Non-symmetric correspondence analysis with ordinal variables using orthogonal polynomials

Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials

“…Recent papers that describe the use of orthogonal polynomials for correspondence analysis have shown them to be an important tool for identifying sources of association that exist in twoway contingency tables with one ordinal variable (Beh 2001;Lombardo et al 2011) or with two ordinal variables (Beh 1997(Beh , 1998Lombardo et al 2007), in three-way contingency tables (Beh and Davy 1998;Beh et al 2007) and in multiple contingency tables (Lombardo and Meulman 2010).…”

“…• j ) and χ 2 α with (J − 1) (or (I − 1)) degrees of freedom; see Lombardo et al (2007) and D'Ambra, Beh and Amenta (2005).…”

“…In particular, we will use Emerson's (1968) orthogonal polynomials for this purpose. These polynomials have been used earlier in the literature to quantify ordered variables; see Beh (1997Beh ( , 2001, Best and Rayner (1996), D'Ambra, Lombardo and Amenta (2002), Corbellini, Riani and Donatini (2008), Lombardo et al (2007), Lombardo and Meulman (2010), Lombardo and Beh (2010) and Manté, Bernard, Bonhomme and Nerini (2013). To enhance the interpretation of orthogonal polynomials, we will also propose a new graphical display, the polynomial biplot.…”

“…One may consult, for example, Agresti (1996Agresti ( , 2010 for a description of many of these. Similarly, graphing the association structure between the categorical variables has received just as much attention; see, for example, Beh (1997Beh ( , 1998, Beh and Lombardo (2014), Gifi (1990), Greenacre (1984Greenacre ( , 2007, Lebart, Morineau and Warwick (1984), Lombardo, Beh and D'Ambra (2007), Lombardo and Meulman (2010), Meulman, Van der Kooij and Heiser (2004), Nishisato (1980Nishisato ( , 2007 and Ter Braak (1986). The use of external information (concomitant variables) in the form of constraints has resulted in simplified multidimensional displays in correspondence analysis; see , Takane, Yanai, and Mayekawa (1991), and Takane and Jung (2009).…”

“…Both these types can have zero, one, or two ordered variables, but we will restrict ourselves here to those cases when at least one of the variables has ordered categories, and such ordered variables will always be modelled by orthogonal polynomials. When one of the variables has ordered categories and the other has not, we will consider the hybrid decomposition (HD) (Beh 2001, Lombardo, Beh & D'Ambra 2011, and when both have ordered categories we will use the bivariate moment decomposition (BMD) (Beh 1997;Lombardo et al 2007). …”