Marker-Effect and GRM Representations

Why Two Representations Describe the Same Genetic Covariance

An additive genetic model can be written in terms of one random effect for each marker, or in terms of a single vector of correlated genetic effects across individuals. The first representation highlights how SNP effects combine; the second summarizes how genetically similar individuals are expected to have similar genetic deviations. Under specific assumptions about marker effects and genotype scaling, the two representations imply exactly the same covariance matrix.

The equivalence is useful because a model with many marker effects can be understood through a relationship matrix over individuals. It is not a claim that every marker-based model has the same covariance, nor that the matrix determines the realized effects uniquely. The equality follows from the assumed distribution of the marker effects and the chosen genotype matrix.

Marker-Effect Model and Genotype Scaling

Let \(n\) be the number of individuals and \(m\) the number of SNPs. Write \(Z\) for the \(n \times m\) genotype matrix, with row \(i\) describing individual \(i\) and column \(j\) describing SNP \(j\). In an additive model, the genetic value for individual \(i\) is the sum of that individual’s genotype values multiplied by the corresponding marker effects.

A genotype entry may be a raw allele dosage, usually coded 0, 1, or 2 for the number of copies of a specified allele, or a transformed value. A common scaling is to center each dosage using its allele frequency \(p_j\), then divide by its population standard deviation under Hardy–Weinberg sampling, \(\sqrt{2p_j(1-p_j)}\). In that convention, the transformed entry for individual \(i\) at SNP \(j\) is \((g_{ij}-2p_j)/\sqrt{2p_j(1-p_j)}\). This expresses genotype in standard-deviation units. Other centering and standardization conventions are possible, but the matrix used in the model and the GRM must be the same.

For the main derivation, define the GRM using the chosen genotype matrix as \(K=ZZ'/m\). This convention divides the genotype cross-product by the number of markers. Some analyses use a different denominator or omit the division; that changes the numerical variance multiplier, not the underlying covariance identity.

Deriving the GRM Representation

The key step is to calculate the covariance of a linear transformation of the random marker-effect vector. For fixed \(Z\), the mean genetic value is \(E(u\mid Z)=ZE(\alpha)=0\). Its covariance is:

The second line follows because multiplying a random vector by a fixed matrix on the left and its transpose on the right transforms its covariance in this way. The third line uses the assumed covariance \(\operatorname{Cov}(\alpha)=\sigma_\alpha^2 I_m\); the last line uses \(K=ZZ'/m\). Thus, if we define \(\sigma_u^2=m\sigma_\alpha^2\), the marker model implies \(\operatorname{Cov}(u\mid Z)=\sigma_u^2K\).

If the observed phenotype also contains fixed effects and an independent residual, the corresponding model can be written \(y=Xb+u+\varepsilon\), where \(Xb\) is the fixed-effect mean and \(\varepsilon\sim N(0,\sigma_e^2I_n)\). Under the stated normal marker-effect assumption, and assuming the residual is independent of the marker effects, this gives:

This equation specifies the covariance implied by the marker model; it does not require fitting the fixed effects or estimating the variance components. The scalar multiplying \(K\) is tied to the definition of \(K\), so it should not be interpreted without stating that definition.

What the Assumptions Do—and Do Not—Say

The covariance identity depends on how marker effects vary across SNPs. Independent effects with a common variance yield \(\sigma_\alpha^2ZZ'\). If different markers have different variances, collected in a diagonal matrix \(D\), then the covariance is \(ZDZ'\), which is not generally a scalar multiple of \(ZZ'\). If marker effects are correlated, their full covariance matrix \(\Sigma_\alpha\) gives \(Z\Sigma_\alpha Z'\). A GRM proportional to the unweighted cross-product represents the common-variance, independent-effects case, not every possible pattern of marker effects.

Normality is needed for the stated multivariate normal distribution of \(u\) and \(y\), but not for the covariance multiplication itself. If the effects are independent, mean zero, and have common finite variance \(\sigma_\alpha^2\), then \(\operatorname{Cov}(u\mid Z)=\sigma_\alpha^2ZZ'\) still holds. The distribution of \(u\) need not be exactly normal in that case.

The choice of genotype scaling determines what “common variance of marker effects” means. With standardized genotypes, a common \(\sigma_\alpha^2\) is the variance of an effect per one standard-deviation increase in genotype. With centered raw dosages, it is the variance per additional allele copy. Those are different assumptions about the genetic architecture: standardized scaling gives the same assumed effect variance across standardized markers, while raw dosage scaling gives the same assumed effect variance per allele-copy unit.

Changing the GRM Normalization

The factor multiplying a relationship matrix depends directly on its denominator. More generally, if \(K_c=ZZ'/c\) for a positive constant \(c\), then:

For \(c=m\), the multiplier is \(m\sigma_\alpha^2\). If the matrix is defined as \(ZZ'\) without division, the multiplier is just \(\sigma_\alpha^2\). If a different scaling constant is used, the multiplier changes accordingly. The product, which is the implied covariance matrix, remains the same when the definition and multiplier are adjusted consistently.

This is why variance-component values cannot be compared across GRM conventions without checking the normalization. The matrix entries may differ by a constant factor even though the model’s implied covariance is identical. Likewise, rescaling genotypes changes \(ZZ'\), so it changes the variance multiplier needed to represent the same covariance.

Common Mistake

A common error is to write \(\operatorname{Cov}(u)=\sigma_\alpha^2K\) immediately after defining \(K=ZZ'/m\). That drops the factor \(m\): since \(\sigma_\alpha^2ZZ'=m\sigma_\alpha^2K\), the stated covariance would be too small by a factor of \(m\). The correction is to either include the factor \(m\), or define a variance multiplier \(\sigma_u^2=m\sigma_\alpha^2\) for this GRM convention. Another mistake is to treat standardization as a harmless relabeling: it changes the units of the marker effects and the assumptions encoded by a common effect variance.

The result matters because it gives a precise bridge between marker-level assumptions and covariance across individuals. It also provides a practical check: before interpreting a variance multiplier, write down the genotype matrix scaling, the GRM denominator, and the marker-effect variance assumption. Only then is the implied covariance—and the meaning of its scale—clear.