Why LD Matters for Colocalization
A locus can contain several correlated variants, even when only one variant directly affects a trait. Because of linkage disequilibrium (LD), a nearby variant can be associated with the trait simply because its genotype tends to occur with the causal genotype. The association pattern across variants therefore reflects both the causal effect and the LD structure.
Colocalization asks whether two traits’ association signals are consistent with a shared causal variant or with distinct causal variants. LD makes that comparison informative in some settings and difficult in others. If two candidate variants are weakly correlated, their association patterns may help distinguish the causal locations. If they are highly correlated, different causal variants can generate nearly indistinguishable patterns. LD also differs across populations, so an LD reference from a mismatched ancestry may give an inaccurate picture of which variants are expected to track one another.
The distinction between signed \(r\) and \(r^2\) matters. A correlation of \(0.8\) and a correlation of \(-0.8\) have the same \(r^2=0.64\), but their marginal effect estimates have opposite signs when alleles are coded consistently. Allele harmonization is therefore essential when comparing signed association patterns.
How a Causal Variant Creates Associations at Proxies
Consider a simple single-causal-variant model for one quantitative trait. Genotypes are standardized to mean zero and variance one, and \(G_1\) is the causal genotype. Suppose \(Y=\beta G_1+\epsilon\), where \(\epsilon\) has mean zero, variance \(\tau^2\), and is independent of the genotypes. For another variant \(G_2\), let \(r=\operatorname{Corr}(G_1,G_2)\). These assumptions describe a simplified association model; population structure, confounding, multiple causal variants, or genotype measurement problems can alter the pattern.
The marginal regression of \(Y\) on \(G_2\) does not condition on \(G_1\). Its population slope is the covariance of \(G_2\) and \(Y\), because \(G_2\) has variance one:
Thus, a proxy with \(r=0.8\) has an expected marginal slope 80% of the causal-variant slope, while a proxy with \(r=-0.8\) has an expected slope with the opposite sign. Under the model, the residual variance from regressing \(Y\) on \(G_2\) is \(\tau^2+\beta^2(1-r^2)\): the part of \(G_1\) not linearly predicted by \(G_2\) remains in the residual, along with \(\epsilon\). For a sample of size \(n\), the approximate noncentrality of the marginal association statistic is therefore
This expression explains why a strong association at a noncausal variant is not, by itself, evidence that the variant is causal. When effects are small relative to residual noise, the denominator is close to \(\tau\), so the association statistic is approximately proportional to \(r\). Across a region, the association profile is a projection of causal effects through the LD structure. In matrix notation, for standardized genotypes with LD matrix \(\mathbf R\) and a vector of causal effects \(\boldsymbol{\beta}\), the vector of marginal slopes is \(\mathbf R\boldsymbol{\beta}\), under the same linear model assumptions.
Worked Example: A Proxy Variant Inherits Association
Suppose a hypothetical study has \(n=10{,}000\) individuals. Variant 1 is causal, its standardized effect is \(\beta=0.10\) outcome standard deviations per genotype standard deviation, and the residual standard deviation is \(\tau=1\). A nearby variant has signed LD correlation \(r=0.80\) with the causal variant. Calculate its expected marginal slope and approximate association noncentrality.
The marginal slope is \(\beta r\), so \(0.10(0.80)=0.0800\) outcome standard deviations per genotype standard deviation. The residual variance in the proxy regression is \(1^2+0.10^2(1-0.80^2)=1+0.01(0.36)=1.0036\).
Since \(\sqrt{n}=100\), the approximate noncentrality is \[ \frac{100(0.10)(0.80)}{\sqrt{1.0036}} =\frac{8}{1.0018}\approx 7.986. \] The proxy can therefore show a substantial association even though, by construction, it has no direct effect in this model. This calculation is an expected signal measure, not a claim that every observed statistic equals 7.986.
Why Similar Profiles Can Mean Different Things
For two traits, a shared causal variant is one explanation for overlapping association patterns. But distinct causal variants can also produce similar patterns when those variants are in strong LD. The evidence is most discriminating when the candidate variants have sufficiently different correlation patterns with the other variants in the region, and the studies have enough information to estimate those patterns.
Worked Example: High LD Blurs Distinct Signals
Consider two standardized variants with \(r=0.95\). For simplicity, suppose each trait has a single causal variant with effect chosen so that the expected association noncentrality at its own causal variant is 10. Ignoring the small residual-variance adjustment, if variant 1 is causal the expected pair of statistics is approximately \((10,9.5)\); if variant 2 is causal it is approximately \((9.5,10)\). These are expectations under alternative causal locations, not observed data.
If trait A is caused by variant 1 and trait B by variant 2, each trait is expected to associate strongly at both variants. The two profiles differ by only 0.5 statistic units at either position. Sampling variation can readily obscure that difference, so the association evidence may not sharply separate a shared-variant explanation from a distinct-variant explanation.
For contrast, if the variants had \(r=0.20\), the corresponding approximate profiles would be \((10,2)\) and \((2,10)\). Their locations are much easier to distinguish in principle. This comparison is not a universal threshold rule: informativeness also depends on sample sizes, effect sizes, the full LD pattern, and uncertainty in the association estimates.
A colocalization analysis compares evidence under causal explanations; it does not turn LD into proof of a shared biological mechanism. In common single-signal approaches, LD shapes the observed marginal statistics even when the procedure uses summary association results rather than an explicit LD matrix. Other approaches use LD information directly. In either case, a dense cluster of small association p-values can arise from one signal spreading across correlated variants, rather than from many independent causal effects.
Ancestry-Matched LD and a Numerical Illustration
LD is a property of a population’s haplotype distribution. Correlations can differ across ancestry groups because allele frequencies and historical recombination patterns differ. Consequently, a reference panel from another population may misrepresent which variants track the causal variant in the GWAS or eQTL sample. The problem is especially consequential when two candidate variants are highly correlated in one population but less correlated in another.
The following calculation illustrates the effect of changing the correlation in the model above. Let \(\beta=0.15\), \(\tau=1\), and \(n=3{,}600\), so \(\sqrt{n}=60\). The expected proxy statistic under the model is \(60\beta r/\sqrt{1+\beta^2(1-r^2)}\).
Worked Example: A Mismatched LD Correlation Changes the Expected Signal
Suppose the study-population correlation between a causal variant and a proxy is \(r=0.80\). The expected proxy statistic is \[ \frac{60(0.15)(0.80)} {\sqrt{1+0.15^2(1-0.80^2)}} = \frac{7.2}{\sqrt{1.0081}} \approx \frac{7.2}{1.0040} \approx 7.171. \]
Now suppose a different-ancestry reference estimates \(r=0.40\). Substituting that correlation gives \[ \frac{60(0.15)(0.40)} {\sqrt{1+0.15^2(1-0.40^2)}} = \frac{3.6}{\sqrt{1.0189}} \approx \frac{3.6}{1.0094} \approx 3.566. \]
The two expectations differ substantially because the assumed proxy correlation differs. This does not mean that an analyst should replace the observed association statistic with either expectation. It shows why a reference panel with inappropriate LD can mislead interpretation of which variants could plausibly generate the observed pattern.
Assumptions, Diagnostics, and Alternatives
The simple derivation assumes a correctly specified linear model, standardized genotypes, a single causal variant, and errors independent of genotypes. Its approximate association-statistic calculation also relies on a sufficiently large sample and a suitable variance estimate. Real loci may violate these assumptions through multiple causal signals, population structure, imputation error, or differences between the analyzed and reference populations. The association patterns may then be more complicated than one LD-projected signal.
Useful checks include comparing allele frequencies and ancestry composition between the study and LD reference; verifying variant identity, genome build, effect-allele alignment, and strand handling; and checking whether the supplied LD matrix is compatible with the variants and sample being analyzed. A reference panel should not be treated as interchangeable merely because it contains the same variant names. When possible, use LD estimated in the relevant study sample or a suitably matched reference and report its source and limitations.
| Approach or situation | What it contributes | Main caution |
|---|---|---|
| Marginal association profiles | Show how evidence is distributed across variants in each trait. | Correlated proxies can look like additional signals or obscure causal location. |
| LD-informed fine-mapping | Uses correlation structure to assess which variants can explain the regional evidence. | Results can be sensitive to LD mismatch and model assumptions. |
| Study-matched or ancestry-matched LD | Better represents the genotype correlations relevant to the association data. | May be unavailable, imprecise in small samples, or unsuitable for admixed samples if handled simplistically. |
When the sample is admixed or the relevant studies have different ancestry compositions, a single external LD matrix may not represent either association pattern well. Variant availability also matters: if a causal variant is absent or poorly imputed, nearby proxies may dominate the profile. These limitations should temper conclusions about whether signals are shared, distinct, or unresolved. The posterior comparison among colocalization hypotheses depends on the input evidence and assumptions; LD alone does not identify a causal variant or establish that a gene mediates a trait.
Common Mistakes and a Practical Interpretation
- Calling every associated proxy causal. A marginal association can be induced by correlation with a causal genotype. Distinguish association from direct causal effect.
- Treating high overlap as decisive proof of a shared variant. Distinct causal variants in strong LD can produce overlapping profiles. Ask whether the observed pattern can distinguish their locations.
- Using \(r^2\) as if it contained direction. \(r^2\) has no sign. Signed effect patterns require consistent allele coding and signed correlations.
- Assuming LD is the same across populations. A correlation estimated in an unmatched reference may not describe the study sample. Check ancestry and describe uncertainty in the reference.
- Reading a locus-wide cluster as many independent effects. One causal signal can spread across correlated variants. The cluster alone does not count the number of causal signals.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If a causal variant has standardized effect \(0.12\) and a proxy has signed correlation \(-0.5\) with it, what is the proxy’s expected marginal slope under the single-variant model?
- Why can two traits with distinct causal variants produce similar association profiles when the variants are in high LD?
- What information is lost when signed \(r\) is replaced by \(r^2\), and why can that matter?
- Name two checks that help assess whether an LD reference is appropriate for a colocalization analysis.
Solutions. 1. The expected proxy slope is \(\beta r=0.12(-0.5)=-0.060\) outcome standard deviations per genotype standard deviation; the negative sign reflects the allele coding and signed correlation. 2. Each causal genotype predicts the other well, so either causal variant can generate a strong marginal association at both positions; their profiles may differ less than sampling variability. 3. Squaring removes the sign, so \(r=0.8\) and \(r=-0.8\) both give \(r^2=0.64\), although the expected marginal slopes have opposite directions. 4. Check ancestry composition and allele frequencies, and verify variant identity, effect-allele alignment, genome build, and compatibility of the LD matrix with the analyzed variants.