Leave-One-Chromosome-Out Relationship Matrices

Why Use a Chromosome-Specific Relationship Matrix?

A genomic relationship matrix summarizes how genetically similar the sampled individuals are. In a linear mixed model (LMM), it supplies the covariance structure for a random genetic effect. In a genome-wide association scan, however, a tested variant is also a genetic measurement of those same individuals. If the variant contributes to the relationship matrix used in its own test, its signal can partly enter the model's random-effect covariance. A leave-one-chromosome-out (LOCO) construction avoids that direct reuse by leaving out every marker on the chromosome containing the tested variant.

The practical rule is simple: for a variant on chromosome $c$, use the relationship matrix constructed without chromosome $c$. The matrix still uses markers on the other chromosomes to represent genome-wide relatedness. LOCO is therefore a chromosome-level exclusion, not a rule to remove only the tested marker or a small region around it.

Definition: Let $z_j$ be the $n$-element vector of standardized genotypes for marker $j$, and let $c(j)$ denote its chromosome. If $M_{-c}$ eligible markers are on chromosomes other than $c$, the LOCO relationship matrix for chromosome $c$ is $$ K^{(-c)}=\frac{1}{M_{-c}}\sum_{j:\,c(j)\ne c}z_jz_j^{T}. $$ It is the average outer product of the genotype vectors retained after excluding chromosome $c$.

Here, $n$ is the number of individuals and $z_j$ is usually centered and scaled so that markers with different allele frequencies are on a comparable scale. For a diploid additive genotype $g_{ij}\in\{0,1,2\}$, one common standardization is $z_{ij}=(g_{ij}-2p_j)/\sqrt{2p_j(1-p_j)}$, where $p_j$ is the allele frequency used for marker $j$. The exact genotype processing and missing-data rules must be applied consistently across the matrices. Markers with no usable genotype variation cannot contribute a valid standardized column and are not counted in $M_{-c}$.

With this definition, $K^{(-c)}$ is an $n\times n$ matrix. Its $(i,k)$ entry summarizes the relationship of individuals $i$ and $k$ using the retained markers. The matrix is positive semidefinite: for any vector $a$, $$ a^T K^{(-c)}a =\frac{1}{M_{-c}}\sum_{j:\,c(j)\ne c}(a^Tz_j)^2\geq 0. $$ This matters because the matrix is used as part of a covariance model.

Using the Matching Matrix in an Association Test

For a tested variant $j$ on chromosome $c$, a basic LOCO association model is $$ y=X\beta+x_jb_j+g^{(-c)}+\varepsilon,\qquad g^{(-c)}\sim N(0,\sigma_{g,c}^{2}K^{(-c)}),\qquad \varepsilon\sim N(0,\sigma_{e,c}^{2}I). $$ Here $y$ is the phenotype vector, $X$ contains the fixed covariates, $x_j$ is the tested genotype vector, and $b_j$ is its fixed-effect coefficient. The random effect and residual are assumed independent. Under the null hypothesis $H_0:b_j=0$, the covariance is $V^{(-c)}=\sigma_{g,c}^{2}K^{(-c)}+\sigma_{e,c}^{2}I$. A Wald, score, or likelihood-ratio association test can then be formed under the corresponding fitted null model.

The key selection decision is made using the tested variant's chromosome, not the variant's position within that chromosome: all variants on chromosome $c$ use $K^{(-c)}$. The fixed covariates, phenotype, sample, and genotype quality-control criteria should otherwise be handled consistently. Many analyses estimate variance components for each chromosome-specific null model. Some computational workflows reuse variance-component estimates across chromosomes; if so, that choice should be explicit, since the fitted covariance may not be exactly the covariance implied by refitting with each $K^{(-c)}$.

Worked Example: Constructing a LOCO Matrix

Suppose four hypothetical individuals have two standardized genotype vectors on chromosome 2, $z_1=(1,-1,1,-1)^T$ and $z_2=(1,1,-1,-1)^T$, and the tested variant is on chromosome 1. Assume these are the only eligible markers outside chromosome 1. Each vector is centered and has mean square 1, so they are on the stated standardized scale.

The chromosome 1 LOCO matrix averages their outer products: $$ K^{(-1)}=\frac{z_1z_1^T+z_2z_2^T}{2} =\frac{1}{2} \begin{pmatrix} 2&0&0&-2\\ 0&2&-2&0\\ 0&-2&2&0\\ -2&0&0&2 \end{pmatrix} = \begin{pmatrix} 1&0&0&-1\\ 0&1&-1&0\\ 0&-1&1&0\\ -1&0&0&1 \end{pmatrix}. $$ For example, the $(1,4)$ entry is $\{(1)(-1)+(1)(-1)\}/2=-1$; the $(1,2)$ entry is $\{(1)(-1)+(1)(1)\}/2=0$.

The matrix uses chromosome 2 markers because chromosome 1 is being tested. It is positive semidefinite by the outer-product argument above, although it is rank deficient because only two genotype vectors contribute. This toy calculation illustrates matrix construction; in an actual analysis, marker count, sample size, and genotype processing determine the matrix used in the fitted model.

Why the Exclusion Works, and What It Does Not Do

The construction can be understood by splitting a full-genome matrix into the contribution from chromosome $c$ and the contribution from all other chromosomes. Let $M_c$ be the number of eligible markers on $c$, and let $K^{(c)}$ and $K^{(-c)}$ be the averages within and outside that chromosome, respectively. If $M=M_c+M_{-c}$, then the full matrix based on the same standardized genotypes is $$ K=\frac{M_c}{M}K^{(c)}+\frac{M_{-c}}{M}K^{(-c)}. $$ Thus, a tested chromosome contributes directly to the full matrix, while it contributes nothing to $K^{(-c)}$. Removing that contribution is the defining LOCO safeguard. The exclusion does not prove that all possible dependence between a marker and the remaining genetic background disappears; it ensures only that markers on the tested chromosome are not used to construct that chromosome's random-effect relationship matrix.

This distinction is important. LOCO does not remove the tested variant from the fixed-effect test, and it does not remove other chromosomes from the random-effect model. Nor is it equivalent to excluding only a nearby window: every marker on the tested chromosome is omitted, including markers far from the tested site. The broader exclusion makes the matrix choice straightforward at scan scale, but it also means that each chromosome has a slightly different set of markers and potentially a different estimated covariance.

Worked Example: Selecting a Matrix Across a Scan

Suppose a hypothetical genome-wide scan uses autosomes 1, 2, and 3, with one LOCO matrix prepared for each. The tested variants are $v_A$ on chromosome 1, $v_B$ on chromosome 2, and $v_C$ on chromosome 1. Apply the selection rule.

For $v_A$, use $K^{(-1)}$, which excludes all chromosome 1 markers. For $v_B$, use $K^{(-2)}$. For $v_C$, also use $K^{(-1)}$; the position of $v_C$ within chromosome 1 does not change the choice. In this scan, both chromosome 1 variants use the same chromosome 1 LOCO null covariance, while the chromosome 2 variant uses the chromosome 2 LOCO covariance.

If a variant has a missing or ambiguous chromosome assignment, the matrix cannot be selected reliably from this rule alone. Its chromosome assignment should be resolved or the variant should be excluded from the LOCO scan, rather than assigned an arbitrary matrix.

Scaling and Fitting Choices

The denominator in the LOCO definition is $M_{-c}$, the number of markers actually contributing to that matrix. Dividing by the full marker count $M$ after excluding chromosome $c$ would shrink the matrix by a factor of $M_{-c}/M$, changing the covariance scale. Such a matrix is not the same average relationship matrix, and its variance-component scale would differ. A consistent normalization is essential when comparing or fitting the chromosome-specific matrices.

Even with the same normalization rule, the matrices need not have identical diagonals or eigenvalues. Genotype standardization, missingness, allele-frequency estimates, and chromosome-specific marker counts affect their realized values. A practical analysis should document which variants enter each matrix, how genotypes are standardized and imputed, and whether matrix trace or diagonal rescaling is performed. Any rescaling must be applied consistently and reported, since it changes the interpretation of the associated variance component.

Worked Example: The Effect of the Denominator

Consider a hypothetical data set with 100 eligible markers in total: 20 on chromosome 1 and 80 on other chromosomes. Suppose the average chromosome 1 contribution has entry $K^{(1)}_{12}=0.60$, and the average outside-chromosome contribution has entry $K^{(-1)}_{12}=0.10$, for individuals 1 and 2.

Using the full-genome average, that off-diagonal entry is $$ K_{12}=\frac{20}{100}(0.60)+\frac{80}{100}(0.10) =0.12+0.08=0.20. $$ The chromosome 1 LOCO entry is $0.10$, because it averages only the 80 retained markers. If the retained sum were incorrectly divided by 100 rather than 80, the resulting entry would be $(80/100)(0.10)=0.08$. That is a differently scaled matrix, not the stated LOCO average. This arithmetic concerns matrix construction only; it does not by itself determine the association estimate or its uncertainty.

Diagnostics, Alternatives, and Limitations

Several checks help detect a faulty LOCO implementation before interpreting association results:

Chromosome mapping: confirm that each tested variant is paired with the matrix omitting its own chromosome, including variants with special or nonstandard chromosome labels.
Marker accounting: verify that no chromosome $c$ marker enters $K^{(-c)}$, and that $M_{-c}$ counts exactly the eligible retained markers.
Matrix properties: check dimensions, symmetry, finite entries, and positive semidefiniteness up to numerical tolerance. Large negative eigenvalues or unexpected diagonal scaling can indicate data-processing or normalization errors.
Consistency across chromosomes: compare marker counts and diagonal summaries across LOCO matrices. Differences can be real consequences of marker composition, but a dramatic unexplained change warrants checking missingness, allele coding, and exclusions.
Model fitting: record whether variance components were estimated separately for each LOCO matrix or reused, and ensure the same individuals and covariate definitions are used in the chromosome-specific analyses.

A common mistake is to say, “The tested SNP is not in the full relationship matrix, because it is tested as a fixed effect.” That reasoning is incorrect: the full matrix can still include the tested SNP's genotype contribution. The correction is to select $K^{(-c)}$ for a tested variant on chromosome $c$, ensuring that no marker on that chromosome contributes to its random-effect matrix.

The main alternative is a single full-genome relationship matrix used for every test. It is simpler and requires only one matrix, but it retains the tested chromosome's contribution. LOCO requires multiple matrices and can require repeated null-model fitting, but it removes that chromosome's contribution while retaining information from the rest of the genome. A leave-one-marker-out or local-window exclusion is a different construction; it may be more targeted, but it is not the chromosome-wide rule defined here and requires its own explicit matrix and fitting procedure.

LOCO is not a remedy for every model problem. It cannot correct a misspecified phenotype model, poor covariate adjustment, genotyping errors, or inappropriate handling of sample structure. A chromosome with few eligible markers may also yield a less stable or less representative relationship matrix. In those settings, inspect the matrix and fitted model rather than assuming that chromosome exclusion alone guarantees valid association inference.

Key takeaway: For each tested variant on chromosome $c$, use a relationship matrix averaged over eligible markers outside chromosome $c$, with a denominator and genotype-processing scheme that match the markers actually retained. Check chromosome assignment, matrix scaling, and null-model fitting as part of the analysis.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions.

A variant is on chromosome 4. Which markers may contribute to its LOCO relationship matrix, and which must be excluded?
Why is $K^{(-c)}$ positive semidefinite under the outer-product definition?
A matrix is constructed by excluding chromosome $c$, but its retained genotype sum is divided by the full marker count $M$, not $M_{-c}$. What has changed?
Two tested variants are both on chromosome 7 but are far apart. Which LOCO matrices should they use?

Solutions. (1) Markers on every chromosome other than chromosome 4 may contribute; all chromosome 4 markers are excluded. (2) For any vector $a$, $a^TK^{(-c)}a$ is the average of the nonnegative squared quantities $(a^Tz_j)^2$, so it is nonnegative. (3) The matrix is scaled by $M_{-c}/M$ relative to the average over retained markers; it is not the defined LOCO average. (4) Both use $K^{(-7)}$, because matrix selection depends on chromosome, not distance between tested positions.