Conditional and Joint Evidence for Multiple Signals

Why evidence changes when signals are considered together

A variant can have a strong marginal association because it is correlated with a causal variant nearby. Conversely, a genuine effect can look weak marginally when its contribution is offset by another correlated signal. Thus a marginal \(z\)-statistic answers a different question from a conditional \(z\)-statistic: the first measures association without adjustment for other variants; the second measures the evidence remaining after adjustment. A joint model goes further by estimating and evaluating several variant effects together.

These distinctions matter in fine-mapping because the goal is not merely to locate a region associated with a phenotype. The goal is to compare variant-level explanations while accounting for linkage disequilibrium (LD), the correlation among genotypes. The earlier material on LD and variant indistinguishability establishes why highly correlated variants can be difficult to separate; here the focus is how that correlation affects evidence when multiple signals are possible.

Model and assumptions

For a concise derivation, suppose there are \(n\) individuals and \(m\) candidate variants. Let \(X\) be the \(n\)-by-\(m\) matrix of centered, standardized genotypes, and let \(y\) be a centered quantitative phenotype. Assume the columns of \(X\) have squared length \(n\), and write \(R=X^\mathsf{T}X/n\) for their sample correlation matrix. Consider the linear model

Here \(\beta_j\) is the effect on \(y\), in phenotype units per one standard deviation of genotype \(j\). For the summary-statistic relationships below, assume a common residual variance, a correctly specified linear mean model, and an LD matrix \(R\) estimated in the same population as the association statistics. The normal approximation for the marginal \(z\)-statistics is most useful when the sample is sufficiently large and the association statistics are well behaved. If covariates are present, both the association statistics and LD calculations must account for the same covariate adjustment.

Let \(z_j\) denote the marginal association \(z\)-statistic for variant \(j\), obtained by regressing \(y\) on that variant alone. Under the model, its approximate mean is \(\sqrt{n} (R\beta)_j/\sigma\), not simply \(\sqrt{n}\beta_j/\sigma\). The contribution of other effects enters through the correlations in \(R\). For a pair of variants with correlation \(\rho\), the two marginal statistics have approximate covariance matrix \(R\).

Deriving the two-variant conditional evidence

For two variants, write their LD matrix and the vector of marginal statistics as

In the joint regression of \(y\) on both variants, the least-squares coefficient estimate is \(\hat{\beta}=(X^\mathsf{T}X)^{-1}X^\mathsf{T}y\). The marginal score vector is proportional to \(X^\mathsf{T}y\), so the joint coefficient direction is proportional to \(R^{-1}z\). This relationship holds when the statistics share the same scaling; standard errors are also needed to turn coefficient estimates into conditional \(z\)-statistics.

The conditional statistic for variant 1 can also be derived by residualizing. Remove from genotype 1 its linear projection on genotype 2. The residual genotype has variance \(1-\rho^2\). Its association score is proportional to \(z_1-\rho z_2\), because the component predictable from genotype 2 is removed. Standardizing by the residual standard deviation gives the conditional statistic. Reversing the roles gives the corresponding statistic for variant 2.

This calculation assumes \(|\rho|<1\), so the two genotypes are not perfectly collinear. If \(|\rho|=1\), their separate effects are not identifiable from these data. As \(|\rho|\) approaches 1, the denominator approaches zero: conditional estimates can become highly variable, and small changes in the statistics can produce large changes in the estimated effects.

Joint effects can differ sharply from marginal effects

For two variants, the inverse LD matrix is

Consequently, the joint coefficient direction is \(R^{-1}z\). In the standardized linear model, the covariance matrix of the joint coefficient estimates is \(\sigma^2(X^\mathsf{T}X)^{-1}=(\sigma^2/n)R^{-1}\). Thus a joint test uses each coefficient divided by its standard error; it does not use the unscaled coefficient direction as a \(z\)-statistic. This distinction is important when LD is strong, because the diagonal elements of \(R^{-1}\), and hence the standard errors, can be large.

Conditioning and joint causal configurations

A conditional test usually asks a specific question, such as whether variant 2 has an effect after variant 1 is included. A multi-causal configuration model asks which set of variants could carry effects. If \(S\) is a proposed configuration, it specifies that effects outside \(S\) are zero while effects in \(S\) are allowed. A posterior probability for \(S\) combines its marginal likelihood with its prior probability:

This formula makes clear why a conditional \(p\)-value is not itself a posterior probability for a causal configuration. The posterior depends on the likelihood under each configuration and on the prior probabilities of those configurations. For example, under the summary-statistic approximation \(z\mid\beta\sim N(\sqrt{n}R\beta/\sigma,R)\), a configuration restricts which entries of \(\beta\) can be nonzero. If effect sizes are assigned a prior distribution, the configuration likelihood integrates over those possible effects. The resulting comparison accounts for the joint pattern of statistics and LD rather than merely ranking marginal \(z\)-statistics.

Conditioning on a variant as though it were certainly causal can be useful for examining residual evidence, but it is not equivalent to averaging over uncertainty about whether that variant is causal. If the conditioned variant is only a proxy, or if the model omits another signal, the residual evidence can be misleading. Joint configuration analysis represents competing explanations more directly, although it depends on its configuration prior, effect-size assumptions, and LD specification.

Diagnostics, alternatives, and limits

Before interpreting conditional or joint evidence, check whether the data and model support separating the variants. Inspect the relevant LD matrix for near-collinearity, verify that the LD reference and association statistics refer to compatible populations and variant coding, and confirm that covariate adjustment is consistent. Large changes in coefficient signs or magnitudes after adding a correlated variant, large joint standard errors, or unstable results across reasonable LD estimates are warnings that variant-level separation is weak.

Conditional regression and joint configuration analysis are related but serve different purposes. Conditional regression provides a direct test of effects given a specified adjustment set. Joint configuration models compare alternative sets of potentially causal variants and can reflect uncertainty over those sets through posterior probabilities. Marginal analysis is simpler and can be informative for detecting a locus, but it does not isolate effects when several correlated signals may be present.

None of these methods can create information that the genotypes do not contain. Near-perfect LD can make separate effects practically or exactly unidentifiable. Results also depend on the assumed phenotype model, effect-size and configuration priors, and LD accuracy. Unmodeled signals, population structure, phenotype misspecification, or an LD reference mismatch can distort conditional evidence. Finally, statistical evidence for a variant is not by itself proof of a causal molecular mechanism.