Eliciting priors and relaxing the single causal variant assumption in colocalisation analyses

Abstract

Horizontal integration of summary statistics from different GWAS traits can be used to evaluate evidence for their shared genetic causality. One popular method to do this is a Bayesian method, coloc, which is attractive in requiring only GWAS summary statistics and no linkage disequilibrium estimates and is now being used routinely to perform thousands of comparisons between traits. Here we show that while most users do not adjust default software values, misspecification of prior parameters can substantially alter posterior inference. We suggest data driven methods to derive sensible prior values, and demonstrate how sensitivity analysis can be used to assess robustness of posterior inference. The flexibility of coloc comes at the expense of an unrealistic assumption of a single causal variant per trait. This assumption can be relaxed by stepwise conditioning, but this requires external software and an LD matrix aligned to study alleles. We have now implemented conditioning within coloc, and propose a new alternative method, masking, that does not require LD and approximates conditioning when causal variants are independent. Importantly, masking can be used in combination with conditioning where allelically aligned LD estimates are available for only a single trait. We have implemented these developments in a new version of coloc which we hope will enable more informed choice of priors and overcome the restriction of the single causal variant assumptions in coloc analysis.

Fig 1. Each hypothesis for coloc analysis H₀…H₄ may be enumerated by configurations, one configuration per row shown grouped by hypothesis.

Each circle in this figure represents one of n genetic variants, and is shaded orange if causal for trait 1, blue if causal for trait 2. There are different numbers of configurations for each hypothesis, depending on the number of SNPs in a region, and the prior is set according to three prior probabilities so that all configurations within a hypothesis are equally likely.

Fig 2. Effects of varying p₁₂ on the prior for H₄ (coloured lines) compared to H₃ (dashed line) as a function of the number of SNPs in the region.

For all plots p1 = p2 = 10⁻⁴ is constant. The coloured squares highlight points P(H₃) = P(H₄) for different p₁₂.

Fig 3. Determining plausible priors q₁, q₂.

a q_. estimated for eQTLs as the ratio of estimated number of LD-independent significant eQTL variants divided by number of SNPs considered for an eQTL analysis in GTeX whole blood samples in successively larger windows around a gene TSS. Separate lines show findings in 5 equal groups of MAF, with the top and bottom groups labelled. b The number of hits claimed per study according to the GWAS catalog. q_. could be estimated as number of hits / number of common SNPs (∼ 2, 000, 000). c Posterior probability of association at a single SNP as a function of -log10 p values for varying values of q_.. We considered both case/control and quantitative trait designs, and a range of MAF (0.05-0.5) and sample size (2000,5000,10000). The relationship between -log10 p (x axis) and posterior probability of association (y axis) is consistent across all designs, affected only by the prior probability of association (q₁, q₂). The vertical line indicates p = 5 × 10⁻⁸, the conventional genome-wide significance threshold in European populations.

Fig 4. Distribution of expected posterior probabilities across a wide range of simulated data.

In all analyses we fixed p₂ = p₁ = 10⁻⁴ and varied p₁₂. Coloured bar heights represent the average posterior probability for each hypothesis over the set of simulations for a given simulated hypothesis and sample size.

Fig 5. Example of sensitivity analysis on a dataset which shows evidence for colocalisation at a predefined rule of posterior P(H4) > 0.5 only when the prior beliefs in H3 and H4 are approximately equal.

The left hand panels show local Manhattan plots for the two traits, while the right hand panels show prior and posterior probabilities for H0-H4 as a function of p₁₂. The dashed vertical line indicates the value of p₁₂ used in initial analysis (the value about which sensitivity is to be checked). H₀ is omitted from the prior plot to enable the relative difference for the other hypotheses to be seen.

Fig 6. Masking as an alternative strategy to conditioning when attempting to colocalise trait signals with multiple causal variants in a region.

Top panel: input local Manhattan plots, with causal variants for each trait highlighted in red. We can use conditioning (left column) to perform multiple colocalisation analyses in a region. First, lead SNPs for each signal are identified through successively conditioning on selected SNPs and adding the most significant SNP out of the remainder, until some significance threshold is no longer reached. Then we condition on all but one lead SNP for each parallel coloc analysis. Note that when multiple lead SNPs are identified for each trait, eg n and m for traits 1 and 2 respectively, then n × m coloc analyses are performed. When an allele-aligned LD matrix is not available, an alternative is masking (right column) which differs by successively restricting the search space to SNPs not in LD with any lead SNPs instead of conditioning. Multiple coloc analyses are again performed, but setting the per SNP Bayes factor to 1 for hypotheses containing SNPs in LD with any but one of the lead SNPs. Note that for convenience of display, all SNPs in r² > α with the lead SNP are assumed to be in a contiguous block, shaded gray.

Fig 7. Average posterior probabilities for each hypothesis under different analysis strategies when trait 1 has two causal variants, A and B, and trait 2 has just one.

The left column shows the identity of causal variants for each trait and their relative effect sizes under four different models. The right column shows the average posterior that can be assigned to specific comparisons for of variants for trait 1: trait 2. We exploit our knowledge of the identity of the causal variants in simulated data to label each comparison according to LD between the lead SNP for each trait and the simulated causal variants. When labels cannot be unambiguously assigned (r² < 0.8 with any causal variant) we use “?”.

Fig 8. Average posterior probabilities for each hypothesis under different analysis strategies when both traits have two causal variants.

Information is displayed as described in Fig 7.