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Concept

Regressionanalysisisamathematicalmodel.Whenthedependentvariableandtheindependentvariablehavealinearrelationship,itisaspeciallinearmodel.

Thesimplestcaseisone-variablelinearregression,whichconsistsofanindependentvariableandadependentvariablethatareroughlylinearlyrelated;themodelisY=a+bX+ε(Xistheindependentvariable,YisthecauseVariable,εisrandomerror).

Usuallyassumethatthemeanvalueoftherandomerroris0,andthevarianceisσ^2(σ^2﹥0,σ^2hasnothingtodowiththevalueofX).Ifitisfurtherassumedthattherandomerrorfollowsanormaldistribution,itiscalledanormallinearmodel.Generally,iftherearekindependentvariablesand1dependentvariable,thevalueofthedependentvariableisdividedintotwoparts:onepartisaffectedbytheindependentvariable,thatis,expressedasitsfunction,thefunctionformisknownandcontainsunknownparameters;theotherpartisdeterminedbyOtherunconsideredfactorsandrandomeffectsarerandomerrors.

Whenthefunctionisalinearfunctionwithunknownparameters,itiscalledalinearregressionanalysismodel;whenthefunctionisanonlinearfunctionwithunknownparameters,itiscalledanonlinearregressionanalysismodel.Whenthenumberofindependentvariablesisgreaterthan1,itiscalledmultipleregression,andwhenthenumberofdependentvariablesisgreaterthan1,itiscalledmultipleregression.

Regressionanalysiscontent

Themaincontentofregressionanalysisisasfollows:

①Startingfromasetofdata,determinethequantitativerelationshipbetweencertainvariables;Thatis,amathematicalmodelisestablishedandunknownparametersareestimated.Usuallytheleastsquaremethodisused.

②Testthetrustworthinessoftheserelations.

③Intherelationshipbetweenmultipleindependentvariablesaffectingadependentvariable,judgewhethertheindependentvariablehasasignificantimpact,andselectthesignificantimpactintothemodel,andeliminateinsignificantvariables.Stepwiseregression,forwardregression,andbackwardregressionareusuallyused.

④Usetherequiredrelationshiptopredictorcontrolacertainprocess.

Theapplicationofregressionanalysisisveryextensive,andtheuseofstatisticalsoftwarepackagescanmakevariousalgorithmsmoreconvenient.

Typesofregression

Themaintypesofregressionare:linearregression,curvilinearregression,binarylogisticregression,andmultiplelogisticregression.

Applicationofanalysis

Correlationanalysisstudiesthecorrelationbetweenphenomena,thedirectionandclosenessofcorrelation,andgenerallydoesnotdistinguishbetweenindependentvariablesordependentvariables.Regressionanalysisistoanalyzethespecificformsofcorrelationbetweenphenomena,determinethecausalrelationship,andusemathematicalmodelstoexpressthespecificrelationship.Forexample,fromthecorrelationanalysis,wecanknowthatthe"quality"and"usersatisfaction"variablesarecloselyrelated,butwhichvariablebetweenthesetwovariablesisaffectedbywhichvariable,andthedegreeofinfluence,requiresregressionanalysisMethodtodetermine.

Generallyspeaking,regressionanalysisistodeterminethecausalrelationshipbetweendependentvariablesandindependentvariables,establisharegressionmodel,andsolvethevariousparametersofthemodelbasedonthemeasureddata,andthenevaluatewhethertheregressionmodelisItcanfitthemeasureddatawell;ifitcanfitwell,furtherpredictionscanbemadebasedontheindependentvariables.

Forexample,ifyouwanttostudythecausalrelationshipbetweenqualityandusersatisfaction,inapracticalsense,productqualitywillaffectusersatisfaction,sosetusersatisfactionasthedependentvariableandrecorditasY;Qualityistheindependentvariable,denotedasX.AccordingtothescatterplotinFigure8-3,thefollowinglinearrelationshipcanbeestablished:

Y=A+BX+§

where:AandBareundeterminedparameters,andAisregressionTheinterceptofthestraightline;Bistheslopeoftheregressionline,whichrepresentstheaveragechangeofYwhenXchangesbyoneunit;§istherandomerrortermthatdependsonusersatisfaction.

LinearregressioncanbeeasilyimplementedintheSPSSsoftware.Theregressionequationisasfollows:

y=0.857+0.836xTheinterceptoftheregressionlineontheyaxisis0.857andtheslopeis0.836,Thatis,foreveryonepointimprovementinquality,usersatisfactionincreasesby0.836pointsonaverage;inotherwords,thecontributionofeveryonepointimprovementinqualitytousersatisfactionis0.836points.

Theexampleshownaboveisasimplelinearregressionproblemofoneindependentvariable.Duringdataanalysis,thiscanalsobeextendedtomultipleregressionofmultipleindependentvariables.PleaserefertothespecificregressionprocessandmeaningRefertorelevantstatisticsbooks.Inaddition,intheSPSSresultoutput,R2,FtestvalueandTtestvaluecanalsobereported.R2isalsocalledthecoefficientofdeterminationoftheequation,whichindicatesthedegreeofinterpretationofthevariableXtoYintheequation.ThevalueofR2isbetween0and1.Thecloserto1,thestrongertheinterpretationabilityofXtoYintheequation.R2isusuallymultipliedby100%toexpressthepercentageofYchangeexplainedbytheregressionequation.TheFtestisoutputthroughtheanalysisofvariancetable,andthesignificancelevelisusedtotestwhetherthelinearrelationshipoftheregressionequationissignificant.Generallyspeaking,significancelevelsbelow0.05aremeaningful.WhentheFtestpasses,itmeansthatatleastoneoftheregressioncoefficientsintheequationissignificant,butnotallregressioncoefficientsaresignificant,soaTtestisneededtoverifythesignificanceoftheregressioncoefficients.Similarly,theTtestcanbedeterminedbythesignificanceleveloralook-uptable.Intheexampleshownabove,themeaningofeachparameterisshowninTable1-1.

Table1-1linearregressionequationtest