Proximal Contamination and Leave-One-Chromosome-Out GRMs

Why the Relationship Matrix Matters in Association Testing

A single-variant association model often includes a polygenic random effect to account for genetic similarity among individuals. That adjustment is useful, but it creates a potential tension: if the tested variant and nearby variants also contribute to the genomic relationship matrix, the random effect can represent some of the same local genetic pattern that the fixed-effect test is trying to measure. This is called proximal contamination.

A common diagnostic and mitigation strategy is leave-one-chromosome-out (LOCO) analysis. For each tested chromosome, construct the relationship matrix using markers outside that chromosome, then fit the association model using that matrix. Comparing the full-matrix and LOCO results shows whether including the tested chromosome’s markers materially changes the estimated association. It does not, by itself, establish that either estimate is the causal effect.

Model and Definition

Let \(y\) be an \(n\)-vector of quantitative phenotypes, let \(X_0\) contain the fixed covariates (such as an intercept and ancestry covariates), and let \(x_j\) be the genotype dosage vector for the variant being tested. Dosage is measured in allele copies, usually from 0 to 2. A single-variant linear mixed model is

y=X_0\gamma+x_j\beta_j+g+\varepsilon,\qquad g\sim N(0,\sigma_g^2K),\qquad \varepsilon\sim N(0,\sigma_e^2I).

Here \(\beta_j\) is the fixed association effect, measured in phenotype units per additional dosage copy; \(g\) is the polygenic random effect; and \(\varepsilon\) is residual error. The model assumes \(g\) and \(\varepsilon\) are independent, and the covariance matrix \(K\) is constructed from genotype data. Conditional on the genotypes and model parameters, the marginal covariance of \(y\) is \(V=\sigma_g^2K+\sigma_e^2I\). The association test evaluates \(H_0:\beta_j=0\), while accounting for this covariance.

Definition: Proximal contamination is the potential for a polygenic covariance matrix to include marker information from the same genomic region as the tested variant, allowing the random effect to absorb part of the local phenotype pattern and thereby changing the fixed-effect association estimate, its standard error, or its test statistic. The effect is often a loss of association evidence, but its direction and size are not guaranteed.

The issue is not that a random effect literally removes a variant from the data. Rather, the model allows phenotype variation aligned with genetic similarity represented in \(K\) to be explained by \(g\). When the tested variant is correlated with markers contributing to \(K\), the local fixed-effect and random-effect explanations can overlap.

How a Marker Enters the GRM

For illustration, suppose \(Z\) is a matrix of centered and scaled genotypes for \(M\) markers, with one marker per column. A commonly used relationship matrix is

K=\frac{ZZ^\mathsf{T}}{M} =\frac{1}{M}\sum_{k=1}^{M}z_kz_k^\mathsf{T},

where \(z_k\) is the genotype vector for marker \(k\). Scaling conventions vary, so the construction used in an analysis should follow its specified GRM definition. The key point here is the additive contribution: each marker contributes a rank-one matrix \(z_kz_k^\mathsf{T}\), weighted by the scaling factor.

If marker \(j\) is on chromosome \(c\), its direct contribution to the full matrix is \(z_jz_j^\mathsf{T}/M\). Other markers on chromosome \(c\), including markers correlated with \(j\) through linkage disequilibrium, also contribute. Thus, the relevant overlap can be larger than the contribution of the tested marker alone.

Worked Example: A Tested Chromosome’s Contribution to the GRM

Consider four hypothetical individuals and three centered, scaled marker vectors: \(z_1=(1,1,-1,-1)^\mathsf{T}\), \(z_2=(1,-1,1,-1)^\mathsf{T}\), and \(z_3=(1,-1,-1,1)^\mathsf{T}\). Suppose markers 1 and 2 are on the tested chromosome, marker 3 is elsewhere, and \(K\) uses the average of the three marker contributions.

For individuals 1 and 2, the full-matrix entry is

K_{12} =\frac{z_{11}z_{21}+z_{12}z_{22}+z_{13}z_{23}}{3} =\frac{(1)(1)+(1)(-1)+(1)(-1)}{3} =-\frac{1}{3}.

If the tested chromosome is left out, only marker 3 remains in this toy example, so \(K_{-c}=z_3z_3^\mathsf{T}\), and \( (K_{-c})_{12}=(1)(-1)=-1\). The off-diagonal entry changes from \(-1/3\) to \(-1\), because the two excluded markers had partly offset the retained marker’s contribution. This example demonstrates that exclusion changes the covariance pattern; it does not imply that every LOCO entry must become larger in magnitude.

Constructing and Fitting a LOCO Analysis

For each chromosome \(c\), the LOCO matrix \(K_{-c}\) is built from markers not on chromosome \(c\). If the full GRM averages over \(M\) markers, a straightforward chromosome-specific construction averages over the \(M_{-c}\) retained markers:

K_{-c}=\frac{1}{M_{-c}}\sum_{k:\,\operatorname{chr}(k)\ne c}z_kz_k^\mathsf{T}.

Using the retained-marker count keeps the matrix on the same average-marker scale as the full GRM. Other scaling conventions may be appropriate in a particular analysis, but the scale must be handled consistently: changing the scale of \(K\) changes the numerical interpretation of its variance component.

For a variant on chromosome \(c\), fit the single-variant model with \(K_{-c}\) in place of the full \(K\). Keep the phenotype, tested genotype, fixed covariates, and association test definition the same. Variance components are generally re-estimated under the chosen covariance matrix (or estimated using a specified null-model procedure). Consequently, a difference in association results can reflect both the changed covariance pattern and the resulting variance-component estimates.

Identify the tested chromosome.
Determine which chromosome contains the variant whose fixed effect is being tested.

Build its excluded-marker GRM.
Remove all markers on that chromosome and compute \(K_{-c}\) using the analysis’s stated genotype scaling and GRM normalization.

Fit the association model.
Use the same phenotype and fixed-effect specification, and fit the tested genotype as a fixed effect with the LOCO covariance structure.

Compare like with like.
Report both effect estimates and their standard errors, as well as test statistics and p-values, noting the matrix and fitting procedure used for each.

Why Exclusion Can Change the Test

In generalized least squares, the fixed-effect estimate depends on the covariance matrix. For a given \(V\), let \(P\) denote the residual precision matrix after accounting for the fixed covariates \(X_0\):

P=V^{-1}-V^{-1}X_0(X_0^\mathsf{T}V^{-1}X_0)^{-1}X_0^\mathsf{T}V^{-1}.

The single-variant estimate and its model-based standard error are

\hat{\beta}_j=(x_j^\mathsf{T}Px_j)^{-1}x_j^\mathsf{T}Py,\qquad SE(\hat{\beta}_j)=(x_j^\mathsf{T}Px_j)^{-1/2}.

These expressions make the dependence clear: changing \(K\) changes \(V\), which changes \(P\), and can therefore change both the estimate and its standard error. The resulting Wald statistic is \(z=\hat{\beta}_j/SE(\hat{\beta}_j)\); under the usual large-sample null approximation, its two-sided p-value is \(2\{1-\Phi(|z|)\}\), where \(\Phi\) is the standard normal cumulative distribution function.

A useful justification for LOCO follows directly from the marker-sum representation. Splitting the full GRM into markers on chromosome \(c\) and markers elsewhere gives

K_{\mathrm{full}} =\frac{M_{-c}}{M}K_{-c} +\frac{M_c}{M}K_c,

where \(M_c\) is the number of markers on chromosome \(c\), and \(K_c\) is the average contribution among those markers. This identity follows by dividing the full sum into its two disjoint marker sets. The chromosome’s contribution is therefore explicitly present in the full matrix and absent from \(K_{-c}\). If local marker patterns align with the tested genotype, this exclusion can reduce overlap between the random-effect covariance and the fixed-effect test. It does not guarantee a larger estimate, smaller p-value, or unbiased causal effect.

Worked Example: Comparing Full-GRM and LOCO Association Results

Suppose a hypothetical single-variant analysis estimates a dosage effect of \(0.2286\) phenotype units per allele copy using the full GRM, with standard error \(0.1000\). The corresponding LOCO analysis estimates \(0.3000\) phenotype units per allele copy, with standard error \(0.1050\). Use two-sided Wald tests and report the p-values to five decimal places.

For the full-GRM fit,

z_{\mathrm{full}}=\frac{0.2286}{0.1000}=2.286,\qquad p_{\mathrm{full}}=2\{1-\Phi(2.286)\}\approx 0.02223.

For the LOCO fit,

z_{\mathrm{LOCO}}=\frac{0.3000}{0.1050}=2.857,\qquad p_{\mathrm{LOCO}}=2\{1-\Phi(2.857)\}\approx 0.00428.

At \(\alpha=0.05\), both tests reject \(H_0:\beta_j=0\) under their respective model-based tests, and the LOCO fit has stronger evidence against that null. The estimated effect is larger in the LOCO fit, but its standard error is also slightly larger; the smaller p-value comes from the larger ratio of estimate to standard error. Approximate 95% Wald intervals are \(0.2286\pm1.96(0.1000)=[0.0326,0.4246]\) for the full-GRM fit and \(0.3000\pm1.96(0.1050)=[0.0942,0.5058]\) for LOCO, in phenotype units per allele copy. These intervals are model-based approximations, not probabilities that the fixed parameter lies in a particular realized interval.

The difference is consistent with the possibility that the full GRM absorbed some local signal, but it does not prove that this occurred. Different covariance estimates, sampling variation, or model misspecification could also contribute. The two estimates should be described as results under two covariance specifications, not as two independent replications.

Worked Example: Interpreting Little Change Under LOCO

In another hypothetical analysis, a variant has full-GRM estimate \(0.150\) phenotype units per dosage copy with standard error \(0.060\), and LOCO estimate \(0.148\) with standard error \(0.061\). Calculate and interpret the Wald statistics.

The full-GRM statistic is \(z=0.150/0.060=2.50\), giving a two-sided normal p-value of approximately \(0.0124\). The LOCO statistic is \(z=0.148/0.061=2.426\), giving a two-sided p-value of approximately \(0.0153\). Both estimates are positive and similar in magnitude, and their standard errors are also similar. Thus, in this hypothetical comparison, excluding the tested chromosome has little effect on the estimated association. This does not show that proximal contamination is impossible; it indicates only that the association results are not highly sensitive to this particular change in the GRM.

Assumptions, Uses, and Limits

A full-GRM versus LOCO comparison is most interpretable when both fits use the same individuals, phenotype definition, genotype dosage for the tested variant, fixed covariates, and association-test procedure. The GRMs should be constructed with documented marker inclusion, allele-frequency handling, and scaling. The mixed-model assumptions also remain relevant: the chosen covariance structure should adequately represent residual genetic similarity, and the uncertainty calculation should match the estimation and testing procedure.

LOCO is useful because it reduces direct chromosome-level overlap while retaining genome-wide information from other chromosomes to model relatedness and broad genetic structure. It is a sensitivity analysis, not a universal correction. It cannot guarantee that local signal has been removed from every source, nor does it eliminate confounding from population structure, relatedness, phenotype artifacts, or other model problems. The fixed-effect estimate is an association conditional on the model; it should not be described as a causal effect without additional justification.

The comparison can be summarized in context: state the two effect estimates in phenotype units per dosage copy, their standard errors, and the corresponding test results; identify which covariance matrix produced each result; and note whether the substantive conclusion changes. A p-value difference alone is not an adequate comparison because it does not distinguish a change in effect magnitude from a change in precision.

Common Mistake

Common mistake: “The LOCO p-value is smaller, so the full-GRM analysis was definitely biased toward the null, and the LOCO estimate is the true genetic effect.” This conclusion overstates what the comparison establishes. The correction is that LOCO removes markers on the tested chromosome from the GRM, which can reduce proximal contamination, but the estimates can still differ because the covariance and variance components differ. A smaller p-value is evidence against the tested null under that fitted model; it is not proof of causation or proof that the full-GRM estimate was biased.

Key takeaway: A GRM that includes the tested chromosome can represent local genetic patterns also related to the tested genotype. LOCO removes that chromosome’s markers from the covariance matrix and provides a useful comparison, but the resulting changes in association estimates must be interpreted as sensitivity to model specification rather than definitive evidence of bias or causation.

Check Your Understanding

Answer each question using the model and definitions above.

In a single-variant mixed model, distinguish the role of the tested genotype’s fixed effect from the role of the GRM-based random effect.
Why can markers in linkage disequilibrium with the tested variant matter even when the tested marker itself contributes only one term to the GRM?
For a variant on chromosome \(c\), which markers are excluded when constructing \(K_{-c}\), and why should GRM scaling be documented?
If a LOCO analysis has a larger effect estimate but also a larger standard error, what quantities should be compared before concluding that association evidence increased?
Does a smaller LOCO p-value prove that the full-GRM result was biased or that the variant is causal? Explain.