Multivariate Modelling of Binary Responses with Normal and Non–Normal Random Effects

In different fields of study, multivariate binary data is often found, especially when several different qualitative characteristics or attributes are measured in the same unit or from the same person. These bivariate or multivariate responses observed from the same individual or a unit are likely to be correlated. This study aimed to evaluate the influence on the regression estimates of the parameters when binary responses are modeled jointly. The correlation between binary outcomes was captured by incorporating random effects. Normal and bridge distributions were assumed for the random effects. A simulation study was performed to illustrate the impact on the marginal parameter estimates of the joint response model when using the bridge and normal distributions for the random effects. The simulation study revealed that the joint model with either normal or bridge random effects provides a better gain in efficiency in the parameter estimates compared to the individual models which assume responses are independent. Furthermore, the parameter estimates of the joint model are more or less the same under the normal distribution and bridge distribution of the random effects when outcomes are correlated. However, slight differences are noted in the standard errors of the parameter estimates. In addition, when two outcomes are not correlated there is no gain in the fitting joint model over separate univariate models. Finally, these methods were applied to the Bangladesh Demographic and Health Survey 2011 (BDHS 2011) data.