Revelle's Beta: The Wait Is Over-Computation Becomes Possible.

Abstract

Over 45 years ago, William Revelle proposed a reliability measure based on the worst split-half of a test or scale, commonly known as Revelle's beta, to assess the general factor saturation. However, to this day, there is no reliable method for computing this measure, as existing approaches are either computationally infeasible or insufficiently accurate in identifying the worst split-half. This difficulty arises because the number of candidate splits increases exponentially with the number of items. In this article, we show that computing Revelle's beta is conceptually equivalent to divisive ("top-down") hierarchical clustering. This insight allows us to reduce the number of candidate splits to a quadratic problem, making the computation feasible. We specify theoretical conditions under which this approach is guaranteed to recover the worst split-half. To validate the efficiency of our approach, we conduct simulation studies and analyze real-world data. Code implementations accompanying this work are available online, together with Supplementary Material.