Preprint identifies geometric basis for disagreements between standard linkage disequilibrium measures

Linkage disequilibrium (LD) — the non-random association between alleles at different positions in the genome — underpins a wide range of statistical genetics methods, including genome-wide association study fine-mapping, imputation, and haplotype analysis. The most widely used LD metric, r², has known limitations, but the geometric basis for its failures in certain frequency configurations has not been fully characterised.

A preprint posted to bioRxiv applies a Fisher geometry framework to explain why conventional LD measures disagree in what the authors term the "rare–common regime" — configurations where one allele is rare and the other is common, as occurs in nested haplotype structures. The authors demonstrate that r² is both symmetric and quadratic: squaring discards directional (phase) information retained by the signed coefficient D, while symmetric normalisation conceals the asymmetry between the conditional probabilities P(A|B) and P(B|A). They propose that these geometric properties cause r² to misrepresent LD structure in precisely the configurations most relevant to identifying causal variants in association studies.

The work is directly relevant to researchers working on fine-mapping methods, haplotype analysis, and the interpretation of LD patterns across diverse populations. It complements a cluster published in Genetic Current on 2 July 2026 describing a carrier-set topology framework for LD characterisation. As a preprint, the findings have not yet been peer-reviewed.