Scaling Genomic Relationship Matrices

Why Matrix Scale Matters

A genomic relationship matrix (GRM) summarizes pairwise genetic similarity among the individuals in a study. In a Gaussian mixed model, it determines the pattern of covariance assigned to genetic effects. But the numerical values in a GRM depend on how it is normalized. Consequently, a variance component reported alongside the matrix cannot be interpreted without knowing that normalization.

The key distinction is between changing the scale of one fixed matrix and changing the matrix’s pattern of relative similarities. A uniform rescaling changes the numerical value of the corresponding variance component but can leave the model’s covariance unchanged. A different weighting of markers may change the covariance pattern itself, in which case a single variance-component conversion is generally not enough.

Definition: Let \(K\) be an \(n \times n\) genomic relationship matrix for \(n\) individuals. In the Gaussian mixed model \(y=X\beta+g+e\), suppose \(K\) is known and positive semidefinite, \(g\sim N(0,\sigma_g^2K)\), and \(e\sim N(0,\sigma_e^2I)\), with \(g\) and \(e\) independent. Then \(\sigma_g^2\) is the genetic variance component on the scale defined by \(K\), and the marginal covariance of \(y\) is \(V=\sigma_g^2K+\sigma_e^2I\).

Here \(y\) is the vector of outcomes, \(X\beta\) is the fixed-effects mean, \(g\) is the vector of genetic random effects, and \(e\) is residual error. The identity matrix \(I\) has size \(n \times n\). This model requires \(K\) to be positive semidefinite so that \(\sigma_g^2K\) can be a covariance matrix for \(\sigma_g^2\geq 0\). The GRM is often dimensionless, while \(\sigma_g^2\) and \(\sigma_e^2\) have units of outcome squared. The component’s numerical value depends on how \(K\) is scaled.

A General Rule for Scalar Rescaling

Suppose two conventions use matrices related by \(K^*=aK\), where \(a>0\) is a constant. Their entries have the same relative pattern, but the entries in \(K^*\) are all \(a\) times those in \(K\). To represent the same genetic covariance, the variance component must change in the opposite direction.

\sigma_{g,*}^2K^*=\sigma_g^2K \quad\Longleftrightarrow\quad \sigma_{g,*}^2=\frac{\sigma_g^2}{a}.

The residual component does not need to change under this conversion because it multiplies \(I\), not \(K\). Thus, when both models use the same fixed effects and residual covariance, the total covariance \(V\) is preserved. This is an algebraic equivalence of parameterizations, not a claim that every estimation procedure will return identical numerical results after rounding or numerical tolerances.

Key result: If \(K^*=aK\) for a positive scalar \(a\), then use \(\sigma_{g,*}^2=\sigma_g^2/a\) to preserve the genetic covariance. If the matrices are not scalar multiples of one another, this one-number conversion does not generally preserve covariance.

Worked Example: Doubling Every GRM Entry

Suppose a hypothetical two-individual model uses \(K=\begin{pmatrix}1&0.2\\0.2&1\end{pmatrix}\), \(\sigma_g^2=0.30\) outcome-units squared, and \(\sigma_e^2=0.50\) outcome-units squared. A second convention reports \(K^*=2K\). Find the genetic variance component under the second convention and verify that the covariance is unchanged.

Since \(a=2\), the conversion rule gives \(\sigma_{g,*}^2=0.30/2=0.150\) outcome-units squared. Under the original convention, \(\sigma_g^2K=0.30K=\begin{pmatrix}0.30&0.06\\0.06&0.30\end{pmatrix}\). Under the second convention, \(\sigma_{g,*}^2K^*=0.150(2K)=0.30K\), which is the same matrix.

Adding the unchanged residual covariance gives \(V=\begin{pmatrix}0.80&0.06\\0.06&0.80\end{pmatrix}\) outcome-units squared under either convention. The matrix entries were doubled, but the genetic covariance did not change because the variance component was halved.

Common Normalization Choices

A GRM may be constructed from centered genotype dosages and then divided by a chosen quantity. For illustration, let \(x_{ij}\) denote the dosage for individual \(i\) at marker \(j\), and let \(p_j\) denote the allele frequency used for that marker. A common per-marker standardization uses entries formed from sums of products of centered dosages, with each marker divided by its genotype variance \(2p_j(1-p_j)\), then averages across the \(m\) markers. Such a construction gives each standardized marker equal weight in that average.

Another construction divides the sum of the centered-dosage cross-products by the total genotype variance across markers, \(2\sum_j p_j(1-p_j)\). This pools the marker contributions before dividing. These two definitions are both used in practice, but when allele frequencies vary, they do not generally differ by a single constant: the pooled version gives relatively more weight to markers with larger \(2p_j(1-p_j)\). Therefore, treating them as interchangeable scalar normalizations can be incorrect.

A separate, simple convention is to rescale a given GRM so its mean diagonal is one. The mean diagonal is \(\operatorname{tr}(K)/n\), where \(\operatorname{tr}(K)\) is the sum of the diagonal entries. If this mean is \(d>0\), define \(K_{\mathrm{unit}}=K/d\). This is a scalar rescaling, so the conversion rule applies exactly. It makes the average diagonal equal to one, but it does not force every individual diagonal entry to equal one.

Worked Example: Converting to Mean-Diagonal-One Scale

Consider the hypothetical matrix \(K=\begin{pmatrix}2&0.4\\0.4&1\end{pmatrix}\), with \(\sigma_g^2=0.12\) outcome-units squared. Its mean diagonal is \(d=(2+1)/2=1.5\). Convert it to mean-diagonal-one scale and find the corresponding variance component.

The normalized matrix is \(K_{\mathrm{unit}}=K/1.5\), so \(a=1/1.5=2/3\) in \(K_{\mathrm{unit}}=aK\). The conversion rule gives \(\sigma_{g,\mathrm{unit}}^2=0.12/(2/3)=0.180\) outcome-units squared.

The original genetic covariance is \(0.12K=\begin{pmatrix}0.24&0.048\\0.048&0.12\end{pmatrix}\). The normalized matrix is \(K_{\mathrm{unit}}=\begin{pmatrix}1.333&0.2667\\0.2667&0.6667\end{pmatrix}\), and multiplying by \(0.180\) gives the same genetic covariance, up to rounding. Its mean diagonal is one, while its two diagonal entries remain different.

When Normalizations Change More Than Scale

To see why a general conversion is not always possible, compare two marker-based constructions using the same centered genotype data but different weights across markers. If one matrix is a scalar multiple of the other, every corresponding entry has the same ratio. If the ratios differ across entries, no single variance component can make the two genetic covariance matrices identical.

Worked Example: Two Marker Normalizations Are Not Scalar Multiples

Consider three hypothetical individuals and two markers with dosage columns \((1,0,0)\) and \((0,1,2)\). Their allele frequencies in this sample are \(p_1=1/6\) and \(p_2=1/2\), so the corresponding genotype variances are \(q_1=2p_1(1-p_1)=5/18\) and \(q_2=2p_2(1-p_2)=1/2\). Let \(A\) and \(B\) be the outer-product matrices of the two centered dosage columns. The centered columns are \((2/3,-1/3,-1/3)\) and \((-1,0,1)\), respectively.

The per-marker average is \(K_{\mathrm{avg}}=\frac12(A/q_1+B/q_2)\). Substituting the centered columns gives \(K_{\mathrm{avg}}=\begin{pmatrix}1.8&-0.4&-1.4\\-0.4&0.2&0.4\\-1.4&0.4&1.8\end{pmatrix}\). The pooled normalization is \(K_{\mathrm{pool}}=(A+B)/(q_1+q_2)\), which gives \(K_{\mathrm{pool}}=\begin{pmatrix}13/7&-2/7&-11/7\\-2/7&1/7&1/7\\-11/7&1/7&13/7\end{pmatrix}\).

The ratio of the first diagonal entries is \((13/7)/1.8\approx1.032\), while the ratio of the second diagonal entries is \((1/7)/0.2\approx0.7143\). Since these ratios are not equal, \(K_{\mathrm{pool}}\) is not a scalar multiple of \(K_{\mathrm{avg}}\). No one variance-component multiplier can make their genetic covariance matrices equal for all entries. This difference reflects a change in marker weighting, not merely a change in units.

Deriving the Conversion and Checking It

The scalar conversion follows directly from the covariance model. If \(K^*=aK\), the new genetic covariance is \(\sigma_{g,*}^2K^*=\sigma_{g,*}^2aK\). For it to equal the original covariance \(\sigma_g^2K\), the coefficients of \(K\) must match, so \(a\sigma_{g,*}^2=\sigma_g^2\). Since \(a>0\), division by \(a\) yields \(\sigma_{g,*}^2=\sigma_g^2/a\). The total covariance is then unchanged:

V^*=\sigma_{g,*}^2K^*+\sigma_e^2I =\left(\frac{\sigma_g^2}{a}\right)(aK)+\sigma_e^2I =\sigma_g^2K+\sigma_e^2I =V.

This equality also clarifies what remains invariant: the implied genetic covariance, its diagonal variances, its off-diagonal covariances, and the total covariance matrix. The numerical value of the genetic variance component itself is not invariant. It is tied to the chosen matrix scale. If the matrix is not scalar-rescaled, equality would instead require \(\sigma_{g,*}^2K^*=\sigma_g^2K\) entry by entry, which generally cannot be achieved by changing just one scalar.

Interpretation and Common Pitfalls

When reading a variance-component result, identify the precise GRM construction, including its centering, marker weighting, and denominator. A component of \(0.18\) under a mean-diagonal-one convention is not automatically comparable to a component of \(0.18\) under a differently scaled matrix. The product \(\sigma_g^2K\), rather than the variance-component number alone, specifies the genetic covariance under the model.

Pitfall: assuming a larger component means a larger genetic covariance. That conclusion is not justified if the GRM scale differs. For example, doubling \(K\) while halving \(\sigma_g^2\) leaves the covariance unchanged.
Pitfall: converting between any two GRMs with one ratio. This works only when the matrices are scalar multiples. Changed marker weights can alter their entry-wise pattern and require a different model comparison.
Pitfall: interpreting mean-diagonal normalization as making every diagonal equal to one. It fixes the average diagonal only; individual diagonal entries can still differ.
Pitfall: comparing variance components without units or construction details. State the outcome scale and the GRM normalization so the reported component has a clear interpretation.

Key takeaway: A GRM defines the scale and pattern of genetic covariance. Under uniform positive rescaling, divide the variance component by the same factor to preserve that covariance. When normalization changes relative marker weights, do not assume a scalar conversion exists.

Check Your Understanding

Use the covariance model and scalar-conversion rule to answer these questions.

If \(K^*=0.5K\), what value of \(\sigma_{g,*}^2\) preserves the genetic covariance when the original component is \(0.24\)?
Why does normalizing a GRM to mean diagonal one not imply that every diagonal entry equals one?
Two matrices have different ratios between corresponding diagonal entries. Can one variance-component multiplier make them identical as genetic covariance matrices? Explain.
Does the residual variance component change when a GRM is multiplied by a positive scalar and the genetic component is converted accordingly?
Why should a reported genetic variance component be interpreted together with the GRM construction?