Deriving the Gaussian Summary-Statistic Likelihood

Why the Likelihood Must Account for LD

Marginal association estimates are computed one variant at a time, but they are not statistically independent when variants are correlated through linkage disequilibrium (LD). A signal at one variant can therefore produce association estimates at nearby correlated variants. A likelihood for these estimates must represent both their expected values under a proposed vector of joint effects and their correlated sampling error.

This tutorial derives a Gaussian approximation for that likelihood. It is a sampling model for the vector of marginal estimates, not a claim that the observed estimates are independent or that the LD matrix alone identifies a causal variant. The likelihood is useful only when the effect scale, standard errors, and LD matrix are aligned with the same variants and analysis.

Model, Notation, and Assumptions

Consider \(m\) variants. Let \(R\) be their \(m\times m\) LD correlation matrix, with \(r_{jk}\) the correlation between genotypes at variants \(j\) and \(k\). Let \(\beta_j\) denote the joint effect of variant \(j\), conditional on the other variants in this set. For the simplest derivation, genotypes are centered and scaled to variance one, and the quantitative phenotype is modeled as

where \(X\) has one standardized genotype column per variant and \(\varepsilon\) has mean zero, independent components, and common variance \(\sigma^2\). The approximation treats the observed LD matrix as the population or sample correlation matrix relevant to the association analysis. In practice, covariate adjustment requires LD calculated on genotypes residualized consistently with that analysis, or a suitable approximation to it.

The stated mean uses effects and genotypes on a consistent standardized scale. With standardized genotypes, the marginal regression coefficient at variant \(j\) has population mean \(\sum_k r_{jk}\beta_k\): it collects the contributions of all joint effects weighted by their correlations with variant \(j\). If effects are reported on a different scale, the mean and standard errors must be transformed consistently rather than mixing scales.

The covariance \(SRS\) is an approximation when standard errors differ across variants. It says that the sampling correlation between \(\hat b_j\) and \(\hat b_k\) is approximately \(r_{jk}\). This is appropriate in a common-sample setting with a compatible LD matrix and similar association procedures. It is not guaranteed merely because marginal standard errors have been supplied.

Deriving the Mean and Covariance

For centered, variance-one genotype columns, the sample cross-product matrix satisfies \(X^\mathsf{T}X/n\approx R\), where \(n\) is the sample size. The vector of univariate marginal regression estimates can be written, under this scaling, as \(\hat b=X^\mathsf{T}y/n\). Substituting the phenotype model gives

Conditional on the genotypes, the first term is the expected marginal estimate. For the error term, independence and variance \(\sigma^2\) imply

Thus the idealized covariance is a scalar times \(R\). For standardized genotypes and one common analysis, the marginal standard errors are approximately equal, \(s_j\approx \sigma/\sqrt{n}\). More generally, observed standard errors can differ because of allele frequencies, missingness, covariate adjustment, or other features. Replacing the common scale by the reported standard errors yields the working approximation \(SRS\). The Gaussian form follows exactly in the fixed-design linear model with Gaussian errors and known variance; with estimated variances or non-Gaussian outcomes it is generally an asymptotic approximation.

The determinant term matters when comparing likelihoods with different covariance assumptions; for a fixed \(R\) and \(S\), it is constant as \(\beta\) varies. If \(z=S^{-1}\hat b\), the standardized form is \(z\approx N(S^{-1}R\beta,R)\). The mean in this form is not generally \(R\beta\), because standardizing the estimates also transforms their expected values.

Using the Likelihood and Checking Its Inputs

A likelihood calculation evaluates how far the observed estimates are from the mean predicted by a specified \(\beta\), accounting for the covariance among estimates. The relevant squared distance is the quadratic form \((\hat b-R\beta)^\mathsf{T}\Sigma^{-1}(\hat b-R\beta)\). It is not generally the sum of squared, individually standardized residuals: that simpler sum would discard the off-diagonal covariance.

Before using the likelihood, verify that the variant order in \(R\), \(\hat b\), and \(S\) is identical; that effect alleles and effect directions are harmonized; and that the LD reference is appropriate for the association sample. A sign reversal for an allele changes the signs of corresponding correlations and estimates, so inconsistent allele alignment can change the likelihood substantially. Standard errors should correspond to the same effect scale and association model as the estimates.

The matrix \(R\) must be a valid correlation matrix: it is symmetric, has unit diagonal, and is positive semidefinite. The likelihood formula with \(\Sigma^{-1}\) requires positive definiteness. Perfectly correlated variants can make it singular; near-perfect correlation can make the inverse numerically unstable. In such cases, the data do not provide independent directions of information for all variants. Reducing redundant variants or using a method explicitly designed for singular or regularized covariance may be necessary; silently inverting an invalid matrix does not fix the model.

Scope of the Approximation and Common Pitfalls

The derivation is most direct for quantitative traits under a linear model, a common set of individuals, and appropriately scaled genotypes. For binary traits analyzed by logistic regression, a Gaussian approximation to estimates or score statistics may be useful in large samples, but the covariance can depend on the analysis and is not automatically the raw genotype LD matrix. Relatedness, sample overlap patterns, variant-specific missingness, imputation uncertainty, or substantial differences between the analysis and LD-reference populations can also undermine \(SRS\).

A useful comparison is with treating marginal estimates as independent. That model uses a diagonal covariance, effectively setting all off-diagonal correlations to zero. It can be reasonable when LD is negligible, but in a correlated region it misstates the joint sampling variation and ignores the LD-induced pattern in expected marginal estimates. The Gaussian LD likelihood retains that dependence, but relies on accurate inputs and a suitable sampling approximation; it does not remove ambiguity when variants are highly correlated.

• Common mistake: Setting the expected marginal estimate at variant \(j\) equal to its joint effect \(\beta_j\). Correction: Under the standardized linear model, its expected marginal estimate is \((R\beta)_j=\sum_k r_{jk}\beta_k\).
• Common mistake: Using \(R\) itself as the covariance of estimates measured in phenotype units. Correction: Scale by their standard errors: \(\Sigma=SRS\). In standardized \(z\)-score units, the approximate covariance is \(R\).
• Common mistake: Assuming a supplied LD matrix is suitable because its entries are correlations. Correction: Check positive semidefiniteness, variant order, allele orientation, and compatibility with the association analysis.

The likelihood is a model for the joint sampling distribution of summary estimates conditional on the proposed effect vector and covariance assumptions. Its reliability depends on how well that model represents the data. In particular, high LD can make the covariance nearly singular and can leave distinct joint-effect patterns producing similar marginal summaries.