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2.4.3 Le coefficient de corrélation multiple (ou coefficient de

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predictors). Let's fit a linear model for the prediction of trait anxiety using all five of our personality factors: model1.trait <- lm 



Le rapport de corrélation multiple et ses applications

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Using maximum likelihood estimation as described by R. A. Fisher (1912) a new estimator for the population squared multiple correlation was developed. This.



[PDF] Multiple correlation and multiple regression - The Personality Project

This solution may be generalized to the problem of how to predict a single variable from the weighted linear sum of multiple variables (multiple regression) or 



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[PDF] Multiple correlation and multiple regression - The Personality Project

Chapter5

Multiplecorrelationand multipleregression

Theprevi ouschapterconsideredhowt odeterminetherelat ionshipbetweentwovariables andhowt opredicton efromt heother.Thegeneralsolution wastoc onsiderthe ratioof thecovarian cebetweentwovariablestoth evarianceofthepredictorv ariable(regression) orth eratioofthe covariancetoth es quarerootoft heproductthevariances(correlation). Thissolutionma ybegeneralizedtothe problemofho wtopre dictasinglevariablefromthe weightedlinearsumofmult iplevariables(multipleregression)ort ome asureth estrengthof thisrelationsh ip(multiplecorrelatio n).Aspar toft heproblemof find ingthewe ights,the conceptsofpartialcovarianceandpartialcorrelationwillbeintro duced. Todoallofthiswill requirefindingthevarianc eofacompositescore,and thecovarian ceofthiscompositewit h anotherscore,which mightitselfbeac omposite. Muchofpsychome tri ctheoryismerelyanextension,anelaborati on,orageneralization ofth eseconcepts.Almost alltestsarecompositesofitems orsubtests.An unders tanding howtodecom posetes tvarianceintoitscomponen tparts,andconv ersely,anunderstandi ng howtoanaly zetest sascompositesofitem s,allowsustoanaly zet hemeaningoftests.But testsarenotmerely composit esofit ems.Testsrelatetooth ertests.Adeepappreciationof thebasicPe arsoncorrelationc oe ffi cientfacilitatesan understandingofitsgeneralization to multipleandpartialcorrelation ,tofact oranalysis,andtoque stionsofvalidity.

5.1Thevar ianceofco mposites

Ifx 1 andx 2 arevect orsofNobservationscente re darou ndtheirmean(thatis,deviation scores)theirvariances areV x1 =∑x 2 i1 /(N-1)and V x2 =∑x 2 i2 /(N-1),or,in matrix terms V x1 =x 1 x 1 /(N-1)and V x2 =x 2 x 2 /(N-1).Thev arianceofth ecompositemadeupofthe sum ofth ecorrespon dingscores,x+yisjus t V (x1+x2) ∑(x i +y i 2 N-1 ∑x 2 i +∑y 2 i +2∑x i y i N-1 (x+y) (x+y) N-1 .(5.1) Generalizing5.1tothec aseofnxs,th ecomposite matr ixof theseisju st N X n withdimension s ofNr ows andncolumns.Th ematri xofv ariancesandcovariancesofth eindividu alitemsof thiscompositei swrittenasSasit isasamp leesti mate ofthepopulationvari ance-covariance matrix,Σ.It isperh apshelp fultoviewSinterm sofitselements,n ofwh icharevari ances 127

1285Multiple correlation andmultipleregression

andn 2 -n=n?(n-1)are covarian ces: S= v x1 c x1x2

···c

x1xn c x1x2 v x2 c x2xn c x1xn c x2xn

···v

xn ThediagonalofS=diag(S)is justt hevectorofindi vidualvari ances.ThetraceofSis thesumoft hediagonalsand wil lbeuseda greatdealwhenconsid ering howtoestimate reliability.Itisconvenienttorepre sentth esumofal loftheelementsinthematrix,S,ast he varianceofthecompositem atr ix. V X X X N-1 1 (X X)1 N-1

5.2Multiple regression

Theproblem oftheoptimallinearpre dic tionofˆyinterm sofxmaybegener alizedt othe problemoflinearlypred ic tingˆyinterm sofacompositevariabl eXwhereXismadeu pof individualvariablesx 1 ,x 2 ,...,x n .Ju stasb y .x=cov xy /var x istheop timalslopef orpredicting y,soi tisposs ibleto findase tofweights(βweightsinthes tandardized case,bweightsin theunstan dardizedcase)foreachoftheindividualx i s.

Considerfirsttheproblemof twopredictor s,x

1 andx 2 ,we wantto findthefind weights , b i ,th atwhenmul tipliedbyx 1 andxquotesdbs_dbs2.pdfusesText_2
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