We discuss study designs giving rise to a dichotomous outcome measured in each eye of a study population (e.g. presence of cataract, glaucoma, or diabetic retinopathy), either in a cross-sectional prevalence study, or after follow-up of a fixed time period in a prospective cohort study or clinical trial. We first consider studies with a single, dichotomous predictor variable, such as treatment assignment in a randomized trial, and distinguish situations where the eye and the person are the unit of randomization. Next, we describe methods with multiple independent variables, possibly including both continuous and categorical characteristics. A main focus will be on fitting and interpretation of logistic regression models that appropriately account for the correlation between the outcome in the two eyes of the same person. Real data examples will consider both eye-specific and person-specific covariates as well as the situation when some subjects contribute information from a single eye.