Modeling paired binary data by a new bivariate Bernoulli model with flexible beta kernel correlation

Paired binary data often appear in studies of subjects with two sites such as eyes, ears, lungs, kidneys, feet and so on. Three popular models [i.e., (Rosner in Biometrics 38:105-114, 1982) R model, (Dallal in Biometrics 44:253-257, 1988) model and (Donner in Biometrics 45:605-661, 1989) model] were proposed to fit such twin data by considering the intra-person correlation. However, Rosner’s R model can only fit the twin data with an increasing correlation coefficient, Dallal’s model may incur the problem of over–fitting, while Donner’s model can only fit the twin data with a constant correlation. This paper aims to propose a new bivariate Bernoulli model with flexible beta kernel correlation (denoted by $$\hbox {Bernoulli}_2^{\textrm{bk}}$$ Bernoulli 2 bk ) for fitting the paired binary data with a wide range of group–specific disease probabilities. The correlation coefficient of the $$\hbox {Bernoulli}_2^{\textrm{bk}}$$ Bernoulli 2 bk model could be increasing, or decreasing, or unimodal, or convex with respect to the disease probability of one eye. To obtain the maximum likelihood estimates (MLEs) of parameters, we develop a series of minorization–maximization (MM) algorithms by constructing four surrogate functions with closed–form expressions at each iteration of the MM algorithms. Simulation studies are conducted, and two real datasets are analyzed to illustrate the proposed model and methods.