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 Since such an investigation is a nonsense when the Q variables are independent, we reuse a previous unpublished work about the chi-squared independence test for Burt's tables.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.

 抽象的

我们提出了一种在完全析取表中检测古特曼效应的方法 带有Q问题。由于当Q变量独立时，这样的研究是无意义的，因此我们重用了之前未发表的关于 Burt 表的卡方独立性检验的工作。然后，我们引入了一种两步方法，包括从初步对应分析（CA）中插入第一个奇异向量 作为随后的单序序数对应分析 (OCA) 的分数x 。 OCA 主要在于通过一系列正交多项式来完成x ，取代 CA 的经典因子。因此，在存在纯格特曼效应的情况下，我们原则上应该让第二个奇异向量与 2 次多项式一致，等等。关联中使用的皮尔逊卡方统计量（由 OCA 产生）的混合分解通过排列测试可以揭示这种关系，即结构中存在格特曼效应 ，并确定其程度 - 精度取决于信噪比。 所提出的方法相继在人工数据（或多或少有噪声）、众所周知的基准和土壤样品的同步加速器 X 射线衍射数据上进行了测试。