Canonical Correlation Analysis

The basic idea of canonical correlation analysis is clearly illustrated by Russett (1969), who analyzed the association between some economic and political characteristics of 47 countries. Five indicators were used for measuring economic inequality: the division of farmland, the Gini coefficient, the percentage of tenant farmers, the gross national product (GNP) per capita, and the percentage of farmers. Russett measured political instability with four indicators: the instability of leadership, the level of internal group violence, the occurrence of internal war, and the stability of democracy. The research hypothesis was that Alexis de Tocqueville was right: There is not one nation that is capable of maintaining a democratic form of government for an extensive period of time if economic resources are unevenly distributed among its citizens. In other words, there is a significant association between the economic in equality and the political in stability of nations.

The two theoretical concepts in this research problem are “economic inequality” (X*) and “political instability” (Y*). They are called canonical variables with an expected high correlation, known as canonical correlation. The first canonical variable, X*, is measured by p = 5 indicators, X1 to X5, and we will consider X* as a linear combination (a weighted sum) of these X variables. In an analogous fashion, Y*, the second canonical variable, is a linear combination of the q = 4 indicators, Y1 to Y4. In the most general case, in which the X set contains p variables and the Y set q variables, the diagram can be represented as in Figure 1.

Figure 1 Canonical Correlation

In Russett's (1969) research problem, the canonical correlation between the canonical variables X* and Y * is in fact a causal relationship because he asserts that nations with greater economic inequality display greater political instability. The arrow in the other direction, according to which the political characteristics would influence the economic characteristics, is not implied in de Tocqeville's or Russett's view. However, considering the fact that a canonical correlation is a “correlation,” the statistical analysis is not asymmetrical. For this reason, we do not draw a causal arrow from X* to Y * in the figure but a doublesided curved arrow, indicating that the question of causality remains open.