtheywereassigned.
• Hencetheassignmentdoesn’tmatterandwecanjustrandomlyassignthesubjects’resultstothetwogroupstoseewhatwouldhappenunderatruenullhypothesis.
![Page 38: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/38.jpg)
SwimmingwithDolphins• Wecansimulatethiswithcards– 13cardstorepresenttheimprovers– 17cardsrepresentthenon-improvers
• Shufflethecards– put15inonepile(dolphintherapy)– put15inanother(controlgroup)
![Page 39: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/39.jpg)
SwimmingwithDolphins• ComputetheproportionofimproversintheDolphinTherapygroup
• ComputetheproportionofimproversintheControlgroup
• Thedifferenceinthesetwoproportionsiswhatcouldjustaswellhavehappenedundertheassumptionthereisnoassociationbetweenswimmingwithdolphinsandsubstantialimprovementindepression.
![Page 40: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/40.jpg)
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
![Page 41: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/41.jpg)
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
![Page 42: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/42.jpg)
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
![Page 43: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/43.jpg)
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
![Page 44: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/44.jpg)
SwimmingwithDolphins• Wedid1000repetitionstodevelopanulldistribution.
![Page 45: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/45.jpg)
SwimmingwithDolphins
• 13outof1000resultshadadifferenceof0.467orhigher(p-value=0.013).
• 0.467is(.*+,-((../0
≈ 2.52 SDabovezero.� Usingeitherthep-valueorstandardizedstatistic,wehavestrongevidenceagainstthenullandcanconcludethattheimprovementduetoswimmingwithdolphinswasstatisticallysignificant.
![Page 46: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/46.jpg)
SwimmingwithDolphins
� A95%confidenceintervalforthedifferenceintheprobabilityusingthestandarddeviationfromthenulldistributionis0.467+ 2(0.185)=0.467+ 0.370or(0.097to0.837)
• Weare95%confidentthatwhenallowedtoswimwithdolphins,theprobabilityofimprovingisbetween0.097and0.837higherthanwhennodolphinsarepresent.
• Howdoesthisintervalbackupourconclusionfromthetestofsignificance?
![Page 47: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/47.jpg)
SwimmingwithDolphins
• Canwesaythatthepresenceofdolphinscaused thisimprovement?– Sincethiswasarandomizedexperiment,andassumingeverythingwasidenticalbetweenthegroups,wehavestrongevidencethatdolphinswerethecause
• Canwegeneralizetoalargerpopulation?– Maybemildtomoderatelydepressed18-65yearoldpatientswillingtovolunteerforthisstudy
– Wehavenoevidencethatrandomselectionwasusedtofindthe30subjects."Outpatients,recruitedthroughannouncementsontheinternet,radio,newspapers,andhospitals."
![Page 48: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/48.jpg)
7.Comparingtwoproportions:Theory-BasedApproach,andsmokingandgenderexample.
Section5.3
![Page 49: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/49.jpg)
Introduction
• Justaswithasingleproportion,wecanoftenpredictresultsofasimulationusingatheory-basedapproach.
• Thetheory-basedapproachalsogivesasimplerwaytogenerateaconfidenceintervals.
• ThemainnewmathematicalfacttouseistheSEforthedifferencebetweentwoproportionsis
�̂�(1 − �̂�) .9:+ .
9<
� .
![Page 50: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/50.jpg)
Parents’SmokingStatusandtheirBabies’Gender
Example5.3
![Page 51: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/51.jpg)
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)
![Page 52: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/52.jpg)
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
![Page 53: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/53.jpg)
SmokingandGender
NullHypothesis:• Thereisnoassociationbetween
smokingstatusofparentsandsexofchild.
• Theprobabilityofhavingaboyisthesameforparentswhosmokeanddon’tsmoke.
• 𝜋smoking - 𝜋nonsmoking =0
![Page 54: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/54.jpg)
SmokingandGender
AlternativeHypothesis:• Thereisanassociationbetweensmokingstatusofparentsandsexofchild.
• Theprobabilityofhavingaboyisnotthesameforparentswhosmokeanddon’tsmoke
• 𝜋smoking - 𝜋nonsmoking ≠0
![Page 55: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/55.jpg)
SmokingandGender
• Whataretheobservationalunitsinthestudy?• Whatarethevariablesinthisstudy?• Whichvariableshouldbeconsideredtheexplanatoryvariableandwhichtheresponsevariable?
• Whatistheparameterofinterest?• Canyoudrawcause-and-effectconclusionsforthisstudy?
![Page 56: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/56.jpg)
SmokingandGender
Usingthe3SStrategytoassesthestrength1.Statistic:• Theproportionofboysborntononsmokersminustheproportionofboysborntosmokersis0.548– 0.451=0.097.
![Page 57: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/57.jpg)
SmokingandGender
2.Simulate:• Manyrepetitionsofshufflingthe2230boysand1937girlstothe565smokingand3602nonsmokingparents
• Calculatethedifferenceinproportionsofboysbetweenthegroupsforeachrepetition.
• Shufflingsimulatesthenullhypothesisofnoassociation
![Page 58: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/58.jpg)
SmokingandGender3.Strengthofevidence:• Nothingasextremeas
ourobservedstatistic(≥0.097or≤−0.097)occurredin5000repetitions,
• HowmanySDsis0.097abovethemean?Z=0.097/0.023=4.22usingsimulations.Whataboutusingthetheory-basedapproach?
![Page 59: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/59.jpg)
SmokingandGender
• Noticethenulldistributioniscenteredatzeroandisbell-shaped.
• Thiscanbeapproximatedbythenormaldistribution.
![Page 60: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/60.jpg)
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%.
![Page 61: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/61.jpg)
SmokingandGender
• Fukudaetal.(2002)foundthefollowinginJapan.– Outof3602birthswherebothparentsdidnotsmoke,
1975wereboys.This54.8%boys.– Outof565birthswherebothparentssmokedmorethan
apackaday,255wereboys.Thisis45.1%boys.– Intotal,outof4170births,2230wereboys,whichis
53.5%boys.
![Page 62: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/62.jpg)
Formulas• Howdowefindthemarginoferrorforthedifferencein
proportions?
𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 ⨯�̂�.(1 − �̂�.)
𝑛.+�̂�@(1 − �̂�@)
𝑛@
�
• Themultiplierisdependentupontheconfidencelevel.– 1.645for90%confidence– 1.96for95%confidence– 2.576for99%confidence
• Wecanwritetheconfidenceintervalintheform:– statistic± marginoferror.
![Page 63: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/63.jpg)
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.
![Page 64: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/64.jpg)
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.
![Page 65: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/65.jpg)
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.
![Page 66: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/66.jpg)
SmokingandGender• Writtenasthestatistic± marginoferror,the99%CIforthedifferencebetweenthetwoproportionsis
0.097± 0.058.• Marginoferror– 0.058forthe99%confidenceinterval– 0.044forthe95%confidenceinterval
![Page 67: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/67.jpg)
SmokingandGender
• Howwouldthe95%confidenceintervalchangeifwewereestimating
𝜋smoker – 𝜋nonsmoker
insteadof𝜋nonsmoker – 𝜋smoker?
![Page 68: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/68.jpg)
SmokingandGender
• (−0.141,−0.053)or−0.097± 0.044insteadof
• (0.053,0.141)or 0.097± 0.044.
• Thenegativesignsindicatetheprobabilityofaboyborntosmokingparentsislowerthanthatfornonsmokingparents.
![Page 69: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/69.jpg)
SmokingandGender
ValidityConditionsofTheory-Based• Sameaswithasingleproportion.• Shouldhaveatleast10observationsineachofthecellsofthe2x2table.
SmokingParents Non-smokingParents
Total
Male 255 1975 2230Female 310 1627 1937Total 565 3602 4167
![Page 70: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/70.jpg)
SmokingandGender• Thestrongsignificantresultinthisstudyyieldedquiteabitofpresswhenitcameout.
• Soonotherstudiescameoutwhichfoundnorelationshipbetweensmokingandgender(Parazinni etal.2004,Obel etal.2003).
• James(2004)arguedthatconfoundingvariableslike socialfactors,diet,environmentalexposureorstresswerethereasonfortheassociationbetweensmokingandgenderofthebaby.Theseareallconfoundedsinceitwasanobservationalstudy.Differentstudiescouldeasilyhavehaddifferentlevelsoftheseconfoundingfactors.
![Page 71: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/71.jpg)
8.Fivenumbersummary,IQR,andgeysers.
6.1:ComparingTwoGroups:QuantitativeResponse6.2:ComparingTwoMeans:Simulation-BasedApproach6.3:ComparingTwoMeans:Theory-BasedApproach
![Page 72: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/72.jpg)
Section 6.1ExploringQuantitativeData
![Page 73: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/73.jpg)
Quantitativevs.CategoricalVariables
• Categorical– Valuesforwhicharithmeticdoesnotmakesense.– Gender,ethnicity,eyecolor…
• Quantitative– Youcanaddorsubtractthevalues,etc.– Age,height,weight,distance,time…
![Page 74: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/74.jpg)
GraphsforaSingleVariable
Categorical
Quantitative
BarGraph DotPlot
![Page 75: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/75.jpg)
ComparingTwoGroupsGraphically
Categorical
Quantitative
![Page 76: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/76.jpg)
NotationCheck
Statistics� �̅� Samplemean� �̂� Sampleproportion.
Parameters� 𝜇 Populationmean� 𝜋 Population
proportionorprobability.
Statisticssummarizeasampleandparameterssummarizeapopulation
![Page 77: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/77.jpg)
Quartiles
• Suppose25%oftheobservationsliebelowacertainvaluex.Thenxiscalledthelowerquartile(or25th percentile).
• Similarly,if25%oftheobservationsaregreaterthanx,thenxiscalledtheupperquartile (or75thpercentile).
• Thelowerquartilecanbecalculatedbyfindingthemedian,andthendeterminingthemedianofthevaluesbelowtheoverallmedian.Similarlytheupperquartileismedian{xi :xi> overallmedian}.
![Page 78: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/78.jpg)
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.
![Page 79: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/79.jpg)
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.
![Page 80: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/80.jpg)
GeyserEruptions
Example6.1
![Page 81: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/81.jpg)
OldFaithfulInter-EruptionTimes
• Howdothefive-numbersummaryandIQRdifferforinter-eruptiontimesbetween1978and2003?
![Page 82: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/82.jpg)
OldFaithfulInter-EruptionTimes
• 1978IQR=81– 58=23• 2003IQR=98– 87=11
![Page 83: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/83.jpg)
Boxplots
MinQlower MedQupper Max
![Page 84: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/84.jpg)
Boxplots(Outliers)• Adatavaluethatismorethan1.5× IQRabovetheupperquartileorbelowthelowerquartileisconsideredanoutlier.
• Whentheseoccur,thewhiskersonaboxplotextendouttothefarthestvaluenotconsideredanoutlierandoutliersarerepresentedbyadotoranasterisk.
![Page 85: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/85.jpg)
CancerPamphletReadingLevels
• Shortetal.(1995)comparedreadinglevelsofcancerpatientsandreadabilitylevelsofcancerpamphlets.Whatisthe:– Medianreadinglevel?– Meanreadinglevel?
• Arethedataskewedonewayortheother?
![Page 86: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/86.jpg)
• Skewedabittotheright• Meantotherightofmedian
![Page 87: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/87.jpg)
ComparingTwoMeans:Simulation-BasedApproachand
bicyclingtoworkexample.Section 6.2
![Page 88: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/88.jpg)
Comparisonwithproportions.
• Wewillbecomparingmeans,muchthesamewaywecomparedtwoproportionsusingrandomizationtechniques.
• Thedifferencehereisthattheresponsevariableisquantitative(theexplanatoryvariableisstillbinarythough).Soifcardsareusedtodevelopanulldistribution,numbersgoonthecardsinsteadofwords.
![Page 89: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/89.jpg)
BicyclingtoWorkExample6.2
![Page 90: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/90.jpg)
BicyclingtoWork• Doesbicycleweightaffectcommutetime?• BritishMedicalJournal(2010)presentedtheresultsofa
randomizedexperimentdonebyJeremyGroves,who wantedtoknowifbicycleweightaffectedhiscommutetowork.
• For56days(JanuarytoJuly)Grovestossedacointodecideifhewouldbikethe27milestoworkonhiscarbonframebike(20.9lbs)orsteelframebicycle(29.75lbs).
• Herecordedthecommutetimeforeachtrip.
![Page 91: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/91.jpg)
BicyclingtoWork
• Whataretheobservationalunits?– Eachtriptoworkonthe56differentdays.
• Whataretheexplanatoryandresponsevariables?– ExplanatoryiswhichbikeGrovesrode(categorical–binary)
– Responsevariableishiscommutetime(quantitative)
![Page 92: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/92.jpg)
BicyclingtoWork
• Nullhypothesis: Commutetimeisnotaffectedbywhichbikeisused.
• Alternativehypothesis: Commutetimeisaffectedbywhichbikeisused.
![Page 93: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/93.jpg)
BicyclingtoWork• Inchapter5weusedthedifferenceinproportions of
“successes”betweenthetwogroups.• Nowwewillcomparethedifferenceinaverages between
thetwogroups.• Theparametersofinterestare:– µcarbon =Longtermaveragecommutetimewithcarbonframedbike
– µsteel =Longtermaveragecommutetimewithsteelframedbike.
![Page 94: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/94.jpg)
BicyclingtoWork
• µisthepopulationmean.Itisaparameter.• Usingthesymbolsµcarbon andµsteel,wecanrestatethehypotheses.
• H0: µcarbon =µsteel• Ha: µcarbon ≠µsteel .
![Page 95: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/95.jpg)
BicyclingtoWork
Remember:• Thehypothesesareaboutthelongterm associationbetweencommutetimeandbikeused,notjusthis56trips.
• Hypothesesarealwaysaboutpopulationsorprocesses,notthesampledata.
![Page 96: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/96.jpg)
BicyclingtoWork
Samplesize Samplemean SampleSD
Carbonframe 26 108.34min 6.25min
Steelframe 30 107.81min 4.89 min
![Page 97: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/97.jpg)
BicyclingtoWork
• Thesampleaverageandvariabilityforcommutetimewashigherforthecarbonframebike
• Doesthisindicateatendency?• Orcouldahigheraveragejustcomefromtherandomassignment?PerhapsthecarbonframebikewasrandomlyassignedtodayswheretrafficwasheavierorweathersloweddownDr.Grovesonhiswaytowork?
![Page 98: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/98.jpg)
BicyclingtoWork
• Isitpossible togetadifferenceof0.53minutesifcommutetimeisn’taffectedbythebikeused?
• ThesametypeofquestionwasaskedinChapter5forcategoricalresponsevariables.
• Thesameanswer.Yesit’spossible,howlikelythough?
![Page 99: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/99.jpg)
BicyclingtoWork
• The3SStrategyStatistic:
• Chooseastatistic:• Theobserveddifferenceinaveragecommutetimes�̅�carbon – �̅�steel =108.34- 107.81
=0.53minutes
![Page 100: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/100.jpg)
BicyclingtoWork
Simulation:• Wecanimaginesimulatingthisstudywithindexcards.–Writeall56timeson56cards.
• Shuffleall56cardsandrandomlyredistributeintotwostacks:– Onewith26cards(representingthetimesforthecarbon-framebike)
– Another30cards(representingthetimesforthesteel-framebike)
![Page 101: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/101.jpg)
BicyclingtoWork
Simulation(continued):• Shufflingassumesthenullhypothesisofnoassociationbetweencommutetimeandbike
• Aftershufflingwecalculatethedifferenceintheaveragetimesbetweenthetwostacksofcards.
• Repeatthismanytimestodevelopanulldistribution• Let’sseewhatthislookslike
![Page 102: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/102.jpg)
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
![Page 103: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/103.jpg)
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
![Page 104: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/104.jpg)
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
![Page 105: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/105.jpg)
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
![Page 106: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/106.jpg)
BicyclingtoWork• Using1000simulations,weobtainap-valueof72%.• Whatdoesthisp-valuemean?• Ifmeancommutetimesforthebikesarethesamein
thelongrun,andwerepeatedrandomassignmentofthelighterbiketo26daysandtheheavierto30days,adifferenceasextremeas0.53minutesormorewouldoccurinabout72%oftherepetitions.
• Therefore,wedonothavestrongevidencethatthecommutetimesforthetwobikeswilldifferinthelongrun.ThedifferenceobservedbyDr.Grovesisnotstatisticallysignificant.
![Page 107: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/107.jpg)
BicyclingtoWork
• HaveweproventhatthebikeGroveschoosesisnotassociatedwithcommutetime?(Canweconcludethenull?)– No,alargep-valueisnot“strongevidencethatthenullhypothesisistrue.”
– Itsuggeststhatthenullhypothesisisplausible– Therecouldbeasmalllong-termdifference.Buttherealsocouldbenodifference.
![Page 108: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/108.jpg)
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?
![Page 109: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/109.jpg)
BicyclingtoWork
• Weare95%confidentthatthetruelongtermdifference(carbon– steel)inaveragecommutingtimesisbetween-2.41and3.47minutes.Thecarbonframedbikeisbetween2.41minutesfasterand3.47minutesslowerthanthesteelframedbike.
• Doesitmakesensethattheintervalcontains0,basedonourp-value?
![Page 110: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/110.jpg)
BicyclingtoWork
Scopeofconclusions• Canwegeneralizeourconclusiontoalargerpopulation?
• TwoKeyquestions:–Wasthesamplerandomlyobtainedandrepresentativeoftheoverallpopulationofinterest?
–Wasthisanexperiment?Weretheobservationalunitsrandomlyassignedtotreatments?
![Page 111: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/111.jpg)
BicyclingtoWork
• Wasthesamplerepresentativeofanoverallpopulation?
• WhataboutthepopulationofalldaysDr.Grovesmightbiketowork?– No,Grovescommutedonconsecutivedaysinthisstudyanddidnotincludeallseasons.
• Wasthisanexperiment?Weretheobservationalunitsrandomlyassignedtotreatments?– Yes,heflippedacoinforthebike.–Wecanprobablydrawcause-and-effectconclusionshere.
![Page 112: Stat 13, Intro. to Statistical Methods for the Life and ...frederic/13/sum17/day06.pdf · Stat 13, Intro. to Statistical Methods for the Life and Health ... , since in that explanation,](https://reader033.fdocuments.us/reader033/viewer/2022052710/5aafdf737f8b9a07498ddfa1/html5/thumbnails/112.jpg)
BicyclingtoWork
• WecannotgeneralizebeyondGrovesandhistwobikes.
• Alimitationisthatthisstudyisnotdouble-blind– Theresearcherandthesubject(whichhappenedtobethesamepersonhere)werenotblindtowhichtreatmentwasbeingused.
– Dr.Grovesknewwhichbikehewasriding,andthismighthaveaffectedhisstateofmindorhischoiceswhileriding.