Why Linkage Disequilibrium Must Enter Annotation Models

Why Association at a SNP Does Not Locate the Causal Effect

A genome-wide association study tests variants one at a time, but a test at one SNP can reflect effects at other SNPs. The reason is linkage disequilibrium (LD): genotypes at nearby variants can be correlated. If a tested SNP tends to occur with a causal variant, its association statistic can be elevated even when the tested SNP has no causal effect itself.

This distinction matters when asking whether a functional annotation explains heritability. An annotation belongs to a variant—for example, whether a SNP lies in a specified genomic feature. Association strength, by contrast, is measured at a tested SNP and may summarize effects from several correlated causal variants. Treating the test SNP’s annotation as though it necessarily describes the sources of its signal can therefore assign heritability to the wrong category.

Definition: Let \(x_j\) denote the centered, standardized genotype at SNP \(j\), and let \(r_{jk}=\operatorname{Corr}(x_j,x_k)\) be the LD correlation between SNPs \(j\) and \(k\). A tested SNP is the SNP whose marginal association is measured. A causal variant here means a SNP with a nonzero effect in the genetic model, whether or not its own association test is the one being interpreted.

How LD Carries a Causal Effect into a Marginal Association

Consider a quantitative phenotype modeled as \(y=\sum_k x_k\beta_k+\varepsilon\), where \(\beta_k\) is the effect of SNP \(k\), and \(\varepsilon\) is residual variation. With standardized genotypes, the population marginal regression coefficient at tested SNP \(j\) is the covariance between \(x_j\) and \(y\). Under the model, this covariance is:

\operatorname{Cov}(x_j,y)=\sum_k r_{jk}\beta_k.

Thus, the marginal coefficient at \(j\) can be nonzero even if \(\beta_j=0\). Correlated genotypes let SNP \(j\) tag effects at other SNPs. For a single causal variant \(k\), the marginal coefficient at \(j\) is \(r_{jk}\beta_k\); its squared signal is proportional to \(r_{jk}^2\beta_k^2\). The sign of the LD correlation changes the direction of the marginal coefficient, but not this squared contribution.

A commonly used approximation makes the implication for association statistics explicit. Suppose effects at distinct SNPs are independent, have mean zero, and have variances specified by annotations. Assume also that the phenotype and genotypes are appropriately standardized, the tested individuals are independent, and the association statistic is calibrated under the null. For a sample of size \(N\), the expected squared association statistic at \(j\) is approximately:

E[\chi_j^2]\approx 1+N\sum_k r_{jk}^2\operatorname{Var}(\beta_k).

The 1 is the sampling contribution expected when there is no genetic signal. The remaining terms are the association signal tagged at \(j\). Under an additive annotation variance model, write \(\operatorname{Var}(\beta_k)=\sum_c a_{kc}\tau_c\), where \(a_{kc}\) records whether or to what degree SNP \(k\) belongs to annotation \(c\), and \(\tau_c\) is its variance contribution per annotation unit. Substitution gives:

E[\chi_j^2]-1\approx N\sum_c \tau_c\left(\sum_k r_{jk}^2a_{kc}\right).

The quantity in parentheses includes annotations of SNPs linked to the tested SNP, not only the annotation at \(j\). This is the central reason LD must enter annotation models. The expression describes an expected relationship under stated assumptions; it is not a claim that a single observed association statistic identifies which correlated SNP caused the signal.

Worked Example: An Unannotated SNP Tags an Annotated Effect

Suppose standardized genotypes at SNPs 1 and 2 have correlation \(r_{12}=0.8\). SNP 2 has effect \(\beta_2=0.10\) on a standardized quantitative phenotype, while SNP 1 has no direct effect, so \(\beta_1=0\). Suppose SNP 2 belongs to annotation A and SNP 1 does not.

Marginal effect at SNP 1. Since SNP 2 is the only causal variant in this example, the population marginal coefficient at SNP 1 is:

r_{12}\beta_2=0.8(0.10)=0.08.

The SNP 1 association is therefore nonzero even though SNP 1 has no direct effect and is outside annotation A. Under the single-effect model, the noncentrality contribution at SNP 1 is approximately \(N(0.08)^2\). For \(N=10{,}000\), this is \(10{,}000(0.0064)=64\), giving an expected \(\chi^2\) of about \(1+64=65\). The large expected statistic is evidence of tagging in this hypothetical setup, not evidence that SNP 1 itself is causal.

The genotypes are standardized, the phenotype is standardized, and the example assumes one effect and independent sampled individuals. The arithmetic shows why assigning all of SNP 1’s association signal to its own annotation would overlook the annotated causal variant that it tags.

From Tagged Signal to Annotation-Based Attribution

The variance model concerns effects at SNPs \(k\); marginal association statistics are observed at SNPs \(j\). These are different roles. For a tested SNP \(j\), LD determines how much of the effect variance at each \(k\) contributes to its expected association strength. If annotations overlap, a causal SNP can contribute through more than one annotation according to the specified coding. If annotations are disjoint, the same logic applies with each SNP assigned to its relevant category.

A direct regression of \(\chi_j^2\) on only the annotation value \(a_{jc}\) treats SNPs as if the annotation at the test position were sufficient to describe the genetic variance generating their association. But an annotation value at \(j\) does not reveal the annotations of all SNPs linked to \(j\), or the strength of those links. Consequently, the direct regression can dilute a category’s signal when its causal variants are tagged by unannotated SNPs, and can make a category appear associated when its SNPs tag effects from another category.

Worked Example: Several Annotated Sources Contribute to One Test

Consider a tested SNP 1 and two possible causal SNPs, SNPs 2 and 3. Let \(r_{12}=0.6\) and \(r_{13}=0.5\). Suppose SNP 2 belongs to annotation A and has effect variance \(0.0025\), while SNP 3 belongs to annotation B and has effect variance \(0.0016\). Assume the two effects are independent and mean zero. The correlations are compatible with a joint genotype distribution; for example, \(r_{23}=0.2\) gives a positive determinant for the three-SNP correlation matrix.

The expected excess association statistic at SNP 1 is approximately the sum of each variance contribution weighted by squared LD:

E[\chi_1^2]-1\approx N\left[(0.6)^2(0.0025)+(0.5)^2(0.0016)\right].

The annotation A contribution per individual is \(0.36(0.0025)=0.0009\). The annotation B contribution is \(0.25(0.0016)=0.0004\). Their sum is \(0.0013\). At \(N=20{,}000\), the expected excess is \(20{,}000(0.0013)=26\), so \(E[\chi_1^2]\approx27\).

SNP 1’s statistic reflects both annotations despite the example not assigning SNP 1 to either one. A model that relates the statistic only to SNP 1’s annotation has no direct way to represent these two distinct tagged contributions. The calculation relies on independent effects; correlated effects would add covariance terms and require a correspondingly different expectation.

A Short Justification for the Squared-LD Weight

The squared-LD weighting follows from the marginal coefficient and the assumption that distinct SNP effects are independent and mean zero. Squaring \(\sum_k r_{jk}\beta_k\) produces terms for pairs of effects. For \(k\ne \ell\), independence and zero means imply \(E[\beta_k\beta_\ell]=0\); for each SNP \(k\), \(E[\beta_k^2]=\operatorname{Var}(\beta_k)\). Therefore:

E\left[\left(\sum_k r_{jk}\beta_k\right)^2\right] =\sum_k r_{jk}^2E[\beta_k^2] =\sum_k r_{jk}^2\operatorname{Var}(\beta_k).

With standardized genotypes, independent sampled individuals, and a well-calibrated association test, multiplying this genetic signal by \(N\) gives the approximate noncentrality contribution to \(\chi_j^2\); adding the unit sampling baseline yields the expectation stated above. This justification also shows the limits of the expression: if effects are correlated, cross-products need not vanish; if the test is miscalibrated, the baseline need not be 1; and finite-sample or model departures can affect the approximation.

Worked Example: A Tag SNP Can Carry Another Annotation’s Signal

Suppose SNP A is the only causal variant in a two-SNP region and belongs to annotation A. SNP B belongs to annotation B, has no direct effect, and has LD correlation \(r_{AB}=0.7\) with SNP A. Let \(\operatorname{Var}(\beta_A)=0.01\), and take \(N=5{,}000\). Assume independent individuals, standardized genotypes and phenotype, and a calibrated test.

The expected excess association at the tag SNP B is:

E[\chi_B^2]-1\approx N r_{AB}^2\operatorname{Var}(\beta_A) =5{,}000(0.7)^2(0.01)=5{,}000(0.0049)=24.5.

Thus \(E[\chi_B^2]\approx25.5\), although SNP B is not causal and the variance in this example originates at SNP A. A method that interprets the test statistic solely through SNP B’s annotation could associate the tagged signal with annotation B. The computation does not say that a whole-genome regression must assign all signal to B; it shows why the test SNP’s annotation alone cannot establish where the underlying effect variance resides.

What LD-Aware Annotation Models Address—and What They Do Not

An LD-aware model connects association strength at each tested SNP to the annotations of the SNPs it tags, weighted by their LD. In the expectation above, that connection is the sum \(\sum_k r_{jk}^2a_{kc}\) for annotation \(c\). This allows an association test at an unannotated marker to provide information about nearby annotated variants, while also accounting for the possibility that an annotated marker tags effects from other categories.

This adjustment is essential for attribution, but it does not make the result a causal experiment. Annotation definitions may be incomplete or overlapping; LD estimates can be inaccurate when the reference genotypes do not match the study population; and annotations can be correlated with unmodeled features. The expected relationship also depends on its genetic-effect and sampling assumptions. An LD-aware model addresses the mismatch between test location and effect location; it does not by itself prove a molecular mechanism or remove every source of bias.

Common Mistake

Common mistake: “The association statistic is measured at SNP \(j\), so any heritability signal it carries belongs to the annotation of SNP \(j\).”

Correction: The statistic at \(j\) can reflect effects at every SNP \(k\) correlated with it. Under the independent-effect approximation, the contribution from \(k\) is weighted by \(r_{jk}^2\operatorname{Var}(\beta_k)\). Annotation attribution must therefore account for the annotations of linked variants, not just the test SNP’s own label.

Key takeaway: An association statistic is a measure of tagged signal, not a label identifying the causal SNP’s annotation. LD connects each test to a set of potentially causal variants, so annotation models need that connection to attribute heritability coherently.

Check Your Understanding

Answer each question using the distinction between a tested SNP and the variants whose effects it may tag.

If a tested SNP has no direct effect but is correlated with a causal SNP, can its marginal association be nonzero? Explain using the marginal-coefficient formula.
Under independent, mean-zero effects, why does the expected signal involve squared LD rather than signed LD?
A tested SNP is outside annotation A but has high LD with a causal SNP in annotation A. Why can its association statistic still inform an LD-aware model about annotation A?
What assumption about effect dependence is needed for the short squared-LD justification given here?
Give one reason an LD-aware annotation model may still yield biased or uncertain attribution.