Beta-binomial ANOVA for multivariate randomized response data

There is much empirical evidence that randomized response methods improve the cooperation of the respondents when asking sensitive questions. The traditional methods for analysing randomized response data are restricted to univariate data and only allow inferences at the group level due to the randomized response sampling design. Here, a novel beta-binomial model is proposed for analysing multivariate individual count data observed via a randomized response sampling design. This new model allows for the estimation of individual response probabilities (response rates) for multivariate randomized response data utilizing an empirical Bayes approach. A common beta prior specifies that individuals in a group are tied together and the beta prior parameters are allowed to be cluster-dependent. A Bayes factor is proposed to test for group differences in response rates. An analysis of a cheating study, where 10 items measure cheating or academic dishonesty, is used to illustrate application of the proposed model.