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The correlation matrix below is from the assessment of 5 ratings of digitally enhanced face images based on 5 attributes kindness, intelligence, happiness, Iikability and just/fairness. The following 5 scores constitute the data: X, = Kind, X2 2 Intelligent, X3 2 Happy, X4 = Likeable, X5 = Justj'Fair, with corresponding correlation matrix R as below, 1.000 .296 .881 .995 .545 .296 1.000 .022 .326 .837 R: .881 .022 1.000 .867 .130 .995 .326 .367 l .000 .544 .545 .837 .130 .544 1.000 Eigenvalues of R are 3.263, 1.538, .168, .031, and 0.0001. The first 2 eigenvectors of R are given by, .53? .136 .288 .651 el = .434 and e: = .473 .537 .169 .390 .538 The maximum likelihood (ML) procedure the following estimated factor loadings were extracted: Loadings Variables 3 I j l2} Kind .927 .367 Intelligent .03'.r' .959 Happy .930 .03 1 Like-able .916 .385 Just . 194 .950 a) Write out the formulation of each factor above. (3 marks] to} In your own words, interpret each of the 2 factors above. (4 marks) c) Using the estimated factor loadings and the information regarding the eigenvalues and eigenvectors of the correlation matrix R, obtain the specific variances. Show your formulae and working. (3 marks] d} Using the estimated factor loadings, obtain the communalities. Show your formulae and working. (4 marks] e} Using the estimated factor loadings, obtain the proportion of variance explained by each factor. [3 marks} f) Using the estimated factor loadings, obtain the corresponding cumulative proportions. (3 marks) a! 1') Using the estimated factor loadings, and your answers in b) and c) obtain the residual matrix, given by R _ LL'_ '1' . (5 marks) Does the factor model account well for the variation in the data, based on your computed residual matrix in g)? Justify your answer. [3 marks) Does the factor model account for the variation in the data well, based on your residual matrix? {3 marks) How does Factor Analysis (FA) differ to Principal component analysis {PCA}? What are the similarities between PCA and FA? Answer in 34 sentences. {4 marks)