A covariate nonrandomized response model for multicategorical sensitive variables

The diagonal model (DM) is a recently published nonrandomized response (NRR) survey method to collect data on categorical sensitive characteristics Y ∗. Based on DM data, the distribution of Y ∗ can be estimated. In contrast to randomized response (RR) techniques, NRR schemes avoid the use of a randomization device. Due to this fact, survey complexity and study costs decrease. In this article, we assume that not only Y ∗, but also nonsensitive characteristics X∗ 1 , ..., X ∗ p are involved in the survey. Then, the aim of this paper is to provide methods to investigate the dependence of Y ∗ on X∗ = (X∗ 1 , ..., X ∗ p ). For instance, the influence of gender and profession on income (recorded in income classes) may be under study. In particular, we describe two estimation procedures: Stratum-wise estimation and LR-DM estimation. Stratum-wise estimation is suitable if only few covariate levels appear in the sample. LR-DM estimation is based on a logistic regression model for the relation between Y ∗ and X∗ and requires several techniques for generalized linear models (e.g., Fisher scoring). In simulations, we first investigate the convergence behavior of the Fisher scoring algorithm. Subsequently, we illustrate the connection between efficiency of the LR-DM estimation and the degree of privacy protection. Finally, the efficiency of the LR-DM estimation is compared with the efficiency of the stratum-wise estimation.