MultivariateAnalysisofVariance~MANOVASPSSSPSSMANOVA~thehistory�DevelopedasatheoreticalconstructbyS.S.Wilksin1932�PublishedinBiometrika�WideavailabilityofcomputersmadethesemethodspracticalforresearchersMANOVA~thedefinition�Techniqueforassessinggroupdifferencesacrossmultiplemetricdependentvariables(DV’s)simultaneously,basedonasetofcategorical(non-metric)variablesactingasindependentvariables(IV’s)ANOVAvs.MANOVA�ANOVA~only1dependentvariable�MANOVA~2ormoredependentvariables�Bothareusedwithexperimentaldesignsinwhichresearchersmanipulateorcontroloneormoreindependentvariablestodeterminetheeffectonone(ANOVA)ormore(MANOVA)dependentvariablesEquations�ANOVAY1=X1+X2+X3+...+Xn(metricDV)(non-metricIV’s)�MANOVAY1+Y2+...+Yn=X1+X2+X3+...+Xn(metricDV’s)(non-metricIV’s)MANOVAandRegression�Notethedifferentterminology�Inmultipleregression,univariateandmultivariate/multiplerefertothenumberofIV’s�InANOVAandMANOVAdiscussions,univariateandmultivariaterefertothenumberofDV’sUnivariateResearchExampleAdappeal?�Subjectsshowndifferentadvertisingmessages�EmotionalorInformationalor??�Viewersrateappealofthemessageusingscoresfrom1to10UnivariateReview~tTest�Twocommercialsshown(emotional~informational)�Singletreatment/factorwithtwolevels�UseatTest:OneIV’s,oneDV,twotreatmentgroups�Thetstatistic=M1-M2----------------------SEM1M2UnivariateReview~ANOVA�Twoormorecommercials(emotional~informational~funny~etc)�Singletreatment/factorwithtwoormorelevels�UseANOVA:MultipleIV’s,oneDV,twoormoretreatmentgroups�Fstatistic=MSB------MSWUnivariateHypothesisTesting�NullHypothesis(H0)~ThatthereisnodifferencebetweentheDVmeansofthetreatmentgroups�AlternateHypothesis(HA)~ThatthereisastatisticallysignificantdifferencebetweentheDVmeansofthetreatmentgroupsMultivariateResearchExample�Subjectsshowndifferentadvertisingmessages�EmotionalorInformationalor??�Viewersrateappealofthemessageusingscoresfrom1to10Adappeal?WillIbuy?MultivariateProcedures�Hotelling’sT2OneIV,multipleDV’s,twogroups�ThekGroupCase:MANOVAMultipleIV’s,multipleDV’s,morethantwotreatmentgroups�NullHypothesis~thatthereisnodifferencebetweenvectorsofmeansofmultipleDV’sacrossthetreatmentgroupsNullHypothesisTestingANOVA�H0:M1=M2=...Mk�H0:Allthegroupmeansareequal,thatis,theycomefromthesamepopulationsMANOVAM11M21Mp1M12M22Mp2M1kM1kMpk=...==�H0:Allthegroupmeanvectorsareequal,thatis,theycomefromthesamepopulations心理学案例——问题的提出�考查不同年级、不同性别学生的学习成绩和自我概念的差异。�其中，�学习成绩指语文和数学成绩；�自我概念包括3个大的维度：学业自我，身体自我和社会自我，这3个维度是有相关的。问题中的变量�自变量有两个�性别�年级（以考察高一、高二、高三为例）�因变量也有两个�学业成绩�自我概念不同年级、性别的学生自我概念的差异比较——MANOVA�为什么用MNOVA？�由于一个人对自己的看法是相互联系的，因此要同时考察青少年在学习、身体和社会交往等方面自我评价的差异就必须用MANOVA�多元方差分析的结果要分3步进行考察。学生自我概念的差异比较（MANOVA）－步骤（第一步）�第一步考察�各自变量（年级、性别）的主效应是否显著�交互作用是否显著。�要回答的问题�总体上看，青少年自我概念是否有年级和性别差异？�年级和性别之间是不是有交互作用？MANOVA演示.sav学生自我概念的差异比较（MANOVA）－OUTPUT（第一步）MultivariateTestsc.9896133.281a3.000206.000.000.0116133.281a3.000206.000.00089.3206133.281a3.000206.000.00089.3206133.281a3.000206.000.000.0752.683...
