Bayesian non-randomized response models for surveys with sensitive questions

Three non-randomized response (NRR) models (namely, the non-randomized triangular, crosswise and hidden sensitivity models) have recently been developed for analyzing dichotomous sensitive questions. Unlike existing randomized response (RR) models, no randomizing device is required for NRR models. This helps to reduce the cost, increase the efficiency, ensure the reproducibility, widen the applicability and encourage the cooperation. However, in applications (e.g., estimating the proportion of a rare sensitive attribute in a population) with highly skewed likelihood functions, classical asymptotic methods based on maximum likelihood estimates and their asymptotic standard errors may not be adequate. The purposes of this article are two folds. First, we develop Bayesian approaches for analyzing dichotomous sensitive questions based on the aforementioned NRR models. For both the non-randomized triangular and crosswise models, we obtain the exact posterior distribution and its explicit posterior moments, derive posterior mode via the EM algorithm and provide procedure for generating i.i.d. posterior samples. For the hidden sensitivity model, we consider Bayesian analysis under the commonly used conjugate Dirichlet prior. Second, noting that the covariance structure associated with the Dirichlet distribution is completely nonpositive, we propose three new joint priors for modeling independence structure with restrictions, negative correlation structure and positive correlation structure, respectively. A new hierarchical modeling strategy is provided. Importance sampling and data augmentation algorithm are employed to compute posterior moments and generate posterior samples. Three data sets from a sensitive sexual behavior study, an induced abortion study and a HIV study are used to illustrate the proposed methodologies.