Why Linkage Disequilibrium Changes Marginal GWAS Effects

Why a GWAS Coefficient Can Differ From a Variant’s Own Effect

A GWAS commonly estimates the association of each variant with a phenotype one variant at a time. When variants are correlated through linkage disequilibrium (LD), a variant’s marginal association can reflect effects at other correlated variants as well as its own. Consequently, a strong marginal association does not by itself establish that the tested variant has a direct effect in a joint model.

This distinction matters when constructing a polygenic risk score (PRS) from summary statistics. A set of marginal GWAS coefficients reflects both the underlying genetic effects and the LD pattern in the GWAS population. The coefficients are therefore not generally interchangeable with joint effects, and their values can change when the LD pattern changes, even if the underlying joint effects do not.

The Relationship Between Marginal and Joint Effects

First consider a linear additive model after centering the variables and adjusting for any required covariates. Write the genotype variables in standardized form, so each \(X_j\) has mean 0 and variance 1. The joint model is

where \(\beta_k\) is the joint effect for variant \(k\), measured in phenotype units per one standard deviation of its genotype, and the residual \(\varepsilon\) has mean 0 and is uncorrelated with every included genotype. Let \(R\) be the genotype correlation matrix, with entries \(r_{jk}=\operatorname{Corr}(X_j,X_k)\). Under these assumptions, the population marginal slope for variant \(j\) is

The second equality uses \(\operatorname{Var}(X_j)=1\). The last equality follows by substituting the joint model and using \(\operatorname{Cov}(X_j,\varepsilon)=0\). In vector form, the relation is \(\boldsymbol{\alpha}=R\boldsymbol{\beta}\). Thus, a marginal coefficient combines joint effects, weighted by correlations with the tested variant. The tested variant’s own joint effect contributes because \(r_{jj}=1\); correlated variants contribute too.

For unstandardized dosages, if the joint model uses raw \(G_k\) dosages and coefficients \(\beta_k\) per effect-allele copy, the corresponding relation is

Here \(\alpha_j\) and \(\beta_k\) are in phenotype units per dosage copy. The standardized and dosage-scale formulas express the same principle, but their coefficients have different units. This tutorial assumes alleles are coded consistently: changing the coded allele changes the sign of the corresponding genotype and effect.

Deriving the Summary-Statistic Relationship

The population formula also explains what a vector of marginal GWAS estimates measures. Suppose \(n\) observations have centered, standardized genotype columns, collected in a matrix \(X\), and the phenotype follows \(Y=X\boldsymbol{\beta}+\boldsymbol{\varepsilon}\). For the simple marginal regression of \(Y\) on column \(j\), the estimated slope is \(\hat{\alpha}_j=X_j^\mathsf{T}Y/n\), because \(X_j^\mathsf{T}X_j=n\). Collecting these slopes gives

where \(\hat{R}=X^\mathsf{T}X/n\) is the sample genotype correlation matrix under this standardization. The first term is the LD-weighted signal; the second is sampling noise. If the residuals are independent with variance \(\sigma^2\), then, conditional on \(X\), the noise vector has covariance \(\sigma^2\hat{R}/n\). Thus LD affects both the expected marginal coefficients and, in this simplified model, the covariance among their estimation errors.

In a real GWAS, genotypes may be differently scaled, sample sizes and standard errors can vary across variants, and covariates are usually included. The exact sample calculation must reflect that analysis. The core population relation still applies to the appropriately adjusted variables under the linear-model assumptions. For a binary phenotype analyzed by logistic regression, the linear identity is not generally exact for the reported marginal log-odds coefficients; interpretation depends on the fitted model and effect scale.

What LD Means for PRS Weights

A PRS formed from standardized genotypes can be written \(S=\boldsymbol{w}^{\mathsf{T}}\boldsymbol{X}\), where \(w_j\) is the weight assigned to variant \(j\). If all relevant causal variants are included and the additive linear model is correct, then the model’s genetic component is \(\boldsymbol{\beta}^{\mathsf{T}}\boldsymbol{X}\). Marginal GWAS coefficients instead estimate \(\boldsymbol{\alpha}=R\boldsymbol{\beta}\). Using them directly as weights therefore makes the score reflect the LD-weighted effects, not generally the joint effects.

This does not mean marginal effects are useless as score weights. They may carry predictive information, especially when effects are estimated with uncertainty or the score is evaluated in a particular population. It does mean their interpretation is conditional on the LD pattern and the GWAS model. In an idealized setting where \(R\) is known and invertible, \(\boldsymbol{\beta}=R^{-1}\boldsymbol{\alpha}\); however, when variants are highly correlated, \(R\) can be close to singular, so small changes in marginal estimates can produce large changes in an inferred joint-effect vector. This algebraic observation is not by itself a recommendation to invert an estimated LD matrix.

Assumptions, Diagnostics, and Alternatives

The identity \(\boldsymbol{\alpha}=R\boldsymbol{\beta}\) relies on a correctly specified additive linear model for the phenotype scale being analyzed, consistent genotype coding, and residuals uncorrelated with the included genotypes after adjustment. The LD matrix and effects must refer to the same population and the same set of variants. If a model omits relevant variants, the marginal effect can also reflect their correlation with the included variants. Dominance, interactions, nonlinearity, population structure not adequately controlled, or a different phenotype scale can make the simple interpretation incomplete.

Useful checks include verifying effect-allele alignment, inspecting genotype correlations among nearby variants, and comparing marginal estimates with estimates from a joint model when that comparison is meaningful. Large marginal effects at many highly correlated variants may represent one shared signal rather than many independent effects. Differences between marginal and joint coefficients are not, on their own, proof that either estimate is erroneous: the two coefficients answer different questions. A change in the target population’s LD can also change marginal effects, even if the underlying joint effects are unchanged.

The main alternatives address different aims. A marginal analysis estimates one-variant-at-a-time associations. A joint regression estimates effects conditional on the other included variants, but strong LD can make those estimates unstable or difficult to distinguish. LD-aware summary-statistic approaches use information about correlations to relate marginal estimates to a joint-effect model. They rely on additional assumptions and appropriate LD information; choosing and auditing that information is a separate practical task.

Common Mistake and Key Takeaway

Wrong reasoning: “Variant 2 has a nonzero marginal GWAS coefficient, so variant 2 must have a nonzero joint effect.” The first worked example disproves this: \(\alpha_2=0.09\) even though \(\beta_2=0\). The correction is to distinguish the association from a one-variant regression from the effect conditional on other variants. Under the stated linear-model assumptions, the former is a row of \(R\boldsymbol{\beta}\), not simply the corresponding entry of \(\boldsymbol{\beta}\).