醫(yī)學(xué)統(tǒng)計(jì)學(xué)課件(英文版)

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