Cost-Sensitive Ordinal Gini: Formulation, Structural Properties, and Empirical Evaluation

Abstract

Ordinal classification arises whenever target categories possess an inherent order, such as in healthcare, risk assessment, and institutional quality assessment. Existing impurity measures for ordinal decision trees, including Ordinal Gini (OGini), assign the same weight to every boundary between adjacent classes, so the splitting criterion cannot express that errors crossing different boundaries may carry different consequences. This study formulates Cost-Sensitive Ordinal Gini (CS-OGini), a generalisation of OGini in which each ordinal boundary is assigned a configurable cost, and shows that OGini is recovered exactly when all boundary weights are equal. Two families of weighting rules are considered: weights specified as a function of the boundary index, and weights derived from the cumulative class proportions of the training data. For the second family we prove that $\beta$ = 1 equalises the contribution of all ordinal boundaries to the impurity, and is the unique exponent doing so whenever those contributions are not already equal, while $\beta$ = 0 recovers OGini.We further establish three structural properties that delimit what boundary weighting can achieve. The accumulated boundary cost is monotone in the ordinal distance between classes; the cost-minimising prediction at a leaf is the median of the class distribution irrespective of the boundary weights, so the weights influence the model only through the tree structure; and any impurity defined as the expected cost between two independent labels is invariant to the antisymmetric component of the cost matrix, so asymmetric ordinal costs cannot be represented within this class of criteria.The empirical study compares nine splitting criteria, including CART, ID3, Ranking Impurity, class-weighted Gini, an ordinal entropy counterpart of OGini, the Frank and Hall decomposition, OGini, and two CS-OGini variants, on fifteen ordinal datasets under an identical training and evaluation protocol. The criteria are statistically indistinguishable: mean accuracy spans only 1.00 percentage point across all nine, five of six omnibus tests are non-significant, no pairwise comparison against OGini survives correction for multiple testing, and the single significant omnibus result is attributable to the two criteria lying outside the closely grouped set rather than to any criterion outperforming the others. Paired confidence intervals bound the difference between CS-OGini and OGini within [-0.82, +0.30] percentage points of accuracy. The contribution of this study is therefore a formal framework for boundary-weighted ordinal impurity together with provable limits on what such weighting can accomplish, supported by evidence that these limits are binding in practice.