Why Association Evidence Is Shared
A genomic association peak can contain several variants with similar evidence. This does not necessarily mean that each variant independently affects the trait. If two variants tend to occur together, their genotype measurements carry overlapping information: a variant that is not causal can still predict the trait because it tags the genotype of a causal variant. Linkage disequilibrium (LD) describes this correlation between genetic variants.
For fine-mapping, the crucial question is not only whether a variant is associated, but whether the observed genotype data contain enough distinct variation to distinguish it from nearby alternatives. LD can make the evidence broad even when there is a single causal variant. In the limiting case of perfect LD, distinct variants can be statistically indistinguishable using those genotypes alone.
Alleles must be oriented consistently when interpreting the sign. If the effect allele at one variant is replaced by its alternate allele, its dosage is reversed relative to the original coding, and the correlation's sign changes. Thus negative correlation does not mean weak LD: \(r=-0.95\) represents very strong LD, with \(r^2=0.9025\). The sign matters for the direction of a transferred marginal effect, while the magnitude and square describe strength.
How a Tag Variant Inherits an Association
The following simple model makes evidence sharing precise. Assume a quantitative outcome \(Y\), centered and standardized genotype columns \(X_k\) and \(X_j\) with variance 1, and a single causal genotype \(X_k\). Write
where \(\beta\) is the change in the mean outcome per one-standard-deviation increase in \(X_k\), and the residual \(\varepsilon\) has mean zero and is uncorrelated with both genotype columns. The model is an illustration of the association geometry; it is not a claim that every trait or genetic architecture follows a single-variant linear model. Under these assumptions, the marginal slope from regressing \(Y\) on \(X_j\) alone is
The second equality uses \(\operatorname{Cov}(X_j,\varepsilon)=0\), \(\operatorname{Var}(X_j)=1\), and \(\operatorname{Cov}(X_j,X_k)=r_{jk}\). A correlated tag variant therefore has an expected marginal slope with the causal variant's sign if \(r_{jk}>0\), and with the opposite sign if \(r_{jk}<0\). Under this idealized model, the square of that slope is \(\beta^2r_{jk}^2\); this is a relation between expected effects, not a guarantee about a particular noisy sample's test statistics or p-values.
Worked Example: A Correlated Tag Variant
Suppose standardized genotype \(X_1\) affects a standardized quantitative trait with \(\beta=0.50\) outcome standard deviations per genotype standard deviation. A nearby variant has genotype correlation \(r_{12}=0.80\) with \(X_1\). Assume the single-causal-variant model above and residuals uncorrelated with both genotypes. What marginal slope is expected for variant 2?
Using the derived relation,
Thus variant 2 is expected to show a positive marginal association of 0.40 trait standard deviations per genotype standard deviation, despite having no direct effect in the stated model. The LD strength is \(r_{12}^2=(0.80)^2=0.64\). This calculation shows why an association at a tag variant is not, by itself, evidence that the tag is causal.
For association tests, the same geometry often makes nearby statistics look similar: in a large sample and under comparable standard errors, the expected signed association signal at a tag tends to track the signed correlation with the causal genotype. This is an approximation to the behavior of association statistics, not an identity for observed statistics. Sampling variation, allele frequencies, missingness, phenotype modeling, and standard errors can make the realized values differ.
Perfect LD and Indistinguishable Variants
Two standardized genotype columns are perfectly correlated when \(r_{12}=1\), and perfectly negatively correlated when \(r_{12}=-1\). In either case, one column is an exact linear function of the other: \(X_2=X_1\) or \(X_2=-X_1\). In the single-variant model, the same fitted genetic contribution can then be written using either variant, with a sign change in the effect when the columns are opposites. The outcome data cannot tell which of those perfectly linked variants was the causal one.
This is an identifiability issue, not merely a matter of low statistical power. A design containing exactly dependent genotype columns has no independent genotype variation with which to separate their effects. If two candidate causal explanations imply the same fitted genetic contribution for every observed individual, an association analysis of those observations cannot choose between them on the basis of the genotype patterns.
Worked Example: Perfect Positive and Negative LD
Consider four individuals with centered, unit-variance genotype columns represented by \(X_1=(-1,1,-1,1)\). First suppose \(X_2=(-1,1,-1,1)\) as well. The columns are identical, so their correlation is \(r_{12}=1\). Under a model in which variant 1 contributes \(0.70X_1\) to the outcome, the genetic contribution is also \(0.70X_2\). The observed outcomes cannot distinguish these two descriptions.
Now instead code the second variant so that \(X_2=(1,-1,1,-1)=-X_1\). The correlation is \(r_{12}=-1\). The contribution \(0.70X_1\) can still be written using variant 2, because
The effect coefficient changes sign, but the predicted genetic contribution for every individual is unchanged. Therefore both perfect positive and perfect negative correlation make the variants indistinguishable from these genotype data alone. Merely observing different coded columns is not enough to establish distinguishability; the relevant condition is whether the columns are perfectly linearly dependent.
Near-Perfect LD: Distinguishable in Principle, Difficult in Practice
If \(|r_{12}|<1\), the standardized columns are not perfectly linearly dependent. There is at least some genotype variation that differs between them, so they are distinguishable in principle. But when \(|r_{12}|\) is close to 1, that distinct variation is limited. A finite study may contain too little information to tell which variant better explains the association. It is important to keep these statements separate: exact indistinguishability is a property of perfect dependence, whereas poor resolution can occur with very high but imperfect LD.
Worked Example: Negative LD and Shared Evidence
Suppose the standardized genotype at causal variant 1 has slope \(\beta=0.40\) for a standardized trait. A second variant has \(r_{12}=-0.90\) with it, and the single-causal-variant model assumptions hold. The expected marginal slope at variant 2 is
The negative sign is expected because the coded genotype columns move in opposite directions. The LD strength is \(r_{12}^2=0.81\). Relative to the squared causal slope, the squared expected marginal slope is
Thus a strong association at variant 2 would be compatible with variant 1 being causal and variant 2 merely tagging it. The result does not prove that variant 1 is causal; it describes what the single-causal model predicts for the tag's expected marginal effect.
How to Assess LD-Based Ambiguity
A useful assessment starts with the genotype data or an LD estimate appropriate to the study population. Check which variants are highly correlated with the lead association, whether their allele coding is aligned, and whether the differences between candidate genotype columns are supported by enough observations. The following distinctions help prevent overinterpretation.
| Pattern | What it implies | What it does not imply |
|---|---|---|
| \(r\) near \(1\) or \(-1\) | Strongly shared genotype information and likely difficulty resolving variants | That both variants cause the trait |
| \(|r|=1\) | Exact linear dependence of the two standardized columns; no separation from those genotype patterns alone | That external functional or experimental evidence cannot distinguish them |
| \(|r|<1\) | Some genotype variation separates the columns in principle | That the sample has enough power to identify the causal variant |
| Opposite signs of marginal effects | Potentially consistent with negative LD and consistent allele coding | Evidence against shared association without checking orientation and LD |
Diagnostics should include allele alignment, allele frequencies, and the source of the LD estimate. A reference panel from a different ancestry or a different variant set may have correlations that do not match those in the association sample. If the LD used to interpret a signal is inaccurate, apparent inconsistencies between association results and genotype correlation can be artifacts of the mismatch rather than evidence for a different causal explanation.
Also inspect whether the correlation is computed on the same individuals and with compatible genotype handling. Missing genotypes, imputation quality, variant filtering, or inconsistent allele coding can change estimated correlations. For an exact collinearity claim, verify the genotype columns themselves or their rank relationship; a rounded correlation reported as 1.00 is not sufficient proof that \(|r|=1\).
What LD Can and Cannot Resolve
Marginal association analysis asks whether each variant, considered on its own, predicts the outcome. Because it does not separate correlated genotypes, it naturally gives shared evidence to tags. A joint or conditional regression asks whether a genotype contributes information after accounting for another genotype. With imperfect LD, that comparison can use the residual genotype variation left after the correlated column is accounted for. With perfect LD, there is no such residual variation, and the separate effects are not identifiable. These are different questions, and neither a strong marginal association nor a small p-value alone identifies a causal variant.
Other evidence can help prioritize among statistically similar variants: functional annotations, molecular assays, or association data from populations with different LD patterns. Such evidence may sharpen biological interpretation, but it does not alter what the original genotype data can identify. Likewise, LD patterns vary across populations; a pair nearly perfectly correlated in one population may be less correlated in another, potentially providing distinguishing variation in the latter.
The simple derivation assumes a single causal genotype, a linear quantitative-trait model, standardized columns, and residuals uncorrelated with the genotypes. With multiple causal variants, confounding, non-linear relationships, or model misspecification, the expected marginal association can reflect several contributions and need not equal a single \(\beta r\). Even in the simple setting, sample estimates fluctuate. Therefore use LD as a principled explanation for shared evidence and limited resolution, not as a causal conclusion by itself.
Common Pitfalls
- Equating association with causation. A correlated tag can inherit a marginal association even when it has no direct effect under the model. Association identifies prediction, not necessarily the causal variant.
- Using only the sign of \(r\). A large negative correlation is strong LD. The sign records coding orientation; \(|r|\) and \(r^2\) describe strength.
- Assuming different genotype columns must be distinguishable. If standardized columns satisfy \(X_2=-X_1\), they differ at every observation but remain perfectly dependent. The correct criterion for two columns is \(|r|<1\).
- Treating near-perfect LD as exact identity. Imperfectly correlated variants are distinguishable in principle, though finite-sample evidence may be inadequate. Confirm exact dependence rather than relying on rounded correlations.
- Assuming a reference-panel correlation necessarily applies to the study. Population, sample, and processing differences can change estimated LD; check that the LD source is appropriate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Under the stated single-causal-variant model, what is the expected marginal slope at a standardized tag genotype with \(r=0.60\) when the causal slope is \(0.50\)?
- Two standardized genotype columns have correlation \(-1\). Are the variants distinguishable from those genotype columns alone? Explain why their different coded values do not settle the question.
- A pair of variants has \(r=0.97\). What can be concluded about distinguishability in principle, and what cannot be concluded about practical fine-mapping resolution?
- Why can a strong association at a noncausal tag variant be consistent with a single causal variant?
- What should be checked before interpreting an LD estimate from a reference panel alongside association results from a study?
Solutions. (1) The expected slope is \((0.50)(0.60)=0.30\) outcome standard deviations per genotype standard deviation. (2) No. Correlation \(-1\) means \(X_2=-X_1\) after standardization; changing the coefficient's sign gives the same fitted contribution, so the columns are perfectly linearly dependent. (3) Since \(|r|=0.97<1\), the columns are distinguishable in principle. The high correlation means the distinct variation may be small, so the sample might still lack adequate resolution. (4) The tag is correlated with the causal genotype and therefore predicts the outcome; in the simple model its expected marginal slope is \(\beta r\). (5) Check allele alignment, ancestry and population match, variant inclusion, and genotype handling, because differences can make the reference LD differ from the study's LD.