Sensitivity to LD Reference Mismatch

Why LD Reference Mismatch Matters

Fine-mapping uses patterns of association across correlated variants to compare explanations for the observed data. When individual-level genotypes are unavailable, a common approach combines association summary statistics with an LD matrix estimated from a reference sample. The central sensitivity question is whether a different, still plausible reference matrix would produce different posterior inclusion probabilities (PIPs) or credible sets.

A reference matrix is not simply a neutral input. In a multi-variant model it describes how a proposed set of causal effects would generate association statistics across the region. If its correlations differ from those in the study, the likelihood can give different support to candidate configurations. However, LD mismatch does not necessarily affect every fine-mapping model in the same way: in the correctly specified single-causal-variant summary-statistic model, LD cancels from the comparison of variants. That special case is an important check on what a sensitivity analysis can establish.

Definition: Let \(R_{\mathrm{study}}\) be the correlation matrix of the variants’ standardized genotypes in the association study, and let \(R_{\mathrm{ref}}\) be the corresponding matrix estimated from a reference sample. LD reference mismatch means that, after variants and alleles have been aligned, \(R_{\mathrm{ref}}\) differs from \(R_{\mathrm{study}}\). A fine-mapping analysis using \(R_{\mathrm{ref}}\) then uses an approximation to the study’s genotype-correlation structure.

Likelihood, Assumptions, and the Sensitivity Target

For a region with \(J\) variants, write \(z\) for the vector of association \(z\)-statistics and \(R\) for the LD matrix supplied to the model. Under the Gaussian summary-statistic approximation, standardized genotypes, and a common sample of size \(N\), the model is

z\mid\beta,R \ \approx\ N\!\left(\sqrt{N}\,R\beta,\ R\right).

Here \(\beta\) is the vector of genetic effects on the standardized scale. The approximation is most appropriate when the score statistics are well approximated by a multivariate normal distribution and the matrix represents the same variants, allele coding, and relevant study population as the summary statistics. In practice, differences in ancestry, imputation, variant filtering, sample overlap, or allele alignment can all make a reference matrix inappropriate. A matrix with incorrect variant order or allele signs is not a small amount of ordinary LD mismatch.

A causal configuration \(C\) is a specified subset of variants allowed to have nonzero effects. For example, in a two-variant region the possible configurations are the empty set, variant 1 alone, variant 2 alone, and both variants. A model assigns each configuration a prior probability and computes its posterior probability by combining that prior with the likelihood. The PIP for variant \(j\) is the sum of posterior probabilities over configurations that include \(j\):

\operatorname{PIP}_j = \sum_{C:\,j\in C} P(C\mid z,R).

For example, with independent normal effect priors \(\beta_C\sim N(0,W I)\) within a configuration, integrating over effects gives

z\mid C,R\ \approx\ N\!\left(0,\ R+NW R_C R_C^{T}\right),

where \(R_C\) is the matrix containing the columns of \(R\) indexed by \(C\). Thus the likelihood used to compare configurations depends on the supplied LD matrix. Replacing \(R_{\mathrm{study}}\) by \(R_{\mathrm{ref}}\) can change configuration probabilities and, consequently, PIPs.

A useful sensitivity analysis changes the LD reference while holding fixed the summary statistics, variant set, effect-size assumptions, configuration prior, and all other analysis choices. This isolates the effect of the matrix choice. The purpose is not to select whichever matrix gives the preferred result, but to learn whether conclusions are stable across defensible references.

Worked Comparison Under Two LD References

Worked Example: PIPs and a 90% Variant Set

Consider a hypothetical two-variant region with \(z=(3,2)^T\). Use the model \(z\mid\eta,R\sim N(R\eta,R)\), where \(\eta=\sqrt{N}\beta\). For transparent arithmetic, specify a point-mass effect prior: each included variant has \(\eta_j=1\), and each excluded variant has \(\eta_j=0\). Give the four configurations equal prior probabilities. Compare two candidate reference matrices \(R(r)=\begin{pmatrix}1&r\\r&1\end{pmatrix}\), with \(r=0.2\) and \(r=0.8\). This is a deliberately simple Bayesian model, not a claim that fixed effect sizes are generally preferable.

For a configuration with mean vector \(m=R\eta\), its likelihood is proportional to \(\exp(-Q/2)\), where

Q=(z-m)^T R^{-1}(z-m),\qquad R^{-1}=\frac{1}{1-r^2} \begin{pmatrix}1&-r\\-r&1\end{pmatrix}.

The determinant term in the normal density is the same for all four configurations at a fixed \(r\), so it cancels when calculating posterior probabilities within that reference analysis. The means for the null, variant 1, variant 2, and both-variant configurations are respectively \((0,0)^T\), \((1,r)^T\), \((r,1)^T\), and \((1+r,1+r)^T\).

Reference correlation \(r=0.2\). Since \(1-r^2=0.96\), for the null configuration

Q_{\varnothing}=\frac{3^2+2^2-2(0.2)(3)(2)}{0.96} =\frac{10.6}{0.96}=11.0417.

For variant 1, the residual is \((2,1.8)^T\), giving \(Q_1=(4+3.24-1.44)/0.96=6.0417\). For variant 2, the residual is \((2.8,1)^T\), giving \(Q_2=(7.84+1-1.12)/0.96=8.0417\). For both variants, the residual is \((1.8,0.8)^T\), giving \(Q_{12}=(3.24+0.64-0.576)/0.96=3.4417\).

Subtracting the largest log likelihood, that of the both-variant configuration, yields relative likelihood weights \(\exp(-3.8)=0.02237\), \(\exp(-1.3)=0.2725\), \(\exp(-2.3)=0.1003\), and \(1\). Their sum is \(1.3952\). With equal configuration priors, the posterior probabilities, in the same order, are approximately \(0.0160,\ 0.1953,\ 0.0719,\ 0.7168\). Therefore,

\operatorname{PIP}_1=0.1953+0.7168=0.9121,\qquad \operatorname{PIP}_2=0.0719+0.7168=0.7886.

Reference correlation \(r=0.8\). Now \(1-r^2=0.36\). The four quadratic forms are \(Q_{\varnothing}=3.4/0.36=9.4444\), \(Q_1=1.6/0.36=4.4444\), \(Q_2=2.32/0.36=6.4444\), and \(Q_{12}=1.096/0.36=3.0444\). Relative likelihood weights are \(\exp(-3.2)=0.04076\), \(\exp(-0.7)=0.4966\), \(\exp(-1.7)=0.1827\), and \(1\). Their sum is \(1.7200\), so the posterior probabilities are approximately \(0.0237,\ 0.2887,\ 0.1062,\ 0.5814\). The resulting PIPs are

\operatorname{PIP}_1=0.2887+0.5814=0.8701,\qquad \operatorname{PIP}_2=0.1062+0.5814=0.6876.

For clarity, define a variant set at target \(0.90\) here as the smallest set \(S\) for which the posterior probability that \(S\) contains at least one causal variant is at least \(0.90\). At \(r=0.2\), the singleton \(\{1\}\) has probability \(0.9121\), so it meets the target. At \(r=0.8\), neither singleton meets the target: the probabilities are \(0.8701\) for \(\{1\}\) and \(0.6876\) for \(\{2\}\). The pair \(\{1,2\}\) has probability \(1-P(\varnothing\mid z,R)=1-0.0237=0.9763\), so both variants are needed. Under this explicitly defined multi-causal set criterion, the alternative reference changes the reported set from \(\{1\}\) to \(\{1,2\}\).

This example uses a small configuration model to make each calculation visible. Its set criterion is not interchangeable with every credible-set construction used in practice; when reporting a credible set, state the target and the posterior event it is intended to cover.

Diagnostics for Alternative Reference Matrices

Before interpreting changes in PIPs, check that the alternatives are valid and comparable. The following checks help separate genuine sensitivity to LD structure from avoidable data or coding errors.

Match variants and alleles. Confirm that rows and columns refer to the same variants in the same order as the summary statistics. Align effect alleles and confirm that the \(z\)-statistics use the same allele coding. An allele flip changes the signs of the corresponding LD entries and association statistic; it must be handled consistently.
Check matrix properties. A correlation matrix should be symmetric, have unit diagonal, and be positive semidefinite. Near-singular matrices can make calculations unstable, especially methods that use a matrix inverse. Any regularization or eigenvalue adjustment should be documented and applied consistently.
Quantify disagreement. Examine correlation differences for variant pairs, not only a single region-wide summary. A large discrepancy for a pair of highly associated variants may matter more than many small differences elsewhere. Also compare the consequences for posterior rankings and sets.
Check whether the reference is plausible. Consider ancestry, sample size, genotyping and imputation, and the way the study and reference variants were filtered. A large reference sample from a poorly matched population may be less suitable than a smaller, better-matched sample.

Worked Example: Quantifying LD Disagreement

Suppose, hypothetically, an independently available study genotype subset estimates the two-variant correlation as \(r_{\mathrm{study}}=0.65\). The alternative reference correlations from the preceding example are \(0.2\) and \(0.8\). All three matrices have unit diagonal and are positive definite: their determinants are \(1-r^2=0.5775,\ 0.96,\) and \(0.36\), respectively. For these two-by-two matrices, the Frobenius distance is

\|R(r_a)-R(r_b)\|_F =\sqrt{2(r_a-r_b)^2}.

For \(r=0.2\), the distance from the study estimate is \(\sqrt{2(0.65-0.2)^2}=\sqrt{0.405}=0.6364\). For \(r=0.8\), it is \(\sqrt{2(0.65-0.8)^2}=\sqrt{0.045}=0.2121\). By this measure, the \(0.8\) reference is closer to the hypothetical study estimate. Yet its fine-mapping output is not thereby established as correct: the study estimate itself may be noisy or otherwise unrepresentative, and a distance alone does not measure posterior error. The calculation motivates prioritizing plausible references for sensitivity analysis, not treating a matrix-distance threshold as a guarantee.

A Limiting Case: One Causal Variant

The single-causal-variant model gives a useful contrast. Suppose exactly one variant is causal and, for the purpose of comparing candidate \(j\), its scaled effect \(\eta\) is fixed. Under the Gaussian likelihood, the mean is \(R_{\cdot j}\eta\), the \(j\)th LD column multiplied by \(\eta\). Because \(R^{-1}R_{\cdot j}\) is the \(j\)th unit vector, the log likelihood ratio against the null is

\log\frac{L_j}{L_0} = -\frac{1}{2}(z-R_{\cdot j}\eta)^TR^{-1}(z-R_{\cdot j}\eta) +\frac{1}{2}z^TR^{-1}z = \eta z_j-\frac{\eta^2}{2}.

The last expression does not contain \(R\). The same cancellation holds when a common normal prior on the effect is integrated out: the variant-specific evidence depends on its own marginal \(z_j\), not on the LD matrix. Thus, under the stated model and consistent allele alignment, simply swapping valid LD matrices should not change posterior variant probabilities in an exactly-one-causal-variant analysis. This is a property of the model, not a claim that LD is irrelevant to fine-mapping generally.

Worked Example: Checking Single-Causal Invariance

Use the hypothetical \(z=(3,2)^T\) and the same fixed scaled effect \(\eta=1\). With equal prior probability for the two candidate variants, the log likelihood ratios against the null are \(1(3)-1^2/2=2.5\) for variant 1 and \(1(2)-1^2/2=1.5\) for variant 2. Their posterior probabilities conditional on exactly one being causal are therefore

P(1\mid z,\text{one causal}) =\frac{e^{2.5}}{e^{2.5}+e^{1.5}} =\frac{e}{e+1} =0.7311, \qquad P(2\mid z,\text{one causal})=0.2689.

These values are identical for \(r=0.2\) and \(r=0.8\), because the LD term cancels from both likelihood ratios. The multi-causal example changed because comparing configurations with multiple included variants does not reduce to that same single-column comparison.

Interpretation, Failure Modes, and Limits

When PIPs or sets change across plausible LD references, report the change as sensitivity of the fine-mapping result to the LD input. Do not interpret a PIP as the probability that a variant is causal regardless of the model: it is a posterior probability conditional on the specified data, likelihood, prior, and supplied LD matrix. A stable result across references is reassuring about this particular source of variation, but does not establish that the model or its other assumptions are correct.

A frequent mistake is to regard the reference with the largest sample size as automatically adequate, or to change several modeling choices at once and attribute the output difference to LD. Instead, verify that each matrix is a defensible approximation to study LD and hold the other choices fixed during the comparison. Another mistake is to expect LD mismatch to change every single-causal analysis. The cancellation above shows why that expectation is wrong under the stated summary-statistic model; sensitivity can arise in multi-causal models or through other departures from its assumptions.

A further limitation is that alternative references do not necessarily span the true study LD. A comparison across two matrices can reveal instability between those inputs, but cannot rule out instability under an unexamined matrix. When study genotypes are unavailable, the study matrix is not directly observed, so ancestry-matched reference panels and transparent assumptions matter. If a substantial discrepancy is found, investigate variant and allele alignment, population match, and matrix conditioning before treating the change in posterior output as biological evidence.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In a multi-causal model, what does the PIP for variant \(j\) sum over?
For the example’s \(r=0.8\) analysis, calculate the posterior probability that at least one variant in \(\{1,2\}\) is causal.
Why does replacing \(R\) not change candidate-variant likelihood ratios in the stated exactly-one-causal model?
Name two checks that should be completed before interpreting a change in PIPs as sensitivity to LD mismatch.

Solutions. (1) Sum \(P(C\mid z,R)\) over all configurations \(C\) containing \(j\). (2) The only configuration with no causal variant has posterior probability approximately \(0.0237\), so the probability of at least one causal variant in \(\{1,2\}\) is \(1-0.0237=0.9763\). (3) For candidate \(j\), the mean is \(R_{\cdot j}\eta\), and \(R^{-1}R_{\cdot j}\) is the \(j\)th unit vector. Expanding the Gaussian log likelihood ratio gives \(\eta z_j-\eta^2/2\), which contains no \(R\). (4) Examples include confirming variant order and allele alignment, checking symmetry and positive semidefiniteness, and assessing ancestry and variant-filtering comparability.