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Abstract:
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We explore the general problem of describing simply and well the structure underlying a correlation matrix. For two variables, their shared correlation is the dot product (and cosine) of two unit-length vectors; for J variables, the associated J×J correlation matrix corresponds to J points on a unit J-sphere. The distance between any two such points is the length of the great circle arc connecting them, calculated as the arc-cosine of their shared correlation. In this way, the correlations among J variables implicitly define a matrix of distances. ¶For spheres and spherical distances, we introduce a measure of central tendency generalizing the mid-range: maximin consensus. This maximizes the minimum dot product, or equivalently, minimizes the maximum great circle distance. ¶Distance matrices have their own family of statistical methods, including multidimensional scaling, clustering, and archetypal analysis. The latter two result in factor loadings with especially clear interpretations. We make the case for archetypal analysis as best fulfilling Thurstone’s five criteria for simple structure. ¶Our running application considers 13 measures of physical frailty, and their correlation
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