Splitting a Predictor at the Upper Quarter or Third and the Lower Quarter or Third

Abstract

A linear regression of y on x can be approximated by a simple difference: the average values of y corresponding to the highest quarter or third of x, minus the average values of y corresponding to the lowest quarter or third of x. A simple theoretical analysis shows this comparison performs reasonably well, with 80%-90% efficiency compared to the linear regression if the predictor is uniformly or normally distributed. Discretizing x into three categories claws back about half the efficiency lost by the commonly-used strategy of dichotomizing the predictor. We illustrate with the example that motivated this research: an analysis of income and voting which we had originally performed for a scholarly journal but then wanted to communicate to a general audience.

Keywords: discretization,linear regression,statistical communication,trichotomizing