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 tablewithQquestions. Since such an investigation is a nonsense when theQvariables are independent, we reuse a previous unpublished work about the chi-squared independence test for Burt's tables.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) ofas a scorexinto a subsequent singly-ordered Ordinal Correspondence Analysis (OCA) of. OCA mainly consists in completingxby 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, 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.

Automated summary

In this study, a method for detecting the Guttman effect in a disjunctive table is proposed. By reusing the chi-squared independence test method, a two-step approach is introduced to determine the presence and degree of the Guttman effect. The method is tested on both artificial and real data, with successful results.