Scaling a Genomic Relationship Matrix for LMMs

Why the Scale of a Relationship Matrix Matters

In a linear mixed model (LMM), a genomic relationship matrix describes the covariance pattern assigned to a random genetic effect across individuals. Its numerical scale matters because the genetic variance parameter multiplies that matrix. If the matrix is doubled while the parameter is left unchanged, the model assigns twice the genetic covariance to every pair of individuals. If both are adjusted consistently, however, the implied covariance can remain exactly the same.

This is important when comparing software output, methods, or analyses that construct the matrix with different normalizations. A component reported as 0.4 is not interpretable on its own: its meaning depends on the scale of the matrix it multiplies. The central task here is to keep the matrix convention and the variance-component convention aligned.

Definition: Let $y$ be the vector of phenotypes for $n$ individuals, $X$ a fixed-effect design matrix, and $K$ a known genomic relationship matrix. In the random genetic effects model, $$ y=X\beta+u+e,\qquad u\sim N(0,\sigma_g^2K),\qquad e\sim N(0,\sigma_e^2I). $$ Here $u$ is the vector of genetic random effects, $e$ is the residual vector, and $I$ is the $n$-dimensional identity matrix. The standard assumptions are that $K$ is symmetric and positive semidefinite, the random effects have mean zero, and $u$ and $e$ are independent. The parameter $\sigma_g^2$ is the genetic variance multiplier for this particular scaling of $K$; it is not automatically the marginal genetic variance of every individual.

For individual $i$, the marginal variance of the genetic effect is $\sigma_g^2K_{ii}$. For two different individuals $i$ and $j$, their genetic-effect covariance is $\sigma_g^2K_{ij}$. Thus diagonal entries control individual-specific genetic variances, while off-diagonal entries encode covariance associated with genomic similarity. If the diagonal entries vary, a single value of $\sigma_g^2$ multiplies all of them; it does not make them equal.

What Rescaling Does to the Variance Component

Suppose a relationship matrix is replaced by $K^*=cK$, where $c$ is a known positive constant. To preserve the model’s genetic covariance, the parameter must be replaced by $\sigma_g^{2*}=\sigma_g^2/c$. The matrix entries change by the factor $c$, and the variance multiplier changes by its reciprocal.

\sigma_g^{2*}K^*=\left(\frac{\sigma_g^2}{c}\right)(cK)=\sigma_g^2K.

This identity is a direct reparameterization: it does not change the covariance assigned to the phenotype. For the equivalence to hold, the scaling must be a positive scalar, and the model must allow the transformed variance parameter. The same fixed effects, residual model, and other modelling choices must be retained. In particular, this argument does not say that changing marker weights, removing markers, or changing the individuals used to construct $K$ is harmless; those operations can change the matrix in ways that are not a scalar multiple.

Worked Example: Converting a Mean-Diagonal-One Matrix

Suppose a hypothetical two-individual analysis uses $$ K=\begin{pmatrix}2&0.6\\0.6&1\end{pmatrix} $$ and estimates $\sigma_g^2=0.30$. The average diagonal is $\mathrm{tr}(K)/n=(2+1)/2=1.5$. To rescale the matrix so its average diagonal is one, multiply it by $c=1/1.5=2/3$.

The new matrix and corresponding variance multiplier are $$ K^*=\frac{2}{3}K=\begin{pmatrix}1.333&0.400\\0.400&0.667\end{pmatrix}, \qquad \sigma_g^{2*}=\frac{0.30}{2/3}=0.450. $$

Under the original convention, the genetic covariance matrix is $$ 0.30K=\begin{pmatrix}0.600&0.180\\0.180&0.300\end{pmatrix}. $$ Under the rescaled convention, it is $$ 0.450K^*=\begin{pmatrix}0.600&0.180\\0.180&0.300\end{pmatrix}. $$ The covariance is unchanged. The reported multiplier differs because the matrices differ in scale; with the mean-diagonal-one matrix, 0.450 is the average marginal genetic variance across these two individuals.

Common Normalization Choices

A genomic relationship matrix is often formed from genotype information across $m$ markers. For illustration, let $Z$ be an $n\times m$ matrix of centered and possibly standardized marker values, with one row per individual. A common construction is $$ K_{\mathrm{avg}}=\frac{ZZ^\mathsf{T}}{m}. $$ This averages marker contributions. Its scale depends on how the columns of $Z$ were centered and scaled. Even if marker columns have a target variance under the construction, the realized average diagonal need not be exactly one in a finite data set. A further convention is to multiply the matrix by $n/\mathrm{tr}(K)$, giving average diagonal one, provided its trace is positive.

Another convention makes the trace equal to one, so the average diagonal is $1/n$. The two conventions are legitimate if the variance parameter is interpreted on the corresponding scale. A kinship-style matrix may also differ from an additive relationship matrix by a constant factor; multiplying one convention by two, for example, requires dividing its variance multiplier by two to preserve covariance. Names alone are not enough to establish the scale: the construction formula and normalization should be checked.

Worked Example: Marker Averaging and Mean-Diagonal Normalization

Consider a hypothetical feature matrix with three individuals and two centered marker columns: $$ Z=\begin{pmatrix}1&0\\-1&1\\0&-1\end{pmatrix}. $$ Using the average-of-markers construction gives $$ K_{\mathrm{avg}}=\frac{ZZ^\mathsf{T}}{2} =\frac{1}{2}\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix} =\begin{pmatrix}0.5&-0.5&0\\-0.5&1&-0.5\\0&-0.5&0.5\end{pmatrix}. $$ Its trace is $0.5+1+0.5=2$, so its average diagonal is $2/3$, not one. The mean-diagonal-one version is obtained using $c=n/\mathrm{tr}(K_{\mathrm{avg}})=3/2$: $$ K_{\mathrm{unit}}=\frac{3}{2}K_{\mathrm{avg}} =\begin{pmatrix}0.75&-0.75&0\\-0.75&1.5&-0.75\\0&-0.75&0.75\end{pmatrix}. $$

If the marker-averaged matrix has $\sigma_g^2=0.40$, the equivalent parameter for $K_{\mathrm{unit}}$ is $0.40/(3/2)=0.267$ (to three significant figures). For example, the first individual’s genetic variance is $0.40(0.5)=0.200$ under the first convention and $0.267(0.75)\approx0.200$ under the second. Negative off-diagonal entries are not a contradiction: they indicate negative covariance under this particular centered construction, while the full matrix remains positive semidefinite because it is a positive multiple of $ZZ^\mathsf{T}$.

Worked Example: Trace-One Versus Mean-Diagonal-One

Suppose a hypothetical four-individual matrix $K_{\mathrm{unit}}$ has average diagonal one, so $\mathrm{tr}(K_{\mathrm{unit}})=4$, and its fitted genetic multiplier is $0.12$. Define a trace-one matrix by $K_{\mathrm{trace}}=K_{\mathrm{unit}}/4$. Here $c=1/4$, so the equivalent multiplier is $$ \sigma_{g,\mathrm{trace}}^2=\frac{0.12}{1/4}=0.48. $$

For any pair $i,j$, the genetic covariance using the trace-one convention is $0.48(K_{\mathrm{unit},ij}/4)=0.12K_{\mathrm{unit},ij}$, exactly the original value. The trace-one parameter is four times larger, not because the genetic covariance is larger, but because the matrix entries are four times smaller. Since the trace-one matrix has average diagonal $1/4$, the average marginal genetic variance remains $0.48(1/4)=0.12$.

Interpreting the Component and Checking the Matrix

When $K$ has average diagonal one, $\sigma_g^2$ equals the average marginal variance of the genetic random effects across the individuals represented by the matrix: $$ \frac{1}{n}\sum_{i=1}^n \mathrm{Var}(u_i) =\frac{1}{n}\sum_{i=1}^n \sigma_g^2K_{ii} =\sigma_g^2\frac{\mathrm{tr}(K)}{n} =\sigma_g^2. $$ If the average diagonal is $a$ instead, the average marginal genetic variance is $\sigma_g^2a$. This is a statement about the covariance model on the analyzed individuals, not by itself a statement about a population-level variance or heritability.

Before interpreting a reported component, check the exact matrix definition and record at least its dimensions, trace, average diagonal, and scaling formula. Confirm that it is symmetric and positive semidefinite, up to small numerical error. Check whether the diagonal is constant or merely has a specified average. Then verify that the variance parameter is reported on the same scale. These checks are especially useful when comparing outputs from different software or when importing a matrix constructed elsewhere.

Rescaling does not necessarily leave every practical analysis unchanged. It preserves the covariance model when the variance component is freely estimated and transformed consistently. But a fixed variance value, a constraint or prior expressed on the component, or a penalty depending on its numerical value can make the convention consequential. Likewise, weighting markers differently or using a different set of markers can change the relative entries of the matrix, not just their common scale.

Common Pitfalls and Alternatives

Comparing variance components without comparing matrix scales. A larger reported component can accompany a smaller relationship matrix and the same fitted covariance. Compare $\sigma_g^2K$, or convert both components to a common matrix convention, before interpreting the numerical difference.
Assuming mean diagonal one makes every diagonal entry one. It fixes $\mathrm{tr}(K)/n$, the average of the diagonal entries. Individual diagonal values can still differ, so their marginal genetic variances can differ.
Treating every construction change as rescaling. Multiplication by a positive constant permits a reciprocal parameter conversion. Changing allele-frequency weights or marker inclusion can change the matrix’s shape and cannot generally be corrected by one scalar.
Confusing a matrix convention with a biological conclusion. A variance component is tied to the covariance model and its matrix scale. A normalization choice alone does not establish a biological effect or a population variance interpretation.

Marker-count averaging is convenient when the goal is to average contributions over the included markers, but its realized diagonal scale depends on the feature construction. Mean-diagonal-one scaling makes the component equal to average marginal genetic variance for the analyzed individuals. Trace-one scaling is also coherent, but its component is on a different numerical scale. No convention is universally correct for every purpose; transparency and consistent conversion are essential.

Key takeaway: If $K^*=cK$ for $c>0$, the same genetic covariance is represented by $\sigma_g^{2*}=\sigma_g^2/c$. Interpret the component only after establishing the matrix’s scale, construction, and diagonal convention.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A relationship matrix is multiplied by 3. What change to its genetic variance multiplier preserves the genetic covariance?
A matrix has $n=5$ and trace 4. What is its average diagonal, and what factor makes that average equal to one?
With mean-diagonal-one scaling, two individuals have diagonal entries 0.8 and 1.2. If $\sigma_g^2=0.5$, what are their marginal genetic variances?
Why can changing marker weights not usually be handled by a reciprocal change in $\sigma_g^2$?

Solutions. (1) Divide the multiplier by 3, since $(\sigma_g^2/3)(3K)=\sigma_g^2K$. (2) The average diagonal is $4/5=0.8$; multiply the matrix by $1/0.8=1.25$. (3) The variances are $0.5(0.8)=0.40$ and $0.5(1.2)=0.60$, in the squared units of the phenotype. (4) Changing marker weights generally changes relative matrix entries, so the new matrix is not necessarily a constant multiple of the old one; a single reciprocal parameter adjustment cannot then preserve every covariance.