The relationship between the phi coefficient and the unidimensionality index H: Improving psychological scaling from the ground up.

Abstract

To study the dimensional structure of psychological phenomena, a precise definition of unidimensionality is essential. Most definitions of unidimensionality rely on factor analysis. However, the reliability of factor analysis depends on the input data, which primarily consists of Pearson correlations. A significant issue with Pearson correlations is that they are almost guaranteed to underestimate unidimensionality, rendering them unsuitable for evaluating the unidimensionality of a scale. This article formally demonstrates that the simple unidimensionality index H is always at least as high as, or higher than, the Pearson correlation for dichotomous and polytomous items (φ). Leveraging this inequality, a case is presented where five dichotomous items are perfectly unidimensional, yet factor analysis based on φ incorrectly suggests a two-dimensional solution. To illustrate that this issue extends beyond theoretical scenarios, an analysis of real data from a statistics exam (N = 133) is conducted, revealing the same problem. An in-depth analysis of the exam data shows that violations of unidimensionality are systematic and should not be dismissed as mere noise. Inconsistent answering patterns can indicate whether a participant blundered, cheated, or has conceptual misunderstandings, information typically overlooked by traditional scaling procedures based on correlations. The conclusion is that psychologists should consider unidimensionality not as a peripheral concern but as the foundation for any serious scaling attempt. The index H could play a crucial role in establishing this foundation. (PsycInfo Database Record (c) 2025 APA, all rights reserved).