Modeling Bivariate Binary Data

The Bernoulli distribution is a very important discrete distribution with extensive applications to real-life problems. This distribution can be linked with univariate distributions such as binomial, geometric, negative binomial, Poisson, gamma, hypergeometric, exponential, normal, etc., either as a limit or as a sum or other functions. On the other hand, some distributions can be shown to arise from bivariate Bernoulli distribution as well. Since the introduction of the generalized linear model and generalized estimating equations, we observed a very rapid increase in the use of linear models based on binary outcome data. However, as the generalized linear models are proposed only for univariate outcome data and GEE is based on the marginal model, the utility of bivariate relationship cannot be explored adequately. It may be noted here that repeated measures data comprise of two types of associations: (i) association between outcome variables, and (ii) association between explanatory variables and outcome variables. Hence, correlated outcomes pose difficulty in estimating parameters of models for outcome and explanatory variables. In this chapter, regression models for correlated binary outcomes are introduced. A joint model for bivariate Bernoulli is obtained by using marginal and conditional probabilities using two approaches. In the first approach, estimates are obtained using the traditional likelihood method and the second approach provides a generalized bivariate binary model by extending the univariate generalized linear model for bivariate data. Tests for independence and goodness of fit of the model are shown. Section 6.2 reviews the bivariate Bernoulli distribution and defines the joint mass function in terms of conditional and marginal probabilities. Section 6.3 introduces the covariate dependence and shows the logit functions for both conditional and marginal probabilities. The likelihood function and estimating equations are shown. Some measures of dependence in outcomes as well as tests for model, parameters, and dependence are presented in Sect. 6.4. A recently introduced generalized bivariate Bernoulli model is discussed in Sect. 6.5. In this section, the bivariate Bernoulli mass function is expressed in an exponential family of distributions and link functions are obtained for correlated outcome variables as well as for association between two outcomes. Estimating equations are shown using a bivariate generalization of GLM and test for dependence is discussed. Section 6.6 summarizes some alternative procedures for binary repeated measures data. Examples are displayed in Sect. 6.7.