Bivariate Genomic Mixed Models For Two Traits

Why Model Two Traits Together?

When two traits are measured on related individuals, their observed association can reflect more than one source of covariance. Individuals can share genetic effects, and measurements on the same individual can also share residual influences. A bivariate genomic mixed model represents these sources separately while allowing each trait to have its own genetic and residual variance.

The model is a specification: it states how the expected values and covariance of the observations are represented. It does not, by itself, guarantee that the components can be estimated precisely in a particular study. Here the focus is the model and the meaning of its covariance blocks.

Model for the Two Trait Vectors

Let \(y_1\) and \(y_2\) be the \(n\)-vectors of observations for traits 1 and 2, in the same individual order. Let \(X_1\) and \(X_2\) be fixed-effect design matrices, and let \(\beta_1\) and \(\beta_2\) be their coefficient vectors. The model is

Here \(u_1,u_2\) are the genetic random-effect vectors, and \(e_1,e_2\) are residual vectors. The fixed effects may differ between traits; for example, their design matrices can include different covariates. The model assumes the fixed-effect means are \(X_1\beta_1\) and \(X_2\beta_2\), while the random effects have mean zero.

The genomic relationship matrix \(K\) is an \(n\)-by-\(n\), known, symmetric positive semidefinite matrix aligned to the individuals in both trait vectors. Its entries encode the genomic relationship structure used by the model. The covariance specification is

and

The \(g\) subscript denotes the genetic component and the \(e\) subscript the residual component. Each variance is trait-specific; \(\sigma_{g12}\) and \(\sigma_{e12}\) are cross-trait covariances. Covariance units are the product of the units of the two traits; a variance has the square of its trait's units.

Covariance Blocks and Their Interpretation

For compactness, define the trait-level genetic and residual covariance matrices

The diagonal entries are the trait-specific variances. The off-diagonal entries are cross-trait covariances: \(\sigma_{g12}\) describes genetic covariance, while \(\sigma_{e12}\) describes residual covariance for the same individual. For the residual model, \(I_n\) means that residuals are uncorrelated across different individuals; the residual covariance between traits applies to the two measurements on the same individual.

Stack the observations by trait as \(y=(y_1^\mathsf{T},y_2^\mathsf{T})^\mathsf{T}\). Under the stated model, the covariance of \(y\) is

The upper-left and lower-right blocks are the marginal covariance matrices within each trait. Each combines a genetic covariance structured by \(K\) and a residual variance on the diagonal. The off-diagonal block is the covariance between the two trait vectors. Its genetic part is proportional to \(K\); its residual part is proportional to \(I_n\). Thus the model distinguishes cross-trait covariance between different individuals, as represented through genomic relationships, from residual cross-trait covariance within the same individual.

This form follows by adding independent genetic and residual contributions. For example, the covariance between \(y_1\) and \(y_2\) is the sum of \(\operatorname{Cov}(u_1,u_2)=\sigma_{g12}K\) and \(\operatorname{Cov}(e_1,e_2)=\sigma_{e12}I_n\), because genetic effects and residuals are assumed independent. The same addition gives each diagonal block.

Assumptions and Admissible Covariance Components

A covariance matrix must be positive semidefinite. Consequently, both \(\Sigma_g\) and \(\Sigma_e\) must be positive semidefinite, in addition to \(K\) being positive semidefinite. For a two-by-two covariance matrix with variances \(a\) and \(b\) and covariance \(c\), this requires \(a\geq0\), \(b\geq0\), and \(c^2\leq ab\). In the model, this gives

with nonnegative trait variances. These constraints prevent an impossible covariance specification. When both trait variances are positive, the corresponding covariance divided by the square root of their product is a correlation and must lie between \(-1\) and \(1\).

The basic specification also assumes that the genetic and residual vectors are mutually independent, that residuals are independent across individuals, and that the same \(K\) structure applies to both traits. The covariance structure can be stated using second moments alone. If a Gaussian linear mixed model is used for likelihood-based analysis, the joint random effects are additionally assumed to be normally distributed. That distributional assumption is distinct from the covariance-block definitions.

Why the Block Structure Matters

The block representation makes the model's interpretation explicit. A nonzero cross-trait genetic covariance connects the traits through the genomic relationship structure, whereas a nonzero residual covariance connects the two measurements within an individual. Neither covariance alone states that one trait causes the other. They describe components of joint variation under the specified model.

The structure also clarifies what changes when data are incomplete. If one trait is unobserved for some individuals, the covariance for the observations that are actually available is the corresponding submatrix of the full covariance matrix. The underlying model still needs a common individual ordering and a clearly defined covariance structure.

A frequent mistake is to read the off-diagonal block as entirely genetic because the model is called a genomic mixed model. That is incorrect: the block is \(\sigma_{g12}K+\sigma_{e12}I_n\), so it contains both genetic and residual covariance. Another mistake is to treat \(\sigma_{g12}\) as the covariance of observed trait values for every pair of individuals. It is the trait-level genetic covariance multiplier; the genetic covariance for individuals \(i\) and \(j\) is \(\sigma_{g12}K_{ij}\), while the residual contribution is \(\sigma_{e12}\) only when \(i=j\).