【复旦大学首批FIST项目传播学研究方法讲义】6-相关分析与简单回归分析

6、相关分析与简单回归分析Correlation&SimpleRegressionAnalysis张伦复旦大学新闻学院2013年FIST课程·传播研究方法Outline本讲内容相关分析o定义与公式o结果解读o相关系数的统计推断检验o扩展学习简单回归分析o定义与公式o简单回归模型建模步骤多元线性回归分析学习目标理解并掌握相关分析/简单回归分析的基本概念能够运用相关分析以及线性回归分析解决实际问题能够对分析结果进行正确、恰当的解读所有公式仅作理解之用，不需背记！2相关分析第一部分如何描述两个变量之间的关系？途径1：散点图散点图表示两个变量之间的关系，有何优点？有何缺陷？o直观o不可比o非标准化途径2：相关系数检验o是否显著？o是否带有普遍性？4-3-2-1012-2-1012(a)xy-2-1012-2-1012(b)xy-2-1012-2-1012(c)xy-3-2-1012302468(d)xyCorrelation简单相关系数，又称皮尔森相关系数（Pearson’sproduct-momentcorrelationcoefficient），测量两个变量线性联系的紧密程度（direction,strength,andsignificance(alongwithsamplesize)）5StatisticalSignificancevs.StrengthofRelationshiprSignificantNon-significantr=1YesImpossible1r≥.7StrongTrivial.7r≥.3ModerateTrivial.3r0WeakTrivialr=0ImpossibleTrivialCaution:readKozak,M.(2009).Whatisstrongcorrelation?TeachingStatistics,31,85-86.原始算法标准化值算法FormalDefinitionCovariancebetweentwovariablesdividedbytheproductoftheirrespectivestandarddeviations12211()()()()niiinniiiixxyyrxxyy2()1()2(1)xyZZrnSource:Cohen,J.,Cohen,P.,West,S.G.,&Aiken,L.S.(1983).Appliedmultipleregression/correlationanalysisforthebehavioralsciences.P278010203040506005101520No.ofPublicationsTimeSincePh.D.(Years)2()r1()2(1)9.614=1()2(151)=.657xyZZn相关系数的检验相关系数的抽样分布随着n的增大越来越接近于自由度为n-2的t分布E.g.,r=.657,n=15odf=15-2=13op.0592t(1)/(2)rrn20.657t(10.657)/(152)3.14Interpretationr:theamountofchangeinzygivenaunitincreaseinzx(orviceversa).r2:theproportionofvarianceinYassociatedwith/causedbyX(orviceversa).PropertiesoftheproductmomentcorrelationcoefficientindependentoftheunitsofmeasurementItsvaluevariesbetweenzero,whenthevariablehavenolinearrelationship,and+1.00or-1.00,wheneachvariableisperfectlyestimatedbytheother.TheabsolutevaluegivesthedegreeofrelationshipPropertiesoftheProductMomentCorrelationCoefficientItssignindicatesthedirectionoftherelationship.oApositivesignindicatesatendencyforhighvaluesofonevariabletooccurwithhighvaluesoftheother,andlowvaluestooccurwithlow.oAnegativesignindicatesatendencyforhighvaluesofonevariabletobeassociatedwithlowvaluesoftheother.Reversingthedirectionofmeasurementofoneofthevariableswillproduceacoefficientofthesameabsolutevaluebutofoppositesign.12相关分析在社会科学研究中的主要用途对变量之间的关系进行描述，判定数据质量oCorrelationMatrix数据分析工具获得新发现的必要手段oGoogleCorrelatebyGoogleLabso:Huffaker,D.Dimensionsofleadershipandsocialinfluenceinonlinecommunities.HumanCommunicationResearch,36(4),593-617.作为数据分析方法15McCombs,M.E.,&Shaw,D.L.(1972).Theagenda-settingfunctionofmassmedia.Publicopinionquarterly,36(2),176-18716AnyObservations?AnyConclusions?简单回归分析第二部分社会科学家理解现实世界的途径理论自然规则：人们试图猜测或近似。“黑匣子”所有的模型都是对黑匣子的各种猜测，且希望这些猜测离真正的规律越近越好模型统计可以根据目前所拥有的信息（数据）来简历人们所关欣的变量和其他有关变量的关系。这种关系成为模型（Model）。因变量=f(自变量,随机噪声,参数)18自然规则XY相关分析与回归分析的区别Noassumptionofcausality：oThroughthecalculationofthecorrelationcoefficient,onecantellwhetherXandYvarylinearlybutCANNOTtellwhetherXaffectsYorYaffectsX.DeterministicModel:oanequationorsetofequationsthatallowustofullydeterminethevalueofthedependentvariablefromthevaluesoftheindependentvariables.o当变量X发生变化时，变量Y发生了多大的变化先来看一个简单的例子：美国某60个著名商学院的数据oX:学生进入MBA学习前的工资oY:学生进入MBA学习后的工资回归模型的基本任务：o寻找到一条直线【拟合】，来适当地【最小二乘法】代表图中散点的趋势。o用这条直线的特点(Y=ax+b)【估计】，来概括自变量与因变量的关系o对这条直线的特点的“显著性”做出判断【显著性检验】o对这条直线与散点的关系进行评判【模型拟合检验，决定系数】203040506080100120140160180SalaryPreMBASalaryPostMBA简单回归模型的定义DependentDependent(Response)(Response)VariableVariable(e.g(e.g.,.,NumberofNumberofPublications)Publications)Independent(Explanatory)Independent(Explanatory)VariableVariable(e.g.,Years(e.g.,Yearssincegraduation)sincegraduation)PopulationPopulationSlopeSlopePopulationPopulationYY--InterceptInterceptRandomRandomErrorErrorYi=0+1Xi+iUnknownRelationshipPopulationPopulationRandomSampleRandomSampleYXiii01图示：如何理解从样本推断总体简单回归模型的定义Simpleregressionline简单回归模型的数据分析步骤ModelSpecification•DefinetheDVandIVs•Hypothesizethenatureofrelationship•Expectedeffects(CoefficientSign)•FunctionalForm(Linear&Nonlinear)•Basedonpreviousresearch/theoriesModelEstimation•Identifypossibleoutliers&Checktherequiredconditions•ModelEstimation•Choosetheappropriateteststatistics•Specifythesignificancelevel(andcriticalvalue)•Estimatetheteststatistics•TestthenullhypothesesModelAssessment•Evaluatetheperformance(i.e.,goodnessofthefit)ofthemodelResultInterpretation•Interpretthe(social)meaningoftheparametersandperformanceofthemodel简单回归模型的数据分析步骤ModelSpecification•DefinetheDVandIVs•Hypothesizethenatureofrelationship•Expectedeffects(CoefficientSign)•FunctionalForm(Linear&Nonlinear)•Basedonpreviousresearch/theoriesModelEstimation•Identifypossibleoutliers&Checktherequiredconditions•ModelEstimation•Choosetheappropriateteststatistics•Specifythesignificancelevel(andcriticalvalue)•Estimatetheteststatistics•TestthenullhypothesesModelAssessment•Evaluatetheperformance(i.e.,goodnessofthefit)ofthemodelResultInterpretation•Interpretthestatistical&socialmeaningoftheparametersandperformanceofthemodelRequiredConditionsofSimpleRegressionNormality（残差