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

We propose a method for detecting a Guttman effect in a complete disjunctive table $\mathbf{U} $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 Since such an investigation is a nonsense when the Q variables are independent, we reuse a previous unpublished work about the chi-squared independence test for Burt's tables.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 $\mathbf{U} $U as a score x into a subsequent singly-ordered Ordinal Correspondence Analysis (OCA) of $\mathbf{U} $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 $\mathbf{U} $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.