Equation 15.6 attached below

15.6 Factor Analysis We will have a look at another important facet of multivariate statistical analysis: FactorAnaIy- sis. The data observations xvi = 1, , n, are assumed to arise from an N(p, 2) distribution. Consider a hypothetical example where the correlation matrix is given by 1 0.39 0.01 0.04 0.023 1 0.02 0.042 0.03 Cor(X)= 1 0.93 0.93 1 0.94 1 Here, we can see that the rst two components are strongly correlated with each other and also appear to be independent of the rest of the components. Similarly, the last three components are strongly correlated among themselves and independent of the rst two components. A natural intuition is to think of the rst two components arising due to one factor and the remaining three due to a second factor. The factors are also sometimes called latent variables. The development in the rest of the section is only related to orthogonal factor model and it is the same whenever we talk about the factor analysis model. For other variants, refer to Basilevsky (1994), Reyment and J'oreskog (1996), and Brown (2006). 15.6.1 The Orthogonal Factor Analysis Model Let x = (x1, x2, . . . ,xp) be a pvector. To begin with, we will assume that there are m factors with m can be written in terms of A and I as E = A'A+V. (15.26) Using the above relationship, we can arrive at the next expression: cov(X, f) = A. We will consider three methods for estimation of the loadings and communalities: (i) The Principal Component Method, (ii) The Principal Factor Method, and (iii) Maximum Like- lihood Function. We omit a fourth important technique of estimation of factors in "Iterated Principal Factor Method"1. Canonical correlation analysis quantifies the correlation between a linear combination of variables in one set with a linear combination of potentially different variables in another set and maximizes such correlation among the space of linear combinations. Use equation (15.6) to give the sample canonical correlations if the sample covariance/variance matrix is: 8 1 2 2 3 S = [syy Syx] = [ 5 1 3 sxy Sxx 3 1 6 2 1 3 2 7