Binary regression models utilizing logit or probit link functions have been extensively employed for examining the relationship between binary responses and covariates, particularly in medicine. Nonetheless, an erroneous specification of the link function may result in poor model fitting and compromise the statistical significance of covariate effects. In this study, we introduce a diagnostic method associated with a novel family of link functions enabling the assessment of sensitivity for symmetric links in relation to their asymmetric counterparts. This new family offers a comprehensive model encompassing nested symmetric cases. Our method proves beneficial in modeling medical data, especially when evaluating the sensitivity of the commonly used logit link function, prized for its interpretability via odds ratio. Moreover, our method advocates a general link based on the logit function when a standard link is unsatisfactory. We employ likelihood-based methods to estimate parameters of the general model and conduct local influence analysis under the case-weight perturbation scheme. Regarding local influence, we emphasize the relevance of employing appropriate perturbations to avoid misleading outcomes. Additionally, we introduce a diagnostic method for local influence, assessing the sensitivity of odds ratio under two perturbation schemes. Monte Carlo simulations are conducted to evaluate both the diagnostic method performance and parameter estimation of the general model, supplemented by illustrations using medical data related to menstruation and respiratory problems. The results confirm the efficacy of our proposal, highlighting the critical role of statistical diagnostics in modeling.