Relating Ranks

Abstract

Sometimes we have variables that at first glance appear to be measurements, but that on further examination reveal themselves to be something less than actual measurements along a scale. Often they really amount to relative rankings rather than true measurements. For example, soil productivity is sometimes rated by producing an index with an arbitrary formula using such values as content of various nutrients, soil depth, capacity for water retention, and other variables that affect soil productivity. The formulas used in these ratings are carefully considered to produce a set of numbers such that we are sure that higher numbers represent more productive soils and lower numbers represent less productive soils. Such scales, for example, would allow us to say that a rating of 8 means more productive soils than a rating of 4. They seldom, however, leave us in position to say that a rating of 8 means soils twice as productive as a rating of 4. It is our inability to make this last statement that keeps such ratings from being true measurements. Instead, they are rankings. Rankings allow us to put things in rank order (most productive soil, second most productive soil, third most productive soil, etc.) but not to say how much more a high ranking thing is than a low ranking thing. The logic of linear regression relies on the measurement principle. (Think of the scatter plots and the regression equations. If X is twice as large it places the corresponding point twice as far over on the scatter plot. If X is twice as large it has twice the effect on the prediction of Y by way of the regression equation.) If X is actually only a ranking rather than a true measurement, then we should feel uncomfortable about using regression. Instead of performing a linear regression and attempting to predict the actual value of Y from X, we might use a rank order correlation coefficient to assess the strength and significance of a rank order relationship.