In educational and psychological research, the logit and probit links are often used to fit the binary item response data. The appropriateness and importance of the choice of  links within the item response theory (IRT) framework has not been investigated yet. In this paper, we present a family of  IRT models with generalized logit links, which include the traditional logistic and normal ogive models as special cases. This family of models are flexible enough not only to adjust the item characteristic curve tail probability by two shape parameters but also to allow us to fit the same link or different links to different items within the IRT model framework. In addition, the proposed models are implemented in the Stan software to sample from the posterior distributions. Using readily available Stan outputs, the four Bayesian model selection criteria are computed for guiding  the choice of the links within the IRT model framework. Finally, a new Bayesian ranking method is introduced to rank item difficulties according to the posterior probabilities. A detailed analysis of the real reading assessment data is carried out to illustrate the proposed methodology.