Directional nature of the product-moment correlation coefficient and some consequences

Product-moment correlation coefficient (PMC) is usually taken as a symmetric measure of the association because it produces an equal estimate irrespective of how two variables in the analysis are declared. However, in case the other variable has or both have non-continuous scales and when the scales of the variables differ from each other, PMC is unambiguously a directional measure directed so that the variable with a wider scale (X) explains the order or response pattern in the variable with a narrower scale (g) and not in the opposite direction or symmetrically. If the scales of the variables differ from each other, PMC is also prone to give a radical underestimation of the association, that is, the estimates are deflated. Both phenomena have obvious consequences when it comes to interpreting and speaking of the results. Empirical evidence shows that the effect of directionality increases by the discrepancy of the number of categories of the variables of interest. In the measurement modelling setting, if the scale of the score variable is four times wider than the scale of the item, the directionality is notable: score explains the order in the item and no other way around nor symmetrically. This is regarded as a positive and logical direction from the test theory viewpoint. However, the estimate of association may be radically deflated, specifically, if the item has an extremely difficult level. Whenever the statistic r(2) or R-2 is used, as is usual in general scatterplots or when willing to express the explaining power of the variables, this statistic is always a directional measure, and the estimate is an underestimate if the scales differ from each other; this should be kept in mind when interpreting r-squared statistics as well as with the related statistic eta squared within general linear modelling.