定量数据分析_Quantitative_data_analysis

,WillGHopkinsAucklandUniversityofTechnologyAucklandNZ,QuantitativeDataAnalysis,SummarizingData:variables;simplestatistics;effectstatisticsandstatisticalmodels;complexmodels.GeneralizingfromSampletoPopulation:precisionofestimate,confidencelimits,statisticalsignificance,pvalue,errors.,Reference:HopkinsWG(2002).Quantitativedataanalysis(Slideshow).Sportscience6,/jour/0201/Quantitative_analysis.ppt(2046words),SummarizingData,Dataareabunchofvaluesofoneormorevariables.Avariableissomethingthathasdifferentvalues.Valuescanbenumbersornames,dependingonthevariable:Numeric,e.g.weightCounting,e.g.numberofinjuriesOrdinal,petitivelevel(valuesarenumbers/names)Nominal,e.g.sex(valuesarenamesWhenvaluesarenumbers,visualizethedistributionofallvaluesinstemandleafplotsorinafrequencyhistogram.Canalsousenormalprobabilityplotstovisualizehowwellthevaluesfitanormaldistribution.Whenvaluesarenames,visualizethefrequencyofeachvaluewithapiechartorajustalistofvaluesandfrequencies.,Astatisticisanumbersummarizingabunchofvalues.Simpleorunivariatestatisticssummarizevaluesofonevariable.Effectoroutcomestatisticssummarizetherelationshipbetweenvaluesoftwoormorevariables.SimplestatisticsfornumericvariablesMean:theaverageStandarddeviation:thetypicalvariationStandarderrorofthemean:thetypicalvariationinthemeanwithrepeatedsamplingMultiplyby(samplesize)toconverttostandarddeviation.Usethesealsoforcountingandordinalvariables.Usemedian(middlevalueor50thpercentile)andquartiles(25thand75thpercentiles)forgrosslynon-normallydistributeddata.Summarizetheseandothersimplestatisticsvisuallywithboxandwhiskerplots.,SimplestatisticsfornominalvariablesFrequencies,proportions,orodds.Canalsousetheseforordinalvariables.EffectstatisticsDerivedfromstatisticalmodel(equation)oftheformY(dependent)vsX(predictororindependent).DependontypeofYandX.Mainones:,Model:numericvsnumerice.g.bodyfatvssumofskinfoldsModelortest:linearregressionEffectstatistics:slopeandintercept=parameterscorrelationcoefficientorvarianceexplained(=100correlation2)=measuresofgoodnessoffitOtherstatistics:typicalorstandarderroroftheestimate=residualerror=bestmeasureofvalidity(withcriterionvariableontheYaxis),sumskinfolds(mm),bodyfat(%BM),Model:numericvsnominale.g.strengthvssexModelortest:ttest(2groups)1-wayANOVA(2groups)Effectstatistics:differencebetweenmeansexpressedasrawdifference,percentdifference,orfractionoftherootmeansquareerror(Cohenseffect-sizestatistic)varianceexplainedorbetter(varianceexplained/100)=measuresofgoodnessoffitOtherstatistics:rootmeansquareerror=averagestandarddeviationofthetwogroups,female,male,strength,sex,MoreonexpressingthemagnitudeoftheeffectWhatoftenmattersisthedifferencebetweenmeansrelativetothestandarddeviation:,Fractionormultipleofastandarddeviationisknownastheeffect-sizestatistic(orCohensd).Cohensuggestedthresholdsforcorrelationsandeffectsizes.Hopkinsagreeswiththethresholdsforcorrelationsbutsuggestsothersfortheeffectsize:,Forstudiesofathleticperformance,percentdifferencesorchangesinthemeanarebetterthanCoheneffectsizes.,Model:numericvsnominal(repeatedmeasures)e.g.strengthvstrialModelortest:pairedttest(2trials)repeated-measuresANOVAwithonewithin-subjectfactor(2trials)Effectstatistics:changeinmeanexpressedasrawchange,percentchange,orfractionoftheprestandarddeviationOtherstatistics:within-subjectstandarddeviation(notvisibleonaboveplot)=typicalerror:conveyserrorofmeasurementusefultogaugereliability,individualresponses,andmagnitudeofeffects(formeasuresofathleticperformance).,pre,post,strength,trial,Model:nominalvsnominale.g.sportvssexModelortest:chi-squaredtestorcontingencytableEffectstatistics:Relativefrequencies,expressedasadifferenceinfrequencies,ratiooffrequencies(relativerisk),orratioofodds(oddsratio)Relativeriskisappropriateforcross-sectionalorprospectivedesigns.riskofhavingrugbydiseaseformalesrelativetofemalesis(75/100)/(30/100)=2.5Oddsratioisappropriateforcase-controldesigns.calculatedas(75/25)/(30/70)=7.0,females,males,30%,75%,rugbyyes,rugbyno,Model:nominalvsnumerice.g.heartdiseasevsageModelortest:categoricalmodelingEffectstatistics:relativeriskoroddsratioperunitofthenumericvariable(e.g.,2.3perdecade)Model:ordinalorcountsvswhateverCansometimesbeanalyzedasnumericvariablesusingregressionorttestsOtherwiselogisticregressionorgeneralizedlinearmodelingComplexmodelsMostreducibletottests,regression,orrelativefrequencies.Example,age(y),heartdisease(%),0,100,30,50,70,Model:controlledtrial(numericvs2nominals)e.g.strengthvstrialvsgroupModelortest:unpairedttestofchangescores(2trials,2groups)repeated-measuresANOVAwithwithin-andbetween-subjectfactors(2trialsorgroups)Note:uselinediagram,notbargraph,forrepeatedmeasures.Effectstatistics:differenceinchangeinmeanexpressedasrawdifference,percentdifference,orfractionoftheprestandarddeviationOtherstatistics:standarddeviationrepresentingindividualresponses(derivedfromwithin-subjectstandarddeviationsinthetwogroups),pre,post,strength,trial,drug,placebo,Model:extrapredictorvariabletocontrolforsomethinge.g.heartdiseasevsphysicalactivityvsageCantreducetoanythingsimpler.Modelortest:multiplelinearregressionoranalysisofcovariance(ANCOVA)Equivalenttotheeffectofphysicalactivitywitheveryoneatthesameage.Reductionintheeffectofphysicalactivityondiseasewhenageisincludedimpliesageisatleastpartlythereasonormechanismfortheeffect.Sameanalysisgivestheeffectofagewitheveryoneatsamelevelofphysicalactivity.Canusespecialanalysis(mixedmodeling)toincludeamechanismvariableinarepeated-measuresmodel.S.,Problem:somemodelsdontfituniformlyfordifferentsubjectsThatis,between-orwithin-subjectstandarddeviationsdifferbetweensomesubjects.Equivalently,theresidualsarenon-uniform(havedifferentstandarddeviationsfordifferentsubjects).Determinebyexaminingstandarddeviationsorplotsofresidualsvspredicteds.Non-uniformitymakespvaluesandconfidencelimitswrong.HowtofixUseunpairedttestforgroupswithunequalvariances,orTrytakinglogofdependentvariablebeforeanalyzing,orFindsomeothertransformation.AsalastresortUseranktransformation:convertdependentvariabletoranksbeforeanalyzing(=non-parametricanalysissameasWilcoxon,Kruskal-Wallisandothertests).,GeneralizingfromaSampletoaPopulation,Youstudyasampletofindoutaboutthepopulation.Thevalueofastatisticforasampleisonlyanestimateofthetrue(population)value.Expressprecisionoruncertaintyintruevalueusing95%confidencelimits.Confidencelimitsrepresentlikelyrangeofthetruevalue.TheydoNOTrepresentarangeofvaluesindifferentsubjects.Theresa5%chancethetruevalueisoutsidethe95%confidenceinterval:theType0errorrate.Interprettheobservedvalueandtheconfidencelimitsasclinicallyorpracticallybeneficial,trivial,orharmful.Evenbetter,workouttheprobabilitythattheeffectisclinicallyorpracticallybeneficial/trivial/harmful.S.,Statisticalsignificanceisanold-fashionedwayofgeneralizing,basedontestingwhetherthetruevaluecouldbezeroornull.Assumethenullhypothesis:thatthetruevalueiszero(null).Ifyourobservedvaluefallsinaregionofextremevaluesthatwouldoccuronly5%ofthetime,yourejectthenullhypothesis.Thatis,youdecidethatthetruevalueisunlikelytobezero;youcanstatethattheresultisstatisticallysignificantatthe5%level.Iftheobservedvaluedoesnotfallinthe5%unlikelyregion,mostpeoplemistakenlyacceptthenullhypothesis:theyconcludethatthetruevalueiszeroornull!Thepvaluehelpsyoudecidewhetheryourresultfallsintheunlikelyregion.Ifp0.05,yourresultisintheunlikelyregion.,Onemeaningofthepvalue:theprobabilityofamoreextremeobservedvalue(positiveornegative)whentruevalueiszero.Bettermeaningofthepvalue:ifyouobserveapositiveeffect,1-p/2isthechancethetruevalueispositive,andp/2isthechancethetruevalueisnegative.Dittoforanegativeeffect.Example:youobservea1.5%enhancementofperformance(p=0.08).Thereforethereisa96%chancethatthetrueeffectisanyenhancementanda4%chancethatthetrueeffectisanyimpairment.Thisinterpretationdoesnottakeintoaccounttrivialenhancementsandimpairments.Therefore,ifyoumustusepvalues,showexactvalues,notp0.05.Meta-analystsalsoneedtheexactpvalue(orconfidencelimits).,Ifthetruevalueiszero,theresa5%chanceofgettingstatisticalsignificance:theTypeIerrorrate,orrateoffalsepositivesorfalsealarms.Theresalsoachancethatthesmallestworthwhiletruevaluewillproduceanobservedvaluethatisnotstatisticallysignificant:theTypeIIerrorrate,orrateoffalsenegativesorfailedalarms.Intheold-fashionedapproachtoresearchdesign,youaresupposedtohaveenoughsubjectstomakeaTypeIIerrorrateof20%:thatis,yourstudyissupposedtohaveapowerof80%todetectthesmallestworthwhileeffect.Ifyoulookatlotsofeffectsinastudy,theresanincreasedchancebeingwrongaboutatleastoneofthem.Old-fashionedstati