Multi-Ancestry Bayesian Fine-Mapping Models

Why Combine Ancestry-Specific Evidence?

A locus can have different association patterns across ancestry-specific datasets even when the underlying causal variant is the same. One reason is that linkage disequilibrium (LD)—the correlation among genotypes at nearby variants—can differ across datasets. Fine-mapping must therefore use each dataset's association statistics together with an LD matrix appropriate to that dataset, rather than treating a stronger marginal association as direct proof of causality.

Bayesian analysis makes the comparison explicit: specify candidate causal configurations, calculate how well each configuration predicts the observed statistics, and combine that evidence with prior probabilities. A central distinction is between sharing a causal variant and sharing an effect size. The same variant can be causal in multiple ancestry-specific datasets even if its effect sizes differ.

Definition: Let \(a=1,\ldots,A\) index ancestry-specific datasets and \(j=1,\ldots,p\) index variants in a locus. For dataset \(a\), let \(z_a\) be its vector of association \(z\)-statistics, \(R_a\) its LD correlation matrix, and \(N_a\) its sample size. A multi-ancestry fine-mapping model combines the datasets' likelihoods for causal-variant configurations, with a stated prior over those configurations. A shared-variant model restricts the causal index to be the same across datasets; an ancestry-specific model allows the index to differ.

A Summary-Statistic Model Across Datasets

A useful approximate likelihood, when genotypes and phenotype are standardized and the local effects are small, is

z_a \mid \beta_a,R_a,N_a \ \approx\ N\!\left(\sqrt{N_a}\,R_a\beta_a,\ R_a\right).

Here \(\beta_a\) is the vector of per-variant effects in dataset \(a\), on the standardized scale. The mean reflects LD: a nonzero effect at one variant can induce association statistics at correlated variants. The covariance \(R_a\) also reflects correlation among the statistics. This is an approximation, not a universal exact likelihood; its adequacy depends on the association-statistic construction, sample size, LD estimate, and modeling assumptions.

Suppose datasets have no overlapping participants and are conditionally independent given their effects and LD matrices. Then their joint likelihood is the product of the dataset-specific likelihoods. If participants overlap, this factorization generally fails unless cross-dataset dependence is modeled. Alleles and effect directions must also be aligned consistently before the statistics are combined; this tutorial assumes that condition is met.

L(\beta_1,\ldots,\beta_A) \ \approx\ \prod_{a=1}^{A} \phi_p\!\left(z_a;\sqrt{N_a}R_a\beta_a,R_a\right),

where \(\phi_p(x;\mu,\Sigma)\) is the \(p\)-variate normal density with mean \(\mu\) and covariance \(\Sigma\). A Bayesian model integrates this likelihood over effect sizes under its prior and sums over configurations using their prior probabilities. For a model \(M\),

P(D\mid M)=\sum_{\gamma\in\Gamma_M}P(\gamma\mid M) \int P(D\mid\gamma,\theta,M)\,P(\theta\mid\gamma,M)\,d\theta,

where \(D\) collects the ancestry-specific summary statistics and LD matrices, \(\gamma\) denotes a causal configuration, and \(\theta\) denotes its effect sizes. The posterior probability of a model is proportional to its marginal likelihood times its prior probability. Thus the evidence depends on both fit and the stated prior, not on the association statistics alone.

Shared Variant Is Not the Same as Shared Effect

For a simple comparison, assume exactly one causal variant per dataset. Under a shared-variant model, all datasets have the same causal index \(j\), but their effect sizes may be distinct, such as \(\beta_{aj}\sim N(0,W)\) independently across datasets. Under an ancestry-specific-variant model, each dataset has its own index \(j_a\). These are different scientific and statistical restrictions.

A stricter model can instead require a common effect size at the shared variant. That restriction is not implied by sharing the causal index. It can be reasonable in some applications, but should be justified rather than silently imposed. Similarly, an ancestry-specific configuration does not establish that causal biology differs: weak evidence, different sample sizes, LD, or model misspecification can all affect the comparison.

The following short calculation shows why independent datasets contribute multiplicatively under a shared-variant configuration. For a fixed candidate \(j\), let \(BF_{aj}\) be the Bayes factor in dataset \(a\) for the hypothesis that \(j\) is causal, versus the null, after integrating over that dataset's effect prior. Conditional independence gives

\frac{P(D_1,\ldots,D_A\mid j\text{ causal in all datasets})} {P(D_1,\ldots,D_A\mid \text{null in all datasets})} = \prod_{a=1}^{A} \frac{P(D_a\mid j\text{ causal})}{P(D_a\mid\text{null})} = \prod_{a=1}^{A}BF_{aj}.

The equality follows by factoring both numerator and denominator into dataset-specific terms. It would not follow if the datasets were dependent and that dependence were ignored. With a prior weight \(w_j\) on each shared candidate, the shared-model evidence relative to the joint null is \(\sum_j w_j\prod_a BF_{aj}\). For independently selected ancestry-specific variants with weights \(w_{aj}\), the corresponding evidence is \(\prod_a\sum_j w_{aj}BF_{aj}\). The different order of summation and multiplication expresses the different configuration assumptions.

Worked Example: Comparing Shared and Dataset-Specific Variant Models

Consider two hypothetical independent datasets and two candidate variants. Assume exactly one causal variant in each dataset, equal prior probability \(1/2\) for each candidate, and independent effect priors across datasets. To calculate the Bayes factors, use the single-variant approximation \(z\mid\theta\sim N(\theta,1)\), \(\theta\sim N(0,V)\), with \(V=1\). The integrated Bayes factor against the null is

BF(z)=\frac{1}{\sqrt{1+V}}\exp\!\left(\frac{z^2V}{2(1+V)}\right).

The hypothetical statistics are \(z=3.0\) and \(z=2.0\) for variants 1 and 2 in dataset A, and \(z=2.5\) and \(z=2.8\) in dataset B. With \(V=1\), \(BF(z)=2^{-1/2}\exp(z^2/4)\). Thus \(BF_{A1}=2^{-1/2}\exp(2.25)=6.708\), \(BF_{A2}=2^{-1/2}\exp(1)=1.922\), \(BF_{B1}=2^{-1/2}\exp(1.5625)=3.373\), and \(BF_{B2}=2^{-1/2}\exp(1.96)=5.020\).

For a shared causal variant, the evidence relative to the joint null is

\frac{1}{2}(BF_{A1}BF_{B1}+BF_{A2}BF_{B2}) = \frac{1}{2}\bigl((6.708)(3.373)+(1.922)(5.020)\bigr) \approx 16.14.

For independently selected causal variants, the evidence is

\left(\frac{6.708+1.922}{2}\right) \left(\frac{3.373+5.020}{2}\right) \approx (4.315)(4.197)=18.11.

The Bayes factor for shared versus dataset-specific variants is therefore \(16.14/18.11\approx0.891\). If the prior odds for the shared model versus the dataset-specific model are \(2:1\), the posterior odds are \(2(0.891)=1.782\), giving posterior model probability \(1.782/(1+1.782)\approx0.641\) for the shared model. This modest preference comes from the specified likelihoods and priors; it is not a probability that either model is true in an absolute sense.

LD Can Change the Association Pattern

The model's LD matrices matter because the same causal effect can produce different expected marginal statistics. In the approximation above, if only variant 1 has effect \(\beta\), the expected \(z\)-vector is \(\sqrt{N_a}\beta\) times the first column of \(R_a\). A high correlation between variants 1 and 2 can therefore create a strong statistic at variant 2 even when variant 1 is causal.

Worked Example: One Causal Variant, Different LD Patterns

Suppose variant 1 has standardized effect \(\beta=0.02\) in two hypothetical datasets, each with \(N=10{,}000\), and variant 2 is noncausal. For two variants, take the LD matrices to be

R_A= \begin{pmatrix} 1&0.9\\ 0.9&1 \end{pmatrix}, \qquad R_B= \begin{pmatrix} 1&0.2\\ 0.2&1 \end{pmatrix}.

Both matrices are valid correlation matrices: their eigenvalues are \(1+r\) and \(1-r\), which are positive for \(r=0.9\) and \(r=0.2\). With \(\beta_A=\beta_B=(0.02,0)^\mathsf{T}\) and \(\sqrt{N}\beta=100(0.02)=2\), the expected statistics are

E(z_A)\approx 2 \begin{pmatrix}1\\0.9\end{pmatrix} = \begin{pmatrix}2\\1.8\end{pmatrix}, \qquad E(z_B)\approx 2 \begin{pmatrix}1\\0.2\end{pmatrix} = \begin{pmatrix}2\\0.4\end{pmatrix}.

The causal variant is the same in both datasets, while the expected association at the correlated noncausal variant differs substantially. These are model-based expectations, not observed data or a guarantee about realized statistics. The example shows why using a single LD matrix for all datasets can distort the likelihood and the resulting posterior.

Effect Heterogeneity Under a Shared Variant

Once a candidate causal index is considered, effect-size assumptions can be compared separately. For example, under a common-effect model, estimates \(y_a\) at that variant have a shared latent effect \(\beta\sim N(0,\tau^2)\); under an independent-effects model, each \(\beta_a\sim N(0,\tau^2)\) independently. Assume \(y_a\mid\beta_a\sim N(\beta_a,s_a^2)\), with known standard errors, and independent measurement errors. Integrating the effects gives different covariance structures for the estimates.

Worked Example: Shared Variant, Different Effect Sizes

At a hypothetical candidate variant, suppose two aligned effect estimates are \(y_1=0.20\) and \(y_2=-0.10\), each with standard error \(0.10\). Let \(\tau=0.10\). Under a shared effect, the vector \(y\) has covariance \(\tau^2\mathbf{1}\mathbf{1}^{\mathsf T}+0.01I\); under independent effects it has covariance \((\tau^2+0.01)I\). Rescaling \(y\) by \(0.10\) gives \(x=(2,-1)^\mathsf{T}\) and covariance matrices

\Sigma_{\mathrm{common}}= \begin{pmatrix}2&1\\1&2\end{pmatrix}, \qquad \Sigma_{\mathrm{independent}}= \begin{pmatrix}2&0\\0&2\end{pmatrix}.

For zero-mean multivariate normal densities, the ratio of common-effect to independent-effect marginal likelihoods is the square root of the determinant ratio multiplied by the exponential of minus one half the difference in quadratic forms. The determinants are \(3\) and \(4\). Also, \(\Sigma_{\mathrm{common}}^{-1}=\frac13\begin{pmatrix}2&-1\\-1&2\end{pmatrix}\), so \(x^\mathsf{T}\Sigma_{\mathrm{common}}^{-1}x=14/3\); the independent quadratic form is \(x^\mathsf{T}x/2=5/2\). Therefore,

\frac{P(y\mid\text{common effect})}{P(y\mid\text{independent effects})} = \sqrt{\frac{4}{3}}\exp\!\left[-\frac12\left(\frac{14}{3}-\frac52\right)\right] \approx 0.391.

Conditional on this variant and these priors, the estimates favor independent effects over a common effect by about \(1/0.391=2.56\) to 1 in marginal likelihood. This comparison concerns effect-size sharing, not whether the causal variant is shared. The opposite-signed estimates may also prompt checks of allele alignment and study compatibility before interpreting heterogeneity.

Assumptions, Interpretation, and a Common Pitfall

A multi-ancestry posterior is conditional on the model. At minimum, the analyst should state the causal-count assumptions, effect-size priors, prior weights on configurations, summary-statistic likelihood, and whether effects are shared or ancestry-specific. LD matrices should correspond to the datasets whose statistics are modeled, and the variants and alleles must be matched correctly. Differences in sample size affect precision, while differences in LD affect the association pattern. Neither difference alone establishes ancestry-specific causality.

A common mistake is to pick the variant with the largest \(z\)-statistic in each dataset and call those variants ancestry-specific causal variants. That reasoning ignores LD: a correlated proxy can have a larger marginal statistic than the causal variant, and its rank can change when LD changes. The correction is to compare candidate configurations through the joint likelihood using each dataset's LD matrix, then report posterior evidence conditional on the model and priors. A high posterior inclusion probability is evidence under that model, not proof of a causal mechanism.

Key takeaway: Combine independent ancestry-specific evidence by multiplying the appropriate likelihood contributions, but preserve each dataset's LD structure. Specify separately whether the causal index and the effect size are shared; these are distinct modeling assumptions.

Check Your Understanding

Answer each question using the model assumptions stated in the tutorial.

Why does a shared-variant configuration multiply dataset-specific Bayes factors for a fixed candidate?
In the two-variant example, why did the expected statistic at the noncausal variant differ between datasets?
Does a shared causal variant require the same effect size in every ancestry-specific dataset? Explain.
What change to the likelihood is needed if participants overlap across datasets?
Why can the variant with the largest marginal \(z\)-statistic differ from the causal variant?