In ophthalmologic or otolaryngologic studies, bilateral correlated data often arise when observations involving paired organs (e.g., eyes, ears) are measured from each subject. Based on the model of Donner (1989), in this paper, we focus on investigating the relationship between the disease probability and covariates (such as ages, weights, gender and so on) via the logistic regression for the analysis of bilateral correlated data. We first propose a new minorization-maximization} (MM) algorithm and a fast quadratic lower bound (QLB) algorithm to calculate the maximum likelihood estimates of the vector of regression coefficients, and then develop three large-sample tests (i.e., the likelihood ratio test, Wald test and score test) to test if covariates have a significant impact on the disease probability. Simulation studies are conducted to evaluate the performance of the proposed fast QLB algorithm and three testing methods. A real ophthalmologic data set in Iran is used to illustrate the proposed methods.