Data Analysis and Applications 2

1 Context-specific Independence in Innovation Study

The study of (in)dependence relationships among a set of categorical variables collected in a contingency table is an ample topic. In this chapter, we focus on the so-called context-specific (CS) independence where the conditional independence holds only in a subspace of the outcome space. The main aspects that we introduce concern the definition in the same model of marginal, conditional and CS independencies, through the marginal models. Furthermore, we investigate how it is possible to test these CS independencies when there are ordinal variables. Finally, we propose a graphical representation of all the considered independencies taking advantages from the chain graph model (CGM). We show the results of an application on “The Italian Innovation Survey” (Istat 2012).

1.1. Introduction

In the field of categorical variables, with the term CS independence we refer to the particular conditional independence that holds only for some modalities of the variable(s) in the conditioning set, but not for all. That is, given three variables X₁, X₂ and X₃ we describe this situation as X₁ ⊥ X₂ | X₃ = c₃, where c₃ is a subset of all possible values of X₃. Among others, Højsgaaard (2004) and Nyman et al. (2016) studied this topic in detail. In this chapter, we want to improve the main results of these works by dealing with CS independencies concerning subsets of all the considered (also ordinal) variables. For this aim, we use the hierarchical ...