Investigating the association of a sensitive attribute with a random variable using the Christofides generalised randomised response design and Bayesian methods

In empirical studies involving sensitive topics, in addition to the problem of estimating the population proportion with a sensitive characteristic, a question arises as to whether or not there is heterogeneity in the distribution of an auxiliary random variable representing the information of subjects collected from a sensitive group and a non-sensitive group. That is, it is of interest to investigate the influence of sensitive attribute on the auxiliary random variable of interest. Finite mixture models are utilised to evaluate the association. A proposed Bayesian method through data augmentation and Markov chain Monte Carlo is applied to estimate unknown parameters of interest. Deviance information criterion and marginal likelihood are employed to select a suitable model to describe the association of the sensitive characteristic with the auxiliary random variable. Simulation and real data studies are conducted to assess the performance of and illustrate applications of the proposed methodology.