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IntroductionKeydefinitionsthestepsformedicalstatisticsBriefhistoryofStatisticsStatisticsThesciencefordatacollection,sorting,andanalysis.
Definition:thesciencethatstudythecollection,sortingandanalysisofmedicaldata.
Characteristics:
1、Usingthequantitytoreflectthequality2、Usingchanceevents(uncertainty)toreflecttheinevitability(rules)MedicalStatisticsLearningobjectives:1、BasicprinciplesandmethodsofStatistics(LearningEmphasis)
2、ApplicationStatistics——(ClinicalMedicine,PreventiveMedicine,andHealthCareManagement)MedicalStatisticsPurpose:atoolformedicalresearchEmphasis:statisticalindicatorsusedforcalculatingorcomparingthequantitativecharacteristicsofpopulationExample:healthexpectation
infantmortalityMedicalStatisticsSection1.KeydefinitionsⅠvariable,individual,sampleandpopulationindividual(observatoryunit):thebasicunitinstatisticalresearch,itdependsonthepurpose.variable(indicator):individualcharacteristics
examples:height、weight、gender、bloodtype、treatmenteffectetc.Variablevalue:thevalueofvariablesExamples:height1.65metersweight52kggenderfemalebloodtype“O”laboratorytestnegativetreatmenteffectbetterData:composedofalotofvariablevalues.
Example:Dataforbloodglucosehomogeneity:commoncharacteristicsforthegivenindividuals
example:theheightsoftheboyswiththeageof7livinginChangsha2004variation:differenceexistingamongthevariablevaluesofhomogeneityindividuals
example:thedifferentheightsoftheboyswiththeageof7livinginChangsha2004Definition:thewholehomogeneityindividualsdeterminedbyspecificpurpose.example:alltheheightsofboysat7thatlivedinChangsha2004finitepopulation:thespace,timeandpopulationforaspecificpopulationhavebeenlimited.infinitepopulation:
notimeandspacelimitsforthepopulation.Suchpopulationsonlyexistinimagination,soitiscalledinfinitepopulation.populationdefinition:thesetofvariablevaluesofsomeindividualssampledfromthepopulationatrandom.Example:theheightsof200boysat7fromChangsha.sampleSamplingstudySampleinformation(statistic)Populationcharacteristics(parameter)inferencenote:samplingisonlythewaytogetinformation,inferringthepopulationisourpurposeⅡ、variableanddata
measurementdata:itisalsocalledasquantitativeornumericaldata.Itsvalueisquantitative.Measurementdataalwayshasmeasurementunits.
example:heightdata,weightdata
enumerationdata:qualitativeorcountdata.Forsuchdata,itneedstoclassifytheobservationunitsbeforeandcountthem.Itsvalueappeardifferentcharacteristicsandsorts.Binomial:gender,liveordeath,yesorno.Multiple:bloodtype,A、B、O、AB.
rankeddata:ordinalorsemi-quantitativedata.Itneedtoclassifyobservatoryunitsintodifferentclassesaccordingtheextentbeforecalculatethefrequenciesofeachgroups.Thereexistsobviousdifferencesamongdifferentclasses.example:toevaluatethetreatmenteffectofonedrugonheartfailure,weusetheindicator(cured,better,worsen,dead)toassessthetreatmenteffect.Choosingofstatisticalmethodsdependsonthedatatypetoagreatextent。
DatatransformationQuantitativedata
rankeddata(multiple)binomialdataexample:WBC(1/m3)countoffivepersons:
300060005000800012000quantitativevariablelowernormalnormalnormalhigherqualitativevariable
Binomialdata:normal3persons;abnormal2personsMultiplecategorydata:lower1person;normal;3persons;higher1personⅢerrordefinition:thedifferencebetweenmeasurementvalueandtruevalue.1、randerror:unstableandchangingatrandom
errorsthatcausedbyuncontrolledfactors.Commonly,randerrorsarereferredtothoseerrorsappearingduringrepeatedmeasurementsandsampling.Often,measurementerrorisextremelylowerthansamplingerror.InStatistics,samplingerroristhemainstudycontents.2.Nonrandomerrorisdividedintosystematicerrorandnonsystematicerror:Systematicerror:itisproducedinexperimentandkeepsconstantorchangesaccordingcertainrules.Usually,itsreasonsareknownandcontrollable.Nonsystematicerror(grosserror):itisalwayscausedbyobviousgrosses.Ⅳ、frequencyandprobability
1.Frequency
Giventhesamecondition,repeatatrialforntimesindependently.Amongntrials,Aappearsformtimes,sotheratioofm/niscalledthefrequencyofrandomeventAamongntrials.
2.probability:thelikelihoodofrandomevents.Giventhesamecondition,repeatatrialforntimesindependently.Amongntrials,Aappearsfor
times,sotheratioof
iscalledthefrequencyofrandomeventA.Asnincreasesgradually,thefrequency
willapproachaconstant.TheconstantiscalledtheprobabilityofrandomeventAandexpressedin.Incommon,itisabbreviatedas.Range:
Frequencyisusedtodescribethesample,whiletheprobabilityforthepopulation.m/nistheestimationof.Astrialsincreases,theestimationvalueismorereliable.smallprobabilityevent:Becausetheconclusionsaremadebasedonacertainsignificancelevel,statisticiansalwaysselectasjudgecriterion.Sosucheventswitharecalledsmallprobabilityevents.Itmeansthatsucheventshappenrarelyandcanberegardedasnonoccurrence.
Section2thestepsforstatisticalworkHere,itmeansstatisticaldesign,themostimportantfactorforasuccessfulresearch.
Itinvolvesthearrangementsfortheprocessofdatacollection,sortingandanalysis.Ⅰdesign
3.controlThreeprinciplesforexperimentdesign1.randomization2.Replicationobjective:togatheraccurateandreliablerawdata
datasources:
①statisticalreporting
②routinerecords
③purposivesurveysorexperiments
④statisticalyearbookandspecialdatabook
requirements:1、randomization
2、sufficientsamplesizeⅡDatacollectionⅢ.DatasortingItistheprocessthatcleansandsystematizesrawdata.Datasortingpreparestherequireddatafornextstep,dataanalysis.ⅣData/statisticalanalysisobjective:toillustratetheruleshiddeninthedata.
Itincludestwoaspects:1.statisticaldescription:itistheprocessofdescribingthecharacteristicsofdatausingstatisticalindicators,statisticalchartsandstatisticaltables.
2.statisticalinference:theprocessofusingsamplestatistictoinferpopulationparameter.Itconsistsof:parameterestimationandhypothesistesting.
35Content1.Samplingerrorandstandarderrorofmean2.t-distribution3.EstimationofPopulationMean4.t-test5.Noticeofhypothesistest
6.Normalitytestandhomogeneityofvariancetest36§1samplingerrorofmeanandstandarderror37Statisticalinference:drawconclusionsofpopulationfromsampleddata.Whenyouusestatisticalinferenceyouareactingasifthedatacomefromarandomsampleorarandomizedexperiment.38①;②samplemeansarenotequal;③thedistributionofsamplemeansaresymmetry;and④variancearegreatlyreducedcomparedtotheoriginalvariable.Characteristicsofsamplemean:391samplingerrorThedifferencebetweenthestatisticandparametercausedbyindividualvarianceandsampling.40Thestatisticalindicatorreferstoquantityofsampleerrorstandarderrorofmean,SEMreferstoquantityofsampleerrorofmean(3-1)2standarderror,SE41Ithasbeenproved:
42WhensamplestandarddeviationSisusedtoestimatepopulationstandarddeviation:
(3-2)43§2t-distribution441tdistribution----degreeoffreedom,df45Foranormaldistributionfollowsastandardnormaldistribution---N(0,1).Whenisunknown,
followsatdistribution.462figureandcharacteristicoftdistributiondistributionhasonlyoneparameter-----degreeoffreedom47
Figure3-3tdistributionwithdifferentdegreeoffreedom481)characteristic492)ttable:One-sidedprobability;:Two-sidedprobability50Example:
51§3Estimationofpopulationmean521parameterestimationestimatingpopulationparameterbysamplestatistic53Havenotconsideraboutthesamplingerror54
Anestimatedrangeofpopulationparameteraccordingtoanappointedprobability(1
).If
=0.05,then95%confidenceinterval;If=0.01,then99%confidenceinterval.2)intervalestimation552calculatingconfidenceintervalofpopulationmean56
1)CIofonepopulationmeanExample3-2575859Example3-3Randomsampled200adults,meanortheirbloodserumcholesterinwas3.64mmol/L,andSdwas1.20mmol/L,estimatethepopulationmean.60
2)CIofthedifferenceoftwopopulationmeans
or613meaningsofconfidenceinterval624differencebetweenCIofpopulationmeanandreferencerange63§4
ttest64situationsthatt-testcouldbeapplied:
Thevariablefollowsanormaldistribution;Samplesizeissmall;Thevariancesareequal.65Example3-5Hbof36workersengagingworkswithplumbumweretested,themeanofHbwas130.83g/L,andstandarddeviationis25.74g/L.ItisknownthattheaverageHbofnormaladultmanis140g/L.IstheredifferenceonHbbetweenworkerwithplumbumandnormaladultman?130.83g/L≠140g/LMaydueto:
A.thetwopopulationmeanaredifferentB.thesamplingerrorQuestion:Whichisthetruth?
--problemofhypothesistest!66Inhypothesistest(significancetest),thequestionofinterestissimplifiedintotwocompetingclaims/hypothesesbetweenwhichwehaveachoice;thenullhypothesis,denotedH0,againstthealternativehypothesis,denotedH1.Thesetwocompetingclaims/hypothesesarenothowevertreatedonanequalbasis,specialconsiderationisgiventothenullhypothesis.Wehavetwocommonsituations:Basictheoryandapproachesofhypothesistest67
Basictheory:
Underthenullhypothesis
Howpossibletooccurthecurrentsituationandevenmoreunfavorablesituationto
?--Calculateaprobability(-value)Ifitislesspossibletooccurthecurrentsituationandevenmoreunfavorablesituationto,thenreject;otherwise,notreject.
--Givenasmall,compareand
(iscalledthesignificancelevelofthetest)68Sethypothesesandthesignificanceleveloftestnullhypothesis(H0):ThestatementbeingtestedinatestofsignificanceAlternativehypothesis(H1):ThestatementwehopeorsuspectistrueinsteadofH0
One-sidedandtwo-sidedalternativesSignificancelevel
,often
=0.0569
Itisarandomvariablewithadistributionthatweknow.
IfXfollowsanormaldistributionThen(2)
Selectanappropriatetestandcalculatetheteststatistic70
Theprobability,computedassumingthatH0istrue,thattheteststatisticwouldtakeavalueasextremeormoreextremethanthatactuallyobservediscalledthePvalueofthetest.
(3)
DeterminePvalue,andmakedecision
71Figure3-5sketchmapofPvalueinexample3-572IfthePvalueisassmallorsmallerthan,wesaythatthedataarestatisticallysignificantatlevel,andrejectH0,altertoH1.731onesample/groupt-test
Totestthehypothesisbasedonansamplesizeofn,andastandarddeviationSfromapopulationwithunknownmean74Forexample3-5,(1)SethypothesesandthesignificanceleveloftestH0:
=
0=140g/L,H1:
≠
0=140g/L
=0.0575(2)calculatetheteststatistic76(3)DeterminePvalue,andmakedecision
77Canbeappliedinthesituationofpaireddesignedquantitativedata.
2paired/matchedt-test78
Example3-610lacticacidbeverageproductswererandomlysampled,twomethodswereusedtodeterminethefatcontent.Istheredifferencebetweenthetwomethods?79Table3-3resultsofthetwomethods(%)
80
(1)SethypothesesandthesignificanceleveloftestH0:
d=0H1:
d≠0
=0.05
(2)calculatetheteststatisticn=10,
d=2.724,
d2=0.8483,
81
(3)DeterminePvalue,andmakedecision
P<0.001,onthelevelof
=0.05,rejectH0,acceptH1,thereisdifferencebetweenthetwomethods.823two-sample/groupt-testcanbeappliedforcomparingoftwomeansofcompletelyrandomdesignedsamples.83Example3-784
§5Notice851typeoneerrorandtypetwoerror86IfwerejectH0(acceptH1)wheninfactH0istrue,thisisaTypeIerror.
Ifwedon’trejectH0(rejectH1)wheninfactH1istrue,thisisaTypeIIerror
Ifisincreased,then
willbedecreasedwithacertain
n.87Thepowerofafixedleveltestagainstaparticularalternativeis1minustheprobabilityofatypeIIerrorforthatalternative.88Figure3-6TypeIerrorandtypeIIerror
holdCriticalvalue892notice1)rigorousresearchdesign2)differenttestmethodsfordifferentkindofdata3)understandingthemeaningof“significant”904)conclusioncannotbeabsolute5)statisticalconclusionshouldbecombinedwiththespecialtyconclusion6)correctlyuseconfidenceintervalandhypothesistest91§5normalitytestandFtestfortwosamplevariancescomparison921normalitytest
1)graphicmethod:P-Pplot,Q-Qplot2)methodofmoment
skewness,
kurtosis3)Wtestmethod4)Dtestmethod932Ftestfortwosamplevariancescomparison
1.Levenetest2.FtestSection1thebasicideaandconditionofapplicationObjective:deduceandcompareseveral(ortwo)populationmeans.Method:analysisofVariance(ANOVA),ieFtestforcomparingseveralsamplemeans.
Basicidea:accordingtothetypeofdesign,thesumofsquaresofdeviationfrommeans(SS)anddegreeoffreedom(df)weredividedintotwoorseveralsections.Exceptthechanceerror,thevariationofeverysectioncanbeexplainedbyacertainorsomefactors.ConditionofApplication:population:normaldistributionandhomogeneityofvariance.Sample:independentandrandomTypesofdesign:TheANOVAofcompletelyrandomdesign;TheANOVAofrandomizedblockdesign;TheANOVAofLatinsquaredesign;TheANOVAofcross-overdesign;ThebasicideaofANOVAofcompletelyrandomdesign
partitionofvariationsumofsquaresofdeviationsfrommean,SS
:1.totalvariation:thedegreeofvariationofallvariablevalues,theformulaasfollowsamendfactor:
2.between-groupvariation:thesumofsquaresofdeviationsfrommeanbetweengroupsmeansandgrandmeanshowtheeffectsoftreatmentandrandomerror,theformula:3.Within-groupVariation:differencesamongvalueswithineachgroup.Theformulaasfollows:
therelationofthreevariation
meansquare,MSTeststatistic:
If, weretheestimatedvalueoftherandomerror,Fvalueshouldbecloseto1.Ifwerenotequal,Fvaluewillbelargerthan1.
Section2TheANOVAofCompletelyRandomDesignAllofobjectswererandomlydistributedtoggroups(levels),andeverygroupgivethedifferenttreatment.Theeffectsoftreatmentwillbededucedbycomparingthegroupsmeansafterexperimentation.
completelyrandomdesign
Example4-1Adoctorwanttoexplorethecliniceffectofanewmedicineforreducingbloodfat,andselects120patientsaccordingtothesamestandard.Allofpatientsweredivideinto4groupsbythecompletelyrandomdesign.Howshouldhedividethegroups?
Themethodsofdividinggroupsofcompletelyrandomdesign
1.serialnumber:120patientswasnumberedfrom1to120(table4-2column1);2.
choosingrandomfigure:youcanbeginfromtheanyroworanycolumnintheappendix15(forexamplebeginningfromthefifthrowandseventhcolumn),andreadthreedigitinturnasarandomnumbertowritedowntheserialnumber,(table4-2,column2)
3.editserialnumber:editserialnumberaccordingtothenumberfromsmalltolarge(thesamenumberaccordingtoearlyorlateorder)(table4-2,column3)
4.defineinadvance:theserialnumbersfrom1-30weredefinedtheAgroup;31-60weretheBgroup;61-90weretheCgroup;91-120weretheDgroup,(table4-2,column4)(2)thechoiceofstatisticmethods1.Ifthedataaccordwithnormaldistributionandhomogeneityofvariance,one-wayANOVAorindependentttestwasused(g=2);2.Ifthedataarenotnormaldistributionorheterogeneityofvariance,thedatumtransformorWilcoxonranksumtestcanbedone.decomposeofvariation
Example4-2Adoctorwantedtoexplorethecliniceffectofanewmedicineforreducingbloodfat,andselected120patientsaccordingtothesamestandard.Hedividedallofpatientsinto4groupsbythecompletelyrandomdesign.Thelowdensitylipoproteinweremeasuredafter6weeksbydoubleblindexperiment,table4-3.Istheredifferenceamongthepopulationmeansoflowdensitylipoproteinof4groups?
Table4-3thelowdensitylipoprotein
valueof4treatmentgroups(mmol/L)三、stepsofanalysisH0:ie.allof4populationmeansareequal.H1:notallofthepopulationmeansareequal2.Calculateteststatistic1.StatethehypothesesandtestcriteriaTable4-5thetableofANOVAofcompletelyrandomdesignlisttheANOVAtable3.Calculatepvalueanddeduceaccordingtoa=0.05level,reject,andaccept,notallof4populationmeansareequal;ie.differentdosemedicineshavedifferenteffectsonldl-c.
attention:iftheresultofANOVAistorejectH0,andacceptH1,itdoesnotmeanthatallofpopulationmeanshavedifferenceeachother.Ifanalysingwhichgroupshavesignificantdifference,wemustcompareamongseveralpopulationmeans(section6).Wheng=2,theANOVAofcompletelyrandomdesignisequaltoindependentttest,ie.
Section3TheANOVAofrandomizedblockdesignrandomizedblockdesign
Firstly,matchtheobjectsastheblocksaccordingtothenon-treatmentfactoraffectingtheresultofexperiment(suchassex,weight,age,occupation,stateofillness,courseofdiseaseetal).
Secondly,theobjectsofeachblockwererandomlydistributedtoeachtreatmentgrouporcontrolgroup.(1)groupingmethodofrandomizedblockdesign
:(2)characteristicof
randomizedblockdesign
Randomdistributionwasrepeatedmanytimesforobjectsoftheblocks.Thenumberofobjectsissameineverytreatmentgroup.SSoftheblockvariationwasseparatedfromSSofthewithin-groupvariationofcompletelyrandomdesign;SSofwithin-group(sumoferrorsquare)wasdecreased,andpoweroftestwasincreased.
example4-3distribute15whitemiceof5blockstothreetreatmentgroups,howtodoit?Groupingmethod:firstly,numberthemicebytheweight,andmatchthe3nearweighmiceasablock(table4-6).Secondly,select2digitasonerandomnumberfromanyroworanycolumnintherandomnumbertable,forexample,fromthe8throwandthirdcolumn(table4-6);andranktherandomnumberfromsmalltolargeineveryblock.Theobjectofserialnumberineachblockis1,2,3willacceptA,B,Ctreatmentrespectively.(table4-6)
table4-7theresultofrandomblockdesign
partitionofvariation(1)Totalvariation:SStotal.(2)Treatment-groupvariation:SStreatment.(3)block-groupvariation:SSblock.(4)Errorvariation:SSerror.
table4-8theANOVAof
randomblockdesign
Stepsofanalysis
example4-4
15miceweredividedinto5blocksbytheweight.thereare3miceineveryblock.theresultshowedintable4-9.istheredifferenceamong3treatmentgroups?
table4-9thevariablevaluesofdifferentgroups(g)
H0:
H1:notofallpopulationmeansareequalaccordingto
1=2、
2=8,checkFvaluetable:
Atα=0.05level,rejectH0,acceptH1,notallofpopulationmeansareequal.
section6
multiplecomparisoncantheaboveexamplebeanalyzedbyttest?
Numbersofttesta=0.05,theprobabilityofnon-typeIerrorforonecomparison:1-0.05=0.95;theprobabilityofnon-typeIerrorforallof6timesanalysis:=0.77;theprobabilityoftypeIerrorfor6timesanalysis:1-0.77=0.23theprobabilityoftypeIerrorwillbeincreasedConditionofapplication:whentheresultofANOVArejectH0,andacceptH1,notallofpopulationmeansareequal.Ifwantingtoknowthedifferencebetweenanytwogroupmeans,weshoulddothemultiplecomparison.LSD-ttest
(leastsignificantdifference)Theformula
example4-7fortheexample4-2data,aretheredifferenceamongthepopulationmeansof2.4g、4.8g、7.2gandplacebogroup?α=0.05Comparingbetween2.4gandplacebogroup:4.8gVSplacebogroup:LSD-t=-4.297.2gVSplacebo:LSD-t=-8.59。
Dunnett-ttest
formula:Dunnett-
example4-8accordingtoexample4-2,compare3populationmeansoftreatmentgroupsandplacebogroup,respectively?
H0:μi=μ0H1:μiμ0α=0.05Dunnett-Dunnett-Dunnett-三、SNK-qtest
(Student-Newman-Keuls)Example4-9accordingto4-4,comparethe3groupmeansbySNK-qtest
H0:μA=μBH1:μA≠μB,α=0.05rankthe3groupmeansfromsmalltolargeandnumberthem
Table4-15thecomparingbetweentwogroupmeansContentRate、proportionandratio
ApplicationofrelativenumbersStandardizationofrateDynamicseriesandanalysisindexSection1RelativeNumbers1、Rate2、Proportion3、Ratio1、Rate
Rate:Todescribethefrequencyorintensionofsomephenomenon.=Numberofindividualoccurredsomethingwithinaperiodoftime
RateThewholenumberoflikelytooccurredsomethinginthesameperiod
Example1
Toinvestigate8589oldpeopleinsomecityin1998,and2823peoplehadhypertension.Morbidityrate:2823/8589
100%=32.87%2、ProportionProportion:Todescribetheratioofnumberofonepartandthewholenumberinthesamething.Formula:100%=NumberofindividualsinonepartProportionThewholenumberofindividualsExample2Calculatethepatientsproportionof5diseasesinonehospitalin1990and1998.
Example2Calculatethepatientsproportionsof5diseasesinonehospitalin1990and1998.Characteristics:(1)Summationofproportionsinonethingis100%.
(2)Proportionsinthesamethingareinteractional.3、RatioRatio:thequotientoftworelatedindexsFormula:(100%)AindexRatio=BindexExample3
Thereare370malenewbornsand358femalenewbornsinahospitalinoneyear,thenThesexratioofnewbornbabies:370/358×100=103Section2Applicationofrelativenumbers
1、Thedenominatorofrelativenumbershouldnotbetoosmall.
2、Proportionshouldnotsubstituterate.3、Tocalculatethetotalratecorrectly.4、Comparisonofrelativenumbers5、Comparisonofsamplerate(proportion)shoulddohypothesistest.Section3
Standardizationofrate
1、DefinitionTocalculatestandardratebyuniforminteriorconstitute.Standardization(oradjustment)ofratesisusedtoenablethevalidcomparisonofgroupsthatdifferregardinganimportanthealthdeterminant(mostcommonlyage).Itisinfactaspecificapplicationofthegeneralmethodstocontrolforconfoundingfactors.2、CalculationMethodDirectstandardizationIndirectstandardizationApproach1.Choosethecorrectmethodbyconditionofdata.2.Choosestandardcomposing.3.Calculatestandardrate.Formula
Directstandardization
IndirectstandardizationExample4Tocalculatestandardcurerateoftwotherapeutics.Approach:1)Diseasecurerateoftwotherapeuticsisknown-Directstandardization2)Choosetotalpatientsnumberoftwotherapeuticsasstandard.3)Calculateanticipatedcurenumber.4)Calculatestandardcurerate.
380100%47.5%800
=StandardcurerateofA
427100%53.4%800
=StandardcurerateofB
Example5Aresearchinvestigatedoldwomen,776inthecityand789inCountryside.Amongthem,322and335sufferedfromprimaryosteoporosis.Thetotalmorbidityratesare41.5and42.5respectively.Becausetheproportionsofageinurbanandruralareasofthisinvestigationformsaredifferent,soweneedtostandardizethetwomorbidityrate.3221.05305SMR=Urbanstandardmorbidityratio42.1%1.05=44.2%
=Urbanstandardmorbidityrate3350.95353SMR=Ruralstandardmorbidityratio42.1%0.95=40.0%
=RuralstandardmorbidityrateAfterstandardization,urbanmorbidityrateishigherthanrural.
3、Application1.TheStandardizationonlyadapttothatinteriorformsaredifferentintwogroups,andmayinfluencethecomparisonofrate.
2.Becauseofdifferentchosenstandardpopulation,standardizedratesaredifferenttoo.So,whilecomparingseveralstandardizedrates,shouldadoptthesamestandardpopulation.3.Standardizedrateisnolongerthelocalreallevelatthattime,itonlyshowstherelativelevelamongthecomparingmaterials.4.Thestandardizedratesoftwosamplesaresamplevalues,thesamplingerrorexists.Whencomparingthestandardizedratesoftwosamples,weshoulddohypothesistestifthesamplesizeissmall.Section4
Dynamicseriesandanalysisindex
Dynamicseries:Aseriesofstatistica
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