Maximum likelihood estimation of sensitive proportion using repeated randomized response techniques

Randomized response techniques are designed to obtain usable data on sensitive issues while protecting the privacy of individuals. In this paper, based on repeating the randomized response technique, a new technique called repeated randomized response is introduced to increase the protection of privacy and efficiency of estimator for proportion of sensitive attribute. By using this technique, the proportion of academic cheating is estimated among students of Shahid Chamran University of Ahvaz, Ahvaz, Iran.     

Improved strategy to collect sensitive data by using geometric distribution as a randomization device

Abstract

In this article, we improvise Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) techniques by introducing a new two-stage randomization response process. Using the proposed new technique, we achieve better efficiency and increasing protection of privacy of respondents than the Kuk (1990 Kuk, A. Y. C. 1990. Asking sensitive questions indirectly. Biometrika 77 (2):436–8.[Crossref], [Web of Science ®] , [Google Scholar]), Singh and Grewal (2013 Singh, S., and I. S. Grewal. 2013. Geometric distribution as a randomization device implemented in the Kuk’s model. International Journal of Contemporary Mathematical Sciences 8:243–8.[Crossref] , [Google Scholar]) and Hussain et al. (2016 Hussain, Z., J. Shabbir, Z. Pervez, S. F. Shah, and M. Khan. 2016. Generalized geometric distribution of order k: A flexible choice to randomize the response. Communications in Statistics: Simulation and Computation 46:4708–21.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) models. The relative efficiency and protection of the respondents of the proposed two-stage randomization device have been investigated through simulation study, and the situations are reported where the proposed estimator performs better than its competitors. The SAS code used to investigate the performance of the proposed strategy are also provided.     

A stratified estimation of a sensitive character by using geometric distribution

Abstract

In this paper, we consider the estimation of a sensitive character when the population is consisted of several strata; this is undertaken by applying Niharika et al.’s model which is using geometric distribution as a randomization device. A sensitive parameter is estimated for the case in which stratum size is known, and proportional and optimum allocation methods are taken into account. We extended the Niharika et al.’s model to the case of an unknown stratum size; a sensitive parameter is estimated by applying stratified double sampling to the Niharika et al.’s model. Finally, the efficiency of the proposed model is compared with that of Niharika et al. in terms of the estimator variance.     

A GENERALIZED TWO-STAGE RANDOMIZED RESPONSE PROCEDURE IN COMPLEX SAMPLE SURVEYS

The randomized response (RR) technique pioneered by Warner, S.L. (1965) [Randomised response: a survey technique for eliminating evasive answer bias. J. Amer. Statist. Assoc. 60 , 63–69] is a useful tool in estimating the proportion of persons in a community bearing sensitive or socially disapproved characteristics. Mangat, N.S. & Singh, R. (1990) [An alternative rendomized response procedure. Biometrika 77 , 439–442] proposed a modification of Warner's procedure by using two RR techniques. Presented here is a generalized two‐stage RR procedure and derivation of the condition under which the proposed procedure produces a more precise estimator of the population parameter. A comparative study on the performance of this two‐stage procedure and conventional RR techniques, assuming that the respondents' jeopardy level in this proposed method remains the same as that offered by the traditional RR procedures, is also reported. In addition, a numerical example compares the efficiency of the proposed method with the traditional RR procedures.     

An improved binary randomized response model using six decks of cards

This article purposes the estimation of population proportion of a sensitive attribute through randomized response technique. An efficient estimator is suggested using six decks of cards to randomise the response. Many existing models can now be viewed as the special case of the proposed model. The superiority of the proposed procedure is established through numerical calculation of percentage relative efficiency with prominent competitors. The proposed procedure is also studied under stratified random sampling protocol. In addition, it is shown that, the proposed stratified estimator, performs better in term of efficiency than its only existing two deck stratified competitor.     

An optional randomized response model for estimating a rare sensitive attribute using Poisson distribution

The crux of this article is to estimate the mean of the number of persons possessing a rare sensitive attribute based on the Mangat (1991 Mangat, N.S. (1991). An optional randomized response sampling technique using non–stigmatized attribute. Statistica. 51(4):595–602. [Google Scholar]) randomization device by utilizing the Poisson distribution in simple random sampling and stratified sampling. Properties of the proposed randomized response (RR) model have been studied along with recommendations. It is also shown that the proposed model is more efficient than that of Land et al. (2011 Land, M., Singh, S., Sedory, S.A. (2011). Estimation of a rare attribute using Poisson distribution. Statistics. 46(3):351–360.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in simple random sampling and that of Lee et al. (2013 Lee, G.S., Uhm, D., Kim, J.M. (2013). Estimation of a rare sensitive attribute in stratified sampling using Poisson distribution. Statistics. 47(3):575–589.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in stratified random sampling when the proportion of persons possessing a rare unrelated attribute is known. Numerical illustrations are also given in support of the present study.     

A stratified randomized response model for sensitive characteristics using non identical trials

Abstract

Stratification provides a powerful tool for improving the efficiency and, being suitable for various sampling situations, it is commonly used in practice. Motivated by the utility of stratified sampling scheme, we focus on studying behavior of the estimator of proportion of a sensitive attribute while dealing with non-identical Bernoulli trials in survey research. The objective is achieved by considering a general randomized response model. Relative efficiency comparisons are presented along with cost analysis by considering different cost functions. Stratified random sampling is observed to be yielding more precise estimators.     

Adjusted Kuk's model using two non sensitive characteristics unrelated to the sensitive characteristic

In this paper, we adjust the Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions indirectly. Biometrika 77(2):436–438.[Crossref], [Web of Science ®] , [Google Scholar]) model for both protection and efficiency by making use of proportions of two non sensitive characteristics which are unrelated to the main sensitive characteristic of interest. Various situations, where the proportions of the two non sensitive characteristics in the population of interest are known and that when these proportions are unknown, have been investigated. We compared the adjusted model and Kuk's model through a simulation study from both the protection and efficiency points of view.     

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.     

Multi-character survery using randomized response technique

In the present paper, we have consisdered the situation of multi–character survey where the study variables, beside being poorly correlated with the selection probabilities are also sensitive in nature. Randomized Response technique (RRT) proposed by Chaudhuri and Adhikary (1990) is used to elicit the information on the sensitive character. The empirical study carried out shows the relative efficiency of the transformation suggested by Basnel and Singh (1985) over the transformations suggested by Rao (1966) and Amahia et al.(1989) under a super population model.     

Estimation of size and mean of a sensitive quantitative variable for a sub-group of a population

In the present paper, a new problem in survey sampling has been considered and an attempt has bcen made to find its solution. The procedures available in literature deal with the estimation of mean of a sensitive quantitative variable for the whole population. Sometimes the investigator may be interested in estimating the mean of a sensitive quantitative variable for a sub-group of the population. At the same time, he/she may also be interested in estimating the size of this sub-group. In this paper, we have obtained estimators of these parameters. Approximate bias and variance expressions are also obtained for these estimators.     

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.     

Estimation of a rare sensitive attribute in probability proportional to size measures using Poisson distribution

We propose new variants of Land et al.’s [Estimation of a rare sensitive attribute using Poisson distribution. Statistics. 2011. DOI: 10.1080/02331888.2010.524300] randomized response model when a population consists of some clusters and the population is stratified with some clusters in each stratum. The estimator for the mean number of persons who possess a rare sensitive attribute, its variance, and the variance estimator are suggested when the parameter of a rare unrelated attribute is assumed to be known and unknown. The clusters are selected with and without replacement. When they are selected with replacement, the selecting probabilities for each cluster are defined depending on the cluster sizes and with equal probability. In addition, the variance comparison between a probability proportional to size (PPS) and PPS for stratification are performed. When the parameters vary in clusters, the stratified PPS has better efficiency than the PPS.     

An improved randomized response model: estimation of mean

In this paper, we suggest a new randomized response model useful for collecting information on quantitative sensitive variables such as drug use and income. The resultant estimator has been found to be better than the usual additive randomized response model. An interesting feature of the proposed model is that it is free from the known parameters of the scrambling variable unlike the additive model due to Himmelfarb and Edgell [S. Himmelfarb and S.E. Edgell, Additive constant model: a randomized response technique for eliminating evasiveness to quantitative response questions, Psychol. Bull. 87(1980), 525–530]. Relative efficiency of the proposed model has also been studied with the corresponding competitors. At the end, an application of the proposed model has been discussed.     

On the comparison of Warner's and unrelated question randomized response plans for estimating sensitive finite population proportions

We consider the problem of estimation of a finite population proportion (P) related to a sensitive attribute under Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) randomized response plan and the unrelated question plan due to Horvitz et al. (1967 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and prove that for a given probability sampling design, given any linear unbiased estimator (LUE) of P based on Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) plan with any given value of the plan parameter, there exists an LUE of P based on the unrelated question plan with a uniformly smaller variance for suitable choices of the plan parameters. Assuming that only the attribute is sensitive but its complement is innocuous, the same is also shown to be true when the plan parameters for the two plans are so chosen so that both offer the same specified level of privacy.     

Violation of respondent's privacy in moors* model- its rectification through a random group strategy

The primary objective of the randomized response techniques introduced by Warner (1965) and Greenberg et al. (1969) was to protect the privacy of the respondents while eliciting information on sensitive issues. However, Moors' (1971) model, which is considered to be the best in the sense of optimality and convenience to use, fails to protect the privacy of respondents in certain cases. The present paper aims at pointing out this serious draw back. An alternative model termed as 'random group method ' is proposed. The proposed model is free from the above mentioned drawback and provides more efficient estimator as compared to the usual Greenberg at al. 's (1969) model.     

A stratified randomized response model for sensitive characteristics using the negative hypergeometric distribution

Taking clue from the pioneer work of Singh and Sedory (2013 Singh, S., Sedory, S.A. (2013). A new randomized response device for sensitive characteristics: an application of the negative hypergeometric distribution. Metron 71:3–8.[Crossref] , [Google Scholar]), we have suggested a new stratified randomized response (RR) model. The properties of the suggested stratified RR model have been studied under proportional and “Neyman” allocations. The study has been also carried out in the presence of crude prior estimates. Numerical illustrations are also given in support of the present study.     

On using non identical and independent Bernoulli trials in human surveys

Unlike the usual randomized response techniques, as a pioneering attempt, this article focuses on using non identical independent Bernoulli trials in sensitive surveys. For this purpose, a general class of randomized response techniques is considered. The usual randomized response techniques are based on a fixed probability of having a yes answer. Contrary to usual techniques, in the proposed technique every respondent has a different probability of reporting a yes answer. With this setting, in most of the situations, the proposed technique is observed performing better in terms of variability. To illustrate and support the superiority of the proposed technique it is compared with models such as Warner (1965), Greenberg et al. (1969), Mangat and Singh (1990), and Mangat (1994) using identical Bernoulli trials. Relative efficiency and privacy protection are studied in detail using Warner (1965) and Mangat (1994) models.     

Estimation of a rare sensitive attribute in a stratified two-stage unrelated randomized response model by using Poisson probability distribution

The present article deals with the estimation of mean number of respondents who possess a rare sensitive character in presence of known and unknown proportion of a rare unrelated non-sensitive attribute by using the Poisson probability distribution in stratified random sampling as well as in stratified random double sampling. The variance of rare sensitive character is also derived under proportional and optimal allocation methods in stratified random sampling when stratum sizes are known and unknown. The properties of the suggested estimation procedures have been deeply examined. The proposed model is found to be dominant over Lee et al. [Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution. Statistics. 2013;47:575–589] model. Numerical illustrations are presented to support the theoretical results. Results are analysed and suitable recommendations are put forward to the survey practitioners.