A Latent Structure for the Simplex

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Abstract
A perfect Guttman scale generates a perfect Guttman simplex. This is capitalized on to get a transformation useful in a kind of non-linear factor analysis applicable to empirical correlations resembling but not necessarily conforming to a perfect simplex.A Gramian factorization—square, symmetric, possibly of low rank but with no negative characteristic roots—of empirical correlations is transformed into latent information about nonlinear regressions of tests on a single underlying dimension. Class intervals on that continuum are defined in terms of their standard score profiles on the tests. Each class interval exhibits “local independence,” so that individual members may vary but may not covary in their profiles.The method is flexible as to choice of diagonals, fitted rank, and exact shape of resulting non-linear regressions, so long as those regressions are essentially monotonic and have a proper progression of curvatures. Construction of the needed transformation for data of any size is outlined, and the problem of metric for the latent continuum is given several solutions.An empirical example is provided with full and low rank solutions involving different diagonal choices.

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