Canonical Correlation: EquationsData for Canonical CorrelationsDataEquationsSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Canonical Correlation: EquationsPsy 524Andrew AinsworthData for Canonical CorrelationsCanCorr actually takes raw data and computes a correlation matrix and uses this as input data.You can actually put in the correlation matrix as data (e.g. to check someone else’s results)DataThe input correlation set up is:xx xyyx yyR RR REquationsTo find the canonical correlations:First create a canonical input matrix. To get this the following equation is applied:1 1yy yx xx xyR R R R R- -=EquationsTo get the canonical correlations, you get the eigenvalues of R and take the square rootci ir l=EquationsIn this context the eigenvalues represent percent of overlapping variance accounted for in all of the variables by the two canonical variatesEquationsTesting Canonical CorrelationsSince there will be as many CanCorrs as there are variables in the smaller set not all will be meaningful.EquationsWilk’s Chi Square test – tests whether a CanCorr is significantly different than zero.2111 ln2 , var var(1 ), , 1, x ymxymm iik kNWhere N is number of cases k is number of x iables andk is number of y iablesLamda is the product of difference between eigenvalues andgecl=� �+ +� �=- - - L� �� �� �� �L = -L� .nerated across m canonical correlationsEquationsFrom the text example - For the first canonical correlation:222(1 .84)(1 .58) .072 2 18 1 ln.072(4.5)( 2.68) 12.04( )( ) (2)(2) 4x ydf k kccL = - - =� �+ +� �=- - -� �� �� �� �=- - == = =EquationsThe second CanCorr is tested as122(1 .58) .422 2 18 1 ln.422(4.5)( .87) 3.92( 1)( 1) (2 1)(2 1) 1x ydf k kccL = - =� �+ +� �=- - -� �� �� �� �=- - == - - = - - =EquationsCanonical Coefficients Two sets of Canonical Coefficients are requiredOne set to combine the XsOne to combine the YsSimilar to regression coefficientsEquations1/ 21/ 2yˆ( ) 'Where ( )' the transpose of the inverse of the "special" matrix ˆform of square root that keeps all of the eigenvalues positive and is a normalized matrix of eigen vectors fy yy yyyB R BR isB--=-1 *x*or yyBWhere is from above dividing each entry by their corresponding canonical correlation.xx xy yy yR R BB B=EquationsCanonical Variate ScoresLike factor scores (we’ll get there later)What a subject would score if you could measure them directly on the canonical variatex xy yX Z BY Z B==EquationsMatrices of Correlations between variables and canonical variates; also called loadings or loading matricesx xx xy yy yA R BA R B==EquationsEquationsRedundancyWithin - Percent of variance explained by the canonical correlate on its own side of the equation121212 2( .74) .79.582xykixcxcixkiycyciyxcapvkapvkpv====- += =��EquationsRedundancyAcross - variance in Xs explained by the Ys and vice versa122 2( )( )( .74) .79(.84) .482cx yrd pv rrd�=� �- += =� ��
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