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Abstract: Binary regression models, such as Poisson and logistic regression, are commonly employed in clinical studies to estimate measures like relative risks (RR) or odds ratios (OR). While RR are preferred for their straightforward interpretation, logistic regression, which models OR, is the most widely used approach. However, it only provides a reliable estimate of RR when the outcome of interest is infrequent. Meanwhile, the Poisson regression can estimate RR directly but can produce fitted probabilities outside the range of zero and one. To address these challenges, Richardson et al. (2017) proposed a novel binary regression model that estimates RR directly via a log odds-product nuisance model. However, this method encountered challenges in high-dimensional and sparse data estimation (p > N). To address these issues, this study introduces an estimator founded on the binary regression model, which is further refined with an algorithm using Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to solve the optimization problem. This algorithm encourages sparsity in the solution and enables variable selection, thereby improving the model’s utility for high-dimensional and sparse data. Finally, we examine the properties of our estimator in a simulation study that focuses on cases when p > N.