facan - A Quick Introduction to Factor Analysis From Data...

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A Quick Introduction to Factor Analysis From Data Analysis: A Statistical Primer for Psychology Students by Edward L. Wike: . . . [T]here are elaborate extensions of r to multivariate data . . . another case is k variables (factor analysis) . Fortunately, for both the student and author, none of these extensions will be considered here. From Linear Algebra and its Applications by Gilbert Strang: . . . [T]he technique is so much needed that, even starting as an unwelcome black sheep of multi- variate analysis, it has spread from psychology into biology and economics and the social sciences. From linear algebra: A “matrix” is a rectangular (in this discussion, usually square) array of numbers; e.g., 3 5 7 1 2 3 - 1 3 0 3 . 5 2 0 2 - 1 We can add two matrices of the same dimensions, by adding corresponding entries; and we can multiply a matrix by a number, by multiplying all the entries in the matrix by that number: 1 2 0 - 1 + - 2 1 2 3 = - 1 3 2 2 3 1 2 0 - 1 = 3 6 0 - 3 A square matrix is a “diagonal matrix” if the only nonzero entries (if any) are on the “main diagonal” (upper left to lower right): 5 0 0 0 - 1 0 0 0 0 The “transpose” of a matrix is the (new) matrix obtained turning the rows into columns (and vice versa): 3 5 7 1 2 3 T = 3 1 5 2 7 3 - 1 3 0 3 . 5 2 0 2 - 1 T = - 1 3 0 3 . 5 2 0 2 - 1 If a (square) matrix A has the property that A T = A (as this one does), then A is called “symmet- ric”. This just amounts to saying that, for each i and j , the entry in row i and column j is equal to the entry in row j and column i , i.e., the matrix is symmetric about its main diagonal. Relevant example: Suppose we have several variables, the scores on 4 exams, one in math, one in physics, one in English, and one in history — one list of 4 scores for each student in a class. Taking the exams 1
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in pairs, we can find a correlation coefficient for each pair. We then form a matrix with the rows and columns labelled M(ath), P(hysics), E(nglish) and H(istory); the entry in the X row and Y column is the correlation coefficient between X and Y. The result is a symmetric matrix (since the correlation between X and Y is the correlation between Y and X) with 1’s on the main diagonal (since the correlation between X and X is 1): M P E H M 1 . 74 . 24 . 24 P . 74 1 . 24 . 24 E . 24 . 24 1 . 74
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