University:
California State University, Northridge
Course:
PSY 524/L | Multivariate Analysis Computer and Lab
Academic year:

2006
Views:

404

Pages:

19
Author:

Nadia Crane

Canonical Correlation: Equations Data for Canonical Correlations { { CanCorr 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) Data { The input correlation set up is: Rxx Rxy Ryx Ryy Equations { To find the canonical correlations: z First create a canonical input matrix. To get this the following equation is applied: −1 yy −1 xx R = R Ryx R Rxy Equations z To get the canonical correlations, you calculate the eigenvalues of R and take the square root rci = λi Equations z In this context the eigenvalues represent percent of overlapping variance accounted for in all of the variables by the two canonical variates { i.e. it is the squared correlation Equations { Testing Canonical Correlations z Since there will be as many CanCorrs as there are variables in the smaller set, not all will be meaningful (or useful). Equations { Wilk’s Chi Square test – tests whether a CanCorr is significantly different than zero.   kx + ky +1 χ = −  N −1−   ln Λm 2    Where N is number of cases, kx is number of x variables and 2 ky is number of y variables m Λm = ∏(1− λi ) i =1 Lamda, Λ, is the product of difference between eigenvalues and 1, generated across m canonical correlations. Equations { From the text example - For the first canonical correlation: Λ 2 = (1 − .84)(1 − .58) = .07   2 + 2 + 1  χ = − 8 − 1 −    ln .07 2    2 χ = −(4.5)(−2.7) = 12.15 2 df = (k x )(k y ) = (2)(2) = 4 Equations { The second CanCorr is tested as Λ1 = (1 − .58) = .42   2 + 2 + 1  χ = − 8 − 1 −    ln .42 2    χ 2 = −(4.5)(−.87) = 3.92 2 df = (k x − 1)(k y − 1) = (2 − 1)(2 − 1) = 1 Equations { Canonical Coefficients z Two sets of Canonical Coefficients are required { One set to combine the Xs { One to combine the Ys { Similar to regression coefficients Equations By = (Ryy−1/2 )' Bˆy Where (Ryy−1/2 )' is the transpose of the inverse of the "special" matrix form of square root that keeps all of the eigenvalues positive and Bˆy is a normalized matrix of eigen vectors for yy Bx = Rxx-1 Rxy B*y Where B*y is By from above dividing each entry by their corresponding canonical correlation. Equations { Canonical Variate Scores z Like factor scores (we’ll get there later) z What a subject would score if you could measure them directly on the canonical variate X = Z x Bx Y = Z y By Equations { Matrices of Correlations between variables and canonical variates; also called loadings or loading matrices Ax = Rxx Bx Ay = Ryy By Equations First Set Second Set TS TC BS BC Canonical Variate Pairs First Second -.74 .68 .79 .62 -.44 .90 .88 .48 Equations { Percent of variance in a single variable accounted for by it’s own canonical variate z z This is simply the squared loading for any variable e.g. The percent of variance in Top Shimmies explained by the first canonical variate is -.742 ≈ 55% Equations { Redundancy z Within – Average percent of variance in a set of variables explained by their own canonical variate kx 2 aixc pvxc = ∑ i =1 k x ky pv yc = ∑ i =1 2 iyc a ky (−.74) + (.79) = = .58 2 2 pvxc1 2 Equations { Redundancy z Across – average percent of variance in the set of Xs explained by the Y canonical variate and vice versa rd = ( pv)(r ) 2 c  (−.74) + .79  =  (.84) = .48 2   2 rd x1 → y 2