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

1. Comparing two proportions using numerical and visual summaries, good or bad year example.

2. Comparing 2 proportions with CIs + testing using simulation, dolphin example.3. Comparing 2 props. with theory-based testing, smoking and gender example.

Read ch5. http://www.stat.ucla.edu/~frederic/13/F18 .Bring a PENCIL and CALCULATOR and any books or notes you want to the exams. The midterm is Tue Nov 6. There is no lecture Thu Nov1 because of faculty retreat. The first exam will be on everything through today. We will do review next class.

For gradegrubbing, see syllabus. If you would like a question reevaluated, submit your work and a WRITTEN explanation of why you think you

deserve more points and how many more points you think you deserve to your TA. The TA will then give it to me, and I will consider it, and then give it back to the TA

to give back to you.

1

Moreaboutconfoundingfactors.

• Aconfoundingfactoristheageofthetwopopulationsingeneral.Leftiesinthe1980swereonaverageyoungerthanrighties.Manyoldleftieswereconvertedtorightiesatinfancy,intheearly20thcentury,butthispracticehassubsided.Thusinthe1980sand1990s,therewererelativelyfewoldleftiesbutmanyyoungleftiesintheoverallpopulation.Thisaloneexplainsthediscrepancy.

Unit2:ComparingTwoGroups

• InUnit1,welearnedthebasicprocessofstatisticalinferenceusingtestsandconfidenceintervals.Wedidallthisbyfocusingonasingleproportion.

• InUnit2,wewilltaketheseideasandextendthemtocomparingtwogroups.Wewillcomparetwoproportions,twoindependentmeans,andpaireddata.

Chapter5:ComparingTwoProportions

5.1:ComparingTwoGroups:CategoricalResponse5.2:ComparingTwoProportions:Simulation-BasedApproach5.3:ComparingTwoProportions:Theory-BasedApproach

3.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.

Comparing 2 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)wewouldhavehad13improversand17non-improversregardlessofthegrouptowhichtheywereassigned.

• Underthenullhypothesis,theassignmentdoesn’tmatterandwemightaswellhavejustrandomlyassignedthesubjectstothetwogroups.Thuswecansimulatedatabyrandomlyassigningeachpersontodolphinorcontrol.

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 SEsabovezero.� Usingeitherthep-valueorstandardizedstatistic,wehavestrongevidenceagainstthenullandcanconcludethattheimprovementduetoswimmingwithdolphinswasstatisticallysignificant.

SwimmingwithDolphins

� A95%confidenceintervalforthedifferenceintheprobabilityusingthestandarderrorof0.185fromthesimulationsis0.467+ 1.96(0.185)=0.467+0.363or(0.104to0.830)

• Weare95%confidentthatwhenallowedtoswimwithdolphins,theprobabilityofimprovingisbetween0.104and0.830higherthanwhennodolphinsarepresent.

• Howdoesthisintervalbackupourconclusionfromthetestofsignificance?

• Seeif0isintheinterval.

SwimmingwithDolphins

• Canwesaythatthepresenceofdolphinscaused thisimprovement?– Sincethiswasarandomizedexperiment,andassumingeverythingwasidenticalbetweenthegroups,wehavestrongevidencethatdolphinswerethecause

• Canwegeneralizetoalargerpopulation?– Maybemildtomoderatelydepressed18-65yearoldpatientswillingtovolunteerforthisstudy

– Wehavenoevidencethatrandomselectionwasusedtofindthe30subjects."Outpatients,recruitedthroughannouncementsontheinternet,radio,newspapers,andhospitals."

Comparing TwoProportions:Theory-BasedApproach,andsmokingandgenderexample.

Section5.3

Introduction

• Justaswithasingleproportion,wecanoftenpredictresultsofasimulationusingatheory-basedapproach.

• Thetheory-basedapproachalsogivesasimplerwaytogenerateconfidenceintervals.

• ThemainnewmathematicalfacttouseistheSEforthedifferencebetweentwoproportionsis45(.-45)89

+ 45(.-45)8;

� fortesting

or 459(.-459)89

+ 45;(.-45;)8;

� forCIs.

Aclarificationonthenewformula• Themarginoferrorforthedifferenceinproportionsis

𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 ⨯ SE,whereSE = 459(.-459)89

+ 45;(.-45;)8;

� .

Intesting,thenullhypothesisisnodifferencebetweenthetwogroups,soweusetheSE

�̂�(1 − �̂�)𝑛.

+�̂�(1 − �̂�)

𝑛Q

�

where�̂� isthepooledproportioninbothgroupscombined.

ButinCIs,weusetheformula 459(.-459)89

+ 45;(.-45;)8;

�

becausewearenotassuming�̂�. =�̂�QwithCIs.

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• Wewillcomparetheproportionstoseeifthedifference

isstatisticallysignificant.

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:• Simulatemanyrepetitionsofrandomlyshufflingthe2230boysand1937girlstothe565smokingand3602nonsmokingparents

• Calculatethedifferenceinproportionsofboysbetweenthegroupsforeachrepetition.

• Shufflingsimulatesthenullhypothesisofnoassociation

SmokingandGender3.Strengthofevidence:• Nothingasextremeas

ourobservedstatistic(≥0.097or≤−0.097)occurredin5000repetitions,

• HowmanySEsis0.097abovethemean?Z=0.097/0.023=4.22usingsimulations.Whataboutusingthetheory-basedapproach?

SmokingandGender

• Noticethenulldistributioniscenteredatzeroandisbell-shaped.

• Thiscanbeapproximatedbythenormaldistribution.

Formulas

• Thetheory-basedapproachyieldsz=4.30.

𝑧 =�̂�. − �̂�Q

�̂�(1 − �̂�) 1𝑛.+ 1𝑛Q

�

• Here𝑧 = .0*/-.*0.

.0X0(.-.0X0) 9YZ[;\

9]Z]

�=4.30.

• Thep-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.+�̂�Q(1 − �̂�Q)

𝑛Q

�

• Herenislargeandobs.areiid,soweusethemultipliersfromthenormaldistribution.1.645for90%confidence1.96for95%confidence2.576for99%confidence

• Wecanwritetheconfidenceintervalintheform:– statistic± marginoferror.

SmokingandGender• Ourstatisticistheobservedsampledifferenceinproportions,

0.097.

• Pluggingin1.96 ⨯ 459(.-459)89

+ 45;(.-45;)8;

� =0.044,

weget0.097± 0.044asour95%CI.• Wecouldalsowritethisintervalas(0.053,0.141).• Weare95%confidentthattheprobabilityofaboybaby

whereneitherfamilysmokesminustheprobabilityofaboybabywherebothparentssmokeisbetween0.053and0.141.