This article considers the statistical estimation and inference for screening studies in which two binary tests are used for screening with a binary disease status verified only for those subjects with at least one positive test result. The challenge encountered in these studies is the non-identifiability because the disease rate is not identifiable for subjects with negative results from both tests without additional assumptions. Different homogeneous association models have been proposed in the literature to circumvent the non-identifiability problem, which were solved using numerical methods. We propose to formulate the problem as a constrained maximum likelihood estimation (MLE) problem. The MLE has a closed-form in general, which can be solved using a unified two-stage estimation approach. We demonstrate the application of the proposed method on a set of homogeneous association models. The homogeneous association assumptions are generally not testable as all models are saturated. Therefore, we propose an association-ratio plot as a visualization tool for model comparisons. The methods are illustrated through three examples.