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Overview of Confidence IntervalsDr. S. A. Rizwan, M.D.
PublicHealthSpecialistSBCM, JointProgram– Riyadh
MinistryofHealth,KingdomofSaudiArabia
Learningobjectives
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Defineconfidenceintervals• Describetheiruseinstatisticalinference• DescribeandapplythestepsincalculatingCI
Statisticalinference
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Statisticalinference- drawingconclusionsaboutapopulationfromsample
• Methods• ConfidenceIntervals- estimatingavalueofapopulationparameter
• Testsofsignificance- assessevidenceforaclaimaboutapopulation
Thoughtexercises
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Estimationofapopulationmean
• Meanscoreobtainedbythisclassinthepretest exam
Thoughtexercises
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
95ofthese100CIswillcontainthepopulationparameter
Thereare100samplemeansand100CIs
Calculatesamplestatisticeg.meanforeachsample
Take100 samplesfromthesamepopulation
Thoughtexercises
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Wedon’tneedtotakealotofrandomsamplesto“rebuild”thesamplingdistribution
• AllweneedisoneSRSofsizenandrelyonthepropertiesofthesamplemeansdistributiontoinferthepopulationmean
Someimportantterms
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Pointestimate• Standarderror• Confidencelevel
Revise:standarddeviation
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
€
ˆ σ = s =(Yi −Y )2∑n −1
• Howmuchyourdataisspreadoutaroundaverage
• Forexample,areallyourscoresclosetotheaverage?Orarelotsofscoreswayabove(orwaybelow)theaveragescore?
ForMeans Forproportions
Revise:standarderror
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Thisisnotthestandarddeviationofthesample,itisthestandarddeviationofthesampledistributionofproportions(ormeans)
ForMeans Forproportions
Revise:standarderror
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
WhyCI?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Apointestimateprovidesnoinformationabouttheprecisionandreliabilityofestimation
• Apointestimatesaysnothingabouthowcloseitmightbetoμ
• Analternativetoreportingasinglesensiblevalueistocalculateandreportanentireintervalofplausiblevalues– aconfidenceinterval(CI)
WhatisCI?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Anintervalgivesarangeofvalues:• Takesintoconsiderationvariationinsamplestatisticsfromsampletosample
• Basedonobservationsfrom1sample• Givesinformationaboutclosenesstounknownpopulationparameters
• Statedintermsoflevelofconfidence.• Canneverbe100%confident• Anintervalofvaluescomputedfromthesample,thatisalmostsuretocoverthetruepopulationvalue
WhatisCI?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
GeneralformatofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• ZvaluesfordifferentConfidencelevels• 90%- 1.64• 95%- 1.96• 98%- 2.33• 99%- 2.58
PointEstimate± (CriticalValue)*(StandardError)
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• In95%ofthesampleswetake,thetruepopulationproportion(ormean)willbeintheinterval
• Weare95%confidentthatthetruepopulationproportion(ormean)willbeintheinterval
• In95%ofallpossiblesamplesofthissizen,µwillindeedfallinourconfidenceinterval
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Inonly5%ofsampleswouldsamplemeanbefartherfromµ
• Tosaythatweare95%confidentisshorthandfor“95%ofallpossiblesamplesofagivensizefromthispopulation willresultinanintervalthatcapturestheunknownparameter.”
• TointerpretaC%confidenceintervalforanunknownparameter,say,“WeareC%confidentthattheintervalfrom_____to_____capturestheactualvalueofthepopulationparameter”
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Aconfidenceintervalprovidesadditionalinformationaboutvariability
• Fora95%confidenceintervalabout95%ofthesimilarlyconstructedintervalswillcontaintheparameterbeingestimated.
• Also95%ofthesamplemeansforaspecifiedsamplesizewill liewithin1.96standarddeviationsofthehypothesizedpopulation
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Ingeneral,weconstructsuchintervalssothat,shouldwerepeattheprocessalargenumberoftimes,then95%,fora95%confidenceinterval,ofsuchintervalsshouldcontainthepopulationparameterbeingestimatedbythepointestimateandtheconfidenceinterval
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Thespecificintervalwecomputeinanygivensituationmayormaynotcontainthepopulation parameter
• Theonlywayforustobesurethatthepopulationparameteriswithintheboundsoftheconfidenceintervalistoknowthetruevalueforthisparameter
• Obviously, ifweknewthetruevalue,wewouldnotbothertogothroughtheprocessofguessingatthetruthwithestimates
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Example:0.05(0.036,0.064)
• Correct:
• Weare95%confidentthattheintervalfrom0.036to0.064actuallydoescontainthetruevalue
• Thismeansthatifweweretoselectmanydifferentsamplesofsize1000andconstructa95%CIfromeachsample,95%oftheresultingintervalswouldcontainthepopulation value
• (0.036,0.064)isonesuchinterval.(Notethat95%referstotheprocedureweusedtoconstructtheinterval;itdoesnotrefertothepopulation value)
• Wrong:Thereisa95%chancethatthepopulation valuefallsbetween0.036and0.064.(Notethatpisnotrandom,itisafixedbutunknownnumber)
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Youhavemeasuredthesystolicbloodpressureofarandomsampleof30employeesofacompany.A95%confidenceintervalforthemeansystolicbloodpressurefortheemployeesiscomputedtobe(122,138).Whichofthefollowingstatementsgivesavalidinterpretationofthisinterval?
a) 95%ofthesampleofemployeeshasasystolicbloodpressurebetween122and138.
b) 95%oftheemployeesinthecompanyhaveasystolicbloodpressurebetween122and138.
c) Ifthesamplingprocedurewererepeated100times,thenapproximately95ofthesamplemeanswouldbebetween122and138.
d) Ifthesamplingprocedurewererepeated100times,thenapproximately95oftheresulting100confidenceintervalswouldcontainthetruemeansystolicbloodpressureforallemployeesofthecompany.
e) Weare95%confidentthesamplemeanisbetween122and138.
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Themeanandstandarddeviationofthebirthweightsofarepresentativesampleof153newbornsare3250gramsand428gramsrespectively.Onthebasisofthesefigures,a95%confidenceintervalforthepopulationmeanbirthweightrunsfrom3181to3319grams.
a) About95%oftheindividual newborn birthweightsarebetween3181and3319g
b) Themeanbirthweightforthese153newborns isprobablybetween3181and3319g
c) Themeanofthepopulation fromwhichthe153newborns cameisbetween3181and3319g
d) Noneoftheabove
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• TheconfidenceleveldoesNOT tellusthechancethataparticularconfidenceintervalcapturesthepopulationparameter.
• WeCANNOT assignprobabilitytothepopulation valuebecauseitisfixedanddoesnotchangedependingonoursamplevalues.
• Widthoftheinterval– indicatesvariabilityinthedata
VariousinterpretationsofCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• WeCAN say:
• Weare95%confidentthattheconfidenceintervalcalculatedfromoursamplewillcontainthepopulation value
• WeCANNOT say:
• Thereisa95%probability orchancethattheconfidenceintervalwillcontainthepopulation value
• Thereisa95%probability orchancethepopulationvaluewill lieinthisconfidenceinterval
• 95%ofthetimethepopulation valuewill lieinthisconfidenceinterval
InterpretationofCIincomparativesituations
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• NullvaluewithinthelimitsoftheCI
• 0fordifferencesand1forratios
InterpretationofCIincomparativesituations
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Themotherwhosmokehadsignificantlyhigherrisk(RR=2.1;1.8,2.6,p=0.01)ofhavingLBWbabiesandcomparedtothosewhodidnotsmoke
• Doestheintervalcontainnullvalue=No;associationissignificant
• Widthoftheinterval- variabilityintheestimatewasless
InterpretationofCIincomparativesituations
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Themotherwhosmokehadsignificantlyhigherrisk(RR=2.1,0.8,4.9,p=0.06)ofhavingLBWbabiesandcomparedtothosewhodidnotsmoke
• Doestheintervalcontainnullvalue=Yes;associationisinsignificant
• Widthoftheinterval=highvariabilityinthesampleestimate
Thoughtexercise
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Seriesof5trials• Equalduration• Differentsamplesizes• Todeterminewhetheranoveldrugisbetter
thanplaceboinpreventingstroke
• Smallesttrialhas8patients• Largesttrialhas2000patients• Halfofthepatientsineachtrial– Newdrug• Alltrials- Relativeriskreductionby50%
Thoughtexercise
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Questions:• Ineachindividual trial,howconfidentcanwe
beregardingtherelativeriskreduction?• Largertrials- moreconfident
• Whichtrialswouldleadyoutorecommendthetreatmentunequivocally toyourpatients?
• CI- Rangewithinwhichthetrueeffectoftestdrugmightplausiblylieinthegiventrialdata
FactorsaffectingCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Factorsthatdeterminethewidthofaconfidenceintervalare:
• Samplesize,n
• Variabilityinthepopulation
• Desiredlevelofconfidence• Thehighertheconfidencelevel,themore
stronglywebelievethatthetruevalueoftheparameterbeingestimatedlieswithintheinterval
FactorsaffectingCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
AssumptionsforCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Random:Thedatashouldcomefromawell-designedrandomsampleorrandomizedexperiment.
AssumptionsforCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Normal:ThesamplingdistributionofthestatisticisapproximatelyNormal.
• Formeans:• ThesamplingdistributionisexactlyNormalifthe
populationdistributionisNormal.• Whenthepopulationdistribution isnotNormal,
thenthecentrallimittheoremtellsusthesamplingdistributionwillbeapproximatelyNormalifnissufficientlylarge(n≥30).
• Forproportions:• WecanusetheNormalapproximation tothe
samplingdistribution aslongasnp≥10andn(1–p)≥10.
AssumptionsforCI
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Independent:• Individual observationsareindependent
HowdoesCIrelatetosamplesize?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Costisdirectlyproportionaltosamplesize,sowegenerallywanttheminimumsampletodothejob
• Estimatingminimumsamplesizeiscommonlydonewithpopulationproportions
• Withpopulationproportions,youdonotneedtomakeseparateguessesaboutthepopulationmeanandstandarddeviation
• Withpopulationproportions,itiseasytoidentifyaconservativemean,andthebiasdoesnotvarymuch
HowdoesCIrelatetosamplesize?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Formean• Whenwechoosethebestsamplesize,wechooseonehalfoftheconfidenceinterval(thetopone)andsolveforn
nszYic ±=..
22/1
22
)..( µσ
−=
topiczn
HowdoesCIrelatetosamplesize?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Forproportion• Whenwechoosethebestsamplesize,wechooseonehalfoftheconfidenceinterval(thetopone)andsolveforn
nzic
)ˆ1(ˆˆ..ππ
π−
±=
22/1
2
)..()1(π
ππ−
−=
topiczn
HowdoesCIrelatetosamplesize?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
HowdoesCIrelatetosignificancelevel?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
ConfidenceLevel
‘z’Value
‘a’/2Value
80% 1.28 .1000
90% 1.64 .0500
95% 1.96 .0250
98% 2.33 .0100
99% 2.58 .0050
99.8% 3.08 .0010
99.9% 3.27 .0005
HowdoesCIrelatetosignificancelevel?
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Takehomemessages
Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Pvalue,criticalvalue,alfa,type1error,confidenceinterval,samplesizeareallrelatedtoeachother