confidence intervals: basic concepts and overview

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


• Statisticalinference- drawingconclusionsaboutapopulationfromsample

• Methods• ConfidenceIntervals- estimatingavalueofapopulationparameter

• Testsofsignificance- assessevidenceforaclaimaboutapopulation

Thoughtexercises


• Estimationofapopulationmean

• Meanscoreobtainedbythisclassinthepretest exam

Thoughtexercises


95ofthese100CIswillcontainthepopulationparameter

Thereare100samplemeansand100CIs

Calculatesamplestatisticeg.meanforeachsample

Take100 samplesfromthesamepopulation

Thoughtexercises


• Wedon’tneedtotakealotofrandomsamplesto“rebuild”thesamplingdistribution

• AllweneedisoneSRSofsizenandrelyonthepropertiesofthesamplemeansdistributiontoinferthepopulationmean

Someimportantterms


• Pointestimate• Standarderror• Confidencelevel

Revise:standarddeviation


€

ˆ σ = s =(Yi −Y )2∑n −1

• Howmuchyourdataisspreadoutaroundaverage

• Forexample,areallyourscoresclosetotheaverage?Orarelotsofscoreswayabove(orwaybelow)theaveragescore?

ForMeans Forproportions

Revise:standarderror


• Thisisnotthestandarddeviationofthesample,itisthestandarddeviationofthesampledistributionofproportions(ormeans)

ForMeans Forproportions

Revise:standarderror


WhyCI?


• Apointestimateprovidesnoinformationabouttheprecisionandreliabilityofestimation

• Apointestimatesaysnothingabouthowcloseitmightbetoμ

• Analternativetoreportingasinglesensiblevalueistocalculateandreportanentireintervalofplausiblevalues– aconfidenceinterval(CI)

WhatisCI?


• Anintervalgivesarangeofvalues:• Takesintoconsiderationvariationinsamplestatisticsfromsampletosample

• Basedonobservationsfrom1sample• Givesinformationaboutclosenesstounknownpopulationparameters

• Statedintermsoflevelofconfidence.• Canneverbe100%confident• Anintervalofvaluescomputedfromthesample,thatisalmostsuretocoverthetruepopulationvalue

WhatisCI?


GeneralformatofCI


• ZvaluesfordifferentConfidencelevels• 90%- 1.64• 95%- 1.96• 98%- 2.33• 99%- 2.58

PointEstimate± (CriticalValue)*(StandardError)

VariousinterpretationsofCI


• In95%ofthesampleswetake,thetruepopulationproportion(ormean)willbeintheinterval

• Weare95%confidentthatthetruepopulationproportion(ormean)willbeintheinterval

• In95%ofallpossiblesamplesofthissizen,µwillindeedfallinourconfidenceinterval



• Inonly5%ofsampleswouldsamplemeanbefartherfromµ

• Tosaythatweare95%confidentisshorthandfor“95%ofallpossiblesamplesofagivensizefromthispopulation willresultinanintervalthatcapturestheunknownparameter.”

• TointerpretaC%confidenceintervalforanunknownparameter,say,“WeareC%confidentthattheintervalfrom_____to_____capturestheactualvalueofthepopulationparameter”



• Aconfidenceintervalprovidesadditionalinformationaboutvariability

• Fora95%confidenceintervalabout95%ofthesimilarlyconstructedintervalswillcontaintheparameterbeingestimated.

• Also95%ofthesamplemeansforaspecifiedsamplesizewill liewithin1.96standarddeviationsofthehypothesizedpopulation



• Ingeneral,weconstructsuchintervalssothat,shouldwerepeattheprocessalargenumberoftimes,then95%,fora95%confidenceinterval,ofsuchintervalsshouldcontainthepopulationparameterbeingestimatedbythepointestimateandtheconfidenceinterval



• Thespecificintervalwecomputeinanygivensituationmayormaynotcontainthepopulation parameter

• Theonlywayforustobesurethatthepopulationparameteriswithintheboundsoftheconfidenceintervalistoknowthetruevalueforthisparameter

• Obviously, ifweknewthetruevalue,wewouldnotbothertogothroughtheprocessofguessingatthetruthwithestimates



• 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)



• Youhavemeasuredthesystolicbloodpressureofarandomsampleof30employeesofacompany.A95%confidenceintervalforthemeansystolicbloodpressurefortheemployeesiscomputedtobe(122,138).Whichofthefollowingstatementsgivesavalidinterpretationofthisinterval?

a) 95%ofthesampleofemployeeshasasystolicbloodpressurebetween122and138.

b) 95%oftheemployeesinthecompanyhaveasystolicbloodpressurebetween122and138.

c) Ifthesamplingprocedurewererepeated100times,thenapproximately95ofthesamplemeanswouldbebetween122and138.

d) Ifthesamplingprocedurewererepeated100times,thenapproximately95oftheresulting100confidenceintervalswouldcontainthetruemeansystolicbloodpressureforallemployeesofthecompany.

e) Weare95%confidentthesamplemeanisbetween122and138.



• Themeanandstandarddeviationofthebirthweightsofarepresentativesampleof153newbornsare3250gramsand428gramsrespectively.Onthebasisofthesefigures,a95%confidenceintervalforthepopulationmeanbirthweightrunsfrom3181to3319grams.

a) About95%oftheindividual newborn birthweightsarebetween3181and3319g

b) Themeanbirthweightforthese153newborns isprobablybetween3181and3319g

c) Themeanofthepopulation fromwhichthe153newborns cameisbetween3181and3319g

d) Noneoftheabove



• TheconfidenceleveldoesNOT tellusthechancethataparticularconfidenceintervalcapturesthepopulationparameter.

• WeCANNOT assignprobabilitytothepopulation valuebecauseitisfixedanddoesnotchangedependingonoursamplevalues.

• Widthoftheinterval– indicatesvariabilityinthedata



• WeCAN say:

• Weare95%confidentthattheconfidenceintervalcalculatedfromoursamplewillcontainthepopulation value

• WeCANNOT say:

• Thereisa95%probability orchancethattheconfidenceintervalwillcontainthepopulation value

• Thereisa95%probability orchancethepopulationvaluewill lieinthisconfidenceinterval

• 95%ofthetimethepopulation valuewill lieinthisconfidenceinterval

InterpretationofCIincomparativesituations


• NullvaluewithinthelimitsoftheCI

• 0fordifferencesand1forratios



• Themotherwhosmokehadsignificantlyhigherrisk(RR=2.1;1.8,2.6,p=0.01)ofhavingLBWbabiesandcomparedtothosewhodidnotsmoke

• Doestheintervalcontainnullvalue=No;associationissignificant

• Widthoftheinterval- variabilityintheestimatewasless



• Themotherwhosmokehadsignificantlyhigherrisk(RR=2.1,0.8,4.9,p=0.06)ofhavingLBWbabiesandcomparedtothosewhodidnotsmoke

• Doestheintervalcontainnullvalue=Yes;associationisinsignificant

• Widthoftheinterval=highvariabilityinthesampleestimate

Thoughtexercise


• Seriesof5trials• Equalduration• Differentsamplesizes• Todeterminewhetheranoveldrugisbetter

thanplaceboinpreventingstroke

• Smallesttrialhas8patients• Largesttrialhas2000patients• Halfofthepatientsineachtrial– Newdrug• Alltrials- Relativeriskreductionby50%

Thoughtexercise


• Questions:• Ineachindividual trial,howconfidentcanwe

beregardingtherelativeriskreduction?• Largertrials- moreconfident

• Whichtrialswouldleadyoutorecommendthetreatmentunequivocally toyourpatients?

• CI- Rangewithinwhichthetrueeffectoftestdrugmightplausiblylieinthegiventrialdata

FactorsaffectingCI


• Factorsthatdeterminethewidthofaconfidenceintervalare:

• Samplesize,n

• Variabilityinthepopulation

• Desiredlevelofconfidence• Thehighertheconfidencelevel,themore

stronglywebelievethatthetruevalueoftheparameterbeingestimatedlieswithintheinterval

FactorsaffectingCI


AssumptionsforCI


• Random:Thedatashouldcomefromawell-designedrandomsampleorrandomizedexperiment.

AssumptionsforCI


• Normal:ThesamplingdistributionofthestatisticisapproximatelyNormal.

• Formeans:• ThesamplingdistributionisexactlyNormalifthe

populationdistributionisNormal.• Whenthepopulationdistribution isnotNormal,

thenthecentrallimittheoremtellsusthesamplingdistributionwillbeapproximatelyNormalifnissufficientlylarge(n≥30).

• Forproportions:• WecanusetheNormalapproximation tothe

samplingdistribution aslongasnp≥10andn(1–p)≥10.

AssumptionsforCI


• Independent:• Individual observationsareindependent

HowdoesCIrelatetosamplesize?


• Costisdirectlyproportionaltosamplesize,sowegenerallywanttheminimumsampletodothejob

• Estimatingminimumsamplesizeiscommonlydonewithpopulationproportions

• Withpopulationproportions,youdonotneedtomakeseparateguessesaboutthepopulationmeanandstandarddeviation

• Withpopulationproportions,itiseasytoidentifyaconservativemean,andthebiasdoesnotvarymuch



• Formean• Whenwechoosethebestsamplesize,wechooseonehalfoftheconfidenceinterval(thetopone)andsolveforn

nszYic ±=..

22/1

22

)..( µσ

−=

topiczn



• Forproportion• Whenwechoosethebestsamplesize,wechooseonehalfoftheconfidenceinterval(thetopone)andsolveforn

nzic

)ˆ1(ˆˆ..ππ

π−

±=

22/1

2

)..()1(π

ππ−

−=

topiczn

HowdoesCIrelatetosignificancelevel?


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?


Takehomemessages


• Pvalue,criticalvalue,alfa,type1error,confidenceinterval,samplesizeareallrelatedtoeachother

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confidence intervals: basic concepts and overview

Health & Medicine