Comparative Assessment of the Classical Half-Slope Ratio and the Normalized Difference Ratio as a Robust Linearity Diagnostic Tools

Abstract

Robust quantitative diagnostics are crucial for assessing linearity in Exploratory Data Analysis (EDA), especially when data contain outliers. This work defines and compares two such diagnostics derived from median-based half-slopes calculated across partitioned bivariate data. We examine the established Classical Half-Slope Ratio (CHR), calculated as the ratio of the right to the left half-slope, which indicates linearity near unity but is unbounded. We contrast this with a proposed Normalized Difference Ratio (NDR), formulated as the normalized difference between the half-slopes. The NDR is inherently bounded within [-1, 1], precisely indicates linearity at zero, and directly signals the direction of data curvature through its sign (+/-). Illustrative examples confirm that NDR’s magnitude quantifies the degree of non-linearity, while its sign offers clear guidance for data transformations. While both CHR and NDR are valuable outlier-resistant tools complementing visual analysis, the NDR’s bounded, centered scale pro- vides distinct advantages for comparative analysis, standardization, and potential algorithmic use.