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Posterior Probability of Full Colocalization (PPFC)

Summary

The Posterior Probability of Full Colocalization (PPFC) is a Bayesian metric used in multi-trait statistical colocalization to quantify the probability that a set of $m$ genetic traits share a single causal variant in a specified genomic locus. It is the primary metric computed by HyPrColoc to identify shared genetic risk factors across multiple traits.

Mathematical Formulation

The exact posterior probability of full colocalization ($H_m$, where all $m$ traits share a causal variant in a region with $Q$ variants) is defined as: $$\text{PPFC} = P(H_m | D) = \frac{\sum_{S \in \mathcal{S}_m} P(D|S) \times p(S)}{p(D)}$$ where: - $D$ is the combined summary statistics for all $m$ traits. - $\mathcal{S}_m$ is the set of configurations where all $m$ traits share a causal variant. - $P(D|S)$ is the likelihood of the data given configuration $S$ (computed using Wakefield's Approximate Bayes Factors or Joint ABF for correlated studies). - $p(S)$ is the prior probability of configuration $S$. - $p(D)$ is the marginal likelihood across all possible configurations.

HyPrColoc Approximation

Calculating $p(D)$ exactly requires evaluating $(Q+1)^m$ configurations, which is computationally prohibitive for $m > 4$. HyPrColoc overcomes this by approximating the PPFC as: $$\widehat{\text{PPFC}} = P_R P_A$$ where: - Regional Association Probability ($P_R$): The posterior probability that all traits share an association with one or more variants in the region: $$P_R = \frac{P(H_m | D)}{P_{\text{scv}}}$$ (with $P_{\text{scv}}$ being the sum of posterior probabilities for configurations with at most one causal variant across the traits). - Alignment Probability ($P_A$): The posterior probability that the shared associations align at a single causal variant: $$P_A = \frac{P(H_m | D)}{P(H_m | D) + P_{(m-1,1)}}$$ (with $P_{(m-1,1)}$ representing the probability that a subset of $m-1$ traits share a causal variant and the remaining trait has a distinct causal variant).

This approximation reduces the computational complexity to $O(mQ^2)$ (as of 2021-02-03).

Citations

  • Foley, C. N., Staley, J. R., Breen, P. G., Sun, B. B., Kirk, P. D. W., Burgess, S., & Howson, J. M. M. (2021). A fast and efficient colocalization algorithm for identifying shared genetic risk factors across multiple traits. Nature Communications, 12(1), 764. DOI: 10.1038/s41467-020-20885-8. Source paper: s41467-020-20885-8.pdf