Model and data conventions

The phenotype model is

\[y = C\gamma + X\beta + \epsilon,\]

where X is raw diploid dosage and C contains the fixed effects, including an intercept. The evolutionary prior supplies a frequency-dependent variance for each focal-trait effect beta_j. The implementation uses

\[q_j = \hat{x}_j(1-\hat{x}_j),\]

with x_hat_j the sample allele frequency.

Simplified evolutionary model

The simplified model fixes rho_ab^2 = 1 and estimates sigma_b^2 and tau:

\[v_j = \frac{\sigma_b^2}{1 + 2\tau q_j}.\]

Full evolutionary model

The full model estimates the coupling in addition to the two scale parameters:

\[v_j = \sigma_b^2\left(1 - \rho_{ab}^2 \frac{2\tau q_j}{1+2\tau q_j}\right).\]

The parameter constraints are sigma_b2 > 0, tau >= 0, and 0 <= rho_ab^2 <= 1. The simplified model is the exact nested rho_ab^2 = 1 boundary of the full model, not a free reparameterization.

Variance components are estimated by profiled AI-REML. Dense inputs provide an exact reference path; GRG inputs use matrix-free projected solves, warm-started conjugate-gradient solves, and fixed seeded Hutchinson trace estimates. XTrace remains available as an explicit alternative.