Cross trait response covariance

Cross-trait genetic response covariance

This model estimates how two traits' genetic responses covary across contexts. For example, it can estimate the covariance between one trait's age response and another trait's BMI response, or compare response correlation with baseline genetic correlation. The cross-trait covariance matrix is generally asymmetric; its transpose describes the reversed trait pair. The usage guide describes the Python workflow.

Model

For trait X, write the context-specific SNP effect as beta_j^X(e) = phi(e)' b_j^X, with Cov(b_j^X,b_j^Y) = Omega_XY / M. The first context coordinate is one. Each SNP annotation has its own covariance matrix and annotation mass. For the two traits' context-weighted, fixed-effect-adjusted genotype features:

F_a^X = P_X D_a G_X
K_ab^XY = F_a^X (F_b^Y)' / M
E[y_X y_Y'] = sum_ab Omega_XY[a,b] K_ab^XY
              + P_X diag_overlap(phi' Psi_XY phi) P_Y

The fit retains all Q² ordered genetic coefficients. Residual covariance acts only on shared participants and identifies symmetric context products. Redundant residual terms, such as a binary exposure and its square, are reduced to an identifiable basis. Genetic estimates remain signed, with no imposed symmetry or positive-semidefinite projection for Omega_XY.

Reference assumptions

The reference and study must use compatible variants, genotype scaling, context definitions, and fixed effects. Saved chromosome summaries supply within-chromosome reference moments; cross-chromosome LD is omitted. Cohort exposure moments use each trait's sample mask and the actual overlap.

Mode	Treatment of different-person reference moments
factorized (default)	Combines reference LD with cohort exposure moments
factorized_plus_residual	Adds the reference's remaining genotype–context dependence, scaled to cohort size
legacy_transport	Transfers the reconstructed ordered reference moment, scaled to cohort size
legacy_transport_exact	Uses additional Z summaries to correct the ordered reference reconstruction before transfer

Reconstructing ordered moments from the saved symmetric summaries is an approximation. The additional Z summaries described below correct part of that reconstruction.

These modes make different assumptions about the genotype–environment distribution in the study and reference. Retaining more reference dependence also retains its sampling and random-vector noise. Agreement among modes provides a sensitivity check for the supplied data.

Same-person moments use saved reference diagonals restricted to the actual overlap. These diagonals approximate the corresponding study diagonals when the projectors differ. Where a target block's diagonal is unavailable, its contribution is allocated by annotation mass. Within-trait fits use the factorized different-person approximation and their own sample rows.

Additional Z summaries

summit reference zpass computes Z summaries from genotypes and saved reference scaling. The study driver can collect them during scoring with --z-output, avoiding another genotype pass.

Z summaries correct the ordered reference reconstruction for a single annotation, including a weighted annotation. Exact correction for arbitrary pairs of different annotations is not supported. The correction also leaves cohort-transfer assumptions, reference sampling error, and random-vector error in place. The legacy_transport_exact name refers to this limited reference correction.

Fitting and uncertainty

solve_cross_trait_normal_equations fits the ordered genetic coefficients and identifiable residual terms. Results include rank, condition number, minimum Gram eigenvalue, solve residual, and deletion diagnostics.

The default uses 200 paired target-SNP blocks with deletion_method="target_moments" and uncertainty_method="delta". The deletion removes target-block moments while holding the source reference fixed. Coefficients retain their full-genome units. This approximation includes annotation-mass allocation of unavailable block diagonals; it does not recompute both SNP axes from genotypes.

The delta method propagates the joint deletion covariance of Omega_XX, Omega_YY, and Omega_XY into derived correlations and their differences. It includes uncertainty from centering, baseline projection, and both variance denominators. --uncertainty-method jackknife propagates the nonlinear deleted estimates directly. Both methods require matching deletion IDs across the within-trait and cross-trait fits.

Intervals are conditional on the supplied reference and transfer mode. They exclude independent reference sampling, probe redraws, and uncertainty in the transfer assumptions. Weak variance denominators can make correlation intervals unreliable; nonpositive denominators produce undefined values. Raw correlations outside [-1,1] are retained and flagged.

Derived quantities

Baseline covariance and correlation use the original context origin. Centered baseline estimates move each trait's intercept to its own mean context. Orthogonal responses remove the genetic association with that centered baseline. If a_X = Omega_XX[0,1:]/Omega_XX[0,0] in the centered basis, the cross-trait orthogonal-response covariance is

H_XY = Omega_XY[1:,1:] - a_X Omega_XY[0,1:]
       - Omega_XY[1:,0] a_Y' + a_X Omega_XY[0,0] a_Y'.

Outputs include the full response and H matrices, per-context response correlations, and the aggregate covariance tr(S H_XY). Its correlation uses the corresponding within-trait traces and one recorded exposure covariance S. Response-minus-baseline differences use paired uncertainty.

Context-specific genetic correlation can also be evaluated at chosen exposure values. Retain the original context means, scales, and covariance metric with the estimates. The model describes cross-sectional genetic covariance; causal or longitudinal interpretations require additional assumptions.