Poisson–Poisson item count techniques for surveys with sensitive discrete quantitative data

This paper proposes a new Poisson–Poisson item count technique (ICT), which is an extension of the existing Poisson ICT of Tian et al. (Stat Methods Med Res. doi: 10.1177/0962280214563345 , 2017) from estimating the proportion associated with a sensitive binary variable to estimating the Poisson mean associated with a sensitive discrete quantitative variable. The Poisson–Poisson ICT can be used to collect and analyze sensitive discrete quantitative data, where an independent non-sensitive Poisson random variable with mean parameter $$\lambda $$ λ is introduced to facilitate the data collection. Specifically, we first propose the single-trial and multiple-trial survey designs for the Poisson–Poisson ICT with known $$\lambda $$ λ and develop the corresponding statistical inference methods. Then, we compare the single-trial design with the multiple-trial design from the viewpoints of relative efficiency and the degree of privacy protection. Furthermore, we propose a new survey design for the Poisson–Poisson ICT with unknown $$\lambda $$ λ . The allocation of sample sizes in two groups is also discussed. Simulation studies are performed to illustrate the proposed methods. In addition, we also consider the regression model based on the Poisson–Poisson ICT with known $$\lambda $$ λ for the single-trial case. Finally, two real surveys on the gift-giving behavior in two different universities are conducted by using the proposed techniques.