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Detecting the Guttman effect with the help of ordinal correspondence analysis in synchrotron X-ray diffraction data analysis

Abstract : We propose a method for detecting a Guttman effect in a complete disjunctive table U with Q questions. Since such an investigation is a nonsense when the Q variables are independent, we reuse a previous unpublished work about the chi-squared independence test for Burt's tables. Then, we introduce a two-steps method consisting in plugging the first singular vector from a preliminary Correspondence Analysis (CA) of U as a score x into a subsequent singly-ordered Ordinal Correspondence Analysis (OCA) of U. OCA mainly consists in completing x by a sequence of orthogonal polynomials superseding the classical factors of CA. As a consequence, in presence of a pure Guttman effect, we should in principle have that the second singular vector coincide with the polynomial of degree 2, etc. The hybrid decomposition of the Pearson chi-squared statistics (resulting from OCA) used in association with permutation tests makes possible to reveal such relationships, i.e. the presence of a Guttman effect in the structure of U, and to determine its degree-with an accuracy depending on the signal to noise ratio. The proposed method is successively tested on artificial data (more or less noisy), a well-known benchmark, and synchrotron X-ray diffraction data of soil samples.
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Submitted on : Friday, September 18, 2020 - 5:26:59 PM
Last modification on : Thursday, June 9, 2022 - 10:00:04 AM
Long-term archiving on: : Thursday, December 3, 2020 - 1:07:06 PM

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Claude Manté, Sophie S. Cornu, D. Borschneck, C. Mocuta, R. van den Bogaert. Detecting the Guttman effect with the help of ordinal correspondence analysis in synchrotron X-ray diffraction data analysis. Journal of Applied Statistics, Taylor & Francis (Routledge), 2020, pp.1-26. ⟨10.1080/02664763.2020.1810644⟩. ⟨hal-02943227⟩

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