# From GWAS Summary Statistics to Credible Sets

### Z scores to PPs

Maller et al. derive a method to calculate PPs from GWAS summary statistics (Supplementary text) from which the following is based on. Let $$\beta_i$$ for $$i=1,...,k$$ SNPs in a genomic region, be the regression coefficient from a single-SNP logistic regression model, quantifying the evidence of an association between SNP $$i$$ and the disease. Assuming that there is only one CV per region and that this is typed in the study, then if SNP $$i$$ is causal, $$\beta_i\neq 0$$ and $$\beta_j$$ (for $$j\neq i$$) is non-zero only through LD between SNPs $$i$$ and $$j$$. Note that no parametric assumptions are required for $$\beta_i$$ yet, so we write that it is sampled from some distribution, $$\beta_i \sim \text{[ ]}$$. The likelihood is then, $$$\begin{split} P(D|\beta_i\sim\text{[ ]},\text{ }i\text{ causal}) & = P(D_i |\beta_i\sim\text{[ ]},\text{ }i\text{ causal}) \times P(D_{-i}|D_i,\text{ }\beta_i\sim\text{[ ]},\text{ }i\text{ causal})\\ & = P(D_i |\beta_i\sim\text{[ ]},\text{ }i\text{ causal}) \times P(D_{-i}|D_i,\text{ }i\text{ causal})\,, \end{split}$$$

since $$D_{-i}$$ is independent of $$\beta_i$$ given $$D_i$$. Here, $$D$$ is the genotype data (0, 1 or 2 counts of the minor allele) for the entire genomic region and $$i$$ is a SNP in the region, such that $$D_i$$ and $$D_{-i}$$ are the genotype data at SNP $$i$$ and at the remaining SNPs in the genomic region, respectively.

Parametric assumptions can now be placed on SNP $$i$$’s true effect on disease. This is typically quantified as log odds ratio, and is assumed to be sampled from a Gaussian distribution, $$\beta_i\sim N(0,W)$$, where $$W$$ is chosen to reflect the researcher’s prior belief on the variability of the true OR. Conventionally $$W$$ is set to $$0.2^2$$, reflecting a belief that 95% of odds ratios range from $$exp(-1.96\times 0.2)=0.68$$ to $$exp(1.96\times 0.2)=1.48$$.

The posterior probabilities of causality for each SNP $$i$$ in an associated genomic region with $$k$$ SNPs can be calculated where, $$$PP_i=P(\beta_i \sim N(0,W),\text{ }i \text{ causal}|D)\,, \quad i \in \{1,...,k\}.$$$

Under the assumption that each SNP is equally likely to be causal, then $$$P(\beta_i \sim N(0,W),\text{ }i\text{ causal})=\dfrac{1}{k}\,, \quad i \in \{1,...,k\}$$$ and Bayes theorem can be used to write \begin{aligned} PP_i=P(\beta_i \sim N(0,W),\text{ }i \text{ causal}|D)\propto P(D|\beta_i\sim N(0,W),\text{ }i\text{ causal}). \end{aligned}

Dividing through by the probability of the genotype data given the null model of no genetic effect, $$H_0$$, yields a likelihood ratio, $$$PP_i\propto \dfrac{P(D|\beta_i \sim N(0,W),\text{ }i \text{ causal)}}{P(D|H_0)},$$$

from which Equation (1) can be used to derive, $$$PP_i\propto \frac{P(D_i|\beta_i \sim N(0,W),\text{ }i \text{ causal})}{P(D_i|H_0)}= BF_i\,,$$$ where $$BF_i$$ is the Bayes factor for SNP $$i$$, measuring the ratio of the probabilities of the data at SNP $$i$$ given the alternative (SNP $$i$$ is causal) and the null (no genetic effect) models.

In genetic association studies where sample sizes are usually large, these BFs can be approximated using Wakefield’s asymptotic Bayes factors (ABFs). Given that $$\hat\beta_i\sim N(\beta_i,V_i)$$ and $$\beta_i\sim N(0,W)$$,

$$$PP_i\propto BF_i \approx ABF_i=\sqrt{\frac{V_i}{V_i+W}}exp\left(\frac{Z_i^2}{2}\frac{W}{(V_i+W)}\right)\,,$$$ where $$Z_i^2=\dfrac{\hat\beta_i^2}{V_i}$$ is the squared marginal $$Z$$ score for SNP $$i$$.

In Bayesian fine-mapping, PPs are calculated for all SNPs in the genomic region and the variants are sorted into descending order of their PP. The PPs are then cumulatively summed until some threshold, $$\alpha$$, is exceeded. The variants required to exceed this threshold form the $$\alpha$$-level credible set.