ABSTRACT
Pioneered by Hotelling (1935), canonical correlation analysis (CCA) focuses on the relation between two sets of variables, each consisting of two or more variables. In some applications, the two sets may be described in terms of independent and dependent variables, although such designations are not necessary. There are a variety of ways to study relations among groups of variables. The general goal of CCA is to uncover the relational pattern(s) between two sets of variables by investigating how the measured variables in two distinct variable sets combine to form pairs of canonical variates, and to understand the nature of the relation(s) between the two sets of variables. CCA has often been conceptualized as a unified approach to many univariate and multivariate parametric statistical testing procedures (Thompson, 1991), and even a unified approach to some nonparametric procedures (Fan, 1996). The close linkage between CCA and other statistical procedures suggests that the association between two sets of variables often needs to be understood in our statistical analyses: “most of the practical problems arising in statistics can be translated, in some form or the other, as the problem of measurement of association between two vector variates X and Y” (Kshirsagar, 1972, p. 281). From this perspective, CCA has been considered as a general representation of the general linear model (Thompson, 1984), unless we consider structural equation modeling (see Chapters 33 and 34 of this volume) as the most general form of the general linear model that takes measurement error into account (Thompson, 2000). Interested readers are encouraged to consult additional sources for more technical treatments of CCA (Johnson & Wichern, 2002, ch. 10), for more readable explanations and discussions of CCA (Thompson, 1984, 1991), for understanding the linkages between CCA and other statistical techniques (Bagozzi, Fornell, & Larcker, 1981; Fan, 1997; Jorg & Pigorsch, 2013; Kim, Henson, & Gates, 2010; Yan & Budescu, 2009), and for recently proposed methodological extensions of CCA (Heungsun, Jung, Takane, & Woodward, 2012; Ognjen, 2014; Takane & Hwang, 2002; Tenenhaus & Tenenhaus, 2011) including methods for modeling ordinal data (Mishra, 2009). Recommended desiderata for studies involving CCA are presented in Table 3.1 and are discussed in the subsequent sections.
