Why Analyze Configurations?
When more than one variant may contribute to a genetic association signal, the inferential object is not just a ranked list of variants. A model must consider configurations: sets of variants proposed to be causal together. Linkage disequilibrium (LD) matters because correlated genotypes can make the association statistics move together, even when the underlying causal set differs. The evidence for a configuration therefore depends jointly on its variants’ summary statistics and their LD.
This worked analysis considers three candidate variants and assumes exactly two are causal. It compares the three possible pairs under a specified LD-aware summary-statistic model. The calculations yield both posterior probabilities for the pairs and marginal posterior inclusion probabilities for individual variants. These summaries answer different questions: a configuration posterior concerns a particular joint set, while a variant’s inclusion probability sums posterior mass over every configuration containing that variant.
Model and Configuration Bayes Factor
Let \(z=(z_1,\ldots,z_m)^\top\) be the vector of standardized association statistics for \(m\) variants, and let \(R\) be their \(m\)-by-\(m\) LD correlation matrix. Assume that \(R\) is known for this analysis, symmetric and positive definite, with diagonal entries equal to 1. Under the null configuration, the summary statistics follow \(z\sim N(0,R)\). Thus their covariance is not the identity when variants are correlated.
For a proposed configuration \(S\), let \(R_{:S}\) denote the columns of \(R\) indexed by \(S\), and let \(R_{SS}\) be the corresponding submatrix. The model specifies \(z\mid \beta_S,S\sim N(R_{:S}\beta_S,R)\), where \(\beta_S\) is the vector of effects for the variants in \(S\). The effect prior is \(\beta_S\mid S\sim N(0,W I)\), with \(W>0\) a fixed prior variance and \(I\) the identity matrix of the appropriate size. The prior assumes independent, mean-zero effects with common variance \(W\); effects outside \(S\) are zero. These assumptions define the model being compared, rather than conclusions established by the observed statistics.
The Bayes factor \(BF_S\) compares the likelihood integrated over this effect prior with the null likelihood. It measures relative evidence for \(S\) versus the null, not the posterior probability of \(S\) by itself. In this model its convenient form is
Here \(z_S\) is the subvector of observed statistics indexed by \(S\), and \(|\cdot|\) denotes a determinant. The expression uses \(z_S\), not \(R_{S:}z\). The distinction is essential when the null covariance is \(R\): the covariance has already been accounted for in the likelihood ratio.
Why the Bayes Factor Has This Form
The form follows by integrating the effect vector out of the stated normal model. Write \(A=R_{:S}\). Conditional on \(S\), the random effect has covariance \(W I\), so the marginal distribution of \(z\) is \(N(0,R+WAA^\top)\). Comparing this distribution with the null \(N(0,R)\), the determinant lemma and the Woodbury identity reduce the likelihood ratio to a determinant and a quadratic form involving \(A^\top R^{-1}A\) and \(A^\top R^{-1}z\).
These equalities follow directly from \(A=R_{:S}\): selecting rows indexed by \(S\) from \(R_{:S}\) gives \(R_{SS}\), and \(R_{S:}R^{-1}\) is the row-selector matrix for those indices. Substitution gives the stated Bayes factor. In particular, replacing \(z_S\) with \(R_{S:}z\) would not implement this likelihood ratio and can substantially change the result.
A Hypothetical Three-Variant Analysis
Suppose a hypothetical locus has three candidate variants with observed statistics \(z=(3,2,0.5)^\top\). Use the LD matrix and effect-prior variance
This matrix is positive definite: its eigenvalues are \(1.8\), \(0.2\), and \(1\), all positive. Variants 1 and 2 have correlation 0.8, while variant 3 is uncorrelated with each of them in this illustrative reference. The model assumes exactly two causal variants, so the candidate configurations are \(\{1,2\}\), \(\{1,3\}\), and \(\{2,3\}\). For the worked comparison, assign each pair prior probability \(1/3\). This equal-prior choice is an explicit modeling assumption, not a consequence of the data.
Worked Example: Bayes Factor for the Correlated Pair
Calculate the Bayes factor for \(S=\{1,2\}\). The selected statistic vector is \(z_S=(3,2)^\top\), and \(R_{SS}=\begin{pmatrix}1&0.8\\0.8&1\end{pmatrix}\).
First, the determinant factor is based on \(I+W R_{SS}=\begin{pmatrix}1.2&0.16\\0.16&1.2\end{pmatrix}\). Its determinant is \(1.2(1.2)-0.16(0.16)=1.4144\), so the factor is \(1/\sqrt{1.4144}\approx0.8408\).
For the exponent, \(W^{-1}I+R_{SS}=\begin{pmatrix}6&0.8\\0.8&6\end{pmatrix}\), whose inverse is \(\frac{1}{35.36}\begin{pmatrix}6&-0.8\\-0.8&6\end{pmatrix}\). Therefore,
Combining the two factors gives \(BF_{\{1,2\}}\approx0.8408\exp(0.9672)\approx2.212\). Under this model, the observed data are about 2.212 times as probable under this pair’s effect-prior model as under the null. This Bayes factor does not yet compare the pair with the other two non-null configurations.
Worked Example: Comparing All Three Configurations
For \(S=\{1,3\}\), the selected statistics are \((3,0.5)^\top\), and \(R_{SS}=I\). Thus the determinant factor is \(1/1.2\), and the exponent is \(\frac{1}{2}\frac{0.2}{1.2}(3^2+0.5^2)=0.7708333333\). Hence \(BF_{\{1,3\}}=(1/1.2)\exp(0.7708333333)\approx1.8013\).
For \(S=\{2,3\}\), the selected statistics are \((2,0.5)^\top\), also with \(R_{SS}=I\). Its exponent is \(\frac{1}{2}\frac{0.2}{1.2}(2^2+0.5^2)=0.3541666667\), giving \(BF_{\{2,3\}}=(1/1.2)\exp(0.3541666667)\approx1.1875\). Both calculations use the selected entries of \(z\); the LD correlations with variants outside the proposed configuration are accounted for by the null covariance and do not justify substituting \(R_{S:}z\) for \(z_S\).
With equal configuration priors, posterior probabilities are proportional to these Bayes factors. Using the unrounded Bayes factors, their sum is approximately \(5.2007\). Normalizing,
The pair \(\{1,2\}\) is the most probable of the three configurations under these assumptions, but its posterior probability is about 0.4253, not a majority of the posterior mass. The other two pairs together retain about 0.5747. Reporting only the leading pair would therefore conceal substantial uncertainty about which joint configuration is supported.
From Configuration Probabilities to Inclusion Probabilities
A variant’s PIP is obtained by summing the posterior probabilities of every allowed configuration containing it. This is a marginal probability over the configuration model; it is not the probability of the most likely pair. For example, variant 1 appears in \(\{1,2\}\) and \(\{1,3\}\). The PIPs for this example are therefore
The PIPs need not equal the posterior probability of the leading configuration, and several variants can each have substantial inclusion probability. Because every candidate configuration here contains exactly two variants, the PIPs sum to two: \(0.7717+0.6536+0.5747=2.0000\), using the displayed values. In general, under a model with exactly \(k\) causal variants, the sum of the candidate variants’ PIPs is \(k\), provided the configurations cover the full posterior model space.
Worked Example: Constructing a Configuration Credible Set
A configuration credible set is a collection of configurations whose posterior probabilities sum to at least a chosen target. To form a 80% configuration credible set here, rank the pairs from highest to lowest posterior probability and accumulate their mass.
The leading pair contributes \(0.4253\), which is below 0.80. Adding the second pair gives \(0.4253+0.3464=0.7717\), still below 0.80. Adding the final pair gives a cumulative posterior probability of 1.0000 (up to rounding). Thus the 80% configuration credible set contains all three pairs: \(\{1,2\}\), \(\{1,3\}\), and \(\{2,3\}\).
This result is a direct consequence of the computed posterior distribution: no subset of the two highest-probability configurations reaches 0.80. The set describes uncertainty over joint causal configurations. It should not be confused with a list formed by sorting variants by PIP; a sum of PIPs counts expected included variants and is not, in general, the posterior probability that a particular configuration or variant set contains the causal variants.
How LD Shapes the Comparison
The pair \(\{1,2\}\) has a different determinant factor and joint quadratic form from the two pairs involving variant 3. Its selected statistics are correlated under the null, and the model accounts for that dependence through \(R_{SS}\). Treating the selected statistics as independent would replace the correct covariance structure and generally alter the Bayes factor. Conversely, using the full-vector expression without the inverse-covariance factors can mistakenly count LD-related information as if it were additional independent evidence.
The calculation also shows why a large single-variant statistic does not settle a multi-causal comparison. A configuration posterior compares the evidence for each proposed set under the same likelihood and prior structure. Changing the assumed effect prior, configuration priors, candidate variants, or LD matrix can change the resulting posterior probabilities. The numerical conclusions above are conditional on the stated choices.
Common Mistake: Multiplying by the LD Matrix Again
A tempting but incorrect calculation is to use \(R_{S:}z\) as the statistic vector in the configuration Bayes factor. For \(\{1,2\}\) in this example, that gives \((4.6,4.4)^\top\), rather than the selected data \((3,2)^\top\), and produces a much larger quadratic form. It effectively fails to use the covariance adjustment required by the null likelihood.
The correction is to start with the stated likelihood \(z\mid\beta_S,S\sim N(R_{:S}\beta_S,R)\), then use the identities \(R_{S:}R^{-1}z=z_S\) and \(R_{S:}R^{-1}R_{:S}=R_{SS}\). The sufficient statistic in the reduced Bayes factor is \(z_S\). A separate calculation using \(R_{S:}z\) is not interchangeable with that expression.
Check Your Understanding
Answer these questions using the model and calculations above.
- Why does the configuration Bayes factor use \(z_S\) rather than \(R_{S:}z\) when the null covariance is \(R\)?
- If the three configuration priors were unequal, what quantity would be normalized to obtain their posterior probabilities?
- Explain why the PIP of variant 1 is the sum of two configuration probabilities in this example.
- Why are all three pairs required for an 80% configuration credible set, even though the leading pair is the most probable?
- Under the fixed two-causal-variant model, what should the sum of the three PIPs equal, before rounding, and why?