We propose in this paper a set-valued (SV) system model, which is a generalized form of logistic (LG) and Probit (Probit) regression, to be considered as a method for discovering genetic variants, especially rare genetic variants in next-generation sequencing studies, for a binary phenotype. We propose a new SV system identification method to estimate all underlying key system parameters for the Probit model and compare it with the LG model in the setting of genetic association studies. Across an extensive series of simulation studies, the Probit method maintained type I error control and had similar or greater power than the LG method, which is robust to different distributions of noise: logistic, normal, or t distributions. Additionally, the Probit association parameter estimate was 2.7-46.8-fold less variable than the LG log-odds ratio association parameter estimate. Less variability in the association parameter estimate translates to greater power and robustness across the spectrum of minor allele frequencies (MAFs), and these advantages are the most pronounced for rare variants. For instance, in a simulation that generated data from an additive logistic model with an odds ratio of 7.4 for a rare single nucleotide polymorphism with a MAF of 0.005 and a sample size of 2,300, the Probit method had 60% power whereas the LG method had 25% power at the α = 10(-6) level. Consistent with these simulation results, the set of variants identified by the LG method was a subset of those identified by the Probit method in two example analyses. Thus, we suggest the Probit method may be a competitive alternative to the LG method in genetic association studies such as candidate gene, genome-wide, or next-generation sequencing studies for a binary phenotype.