Likelihood-based Estimation of the Regression Model with Scrambled Responses

A significant problem in the collection of responses to potentially sensitive questions, such as relating to illegal, immoral or embarrassing activities, is non-sampling error due to refusal to respond or false responses. Eichhorn & Hayre (1983) suggested the use of scrambled responses to reduce this form of bias. This paper considers a linear regression model in which the dependent variable is unobserved but for which the sum or product with a scrambling random variable of known distribution, is known. The performance of two likelihood-based estimators is investigated, namely of a Bayesian estimator achieved through a Markov chain Monte Carlo (MCMC) sampling scheme, and a classical maximum-likelihood estimator. These two estimators and an estimator suggested by Singh, Joarder & King (1996) are compared. Monte Carlo results show that the Bayesian estimator out-performs the classical estimators in all cases, and the relative performance of the Bayesian estimator improves as the responses become more scrambled.     

Efficient estimation of population mean of sensitive variable in presence of scrambled responses

This paper addresses the problem of estimating the population mean of a sensitive variable in presence of scrambled response. We have suggested a randomized response model which uses known values of mean and variance of scrambling variable. We have shown that the proposed randomized response model is always better than that of Gjestvang and Singh's (2009) and Singh and Tarray's (2014) randomized response models. A numerical study for efficiency comparison is also presented.     

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.     

REGRESSION ANALYSIS USING SCRAMBLED RESPONSES

This paper investigates the general linear regression model Y = Xβ+e assuming the dependent variable is observed as a scrambled response using Eichhorn & Hayre's (1983) approach to collecting sensitive personal information. The estimates of the parameters in the model remain unbiased, but the variances of the estimates increase due to scrambling. The Wald test of the null hypothesis H₀: β=β₀, against the alternative hypothesis H_a: β#β₀, is also investigated. Parameter estimates obtained from scrambled responses are compared to those from conventional or direct-question surveys, using simulation. The coverage by nominal 95% confidence intervals is also reported.     

New Modified Scrambled Faure Sequences

In this paper, we introduce two new modified scrambled Faure sequences based on linear digit scrambling method with good two-dimensional projections in consecutive dimensions using two new linear digit scrambling matrices for linear and nonlinear scrambled Faure sequences. Also, we present performance of our two new scrambling methods by two-dimensional projections and test-integrals.     

Estimation of finite population mean in stratified sampling using scrambled responses in the presence of measurement errors

This study focuses on the estimation of population mean of a sensitive variable in stratified random sampling based on randomized response technique (RRT) when the observations are contaminated by measurement errors (ME). A generalized estimator of population mean is proposed by using additively scrambled responses for the sensitive variable. The expressions for the bias and mean square error (MSE) of the proposed estimator are derived. The performance of the proposed estimator is evaluated both theoretically and empirically. Results are also applied to a real data set.     

Utilization of higher order moments of scrambling variables in randomized response sampling

In this paper, a new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced. The proposed estimator makes use of higher order moments of the scrambling variable at the estimation stage. The proposed estimator has been found to be more efficient than the estimator due to Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] and Franklin [1989. A comparison of estimators for randomized response sampling with continuous distributions from a dichotomous population. Comm. Statist. Theory Methods 18, 489–505] type estimators in randomized response sampling. Recently, Guerriero and Sandri [2007. A note on the comparison of some randomized response procedures. J. Statist. Plann. Inference 137, 2184–2190] have shown that the family of randomized response models proposed by Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] is better than the Simmons’ family in terms of efficiency and protection.     

Some estimation procedures of sensitive character using scrambled response techniques in successive sampling

This paper is an attempt to analyze the effect of scrambled response techniques to estimate the current population mean in two-occasion successive sampling when study and auxiliary variables are sensitive in nature. Using additive and multiplicative scrambled response models on both occasions, some generalized estimation strategies of current population mean have been proposed and their properties are examined. Empirical studies are performed on real data (abortion rate in the states of United States) and simulation studies are also carried out to evaluate the performances of the proposed estimators over other estimators. Results have proved the worthiness of the scrambled response techniques in real-life situations.     

A calibration-based approach to sensitive data: a simulation study

In this paper, we discuss the use of auxiliary information to estimate the population mean of a sensitive variable when data are perturbed by means of three scrambled response devices, namely the additive, the multiplicative and the mixed model. Emphasis is given to the calibration approach, and the behavior of different estimators is investigated through simulated and real data. It is shown that the use of auxiliary information can considerably improve the efficiency of the estimates without jeopardizing respondent privacy.     

改进的随机截尾模型

在随机截尾模型的基础上,为了保护被调查者的隐私,文章提出了一种改进的调查数量敏感性问题的随机化回答方法,并把改进模型的精度与随机截尾模型、随机截尾Warner模型进行了比较。     

A composite class of estimators using scrambled response mechanism for sensitive population mean in successive sampling

This paper considers the problem of estimation of population mean of a sensitive characteristics using non-sensitive auxiliary variable at current move in two move successive sampling. The proposed estimator is studied under five different scrambled response models. Various estimators have been elaborated to be the member of the proposed class of estimators. The properties of the proposed estimators have been analysed. Many estimators belonging to the proposed class have been explored under five scrambled response models. In order to identify the scrambled model effect, the proposed composite class of estimators is compared to the direct methods. Respondents privacy protection have also been elaborated under different models. Theoretical results are supplemented with numerical demonstrations using real data. Simulation has been carried out to show the applicability of proposed estimators and hence suitable recommendations are forwarded.     

The Use of Balanced Incomplete Block Designs in Designing Randomized Response Surveys

This paper investigates the block total response method proposed by Raghavarao and Federer for providing accurate estimates of the base rates of sensitive characteristics during surveys. It determines the best balanced incomplete block design to use to estimate the base rates for three, four, five and six sensitive attributes respectively, given a maximum total number of 13 questions. The estimates obtained from this method have smaller variance than estimates obtained using the similar, but more popular, unmatched count technique.     

Logistic regression analysis of randomized response data with missing covariates

Randomized response is an interview technique designed to eliminate response bias when sensitive questions are asked. In this paper, we present a logistic regression model on randomized response data when the covariates on some subjects are missing at random. In particular, we propose Horvitz and Thompson (1952)-type weighted estimators by using different estimates of the selection probabilities. We present large sample theory for the proposed estimators and show that they are more efficient than the estimator using the true selection probabilities. Simulation results support theoretical analysis. We also illustrate the approach using data from a survey of cable TV.     

Spectral analysis for processes with almost periodic covariances

The structure of random processes with almost periodic covariances is described from a spectral perspective. Under appropriate conditions methods for spectral estimation are described for such processes which are neither stationary nor locally stationary. Some spectral mass is then located off the main diagonal in this spectral plane. A method for estimating the support of the spectral mass is described in the Gaussian case. A number of open questions are mentioned.     

New scrambled response models for estimating the mean of a sensitive quantitative character

Moving from the scrambling mechanism recently suggested by Saha [25 Saha, A. 2007. A simple randomized response technique in complex surveys. Metron, LXV: 59–66.  [Google Scholar]], three scrambled randomized response (SRR) models are introduced with the intent to realize a right trade-off between efficiency and privacy protection. The models perturb the true response on the sensitive variable by resorting to the multiplicative and additive approaches in different ways. Some analytical and numerical comparisons of efficiency are performed to set up the conditions under which improvements upon Saha's model can be obtained and to quantify the efficiency gain. The use of auxiliary information is also discussed in a class of estimators for the sensitive mean under a generic randomization scheme. The class includes also the three proposed SRR models. Finally, some graphical comparisons are carried out from the double perspective of the accuracy in the estimates and respondents’ privacy protection.     

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 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.     

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.     

A simple test for equally likely random responses in a sequence of bernoulli trials

A simple versatile combinatorial test is given for the null hypothesis that the two kinds of responses in a sequence of binary random variables are both equiprobable and randomly distributed. It is sensitive to a preponderance of either kind of response in either of two pre-selected halves of the positions of the sequence, e.g., the right half vs. the left, or the center vs. both ends. It combines the nice features of being, on the one hand, relatively assumption free and easy to apply and, on the other hand, sensitive to basic patterns underlying a poor fit of the empirical distribution to a theoretical model.     

Regression models for binary longitudinal responses

Some conditional models to deal with binary longitudinal responses are proposed, extending random effects models to include serial dependence of Markovian form, and hence allowing for quite general association structures between repeated observations recorded on the same individual. The presence of both these components implies a form of dependence between them, and so a complicated expression for the resulting likelihood. To handle this problem, we introduce, as a first instance, what Follmann and Wu (1995) called, in a different setting, an approximate conditional model, which represents an optimal choice for the general framework of categorical longitudinal responses. Then we define two more formally correct models for the binary case, with no assumption about the distribution of the random effect. All of the discussed models are estimated by means of an EM algorithm for nonparametric maximum likelihood. The algorithm, an adaptation of that used by Aitkin (1996) for the analysis of overdispersed generalized linear models, is initially derived as a form of Gaussian quadrature, and then extended to a completely unknown mixing distribution. A large scale simulation work is described to explore the behaviour of the proposed approaches in a number of different situations.