Variant-level tests¤

Score test for association between a variant and an outcome.

For a null (covariate-only) fit, let \(r_y\) be the working residuals and let \(g\) be a variant genotype vector. After residualizing \(g\) against covariates, the per-variant score statistic is \(U = g^{\top} W r_y\), with variance \(V = g^{\top} W g\), and the reported z-statistic is \(z = U / \sqrt{V}\) with two-sided p-value \(p = 2\Phi(-|z|)\) where \(\Phi(\cdot)\) is the Normal CDF.

This implementation requires jaxqtl.infer.FisherInfoError. Sandwich covariance estimators apply to Wald coefficient inference and do not define a robust version of this score statistic.

Raises:

• ValueError: If std_err is not jaxqtl.infer.FisherInfoError.

Wald test for association between a variant and an outcome.

For each variant, this fits a full model including the variant and reports \(\hat\beta / \mathrm{se}(\hat\beta)\). jaxqtl.infer.LinearModel uses a residualized Gaussian fast path and a Student's t reference distribution with the full model's residual degrees of freedom. Generalized linear models use a Normal reference distribution.

Saddlepoint approximation (SPA) score test.

This starts from a score statistic \(S\) for each variant under the null model, then computes a saddlepoint approximation based on the cumulant generating function \(K(t)\) implied by the fitted mean model. A root \(\hat t\) is found such that \(K'(\hat t) = S\), then the Barndorff-Nielsen approximation is used via \(r^* = w + \log(v/w)/w\) with \(w = \mathrm{sign}(\hat t)\sqrt{2(\hat t S - K(\hat t))}\) and \(v = \hat t\sqrt{K''(\hat t)}\), yielding a two-sided p-value via a Normal tail approximation.

This implementation requires jaxqtl.infer.FisherInfoError. Sandwich covariance estimators do not define a robust version of the underlying score statistic or its saddlepoint approximation.

Attributes:

• cgf: Cumulant generating function for the fitted response family. The caller must choose a CGF whose distribution matches model.family.

Raises:

• ValueError: If std_err is not jaxqtl.infer.FisherInfoError.

CGF implementation for a Gaussian mean/variance model.

For \(Y \sim \mathcal{N}(\mu, \sigma^2)\), the cumulant generating function is \(K(t) = \mu t + \frac{1}{2}\sigma^2 t^2\).

CGF implementation for a Negative Binomial mean/dispersion model.

For \(Y \sim \mathrm{NegBin}(\mu, r)\) parameterized by mean \(\mu\) and shape \(r\), the cumulant generating function can be written as \(K(t) = -r\log\left(1 - \frac{\mu}{r}(\exp(t) - 1)\right)\). This implementation uses \(r = 1/\alpha\) where \(\alpha\) is the fitted dispersion.

CGF implementation for a Poisson mean model.

For \(Y \sim \mathrm{Poisson}(\mu)\), the cumulant generating function is \(K(t) = \log \mathbb{E}[\exp(tY)] = \mu(\exp(t) - 1)\). In this implementation, \(\mu\) is taken from the fitted model mean.

Container for per-variant association test results.

For a genotype matrix with m variants, association fields have shape (m,). Convergence, dispersion, and negative log-likelihood may be scalars when a score test reuses one null-model fit, or arrays with shape (m,) when models are fitted per variant.

Attributes:

• beta: Estimated variant effects.
• se: Standard errors of the variant effects.
• p: Two-sided association p-values.
• z: Score or Wald statistics.
• num_iters: Model-fitting iteration counts.
• converged: Model convergence indicators.
• disp: Fitted family dispersion or scale values.
• negloglikelihood: Negative log-likelihood objective at the fitted model.

Compact null-fit state shared across genotype blocks.

Only residuals, weights, and scalar fit diagnostics are retained, so batched permutations do not copy covariates or unused fitted-model arrays.

SPA preparation retaining a compact score fit and CGF-specific state.

Covariate-only residuals and weights for Gaussian coefficient inference.

Outcome and offset for fitting a full GLM for each variant.