Advancing Randomized Response: Logistic Regression with Zero-Truncated Negative Binomial Design

Abstract

Sensitive survey research often suffers from non-response and misreporting due to privacy concerns. Randomized Response Techniques (RRTs) provide a mechanism to elicit truthful answers while preserving confidentiality, but existing designs face challenges when data are heterogeneous or overdispersed. This paper introduces a new extension of the Zero-Truncated Negative Binomial (ZTNB) randomization device to logistic regression with covariates. The proposed framework integrates the ZTNB distribution into the logistic likelihood, thereby accommodating overdispersion while maintaining unbiased estimation of regression parameters. Theoretical results establish the consistency and asymptotic normality of the maximum likelihood estimators. A comprehensive simulation study evaluates the finite-sample properties of the method under varying coefficient settings and sample sizes. Results demonstrate negligible bias, close agreement between empirical and model-based variability, and coverage probabilities near the nominal 95% level. Compared to the Zero-Truncated Poisson device, the ZTNB approach exhibits improved stability and precision, particularly in small to moderate samples. These findings confirm the ZTNB logistic regression model as a flexible and efficient tool for analyzing sensitive survey data, expanding the scope of RRT-based inference.