Conditional Signals Across Ancestries

Why Conditional Signals Depend on Ancestry

A variant can show a strong marginal association because it tags another associated variant through linkage disequilibrium (LD). If a locus contains multiple associated variants, each variant’s marginal association can reflect several signals at once. Consequently, the strongest marginal variant is not necessarily a causal variant, and conditioning on it does not automatically separate all the signals.

Across ancestries, the same variants can have different LD patterns. A marker that is highly correlated with a causal variant in one ancestry may be weakly correlated with it in another. Thus, a lead variant selected in one ancestry may be a useful first conditioning variable there but a poor substitute for an ancestry-matched joint analysis. Conditional association describes evidence after adjustment for specified variants; it does not, by itself, prove that any adjusted variant is causal.

How LD Shapes Marginal Associations

Consider one ancestry, indexed by \(a\), and a quantitative outcome. For the basic relationship below, suppose the genotypes are centered and standardized, the outcome is centered, and the model has no covariates:

Here \(G_a\) is the matrix of standardized genotypes at the locus, \(\boldsymbol{\beta}_a\) is the vector of joint coefficients, and the errors have mean zero conditional on the genotypes. Let \(R_a\) be the ancestry-specific genotype correlation matrix. In a population linear model, the vector of marginal coefficients is

The relationship follows because the marginal coefficient for variant \(j\) is the covariance of its standardized genotype with the outcome. Substituting the joint model gives the sum of its genotype correlations with all variants, weighted by their joint coefficients. This is why a noncausal marker can have a nonzero marginal coefficient: it may correlate with variants that have nonzero joint coefficients.

For a joint linear model with the full-rank genotype correlation matrix \(R_a\), the population joint coefficients can be recovered from the marginal coefficients by solving \(R_a\boldsymbol{\beta}_a=\mathbf{b}_a\). In sample analyses, estimates have uncertainty, and near-collinearity can make this solution unstable. A summary-statistic likelihood or individual-level regression can account for that uncertainty; simply inverting a noisy LD matrix is not always a reliable practical estimator.

Conditioning on One Variant

For two standardized variants \(j\) and \(k\), let their correlation be \(r_{jk}\). In a regression of the outcome on both variants, the coefficient of \(j\) is the association remaining after adjustment for \(k\). It can be written in terms of marginal coefficients as

The numerator is the covariance between the outcome and the part of genotype \(j\) left after its linear projection on genotype \(k\). The denominator is the variance of that residualized genotype. Thus, the conditional coefficient is not generally the simple difference \(b_j-r_{jk}b_k\); the variance adjustment matters. This expression assumes the same observations and model for the coefficients and correlation, and \(|r_{jk}|<1\). If the variants are perfectly correlated, their separate coefficients cannot be identified from these data.

Why a Single-Variant Conditional Scan Can Be Inadequate

Conditioning each remaining variant on one lead variant at a time is not the same as fitting all remaining variants jointly. When several variants are correlated, their individual conditional coefficients can be attenuated, exaggerated, or close to zero even when the joint model assigns nonzero coefficients to them. A conditional scan is useful for examining residual evidence, but interpreting each one-at-a-time coefficient as an independent signal can be misleading.

For a set \(S\) of conditioning variants, the conditional marginal coefficient of candidate \(j\) can be expressed as

Here \(R_{SS}\) is the correlation matrix among variants in \(S\), \(r_{jS}\) is the row of correlations between \(j\) and \(S\), and \(b_S\) is the vector of their marginal coefficients. This is the coefficient for \(j\) in a model adjusting for \(S\) but not necessarily for other unconditioned variants. To estimate all specified signals jointly, fit the joint model for those variants rather than relying on separate one-at-a-time conditional coefficients.

Using Ancestry-Specific LD in Joint Analysis

For ancestry \(a\), the relationship between marginal and joint coefficients uses that ancestry’s \(R_a\), with the covariate-residualization qualification stated above. When the variants’ effects are modeled separately by ancestry, the corresponding joint coefficient vector is \(\boldsymbol{\beta}_a\). If a model instead imposes shared effects, that is an additional modeling assumption; ancestry-specific LD does not itself imply that effects are shared.

Association results and LD must refer to the same variant set, allele orientation, ancestry, and compatible covariate treatment. Allele harmonization is necessary so that coefficients have consistent effect-allele directions. Likewise, if association regressions adjust for covariates such as ancestry principal components, the relevant LD matrix for these calculations is based on genotypes residualized on those covariates, not automatically the raw genotype correlation matrix.

Covariate Adjustment: Use the Right LD Matrix

The LD relationship is only as appropriate as the genotype correlation matrix used in it. In an adjusted linear regression, the Frisch–Waugh–Lovell result says that a coefficient for a genotype can be obtained by first removing the linear effects of the same covariates from the outcome and genotypes, then regressing the residualized outcome on the residualized genotype or genotypes. Therefore, the correlations relevant to the adjusted joint coefficients are among those residualized genotypes.

For example, suppose a hypothetical adjusted analysis uses two residualized, standardized genotypes with correlation \(0.20\), and the adjusted joint effects are \(0.30\) and \(0\). The adjusted marginal coefficients implied by that model are

Using the adjusted correlation matrix recovers the joint vector: solving \(R_{\mathrm{adj}}\boldsymbol{\beta}=\mathbf{b}_{\mathrm{adj}}\) gives \((0.30,0)\). If raw genotype LD were \(0.60\), substituting it would instead predict a second marginal coefficient of \(0.60(0.30)=0.18\), inconsistent with this adjusted model. The raw matrix does not describe the adjusted genotypes in this example.

Common Mistakes and Practical Interpretation

• Equating a lead variant with a causal variant. The lead is the strongest observed marginal association under the analysis, not necessarily the variant with a nonzero joint effect.
• Using the same LD matrix for every ancestry. Conditional and joint calculations require ancestry-matched LD, with the same variant definitions and compatible covariate adjustment.
• Interpreting one-at-a-time conditioning as a full joint analysis. Correlated remaining variants can mask one another in separate conditional regressions. Fit the specified variants jointly when the question concerns their simultaneous effects.
• Conditioning on one ancestry’s lead in all ancestries. Different LD can make that variant a weak proxy elsewhere. Check the ancestry-specific conditional or joint model rather than assuming the lead captures the same signal.
• Calling a conditional association causal proof. Conditional estimates depend on the chosen variants, LD, sample, and model assumptions. Residual association can reflect an omitted signal or model misspecification as well as a distinct causal effect.