We optimized this function using the NLopt C++ package's implementation of the limited-memory BFGS algorithm [38], a quasi-Newton method that only requires the objective and the gradient as input [39]. As stated previously, we fix the non-centrality parameters, , and only optimize over due to the fact that our model would be over-specified otherwise. Specifically, we set the non-centrality parameters at each SNP to the observed Z-score if the absolute Z-score is greater than 3.7 (corresponding to a p-value of 10e-4) or the sign of the observed Z-score times 3.7 otherwise. Simulation results show that our strategy yields high accuracy to detect causal variants among several simulated approaches to approximate (Figure S5).
