Poisson and negative binomial item count techniques for surveys with sensitive question

Abstract

Although the item count technique is useful in surveys with sensitive questions, privacy of those respondents who possess the sensitive characteristic of interest may not be well protected due to a defect in its original design. In this article, we propose two new survey designs (namely the Poisson item count technique and negative binomial item count technique) which replace several independent Bernoulli random variables required by the original item count technique with a single Poisson or negative binomial random variable, respectively. The proposed models not only provide closed form variance estimate and confidence interval within [0, 1] for the sensitive proportion, but also simplify the survey design of the original item count technique. Most importantly, the new designs do not leak respondents' privacy. Empirical results show that the proposed techniques perform satisfactorily in the sense that it yields accurate parameter estimate and confidence interval.

Optimal Allocation of the Sample in the Poisson Item Count Technique

“…Although ML estimation via the EM algorithm (3)- (5) is strongly advocated in the work of Tian et al (2014), no particular analysis for the optimal allocation of the sample size based on ML estimation via the EM algorithm is conducted in the original paper. This analysis will be provided in the next sections of the presented paper.…”

“…When the variance formula for the method of moments estimator of the sensitive proportion is used, the problem of optimal allocation of the sample is quite simple and the optimal allocation can be obtained straightforwardly by minimising formula (2), which was done in Tian et al (2014):…”

“…The optimal allocation based on the variance formula for the moment estimator of the sensitive proportion was analysed in Tian et al (2014), which resulted in the conclusion that a balanced sample is a reasonable choice. In this paper, we analyse the optimal allocation of the sample size based on ML estimation.…”

“…questions about tax evasions, atypical sexual behaviours, bribes, etc. This paper refers to the new indirect method introduced by Tian et al (2014) who propose to randomly assign each respondent in the sample to either control or treatment groups and apply the following procedure. In the control group, respondents are asked one neutral question, e.g.…”

“…This paper refers to the new indirect method introduced by Tian et al (2014) and examines the optimal allocation of the sample to control and treatment groups. If determining the optimal allocation is based on the variance formula for the method of moments (difference in means) estimator of the sensitive proportion, the solution is quite straightforward and was given in Tian et al (2014). However, maximum likelihood (ML) estimation is known from much better properties, therefore determining the optimal allocation based on ML estimators has more practical importance.…”