Validating Multi-Trait Genetic Covariance Matrices

Why a Genetic Covariance Matrix Must Be Valid

When several traits are analysed together, their additive genetic variances and covariances are collected in a matrix. The matrix does more than summarize pairwise relationships: it must describe a coherent joint distribution of genetic effects. If its entries imply that some linear combination of those effects has negative variance, it cannot be a valid covariance matrix, even if each entry looks plausible on its own.

This tutorial develops the key validity condition, positive semidefiniteness, and shows how it constrains both variances and genetic correlations. These checks apply to the population covariance matrix and are also useful when evaluating estimates assembled or fitted from data.

For any coefficient vector \(\mathbf{a}\), the scalar \(\mathbf{a}^\mathsf{T}\mathbf{g}\) is a linear combination of genetic effects. Its variance is

A variance cannot be negative, so \(\mathbf{G}\) must be PSD. If some nonzero linear combination has exactly zero variance, the matrix is singular but can still be PSD. Thus, singularity alone does not make a covariance matrix invalid.

The Pairwise Bound on Genetic Correlation

For two traits with positive genetic variances, write their genetic covariance matrix as

Here \(\sigma_{g,1}^{2}\) and \(\sigma_{g,2}^{2}\) are the genetic variances, and \(\sigma_{g,12}\) is their genetic covariance. The genetic correlation is defined, when both variances are positive, by

The correlation is dimensionless; the covariance has the product of the two traits’ genetic-effect units. The bound on \(r_g\) follows directly from requiring \(\mathbf{G}_2\) to be PSD.

To see the condition, the diagonal entries must be nonnegative because each is a variance. The determinant of the two-trait matrix must also be nonnegative for it to be PSD. That determinant condition gives the covariance bound, and division by the positive product of genetic standard deviations gives the correlation bound.

Equality corresponds to a singular two-trait matrix: one nonzero linear combination of the effects has zero variance. In that case the genetic effects are perfectly linearly related at the covariance level. A valid correlation can therefore equal \(1\) or \(-1\); validity does not require a strict inequality.

Validity for More Than Two Traits

For a multi-trait matrix, checking each genetic variance and each pairwise correlation is necessary, but it is not sufficient. All traits must be jointly compatible: every linear combination must have nonnegative variance. For a symmetric matrix, this is equivalent to having no negative eigenvalues. In three dimensions, PSD can also be checked by verifying that all principal minors are nonnegative: the three diagonal entries, the three \(2\times2\) principal determinants, and the full determinant.

A useful distinction is that the leading-principal-minor test, which checks only the top-left \(1\times1\), \(2\times2\), and full-size determinants, establishes positive definiteness when all are strictly positive. For a general PSD check, do not substitute that strict test for checking all principal minors or the eigenvalues.

What Invalid Estimates Can Indicate

The population genetic covariance matrix must be PSD, but estimated entries need not satisfy the constraint if estimation is performed without enforcing it. Sampling variability can push an estimated covariance beyond the pairwise bound, and separately estimated pairwise quantities can fail the joint multi-trait constraints. Such an estimate is not evidence that the underlying genetic correlation is genuinely outside \([-1,1]\); it signals that the estimates, considered together, do not form a valid covariance matrix.

Model constraints affect how this problem appears. A joint model may restrict its fitted covariance matrix to the PSD cone, preventing an inadmissible fitted matrix. One way to construct a PSD matrix is \(\mathbf{G}=\mathbf{L}\mathbf{L}^\mathsf{T}\). For a \(q\times k\) matrix \(\mathbf{L}\), this construction is PSD because \(\mathbf{a}^\mathsf{T}\mathbf{G}\mathbf{a}=\|\mathbf{L}^\mathsf{T}\mathbf{a}\|^2\geq0\). It is PD if and only if \(\mathbf{L}\) has full row rank; for a square \(\mathbf{L}\), this means it is nonsingular. If \(\mathbf{L}\) is rank-deficient, the result is singular PSD. When \(k<q\), it cannot have full row rank, so the resulting \(q\times q\) matrix is necessarily singular.

A constrained fit can land on the boundary, where one or more eigenvalues are zero. This may reflect a true lower-dimensional covariance structure, limited information, or the effect of the constraint; a boundary estimate is not automatically an error. Conversely, a PSD estimate is not proof that the model is correctly specified or that the data are informative. Validity is a necessary mathematical check, not a complete assessment of fit.

Practical Validation and Common Pitfalls

A useful diagnostic sequence is to check symmetry and the diagonal variances, inspect pairwise bounds, and then assess the full matrix using its eigenvalues or the appropriate principal minors. Small negative eigenvalues can arise from numerical rounding, so distinguish a tiny computational discrepancy from a materially negative eigenvalue and report the tolerance used. If an estimate is invalid, inspect how the entries were obtained, their uncertainty, and whether the fitting procedure constrained the entire matrix jointly.

Common mistake: “Every estimated pairwise genetic correlation is between \(-1\) and \(1\), so the multi-trait matrix must be valid.” The correction is that those bounds check only two-trait submatrices. Joint validity requires nonnegative variance for every linear combination of all traits; a full-matrix check is still necessary.