Why linkage disequilibrium measures disagree: Fisher geometry of rare common haplotype structure

Fisher Geometry Explains Why Standard Linkage Disequilibrium Measures Disagree

In population genetics, scientists rely on linkage disequilibrium (LD)—a measure of how much specific genetic variants at different locations are "linked" or inherited together more often than chance would predict. While several mathematical formulas exist to summarize this relationship, they frequently yield wildly different results for the same pair of genes. This discrepancy is most acute when one variant is common and the other is rare.

For over fifty years, researchers have used various scalar indices. These include $r^2$ (the squared correlation between alleles) and $D'$ (a normalized version of the LD coefficient $D$). Most assume these indices are simply different ways of looking at the same underlying structure. However, this assumption fails in the "rare–common" regime. In this state, a rare mutation might be perfectly nested within a common genetic background. In such cases, $r^2$ might suggest almost no connection. Meanwhile, other measures suggest a total association.

A new paper by Yuki Ichikawa proposes that this disagreement is not a matter of statistical error. Instead, the conflict is a fundamental consequence of the geometry of the space where these variants reside. By applying the principles of information geometry (the study of probability distributions as geometric shapes), the author shows the mathematical "landscape" is not a flat plane. Rather, it is a curved, distorted manifold (a mathematical space that looks flat locally but has intrinsic curvature).

The Structural Failure of Symmetric Normalization

The core problem lies in how standard measures compress complex genetic information into single numbers. When analyzing two loci (specific positions on a chromosome), researchers want to know if allele A makes finding allele B more likely. However, many common metrics are "locus-symmetric." This means they treat both sites as interchangeable.

The paper highlights that $r^2$ suffers from two specific types of information loss. First, because it involves squaring the LD coefficient $D$, it discards the "phase." The phase is the sign that tells us if alleles are being pulled together (coupling) or pushed apart (repulsion). Second, its symmetric normalization causes structural compression in the rare–common regime. If allele A is very rare ($p_A \ll p_B$), the mathematical ceiling for $r^2$ is strictly capped.

The authors report a specific structural bound: $r^2 \le p_A q_B / (q_A p_B)$. This means $r^2$ cannot reach high values in these scenarios. Even if a rare variant is always found on a specific common background, $r^2$ stays low. This creates a predictable blind spot. Significant biological signals are effectively erased by the math used to measure them.

Mapping the Haplotype Simplex

To understand why this happens, the author moves into the realm of Riemannian geometry. The study treats the possible distributions of genetic haplotypes (combinations of alleles on a single chromosome) as existing on a "haplotype simplex." This is a multidimensional space representing all possible frequency combinations.

The researcher employs the Fisher information metric. This tool defines the "distance" between probability distributions based on how much information they carry. By fixing the mean allele frequency and focusing on the relationship between the frequency contrast ($M$) and the LD coefficient ($D$), the author constructs a two-dimensional "leaf" of this simplex.

The mechanism of the study follows three logical steps: 1. Parametrization: The author rewrites haplotype frequencies as functions of $M$ and $D$. This maps biological constraints—like the rule that frequencies cannot be negative—as physical boundaries .

1. Metric Derivation: Using the multinomial Fisher form, the author calculates how the "geometry" of this space changes. He finds the space is highly anisotropic (having different properties in different directions). A small change in $D$ carries much more "information weight" than a change in $M$. At the most symmetric point, the weight of $D$ is eight times greater than $M$ .

2. Decomposition of Asymmetry: To address lost directional information, the author introduces the conditional-probability asymmetry $\Delta = P(A|B) - P(B|A)$. This decomposes into $\Delta = M + C$. Here, $M$ accounts for frequency differences and $C$ is the "LD-coupled" term capturing the actual genetic association.

Curvature and the Rare-Variant Limit

The most striking findings involve the intrinsic shape of this genetic space. The paper reports that the $(M, D)$ leaf is not flat. It possesses positive Gaussian curvature ($K$) that depends heavily on allele frequencies.

The authors derive an exact formula for curvature along the symmetry axis: $K(s) = \frac{1}{4} + \frac{(2s - 1)^2}{4s(1 - s)}$. This result shows that as an allele becomes rarer ($s \to 0$), the curvature diverges. The landscape becomes fundamentally more warped. This is seen in, where curvature is strictly positive and increases toward the edges of the domain.

The paper also identifies "boundary amplification." As a haplotype frequency approaches zero, the Fisher information diverges sharply .

This explains why estimating rare variants is statistically difficult. You are navigating a region of the manifold that is being stretched to an extreme degree.

The study argues that "disagreement" between measures stems from treating this curved manifold as if it were flat. Practitioners often use Euclidean thresholds (treating the space as a flat grid). However, they are applying a uniform ruler to a surface that expands and curves at different rates.

Limitations of the Geometric Framework

There are practical caveats to consider. The author notes that the $(M, C)$ representation is singular on the line where $M = 0$. This occurs when allele frequencies are equal. In this specific case, the coordinate system breaks down. Researchers must switch back to $(M, D)$ coordinates to maintain a complete picture.

This work is primarily a theoretical development of the geometry. While the author mentions a companion paper that tests these predictions using 1000 Genomes data, this specific note focuses on the mathematical derivation. The jump from theory to empirical application is handled in that secondary study.

Finally, the framework describes a two-locus system. While the author suggests higher-dimensional spaces follow a similar "product-of-spheres" logic, the complexity grows quickly. The simple $(M, D)$ leaf analysis cannot be directly scaled to many-locus interactions without further work.

The Verdict: Move Beyond $r^2$

The conclusion is clear. The reliance on $r^2$ in modern genomics is a trade-off. While $r^2$ is efficient and predictable for common alleles, it cannot capture the nuances of rare-variant architecture.

If your research involves fine-mapping rare mutations, $r^2$ may hide your most important signals. The "ceiling" that prevents $r^2$ from detecting certain associations is a mathematical certainty. No increase in sample size can lift this bound.

To gain a complete view, researchers should use a multi-metric approach. Directional measures like the conditional-probability asymmetry ($\Delta$) should be reported alongside $r^2$. This preserves the "phase" and "direction" that squaring erases. Code for reproducing these geometric calculations is available at https://github.com/mountbook-lab/directional-ld-mc.

Figures from the paper