Bayesian estimation of rare sensitive attribute

Randomized response models have been used to estimate a population proportion of a sensitive attribute. A randomized device is typically employed to protect respondent's privacy in a survey. In addition, an unrelated question is asked to improve the statistical efficiency. In this article, we propose Bayesian estimation of rare sensitive attribute using randomized response technique, which includes a rare unrelated attribute. Two cases are considered, the proportion of a rare unrelated attribute is known and unknown. A simulation study is conducted to assess the performance of the models using mean absolute error and coverage probability. The results show that the performance depends on the parameters and is robust to priors.     

Efficiency comparison of unrelated question models based on same privacy protection degree

In this study, the problem of estimating the proportion π _A of people bearing a sensitive attribute A is considered. Three dichotomous unrelated question mechanisms which are alternative to the well-known Simmons’ model are discussed and their performance is evaluated taking into account both efficiency and respondent privacy protection. The variance of the estimators of π _A is compared under equal levels of confidentiality measures introduced by Lanke (1976) and Leysieffer and Warner (1976).     

Mean and sensitivity estimation in optional randomized response models

Optional randomized response models were introduced by Gupta et al. (2002). These models are based on the basic premise that a question may be sensitive for one respondent but may not be sensitive for another. In an optional RRT (randomized response technique) model, a respondent is asked to provide a scrambled response only if the respondent considers the question sensitive. Otherwise, the respondent provides a truthful response. The researcher does not know which type of response is provided. The proportion of respondents who provide a scrambled response is known as the sensitivity level of the question. In this paper, we estimate simultaneously the mean and the sensitivity level of a quantitative-response sensitive question using a two stage optional RRT model. The estimators are unbiased and asymptotically normally distributed. We discuss the advantages and disadvantages of using additive and multiplicative scrambling.     

A new quasi empirical bayes estimate in randomized response sampling

In this article, a new notion of “quasi-empirical” Bayes estimation is developed for estimating the proportion of a sensitive attribute in a population by making use of both a prior distribution of prevalence of the sensitive attribute in addition to the known prior distribution of an unrelated characteristic. The proposed quasi-empirical Bayes estimate is compared with those of the unrelated question model due to Greenberg et al. by means of a simulation study.     

Estimating sensitive proportions by Warner’s randomized response technique using multiple randomized responses from distinct persons sampled

In 1965 Warner pioneered randomized response techniques to estimate the proportion of people bearing a sensitive characteristic. He restricted applying his randomized response device to gather data on sensitive issues from respondents chosen by simple random sampling with replacement (SRSWR). It has spawned numerous ramifications. We present results for the situation where the distinct persons chosen in an SRSWR are identified but each one independently gives a randomized response by Warner’s device, repeated as many times as he/she is selected. Two new estimators are proposed for the sensitive proportion and compared against relevant competitors.     

Using randomized response from a complex survey to estimate a sensitive proportion in a dichotomous finite population

In estimating the proportion θ_A of people in a given community bearing a sensitive characteristic A, in order to protect the respondent's privacy, various techniques of generating randomized response (RR) rather than direct response are available in the literature. But the theory concerning them is developed only for samples selected by ‘simple random sampling’ (SRS) ‘with replacement’ or at most by SRS without replacement method. Illustrating two such RR devices we show how an estimator along with an estimated measure of its error may be developed if the sample of persons may be drawn adopting a complex survey design involving unequal selection probabilities with or without replacement.     

A new unrelated question randomized response model

In this paper, we extend the work of Gjestvang and Singh [A new randomized response model, J. R. Statist. Soc. Ser. B (Methodological) 68 (2006), pp. 523–530] to propose a new unrelated question randomized response model that can be used for any sampling scheme. The interesting thing is that the estimator based on one sample is free from the use of known proportion of an unrelated character, unlike Horvitz et al. [The unrelated question randomized response model, Social Statistics Section, Proceedings of the American Statistical Association, 1967, pp. 65–72], Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), pp. 520–539] and Mangat et al. [An improved unrelated question randomized response strategy, Calcutta Statist. Assoc. Bull. 42 (1992), pp. 167–168] models. The relative efficiency of the proposed model with respect to the existing competitors has been studied.     

Generalized geometric distribution of order k: A flexible choice to randomize the response

This article focuses on the improvement of a well-celebrated randomized response technique of Kuk. A generalized randomized response technique is suggested. In particular, the generalized geometric distribution of order k is introduced as a randomization device for estimating the population proportion of a rare sensitive attribute. The proposed randomized response technique includes Singh and Grewal and Hussain et al. techniques as its special cases. Through numerical illustrations, it is established that the suggested technique is superior to the Kuk, Singh and Grewal, and Hussain et al. techniques. Flexibility of the proposed technique is also discussed.     

A new class of ratio-type estimators for improving mean estimation of nonsensitive and sensitive variables by using supplementary information

ABSTRACT

Motivated by some recent improvements for mean estimation in finite sampling theory, we propose, in a design-based approach, a new class of ratio-type estimators. The class is initially discussed on the assumption that the study variable has a nonsensitive nature, meaning that it deals with topics that do not generate embarrassment when respondents are directly questioned about them. Under this standard setting, some estimators belonging to the class are shown and the bias, mean square error and minimum mean square error are determined up to the first-order of approximation. The class is subsequently extended to the case where the study variable refers to sensitive issues which produce measurement errors due to nonresponses and/or untruthful reporting. These errors may be reduced by enhancing respondent cooperation through scrambled response methods that mask the true value of the sensitive variable. Hence, four methods (say the additive, multiplicative, mixed and combined additive-multiplicative methods) are discussed for the purposes of the article. Finally, a simulation study is carried out to assess the performance of the proposed class by comparing a number of competing estimators, both in the sensitive and the nonsensitive setting.     

A modified unrelated question randomized response device for complex surveys

Singh et al. (Stat Trans 6(4):515–522, 2003) proposed a modified unrelated question procedure and they also demonstrated that the modified procedure is capable of producing a more efficient estimator of the population parameter π _A , namely, the proportion of persons in a community bearing a sensitive character A when π _A  < 0.50. The development of Singh et al. (Stat Trans 6(4):515–522, 2003) is based on simple random samples with replacement and on the assumption that π _B , namely, the proportion of individuals bearing an unrelated innocuous character B is known. Due to these limitations, Singh et al.’s (Stat Trans 6(4):515–522, 2003) procedure cannot be used in practical surveys where usually the sample units are chosen with varying selection probabilities. In this article, following Singh et al. (Stat Trans 6(4):515–522, 2003) we propose an alternative RR procedure assuming that the population units are sampled with unequal selection probabilities and that the value of π _B is unknown. A numerical example comparing the performance of the proposed RR procedure under alternative sampling designs is also reported.     

Estimation of sensitive data from a dichotomous population

This paper considers the problem of estimating the population proportion when the study variable is sensitive in nature. Two implicit unrelated question procedures are proposed, in which the unrelated character used can be chosen arbitrarily. In addition to unbiasedly estimating population proportion and variance, an empirical study is also carried out to examine the relative efficiency aspect of the proposed procedures. The new procedures are shown to be more efficient than the usual randomized response procedures.     

A comparison of two randomized response survey methods with consideration for the level of respondent protection

In recent years randomized response methods have been introduced in an attempt to improve the accuracy and honesty in personalized response surveys of very sensitive questions. Two randomized response methods are compared, taking into account the protection afforded the respondent. In addition, we point out that the estimators, which previous authors have claimed to be the maximum likelihood estimators of the population proportion with the sensitive characteristic, are in fact not the maximum likelihood estimators.     

Advances in estimation by the item sum technique using auxiliary information in complex surveys

To collect sensitive data, survey statisticians have designed many strategies to reduce nonresponse rates and social desirability response bias. In recent years, the item count technique has gained considerable popularity and credibility as an alternative mode of indirect questioning survey, and several variants of this technique have been proposed as new needs and challenges arise. The item sum technique (IST), which was introduced by Chaudhuri and Christofides (Indirect questioning in sample surveys, Springer-Verlag, Berlin, 2013) and Trappmann et al. (J Surv Stat Methodol 2:58–77, 2014), is one such variant, used to estimate the mean of a sensitive quantitative variable. In this approach, sampled units are asked to respond to a two-list of items containing a sensitive question related to the study variable and various innocuous, nonsensitive, questions. To the best of our knowledge, very few theoretical and applied papers have addressed the IST. In this article, therefore, we present certain methodological advances as a contribution to appraising the use of the IST in real-world surveys. In particular, we employ a generic sampling design to examine the problem of how to improve the estimates of the sensitive mean when auxiliary information on the population under study is available and is used at the design and estimation stages. A Horvitz–Thompson-type estimator and a calibration-type estimator are proposed and their efficiency is evaluated by means of an extensive simulation study. Using simulation experiments, we show that estimates obtained by the IST are nearly equivalent to those obtained using “true data” and that in general they outperform the estimates provided by a competitive randomized response method. Moreover, variance estimation may be considered satisfactory. These results open up new perspectives for academics, researchers and survey practitioners and could justify the use of the IST as a valid alternative to traditional direct questioning survey modes.     

Some New Results on the Multinomial Randomized Response Model

ABSTRACT

The randomized response technique is an effective survey method designed to elicit sensitive information while ensuring the privacy of the respondents. In this article, we present some new results on the randomization response model in situations wherein one or two response variables are assumed to follow a multinomial distribution. For a single sensitive question, we use the well-known Hopkins randomization device to derive estimates, both under the assumption of truthful and untruthful responses, and present a technique for making pairwise comparisons. When there are two sensitive questions of interest, we derive a Pearson product moment correlation estimator based on the multinomial model assumption. This estimator may be used to quantify the linear relationship between two variables when multinomial response data are observed according to a randomized-response protocol.     

Hybrid tripartite randomized response technique

In this paper, a new randomized response technique has been proposed that combines the unrelated question randomized response technique and tripartite randomized response technique. The relative efficiency of the newly proposed Randomized Response technique over the existing Randomized Response technique was obtained. It was observed that the relative efficiency of the proposed technique over the tripartite RRT increases with increase in π_A and π_U. The applicability of the proposed technique was also shown. However, it has been shown that the proposed technique performs better than the conventional ones.     

Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution

This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. depending on the parameters and the probability of selecting a question. We showed that our proposed methods have better efficiencies than Land et al.’s randomized response model in some conditions. When the sizes of stratified populations are not given, other estimators are suggested using a stratified double sampling. For the proportional allocation, the difference between two variances in the stratified sampling and the stratified double sampling is given with the known rare unrelated attribute.     

Simultaneous estimation of means of two sensitive variables

In this paper, we introduce a new problem of simultaneous estimation of means of two quantitative sensitive variables by using only one randomized response another pseudo response from a respondent in a sample. The proposed estimators are extended to stratified random sampling, and the relative efficiency values are computed for equal, proportional, and optimum allocation with respect to the newly introduced naïve estimators.     

Randomized response technique in a National Survey

In order to produce estimates of the number of women having abortions during a 12-month period in the conterminous United States, the randomized response technique was used in the 1973 National Survey of Family Growth. The model applied used 2 unrelated questions in separate half-samples, with a coin as the randomizing device. The randomized response technique resulted in a higher estimate for the number of women with abortions that has previously been obtained through direct questions or reporting systems. The overall estimated proportion who had abortions among women who had been married or who had their own children in the household is 3.0% with a standardized error of 0.8 percentage points; however, there is a wide variation in the half-sample estimates of abortion. Differences between the 2 half-samples led to an examination of possible measurement error. 3 types of errors in measurement which may affect the estimate based on the randomized response technique are: 1) error in the answer to the sensitive question on abortion under the randomized response conditions; 2) error in the answer to the innocuous question under randomized response conditions; and 3) error in the answer to the innocuous question when asked directly. Comparisons between data from the different sources for currently married women suggest that all differences are not due to measurement error and that a large number of women had an unreported and/or illegal abortion in 1973. Although the randomized response models have been in use for at least 10 years, there continues to be a need for work on the field administration and subsequent analysis of these models.     

数量特征敏感问题问卷设计

目前,数量特征敏感问题调查主要采用随机化策略,该策略需使用随机化装置,从而需要在现场实施。提出一种问卷设计技术,该技术用无关问题替代随机化装置,因而不需要调查者亲临现场,不受调查规模及调查单位聚散的限制,使得调查更加方便、实用、经济。给出了相应的无偏估计量,推算出估计量的方差和方差的估计量并举例说明。     

Combining answers to direct and indirect questions: An implementation of Kuk’s randomized response model

Abstract

In this article, we propose a two folded approach for estimation of the population proportion of a sensitive attribute. The rationale of proposed technique is to give respondents choice if they want to avail randomized device or not. Thus, our technique is inherently capable of entertaining the responses attained through direct questioning and by employing randomization device, as well. We believe that free to choose policy will be helpful to develop a curtsy between interviewer and respondent (which is highly desirable, especially when sensitive issue is under consideration) and thus enhances the survey reliability. The proposed technique is developed for simple random sampling, at first, and then extended to stratified random sampling. The superior performance of proposed technique in comparison to the Kuk (1990 Kuk, A. Y. 1990. Asking sensitive questions indirectly. Biometrika 77 (2):436–8. doi:10.1093/biomet/77.2.436.[Crossref], [Web of Science ®] , [Google Scholar]) method is demonstrated throughout this article. We provide algebraic, graphical and numerical evidences in support of our proposed technique. Furthermore, we also offer cost analysis considering most commonly used cost functions in the literature of survey research.