test  generating p values P$_{j}$ . The evidence against the null hypothesis (1) is then measured by the rank of P$_{k}$ ′ in the empirical distribution  [ P$_{j}$ ]  generated. This effectively compares the p value at the test SNP k conditional on SNP k ′ to that conditioning on all other SNPs in the region. However  note that because this method summarizes evidence for colocalisation by a rank only  there is no statistical inference attached. Thresholds for interpreting ranks would be expected to depend on SNP density and LD patterns. The proportional approach frames the null hypothesis differently. A set of q SNPs are chosen which are deemed somehow to jointly be good predictors of one or both traits. Regressing Y and Y ′ against these columns of X and X ′ respectively produces estimates  b$_{1}$ and b$_{2}$   of regression coefficients β$_{1}$ and β$_{2}$   with variance-covariance matrices V$_{1}$ and V$_{2}$ respectively. Since sample sizes are large  the combined likelihood may be closely approximated by a Gaussian likelihood for ( b$_{1}$   b$_{2}$ )  assuming V$_{1}$   V$_{2}$ are known and that Cov( b$_{1}$   b$_{2}$ ) = 0. Assuming equal LD in the two cohorts  i.e. that the correlation structure between the SNPs does not differ  Plagnol et al. [2009] show that the regression coefficients should be proportional and proposed testing for a shared causal variant by testing the null hypothesis i.e. β$_{1}$ = 1 $_{η}$β$_{2}$ = β . The chi-squared statistic is derived from Fieller’s theorem [Fieller  1954]  where u = $^{(}$b$_{1}$ − 1 $_{η}$b$_{2}$ ) and V = V$_{1}$ + 1 η $_{2}$V$_{2}$ . If η were known  T ( η ) 2 would have a χ 2 distribution on q degrees of freedom. Plagnol et. al take a profile likelihood approach and replace η by its maximum likelihood estimate  ˆ η   which also minimises T ( η ) $^{2}$. Asymptotic likelihood theory suggests that T (ˆ η ) 2 has a χ 2 distribution on q − 1 degrees of freedom. Alternatively  Wallace et al. [2012] take a Bayesian approach. They begin by reparametrising the likelihood in terms of θ = tan − $^{1}$( η ) and rewriting the null hypothesis as H prop 0 : β$_{1}$ ∝ β$_{2}$   T ( η ) 2 = u $^{T}$V − $^{1}$u ∼ χ 2 (2) H prop 0 : β$_{1}$ = β cos( θ ); β$_{2}$ = β sin( θ ) . 5