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 Since such an investigation is a nonsense when the Q variables are independent, we reuse a previous unpublished work about the chi-squared independence test for Burt's tables.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

Keywords:

• Ordinal correspondence analysis
• detrended correspondence analysis
• randomization
• eigenvalues
• orthogonal polynomials
• synchrotron X-rays diffraction

Acknowledgments

This research was conducted in the framework of the Agriped project (ANR-10-BLANC-605) supported by the French National Research Agency (ANR). We acknowledge C. Lelay (URSol INRA Orléans,France) for the thin sections production. Synchrotron SOLEIL (France) is acknowledged for allocating beamtime for the local-probe X-ray diffraction experiments. We thank the referees for their stimulating comments and bibliographic contributions, as well as J.-P. Durbec for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was conducted in the framework of the Agriped project (ANR-10-BLANC-605) supported by the French National Research Agency (ANR).