Methods

Methods

This page collects the definitions needed to interpret the estimates. Usage examples are in the individual analysis guides.

The binary-trait path has a separate PCGC guide and PCGC methods. It uses ascertainment-aware risks and off-diagonal moments without the quantitative variance row described below. Ordinary beta/SE inputs do not implicitly become PCGC summaries.

Ordered cross-trait G×E equations and their paired uncertainty are described in Cross-trait genetic response covariance. Their directional target-block deletion convention is distinct from the generalized within-trait convention below.

h² and rg summary statistics

For a marginal linear-regression coefficient beta, standard error se, and per-SNP sample size n, SUMMIT reconstructs a score statistic using

z*_j = sqrt(N_max - c - 1) * beta_j
       / sqrt(beta_j² + (n_j - c - 2) * se_j²).

Here c is the non-intercept covariate rank. The current h² implementation sets c=0; rg resolves c from the input metadata. Heritability denominators in rg retain the h² convention. This distinction matters when comparing implementations.

Write N* = N_max - c - 1 and n*_j = n_j - c - 1. Under the linear-regression null, the variance of this reference-scaled score is d_j = N*/n*_j. SUMMIT subtracts this SNP-specific null term:

q_j = z*_j² - d_j
D_jk = N* L_jk / M_k.

The HE equations use annotation columns as estimating instruments. Equivalently, the univariate variance equations receive y_j = 1 + q_j. Keeping the signal on the reference scale allows traits to share the reference calculations even when their SNP sample sizes differ. No average-N approximation is needed.

--weight-mode ldsc solves

h = argmin_h ||sqrt(W) (q - D h)||².

Its iterative working variances use d_j + N* h² L_j/M, with floors applied only to the weights. This is equivalent to regressing local-N score squares minus one against n*_j L_jk/M_k, with the corresponding local-N weights.

For two traits, let b_j = sqrt(d1_j d2_j). The bivariate response is q12_j = z1*_j z2*_j - c_ov b_j, with design sqrt(N1* N2*) L_jk/M_k. Each covariance estimate is divided by its matching h² estimates to obtain rg. The overlap covariance is supplied or estimated in a separate step; the latter uses b_j as its intercept column and refits it in every jackknife replicate. All refits retain the same per-SNP sample scales.

This bivariate model assumes that the overlap covariance on the local score scale is constant across SNPs. Shared genotype missingness with comparable missing fractions in both traits and their overlap satisfies this assumption to the usual large-sample approximation. Different contributing cohorts or trait-specific missingness can violate it. The two N columns alone do not determine SNP-specific overlap. Nonoverlapping studies use c_ov = 0.

The null subtraction follows the OLS partial-correlation identity; it is not an exact finite-sample identity for logistic or arbitrary meta-analysis beta/SE statistics. The usual LD-score signal model and its assumptions still apply.

Working-variance floors affect regression weights. They do not impose nonnegative h² or bounded rg. For overlapping annotations, component estimates are coefficient contributions, not estimates restricted to an annotation's SNP set.

LD-score Monte Carlo variance

For target SNP i, annotation k, and random vector v, write its contribution as Y_ikv. With B independent vectors, the conditional variance estimate is

Var_MC(Lhat_ik) = [sum_v Y_ikv² - (sum_v Y_ikv)² / B] / [B (B-1)].

The finite-sample null subtraction is constant across vectors and does not enter this variance. The annotation diagnostic sums these per-SNP variances. It does not include cross-SNP covariance, reference sampling uncertainty, or imputation uncertainty.

Genetic features and covariance components

Let G be the centered and scaled genotype matrix, Phi the context matrix, and U an orthonormal basis for the fixed-effect design. Projection is P X = X - U (U.T X).

Generalized contextual features use

$$F_q = P\,\mathrm{diag}(\phi_q)G.$$

Every context uses the same genotype scale. Multiplication by context occurs before projection; projecting G first would define a different model.

For annotation weights A_jk with mass M_k = sum_j A_jk, the kernels are

$$K_{k,qq}=F_q A_k F_q^T/M_k, \qquad K_{k,qr}=(F_q A_k F_r^T+F_r A_k F_q^T)/M_k,\quad q<r.$$

Pairs are stored as diagonals first, then lexicographic off-diagonals. Components are annotation-major. Each coefficient is one entry of Ω; the symmetric off-diagonal contribution is included once in the kernel definition. Residual components have the form P diag(d_h) P.

The default one-environment CLI additionally normalizes projected feature columns to squared norm rank(P). Its raw_projected option retains their natural norms. Generalized models use the common scale without separate post-projection normalization.

Generalized per-SNP reference LD scores

Let r = rank(P), and define R_ab(j,m) = f_aj.T f_bm / r. For a context pair p, O(p) contains its two orientations, or one orientation for a diagonal pair. The directional score from target pair p to source annotation/pair (ell,s) is

$$L_{j,p\to(\ell,s)}= \sum_m A_{m\ell}\sum_{(a,b)\in O(p)}\sum_{(c,d)\in O(s)} R_{bc}(j,m)R_{ad}(j,m).$$

For components c=(k,p) and d=(ell,s), sum target rows as DNUM[c,d] = sum_j A_jk L_j,p→d. The reference kernel Gram entry is

$$T_{cd}=\frac{r^2}{M_k M_\ell} \frac{DNUM_{cd}+DNUM_{dc}}{2}.$$

The randomized implementation uses variant-axis Rademacher probes. Pass 1 completes all global source sketches. Pass 2 scores target variants against those completed sketches. Tiling stays within a decoded block, giving exactly two reference traversals.

Pass 2 also computes component-kernel diagonals d_ci. Their product d @ d.T gives the exact same-person matrix. Reference construction accepts no jackknife block assignment. A separate reducer subsequently groups the fixed per-SNP scores into full and delete-block normal equations, updates retained annotation masses, and reuses the full same-person matrix.

Reference transfer and fitting

When reference and study sample the same relevant genotype–context distribution, separate same-person and different-person contributions scale as

$$T_S=\frac{N_S}{N_R}D_R+ \frac{N_S(N_S-1)}{N_R(N_R-1)}(T_R-D_R).$$

These factors use sample counts, not residual rank. Genetic–residual and residual-only moments are computed in the study cohort. Traits are projected and normalized consistently with the chosen estimator. The resulting small normal equations estimate genetic and residual coefficients jointly.

The sample-probe estimator in summit.context.reference_v1 is a separate method. It estimates aggregate kernel actions and stores grouped numerators; its summary deletion is approximate. Its finite-probe outputs and deletion procedure are not interchangeable with the per-variant estimator above.

PGS posterior mean

For each SNP j, let beta_j ~ N(0, Lambda/M). Let R be a positive diagonal residual covariance and Z the fixed-effect design. With elementwise matrix multiplication denoted by ⊙,

$$V=R+(GG^T/M)\odot(\Phi\Lambda\Phi^T).$$

The solver finds u in the orthogonal complement of Z:

$$PVPu=Py,\qquad Pu=u.$$

It uses projected conjugate gradients with preconditioner P diag(V)^(-1) P. Accepted solutions satisfy ||Py - PVu|| <= max(atol, rtol*||Py||), checked with a fresh application of V. Fixed-effect coefficients are recovered from y-Vu in the retained fixed-effect span.

Posterior SNP weights are

$$B=G^T\mathrm{diag}(u)\Phi\Lambda/M.$$

The algorithm requires neither an inverse of Lambda nor an N-by-N matrix. Singular, additive-only, amplification-only, and zero genetic priors are valid.

Amplification and residual response

Partition a positive-baseline covariance as

$$\Omega=\begin{pmatrix}a&b^T\\b&C\end{pmatrix},\qquad \gamma=b/a,\qquad S=C-bb^T/a.$$

Separate prior scaling uses

$$\Lambda=\kappa_A a \begin{pmatrix}1\\\gamma\end{pmatrix} \begin{pmatrix}1&\gamma^T\end{pmatrix} +\kappa_H\begin{pmatrix}0&0\\0&S\end{pmatrix}.$$

The response spectrum is computed from H^(1/2) S H^(1/2), with H the saved reference response metric. Changing its eigenvalues or retained rank changes the prior and requires refitting. Weighting score components after fitting leaves the posterior SNP weights fixed.

If a supplied fitted prior changes its baseline or cross-covariance, its amplification coefficient is derived from that fitted prior. At zero baseline variance, the declared parent anchor supplies the decomposition and the fitted baseline weights are zero. Repeated nonzero eigenvalues cannot be split by a rank cut.