Credible-Set Purity and LD

Why Examine Purity Within a Credible Set?

A credible set summarizes posterior support across variants: its members collectively account for a specified amount of posterior probability under the fine-mapping model. That statement does not tell us whether the variants are strongly correlated with one another in the genotype data. A set can have the intended posterior probability while containing variants with quite different LD relationships. Credible-set purity describes this within-set LD structure.

This distinction matters when interpreting fine-mapping results. If all members of a set are strongly correlated, they may represent alternatives that are difficult to distinguish using the observed data. If some members have weak LD with others, the set is less coherent in that sense. Low purity can flag a signal whose localization is unresolved or a set that may combine different patterns of evidence. It is a diagnostic, however, not a direct test that the set contains more than one causal variant.

Defining and Calculating the Pairwise LD Summaries

For a sample of \(n\) individuals, let \(g_{ai}\) denote the genotype dosage for individual \(a\) at variant \(i\), usually coded as 0, 1, or 2 copies of a specified allele. The sample correlation between variants \(i\) and \(j\) is

Here \(\bar{g}_i\) and \(\bar{g}_j\) are the sample mean dosages for the two variants. This correlation is defined when both genotype columns vary. The sign of \(r_{ij}\) depends on the allele coding: changing the coded allele at one variant reverses the sign. For that reason, the summaries defined here use \(|r_{ij}|\). They measure the strength of linear correlation without treating opposite allele coding as weaker LD.

For the \(m=k(k-1)/2\) pairwise values \(a_{ij}=|r_{ij}|\), define

The median absolute correlation is the median of the same \(m\) values. These summaries are not interchangeable. The minimum reveals whether even the weakest pair is strongly correlated; the mean describes average pairwise correlation but can conceal a weakly correlated pair; the median is less sensitive to an unusually high or low value. When \(k=2\), all three summaries equal the one absolute correlation. For a singleton set, there are no pairs, so these pairwise purity summaries are undefined rather than equal to 1.

The word “purity” is not a universal guarantee of one exact software definition. Some analyses report a minimum absolute correlation, while others also report mean or median values, or use a threshold based on \(r^2\). Always state which quantity is being reported. Since \(r^2\) squares the correlation, a threshold on \(r^2\) is not numerically the same as the same threshold on \(|r|\).

What the Minimum Does—and Does Not—Guarantee

The minimum absolute correlation has a useful direct interpretation. If the minimum is at least a chosen threshold \(t\), then every pair in the set has absolute correlation at least \(t\). This follows from the definition of a minimum: no observed pairwise value can be smaller than the minimum. Conversely, if the minimum is below \(t\), at least one pair falls below that threshold. Neither statement identifies which variant is causal; the summary concerns genotype correlations among the set members.

A short inequality also explains why the mean cannot replace the minimum when the question is whether every pair is well correlated. For pairwise values \(a_1,\ldots,a_m\), each is at least their minimum, so their sum is at least \(m\min(a_1,\ldots,a_m)\). Dividing by \(m\) gives

The mean is therefore at least the minimum, and can be appreciably larger when most pairs are highly correlated but one is not. A high mean alone does not establish that all pairwise correlations are high.

Assumptions and Data Checks

Purity is only as interpretable as the LD estimates behind it. A basic calculation assumes that the variants in the set are aligned to the same individuals or to a suitable reference panel, and that allele coding and variant identity are handled consistently. The genotype columns must have nonzero variation for correlations to be defined. If genotypes are missing, the analyst should also ensure that pairwise correlations are estimated comparably; using a different subset of individuals for each pair can make the collection of values harder to interpret as one coherent LD summary.

Population composition matters because LD is population-dependent. A reference panel that does not represent the fine-mapping study population can give different correlations, and therefore different purity summaries. This is especially important when variants are uncommon or when the sample used for LD estimation is small: correlations can be noisy. A low or high purity estimate should be read as a property of the specified genotype data or reference panel, not as an invariant biological feature of the set.

For \(k\) variants, the number of comparisons is \(k(k-1)/2\). As the set grows, the number of pairs grows quadratically, and a single weakly correlated pair can drive the minimum downward. That is not a computational defect: it is precisely what the minimum records. It does mean that minimum purity can be sensitive to one pair, so reporting mean or median alongside the minimum can help readers understand the overall pattern.

Interpretation, Diagnostics, and Limitations

A credible-set probability and a purity summary answer different questions. The set probability concerns posterior mass under the fine-mapping model. Purity concerns correlations among the variants included in the set. A high-purity set is not automatically a well-calibrated set, and low purity does not automatically invalidate its posterior probability. Purity adds a diagnostic description of the LD geometry; it is not a replacement for the model’s posterior quantities.

Low purity is consistent with several possibilities. A set may include variants that are weakly correlated with one another because the data do not localize support to one tight LD group. It may also reflect a more complex signal structure, including the possibility that evidence from more than one signal is not cleanly separated in the analysis. These are interpretations to investigate, not conclusions that follow mechanically from one low number. A low minimum can be driven by a single outlying pair, so inspect the pairwise matrix and compare minimum, mean, and median before characterizing the whole set.

Useful diagnostics include verifying variant identifiers and allele alignment, checking whether all pairwise correlations use the same reference individuals, and confirming that the reported metric is \(|r|\) rather than \(r^2\). If the LD reference is poorly matched, the summary may reflect reference mismatch as much as the study’s local LD. A purity value also does not establish causation, functional relevance, or the number of causal variants.

The main alternative to a single minimum is to report several summaries or the full pairwise LD matrix. The minimum is simple and conservative but can be dominated by one pair. The mean gives a broad average but can hide weak pairs; the median resists a single extreme value but also does not guarantee that every pair is strong. A matrix shows which pairs are weak, at the cost of requiring more space and interpretation. For a large set, a compact summary plus the matrix or a clear description of outlying pairs is often more informative than any one number alone.