Why vary the LD matrix?
Fine-mapping uses linkage disequilibrium (LD) to relate the observed association evidence at nearby variants to the variants that may be causal. In practice, an analysis uses an estimated LD matrix, and that matrix may differ across plausible reference samples or estimation choices. If the fine-mapping conclusion changes substantially when the LD matrix changes, the conclusion depends on that choice. An LD sensitivity analysis makes that dependence visible.
The aim is not to select whichever matrix produces the most appealing result. It is to compare results under a set of defensible LD matrices while keeping the association data, variants, priors, and other analysis settings fixed. This isolates sensitivity to LD, rather than mixing it with the prior sensitivity considered in earlier material.
A plausible matrix should represent the same variants, in the same order and allele orientation, and should be a valid correlation matrix: it is symmetric, has diagonal entries equal to 1, and is positive semidefinite. Plausibility also depends on the data context, including ancestry and the population represented by the association statistics. Matrices that are inconsistent with the summary statistics or variant coding should not be treated as meaningful sensitivity scenarios.
For a fair comparison, use the same summary statistics, candidate variants, effect-size prior, prior on the number of causal variants, and other model settings in each run. The LD matrix is the intended change. If other inputs must change, record those changes and do not attribute the resulting differences to LD alone.
What to compare across matrices
Let \(R^{(m)}\) denote the LD matrix used in scenario \(m\), and let \(\gamma\) denote a configuration: a specified set of causal variants. For a given matrix, the posterior probability of a configuration is proportional to its likelihood times its prior:
Here \(z\) represents the association evidence. In a summary-statistic fine-mapping model, LD enters the likelihood because correlated variants can produce similar patterns of association evidence. Keeping the prior fixed does not force the posterior to stay fixed: changing \(R^{(m)}\) can change the likelihoods assigned to configurations.
Thus, a PIP sums posterior probabilities over all configurations containing that variant. Compare each variant’s PIP across matrices, not just the top variant. Useful summaries include the minimum and maximum PIP across the specified scenarios, changes in ranking, and whether a variant remains among the credible-set members.
A credible set is a set of variants selected to accumulate a target amount of posterior probability for a signal. In a one-signal analysis, a common construction ranks variants by PIP and includes them until their cumulative PIP reaches the target, such as 0.95. With multiple signals, the relevant quantities are signal-specific probabilities and sets; membership should be compared for the same signal across matrices. Membership is a thresholded summary: a small PIP change near the cumulative cutoff can change membership even when the underlying probabilities changed only slightly.
Purity describes how strongly correlated the least-correlated pair in the set is under that matrix. It is not a measure of posterior coverage, and high purity does not establish that the set contains the causal variant. It is also matrix-dependent: the same set of variants can have different purity under different LD matrices.
Worked comparisons
Worked Example: Stable Set, Changing Top Variant
Suppose a hypothetical one-signal analysis considers four variants and uses a 95% cumulative-PIP credible-set rule. Three plausible LD matrices produce these PIPs:
| Variant | Matrix A | Matrix B | Matrix C |
|---|---|---|---|
| v1 | 0.52 | 0.31 | 0.47 |
| v2 | 0.28 | 0.44 | 0.33 |
| v3 | 0.15 | 0.20 | 0.16 |
| v4 | 0.05 | 0.05 | 0.04 |
The PIPs sum to 1 in each column, as expected for this one-signal illustration. Under Matrix A, the ranked cumulative probabilities are \(0.52\), \(0.52+0.28=0.80\), and \(0.80+0.15=0.95\), so the set is \(\{v1,v2,v3\}\). Under Matrix B, the ranking is v2, v1, v3, v4, and \(0.44+0.31+0.20=0.95\), giving the same membership. Under Matrix C, \(0.47+0.33+0.16=0.96\), again giving \(\{v1,v2,v3\}\).
The set membership is stable across these scenarios, but the top-ranked variant is not: v1 ranks first under A and C, while v2 ranks first under B. The PIP range for v1 is \(0.52-0.31=0.21\), and for v2 it is \(0.44-0.28=0.16\). A careful conclusion is that the 95% set is stable for these matrices, while the ranking and individual PIPs are sensitive to LD. It would be inaccurate to describe v1 as uniquely identified.
Worked Example: Membership Changes and Purity Changes
In a second hypothetical analysis, the same four variants are considered under two plausible matrices. The 95% credible set is constructed separately in each scenario.
| Variant | Matrix A PIP | Matrix B PIP |
|---|---|---|
| a | 0.50 | 0.48 |
| b | 0.30 | 0.30 |
| c | 0.15 | 0.04 |
| d | 0.05 | 0.18 |
For Matrix A, the cumulative PIP for a and b is \(0.50+0.30=0.80\), and including c gives \(0.80+0.15=0.95\). Thus \(C^{(A)}=\{a,b,c\}\). For Matrix B, the ranking is a, b, d, c, and the first three sum to \(0.48+0.30+0.18=0.96\). Thus \(C^{(B)}=\{a,b,d\}\). The shared members are a and b; c and d exchange membership.
Suppose the absolute correlations among the Matrix A set are \(|R_{ab}|=0.80\), \(|R_{ac}|=0.75\), and \(|R_{bc}|=0.25\). Its purity is \(\min(0.80,0.75,0.25)=0.25\). For the Matrix B set, suppose \(|R_{ab}|=0.82\), \(|R_{ad}|=0.78\), and \(|R_{bd}|=0.55\). Its purity is \(\min(0.82,0.78,0.55)=0.55\). These hypothetical results show both membership sensitivity and a purity change. Matrix B’s higher purity does not mean its set is more likely to contain the causal variant; it means its least-correlated pair is more strongly correlated under that matrix.
Worked Example: Scenario Averages Are Not Automatically Posterior Probabilities
Suppose the PIP for variant x is 0.62, 0.40, and 0.58 across three plausible LD matrices, while the PIP for variant y is 0.25, 0.48, and 0.30. The arithmetic means are
These averages can be reported as descriptive summaries of the three scenarios, provided they are labelled as such. They are not automatically posterior inclusion probabilities that account for LD uncertainty. A Bayesian average across matrices would require a probability model or explicit weights for the matrices and a posterior calculation that incorporates those weights. Treating three scenarios as equally probable without justification silently imposes such a weighting choice.
The scenario results are more informative when shown alongside the means: x ranges from 0.40 to 0.62, and y from 0.25 to 0.48. In this example x has the larger PIP in two scenarios, but y has the larger PIP in the other. The average alone would conceal that ranking reversal.
Why the comparison works
The basic logic follows directly from the configuration posterior. With the variant set, association evidence, and prior held fixed, a change in the matrix changes the likelihood term whenever the model uses LD to evaluate how well a configuration explains the observed evidence. Configuration probabilities can therefore change, and PIPs inherit those changes by summing over configurations.
Keep the variants, summary statistics, priors, and model settings the same so the comparison targets LD sensitivity.
Use matrices that correspond to the same variant order and coding and are valid for the model.
Inspect PIP magnitudes and rankings, signal-specific set membership, and purity calculated from each scenario’s own matrix.
Describe conclusions as stable or sensitive across the matrices examined, not across every possible LD matrix.
This comparison is conditional on the specified scenarios. It does not quantify the probability that an LD matrix is correct, and it does not repair a misspecified fine-mapping model. If the matrices are all drawn from a narrow or inappropriate reference, agreement among them may provide false reassurance.
Diagnostics, alternatives, and limitations
Before interpreting a change as LD sensitivity, check that the runs are genuinely comparable. Confirm variant identities and ordering, effect-allele orientation, the set of variants included, and that each matrix is symmetric with unit diagonal and valid correlation structure. Also check that the same input summary statistics and priors were used. A variant-order mismatch can make results look dramatically different without representing a meaningful alternative LD scenario.
Look at several patterns rather than one threshold. A variant may retain similar PIP but move in rank; credible-set membership may change because a cumulative cutoff is crossed; and purity may change because the pairwise correlations change. These summaries answer different questions. In particular, credible-set membership is a discrete decision, while PIP is continuous. A one-variant membership change does not necessarily imply a large shift in posterior evidence, so inspect the actual PIPs near the cutoff.
A useful comparison table reports scenario-specific PIPs for leading variants, whether each variant is in the relevant credible set, the set intersection and union, and purity under the corresponding matrix. If a summary range or average is added, retain the per-matrix results so that reversals are not hidden. There is no universal cutoff for how much PIP variation makes a result “unstable”; state any decision rule used and explain its purpose.
The main alternative is to choose one LD matrix, often the one considered most representative, and report a single fine-mapping analysis. That can be appropriate when the reference is well justified, but it does not reveal dependence on LD choice. A scenario analysis instead shows how conclusions vary across the plausible matrices supplied. Explicitly averaging across matrices is a distinct approach that needs defensible weights or a model for LD uncertainty; an unweighted average of scenario outputs is not a substitute.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Across two matrices, a variant’s PIP changes from 0.72 to 0.46, but it is included in both 95% credible sets. What is stable, and what is sensitive?
- A three-variant credible set has absolute pairwise LD values 0.91, 0.67, and 0.74. Calculate its purity.
- Why should the same variant order, allele coding, and priors be maintained when comparing LD matrices?
- Three scenario-specific PIPs are 0.60, 0.50, and 0.40. What is their arithmetic mean, and why is it not automatically a posterior PIP averaged over LD uncertainty?
Solutions. (1) Membership is stable across the two specified matrices, but the PIP is sensitive: it changes by \(0.72-0.46=0.26\). The result should not be described as having an unchanged posterior probability. (2) Purity is the minimum absolute pairwise correlation, \(\min(0.91,0.67,0.74)=0.67\). (3) Holding those features fixed isolates the effect of changing the LD matrix. Otherwise, differences could arise from variant alignment, coding, prior choices, or other inputs rather than LD. (4) The mean is \((0.60+0.50+0.40)/3=0.50\). It is a descriptive average of the listed scenarios; a posterior average would require justified probabilities or weights for the matrices and an analysis that incorporates them.