Correlation

Generally speaking, the notion of correlation is very close to that of association. It is the statistical association between two variables of interval or ratio measurement level.

To begin with, correlation should not be confused with causal effect. Indeed, statistical research into causal effect for only two variables happens to be impossible, at least in observational research. Even in extreme cases of so-called unicausal effects, such as contamination with the Treponema pallidum bacteria and contracting syphilis, there are always other variables that come into play, contributing, modifying, or counteracting the effect. The presence of a causal effect can only be sorted out in a multivariate context and is consequently more complex than correlation analysis.

Let us limit ourselves to two interval variables, denoted as X and Y, and let us leave causality aside. We assume for the moment a linear model. It is important to note that there is a difference between the correlation coefficient, often indicated as r, and the unstandardized regression coefficient b (computer output: B). The latter merely indicates the slope of the regression line and is computed as the tangent of the angle formed by the regression line and the x-axis. With income (X) and consumption (Y) as variables, we now have the consumption quote, which is the additional amount one spends after obtaining one unit of extra income, so the change in Y per additional unit in X is B = δY/δX. We will see below that there are, in fact, two such regression coefficients and that the correlation coefficient is the geometrical mean of them.