On randomized response surveys for estimating a proportion

In studies about sensitive characteristics, randomized response (RR) methods are useful for generating reliable data, protecting respondents’ privacy. It is shown that all RR surveys for estimating a proportion can be encompassed in a common model and some general results for statistical inferences can be used for any given survey. The concepts of design and scheme are introduced for characterizing RR surveys. Some consequences of comparing RR designs based on statistical measures of efficiency and respondent’ protection are discussed. In particular, such comparisons lead to the designs that may not be suitable in practice. It is suggested that one should consider other criteria and the scheme parameters for planning a RR survey.     

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.     

A new efficient unrelated randomized response model

The present paper suggests an interesting and useful ramification of the unrelated randomized response model due to Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) [A new unrelated question randomized response model. Statistics 46 (1), 99–109] that can be used for any sampling scheme. We have shown theoretically and numerically that the proposed model is more efficient than Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) model.     

Optional randomized response techniques for quantitative characteristics

Gupta et al. and Huang considered optional randomized response techniques where the probability of choosing the randomized (or direct) response is fixed for all the respondents. In this paper the assumption of the constant probability of choosing the option has been relaxed by dividing respondents into two groups: one group provides direct response and the second a randomized response. The method of estimation of the population mean and variances under the modified assumption are obtained. Relative efficiencies of the proposed techniques are compared theoretically and empirically.     

Improved estimation methods for unrelated question randomized response techniques

The randomized response technique (RRT) is an important tool, commonly used to avoid biased answers in survey on sensitive issues by preserving the respondents’ privacy. In this paper, we introduce a data collection method for survey on sensitive issues combining both the unrelated-question RRT and the direct question design. The direct questioning method is utilized to obtain responses to a non sensitive question that is related to the innocuous question from the unrelated-question RRT. These responses serve as additional information that can be used to improve the estimation of the prevalence of the sensitive behavior. Furthermore, we propose two new methods for the estimation of the proportion of respondents possessing the sensitive attribute under a missing data setup. More specifically, we develop the weighted estimator and the weighted conditional likelihood estimator. The performances of our estimators are studied numerically and compared with that of an existing one. Both proposed estimators are more efficient than the Greenberg's estimator. We illustrate our methods using real data from a survey study on illegal use of cable TV service in Taiwan.     

Analyzing continuous randomized response data with an indifference-zone selection procedure

A review of the randomized response model introduced by Warner (1965) is given, then a randomized response model applicable to continuous data that considers a mixture of two normal distributions is considered. The target here is not to estimate any parameter, but rather to select the population with the best parameter value. This article provides a study on how to choose the best population between k distinct populations using an indifference-zone procedure. Also, this article includes tables for the required sample size needed in order to have a probability of correct selection higher than some specified value in the preference zone for the randomized response model considered.     

An extension of Franklin's randomized response technique for multi-proportions

ABSTRACT

In this article, we introduce six estimators, three based on row averages and the remaining three on column averages of population proportions for trichotomous population when randomized response sampling with a normal randomizing distribution is used. The estimators have been obtained using the method of moments. All the proposed estimators are shown to be unbiased and their variances have been worked out. The percent relative efficiencies of the column total based estimators with respect to row total based estimators are investigated through empirical study.     

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.     

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.     

Protection of privacy in efficient application of randomized response techniques

In estimating the proportion of people bearing a sensitive attribute A, say, in a given community, following Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, certain randomized response (RR) techniques are available for application. These are intended to ensure efficient and unbiased estimation protecting a respondent’s privacy when it touches a person’s socially stigmatizing feature like rash driving, tax evasion, induced abortion, testing HIV positive, etc. Lanke (Int Stat Rev 44:197–203, 1976), Leysieffer and Warner (J Am Stat Assoc 71:649–656, 1976), Anderson (Int Stat Rev 44:213–217, 1976, Scand J Stat 4:11–19, 1977) and Nayak (Commun Stat Theor Method 23:3303–3321, 1994) among others have discussed how maintenance of efficiency is in conflict with protection of privacy. In their RR-related activities the sample selection is traditionally by simple random sampling (SRS) with replacement (WR). In this paper, an extension of an essential similarity in case of general unequal probability sample selection even without replacement is reported. Large scale surveys overwhelmingly employ complex designs other than SRSWR. So extension of RR techniques to complex designs is essential and hence this paper principally refers to them. New jeopardy measures to protect revelation of secrecy presented here are needed as modifications of those in the literature covering SRSWR alone. Observing that multiple responses are feasible in addressing such a dichotomous situation especially with Kuk’s (Biometrika 77:436–438, 1990) and Christofides’ (Metrika 57:195–200, 2003) RR devices, an average of the response-specific jeopardizing measures is proposed. This measure which is device dependent, could be regarded as a technical characteristic of the device and it should be made known to the participants before they agree to use the randomization device. The views expressed are the authors’, not of the organizations they work for. Prof Chaudhuri’s research is partially supported by CSIR Grant No. 21(0539)/02/EMR-II.     

Some robust design strategies for percentile estimation in binary response models

For the problem of percentile estimation of a quantal response curve, the authors determine multiobjective designs which are robust with respect to misspecifications of the model assumptions. They propose a maximin approach based on efficiencies which leads to designs that are simultaneously efficient with respect to various choices of link functions and parameter regions. Furthermore, the authors deal with the problems of designing model and percentile robust experiments. They give various examples of such designs, which are calculated numerically.     

A hierarchical Bayesian model for binary data incorporating selection bias

We consider a Bayesian nonignorable model to accommodate a nonignorable selection mechanism for predicting small area proportions. Our main objective is to extend a model on selection bias in a previously published paper, coauthored by four authors, to accommodate small areas. These authors assume that the survey weights (or their reciprocals that we also call selection probabilities) are available, but there is no simple relation between the binary responses and the selection probabilities. To capture the nonignorable selection bias within each area, they assume that the binary responses and the selection probabilities are correlated. To accommodate the small areas, we extend their model to a hierarchical Bayesian nonignorable model and we use Markov chain Monte Carlo methods to fit it. We illustrate our methodology using a numerical example obtained from data on activity limitation in the U.S. National Health Interview Survey. We also perform a simulation study to assess the effect of the correlation between the binary responses and the selection probabilities.     

A stratified Warner''s randomized response model

This paper proposes a new stratified randomized response model based on Warner's (J. Amer. Statist. Assoc. 60 (1965) 63) model that has an optimal allocation and large gain in precision. It also presents a drawback of the Hong et al. (Korean J. Appl. Statist. 7 (1994) 141) model under their proportional sampling assumption. It is shown that the proposed model is more efficient than the Hong et al. (Korean J. Appl. Statist. 7 (1994) 141) stratified randomized response model. Additionally, it is shown that the estimator based on the proposed method is more efficient than the Warner (J. Amer. Statist. Assoc. 60 (1965) 63), the Mangat and Singh (Biometrika 77 (1990) 439) and the Mangat (J. Roy. Statist. SQC. Ser. B 56 (1) (1994) 93) estimators under the conditions presented in both the case of completely truthful reporting and that of not completely truthful reporting by the respondents.     

Respondent hazards in the unrelated question randomized response model

The randomized response technique is a method of interviewing which allows estimation of population proportion or mean of a sensitive characteristic while maintaining anonymity of response. This is accomplished by use of two questions in the interview, one sensitive and one not so, with the respondent applying a random device to determine which question to answer. The statistician knows the parameters of the random device but is kept from knowledge of individual outcomes. When the nonsensitive question bears on a matter quite foreign to the sensitive characteristic, it is termed an unrelated question, and the associated technique carries that designation. Critical in the construction of such a model are the assigned probability p that the respondent will select the sensitive question and the proportion Π_y of the population having the unrelated nonsensitive characteristic. The choice of these parameters is discussed herein with reference to the risks perceived by respondents.     

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.     

A sinuous stratified unrelated question randomized response model

ABSTRACT

This paper considers the use of stratified random sampling with proportional as well as Neyman allocations to unrelated question randomized response strategy. It has been shown that, for the prior information given, our new model is more efficient in terms of variance (in the case of completely truthful reporting) and mean square error (in case of less than completely truthful reporting). Numerical illustrations are also given in support of the present study.     

Admissible and optimal sampling strategy for estimating finite population mean in randomized response surveys with multiple responses

ABSTRACT

We consider the problem of estimation of a finite population mean (or proportion) related to a sensitive character under a randomized response model when independent responses are obtained from each sampled individual as many times as he/she is selected in the sample and prove the admissibility of a sampling strategy in a class of comparable linear unbiased strategies. We prove that the admissible strategy is also optimal in this class under a super-population model.     

Crossover design in clinical trials for binary response

In this paper, we consider a binary response model for the analysis of the two-treatment, two-period and four-sequence crossover design. We have introduced intra-patient drug dependency parameter in the model and provide two tests for the hypothesis of equality of treatment effects. We employ Monte Carlo simulation to compare our tests and a test that works under parallel design on the basis of type I error rate and power. We find that our procedures are dominant over the competitor with respect to power. Finally, we use a data set to illustrate the applicability of our procedure.     

Model selection information criteria in latent class models with missing data and contingency question

Latent class analysis (LCA) has been found to have important applications in social and behavioural sciences for modelling categorical response variables, and non-response is typical when collecting data. In this study, the non-response mainly included ‘contingency questions’ and real ‘missing data’. The primary objective of this study was to evaluate the effects of some potential factors on model selection indices in LCA with non-response data. We simulated missing data with contingency question and evaluated the accuracy rates of eight information criteria for selecting the correct models. The results showed that the main factors are latent class proportions, conditional probabilities, sample size, the number of items, the missing data rate and the contingency data rate. Interactions of the conditional probabilities with class proportions, sample size and the number of items are also significant. From our simulation results, the impact of missing data and contingency questions can be amended by increasing the sample size or the number of items.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.