where ρ ( S, T ) is the correlation between S and T , then the expected proportion of cases in the population conditional on the allele carried at SNP T is and the relative risk is D$_{1}$ D$_{0}$ . For a rare disease such as T1D, relative risks and odds ratios are approximately equal. Bayesian model averaging requires evaluating all possible multiple SNP models in each trait, and conducting colocalisation testing for each model. We began by defining the posterior probability of model j for both traits as, where π i j is the posterior probability of model j for trait i and a model j indicates which SNPs are included in the model. Even when the number of SNPs to be tested is fixed at two, the number of possible models is p ! 2!( p − $_{2)!}$. Whilst testing all models is feasible if computationally expensive for analysis of real data, it is impractical for simulations. To reduce the computation burden, we first evaluated all p single SNP models and indentified the set of SNPs with very low posterior probability ( π$_{j}$ < 0 . 01). We then excluded any two SNP model containing only SNPs from this set. If p$_{0}$ < p such SNPs were identified, this reduced the number of models to test to p ! 2!( p − 2)! − p$_{0}$ ! 2!( p − $_{2)!}$. For the purposes of simulation, we used the profile likelihood approach to generate a χ 2 1 dis- tributed test statistic and averaged the resulting p values, P$_{j}$ , over the model space to calculate an overall posterior predictive p value, ∑ j P$_{j}$π$_{j}$ . For the application to AITD, we integrated the p value associated with the χ 2 distributed test statistic calculated assuming η was known over both the posterior distribution of η given each model, and the posterior of the model space. The latter is formally correct, but comptationally too expensive for simulation, and the profile likelihood p value and the posterior predictive p value have been shown to be very similar for large samples. 32 π$_{j}$ = π 1 $_{j}$π 2 j ∑ k π 1 $_{k}$π 2 k        D$_{1}$ = r ( π$_{S}$π$_{T}$ + ρ $^{′}$) + ((1 − π$_{S}$ ) π$_{T}$ − ρ $^{′}$) π$_{T}$ T is risk allele D$_{0}$ = r ( π$_{S}$ (1 − π$_{T}$ ) − ρ $^{′}$) + ((1 − π$_{S}$ )(1 − π$_{T}$ ) + ρ $^{′}$) 1 − π$_{T}$ T is protective allele Implementation of Bayesian Model Averaging