Study to Alter the Nuisance Effect of Non-Response Using Scrambled Mechanism

Background

In social surveys, obtaining reliable data through direct questioning may be difficult when potentially sensitive questions like sexual indulgence during teenage years, voluntary prostitution, negligence of government rules, drug intake etc. are included in the survey. To avoid these situations, Warner²⁴ introduced the data collection technique that protects anonymity of the respondent known as the randomized response technique (RRT). Whereas, estimation of the mean of quantitative sensitive variables utilizing randomized response models was initiated by Greenberg et al.⁷ Later, Pollock and Bek¹⁵ introduced an additive technique for mean estimation of quantitative sensitive variables. Eichhorn and Hayre⁶ investigated the Pollock and Bek¹⁵ models in depth and introduced the scrambled response method for estimating the population parameters of quantitative sensitive characteristics. Some other developments for different practical situations to estimate the population mean of positive quantitative sensitive characteristics are by Bar-Lev et al.,³ Singh and Tarray,²³ Kim and Elam,¹⁰ Diana and Perri,⁵ Diana and Perri,⁴ Son and Kim,¹⁷ among others.

In real-life problems, there are well-documented situations where sensitive studies need to be monitored over time to understand the problem in a better way. To handle such situations, Jessen,⁹ Priyanka et al.¹⁴ and Singh and Sharma¹⁸ among others used successive sampling. Arnab and Singh¹ and Priyanka and Trisandhya¹² among others used a scrambled response technique to handle sensitive issues on successive occasions.

Motivated by the above cited works, in the present article an attempt has been made to propose efficient scrambled response models. The degree of privacy protection and unified measures for the proposed models have been discussed. Efficacy of the proposed strategy has been justified.

Application of the proposed models under successive sampling has also been discussed. To estimate the population mean of sensitive variables, exponential types of estimators have been proposed. Properties of the proposed estimators have been discussed under the proposed scrambled response models. Empirical studies using a real data set and Monte Carlo simulation studies have been performed under the proposed scrambled response models. Empirical and simulation results indicate the dominance of the proposed estimators over some well-known estimators.

Proposed Technique

The scrambling technique has been used in many randomization devices, with the goal of increasing the respondent’s cooperation. Pollock and Bek¹⁵ have considered the additive model in which the respondent is asked to sum his sensitive attribute by a random value from a known distribution.

Eichhorn and Hayre⁶ have considered another survey model involving a quantitative response variable and proposed an RR technique for it. Such models are very useful in studies involving a measured response variable which is highly sensitive in its nature. The model considered by Eichhorn and Hayre⁶ is known as a multiplicative model in which the respondent multiplies his answers to the sensitive question by a random number from a known distribution. Motivated by Pollock and Bek¹⁵ and Eichhorn and Hayre⁶ we have proposed these models. The proposed scrambled response models are very useful to deal with highly sensitive issues. For example, studies addressing issues such as: i) not only whether or not a woman had an abortion, but in addition, how many abortions she underwent; ii) not only whether the subject used illicit drugs, but also the number of occasions on which drugs were taken; iii) not only if an individual cheated on his income tax report, but also the amount of under-reporting etc. The proposed models will encourage researchers to think more on these lines. The key issue when choosing a model is to find the right trade-off between privacy protection and efficiency in the estimates. In Measure of Privacy Protection we have tested the privacy level of our models, in Efficiency Comparison the efficiency of the models and in Unified Measure of Models Quality we use a unified measure of model quality. The proposed models protect the privacy of the respondents and perform better in terms of unified measure of model quality. Any researcher may think on this line and produce another model by using additive and multiplicative models but they have to test their efficiency level, privacy level and unified measure of model quality. In many cases, it may be possible that a model may be efficient but it is not necessary that the model also protects the respondent's privacy and performs better in terms of unified measure.

Suppose be the finite population and be a quantitative sensitive variable of interest with unknown mean and variance respectively. Let W₁, W₂, S, and U be the scrambling variables independent of Y with means and variances respectively. A random sample of size n respondents was drawn from the population under a simple random sampling with replacement scheme. Each selected respondent in the sample was provided with a random device having three types of cards bearing statements: i) green cards with the statement: report sensitive variable Y; ii) red cards with the statement: report the scrambled response YW₁+W₂; iii) yellow card: report scramble value Z=YS (for model 1) and Z=W₃(Y+U) (for model 2) with probabilities P_1, P₂ and P₃ respectively, such that .

Model 1

In our models, the hypothesis is to develop whether our models are effective or not for estimating the population mean of quantitative sensitive characteristics and in protecting the privacy of respondents.

Let the response be whose distribution is as follows:

Hence, for the observed reported response , the mean and variance of quantitative sensitive variable Y are as follows:

(1)

where

Model 2

Let the response be whose distribution is as follows:

Hence, for the observed reported response , the mean and variance of quantitative sensitive variable Y are given by

(2)

where

Measure of Privacy Protection

Following the work of Diana and Perri⁵ the square of correlation coefficient between observed response and quantitative sensitive variables from models 1 and 2 are denoted by and given as

(3)

and

(4)

Efficiency Comparison

To show the performances of the proposed models, we have compared with the Bar-Lev et al³ model in terms of percent relative efficiencies (PREs) using the formula:

We have considered that scrambling variables follow a Poisson distribution. The data sets used for empirical comparison are given in Table 1.

Table 1 Data Set Used for Efficiency Comparison

From Tables 2 and 3, it may be seen that when we decrease the value of P₁ (probability of reporting the sensitive variable), the values of percent relative efficiencies increase. The values of percent relative efficiencies are high when the probability of selecting the third statement is also high (probability of reporting scrambled value). The level of privacy protection is closer to 1 in maximum cases. Hence, our proposed models perform better to deal with highly sensitive issues than the competitor when the probability of reporting scrambling variables is high.

Table 2 PRE of the Suggested Models 1, 2 for Data Set A with Respect to the Bar Lev et al3 Model Along with PP and

Table 3 PRE of the Suggested Models 1, 2 for Data Set C with Respect to the Bar Lev et al3 Model Along with PP and

In Table 4, for model 1, the values of percent relative efficiencies increase for decreasing values of P₁ (probability of reporting sensitive variable). The values of percent relative efficiencies are high when the probability of selecting the third statement is also high (probability of reporting scrambled value). For model 2, the values of PREs follow the same trend except for the point P₁=0.2. We get the highest value of percent relative efficiency when the probability of selecting the first statement is 0.3 and the probability of selecting the third statement is 0.6. In almost all cases the privacy level is high for both models. Hence, both of our scrambled response models may be useful when dealing with sensitive issues.

Table 4 PRE of the Suggested Models 1, 2 for Data Set B with Respect to the Bar Lev et al3 Model Along with PP and

In Table 5, for model 2, the values of percent relative efficiencies increase for decreasing values of P₁ (probability of reporting sensitive variables). The values of percent relative efficiencies are high when the probability of selecting the third statement is also high (probability reporting scrambled value). For model 1, the values of PREs follow the same trend as in model 2 except for the probabilities P₁=0.3 and 0.2. We get the highest value of percent relative efficiency when the probability of selecting the first statement is 0.4 and the probability of selecting third statement is 0.5. In almost all cases the privacy level is high for both models.

Table 5 PRE of the Suggested Models 1, 2 for Data Set D with Respect to the Bar Lev et al3 Model Along with PP and

Unified Measure of Models Quality

Following the work of Gupta et al⁸ unified measures are calculated for the proposed models and Bar-Lev et al³ models using the formula

for i=1, 2.

where and are the variance and privacy level of the Bar-Lev et al³ model and and are the variance and privacy level of the proposed models respectively.

The graphical representations of the proposed models with respect to the Bar–Lev et al³ model are shown in Figures 1–8.

Figure 1 Unified measure for Model 1 (Set A).

Figure 2 Unified measure for Model 1 (Set B).

Figure 3 Unified measure for Model 1 (Set C).

Figure 4 Unified measure for Model 1 (Set D).

Figure 5 Unified measure for Model 2 (Set A).

Figure 6 Unified measure for Model 2 (Set B).

Figure 7 Unified measure for Model 2 (Set C).

Figure 8 Unified measure for Model 2 (Set D).

Successive Sampling Scheme

Let be a finite population of size N, which has been sampled over two occasions. The character under study is a sensitive variable denoted by x(y) on the first (second) occasion and z is a non-sensitive auxiliary variable available at both occasions. On the first (second) occasion the sensitive variables x(y) are coded to g (h) with the aid of scrambling variables. The scrambling variable may follow a probability distribution.

The following notations have been considered for further use:

: population means of the variables .

: sample means of the respective variables based on sample sizes shown in suffices.

: sample means of the non-sensitive auxiliary variable based on the sample sizes shown in suffices.

: correlation coefficients between the variables shown in suffices.

: population variance of the variables .

Proposed Scrambled Response Models Under Successive Sampling

In this section our hypothesis is to determine whether our models are effective or not for estimating the quantitative sensitive characteristics and for estimating the sensitive population mean when it changes frequently with the passage of time according to its nature.

The proposed randomized devices under the first and second occasions for the ith respondent scrambled response is given as

Model 1:

For estimating the sensitive population mean on two occasions, successive sampling for the sensitive variables x(y) which are coded as g (h) and given as

(5)

Model 2:

The sensitive variables x(y) are coded as g (h) and are given by

(6)

Proposed Estimators

Exponential-type estimators play a vital role in increasing the precision of the estimates and are more stable over the sampling fluctuations. Bahl and Tuteja² were the first who proposed the exponential estimators and discussed certain regularity conditions under which their proposed estimators were better in comparison with mean per unit and ratio and product estimators. Further, they have reduced the sampling variance/MSE of their proposed estimators to the level of regression estimators. Thus, the exponential estimator given by Bahl and Tuteja² may be treated as an alternative of regression estimators. Thereafter, following the lines of Bahl and Tuteja,² various notable authors such as Singh and Homa,¹⁹ Priyanka and Mittal,¹³ Singh et al.,²⁰ Singh and Pal,²¹ Singh et al²² among others have proposed efficient estimators using an exponential function with effective results which are published in reputed journals. Since, exponential-type estimators are the better alternative for increasing the precision of the estimates. Encouraged by these works and motivated to put forward our idea to sample survey audiences, we have proposed exponential-type estimators and used the proposed scrambled response models in this under successive sampling. The proposed estimators found better in terms of percent relative efficiency and may be used when we deal with quantitative sensitive characteristics and for estimating the sensitive population mean when it changes frequently with the passage of time according to its nature.

For estimating the population mean of sensitive variables on the current (second) occasion, two independent estimators are proposed as follows:

(7)

(8)

The estimators defined in Equations (7 and 8) are further reproduced in the functional form as

(9)

The final estimator is a convex linear combination of two estimators and shown as

(10)

where is a constant to be determined under some criteria such that is more precise.

Properties of the Proposed Estimator

To obtain the bias and mean square error of the proposed estimators , we consider the following transformation.

Such that , where i= 0, 1, 2, 3, 4, 5.

Thus, we have the following expressions

Using Taylor series expansions up to the first order, we expand the functional form of the estimator

(11)

Thus, we have

(12)

The bias and mean square error of the estimator T_u aregiven as

(13)

(14)

Substituting the values of g₁, g₂, g₁₁, g₂₂ and g₁₂ in Equations (13 and 14), we get the bias and mean square of the estimator T_u as

(15)

(16)

The function is based on statistics and satisfies the following regularity conditions:

The point assumes the value in the closed convex subset of two-dimensional space containing the point .

The function is continuous and bounded in .

and where is the first-order derivative of g with respect to .

The first-order partial derivatives of exists and is continuous and bounded in

Similarly, the bias and mean square error of the estimator T_m are derived using the following steps

(17)

Following Taylor series expressions, we have

(18)

where

The bias and mean square error of the estimator T_m are given as

(19)

(20)

Substituting the values of in Equations (19 and 20), we get the bias and mean square of the estimator T_m as

(21)

(22)

The function is based on statistics and satisfies the similar regularity conditions as Equation (11).

From Equations (12 and 18) we get the covariance between

(23)

Minimum Mean Square Error of the Estimator

The MSE of the estimator given in Equation (25) is a function of unknown constant , therefore, to obtain the optimum choice of , we minimize Equation (25) with respect to as

(26)

Putting the value of in Equation (25), we have optimum MSE of the estimator as

(27)

Further, substituting the expressions of in Equations (26 and 27), the simplified values of and are obtained as

(28)

where

Optimum Replacement Strategy

To determine the optimum value of µ (fraction of sample to be drawn afresh on the current occasion) so that be estimated with maximum precision and minimum cost, we minimize µ which results in a quadratic equation in µ given as

(29)

Solving Equation (29), the solutions of are given as

(30)

The real value of lies only if . So, using Equation (30) and putting the admissible value of in Equation (28), the optimum value for the mean square error of the estimator T_prop is written as

(31)

Estimators for Sensitive Population Mean at Current Occasion Under Proposed Scrambled Response Models

To obtain the estimator of the sensitive population mean at current occasion estimators the coded response variable on the current move in Equations 1 and 2 is replaced by its estimators respectively and presented in Table 6.

Table 6 Sensitive Population Mean Estimators and Their Mean Square Errors Under the Proposed Scrambled Response Models

Monte Carlo Simulation Study

The simulation study has been carried out by considering 5000 different samples using Monte Carlo simulation for Population-I. The simulated PRE of the proposed estimator with respect to the sample mean estimator, natural successive sampling estimator and Singh and Sharma¹⁸ estimators have been computed using the data set .

The following steps summarize the simulation study as follows:

Step 1. Consider a real population of size N=51 (Population-I), from which 5000 SRSWOR different samples of size n=20 have been selected.

Step 2. From each of the selected samples, m=16 units were retained as a matched portion from sample n.

Step 3. To draw a SRSWOR of size u=4 (fresh) from the remaining part of the population of size N-n=31 as an unmatched portion.

Step 4. Calculate the value of from new unmatched units, u on the current occasion and T_m from the m units retained units.

Step 5. Calculate the value of the estimator based on the value of and .

Step 6. Repeat step (2), (3), (4) and (5), 5000 times. Thus, we obtain 5000 values for the suggested estimator .

Step 7. The MSE of is obtained by .

The efficiency of the estimator with respect to the considered estimator is defined by:

From Table 7, we deal with quantitative sensitive characteristics and for estimating the population mean when it changes frequently with the passing of time according to its nature. The values of percent relative efficiencies are always greater than 100 for all cases but do not follow any specific pattern for decreasing values of P₁ and increasing values of P₂. Therefore, the proposed estimators may be used when we want to monitor sensitive issues which change according to time.

Table 7 Percent Relative Efficiencies of with Respect to and Singh and Sharma.18

Empirical Study

To show the performances of the proposed estimator , we compare with sample mean estimator (when there is no matching), natural successive sampling estimator where and the Singh and Sharma¹⁸ estimator . The estimator suggested by Singh and Sharma¹⁸ is as follows

(32)

where and

The variance of , optimum variance of and the optimum mean square error of Singh and Sharma¹⁸ estimators are as follows

where

Population-I: source Statistical Abstracts of United States.

Let y, x and z be the number of abortions reported in the states of the US during the year 2007, 2005 and 2004 respectively. We consider a real data of N=51 units with the following parameters:

Population–II: source Priyanka and Trisandhya.¹²

Utilizing the above data, the percent relative efficiencies of the proposed estimator with respect to , and the Singh and Sharma¹⁸ estimators are computed using the formula given below and presented in Tables 7 and 8.

Table 8 Optimum Values and PRE of with Respect to , and Singh and Sharma18 for Population-I.

From Table 8, for models 1 and 2, it may be read that the percent relative efficiencies of the proposed estimator are decreasing with decreasing values of P₁ with respect to the sample mean estimator and natural successive estimator. The minimum value of is 0.5352 which indicates that the fraction of fresh sample to be replaced at the current occasion is as low as about 53% of the total sample size, which reduces the cost of the survey. The values of percent relative efficiencies increase with the decreasing values of P₁ with respect to the Singh and Sharma¹⁸ estimator.

From Table 9, for models 1 and 2, it may be read that the percent relative efficiencies of the proposed estimator decrease with the decreasing values of P₁ with respect to the sample mean estimator and natural successive estimator. The minimum value of is 0.5112, which indicates that the fraction of fresh sample to be replaced at the current occasion is as low as about 51% of the total sample size, which reduces the cost of the survey. The values of percent relative efficiencies increase with decreasing value of P₁ with respect to the Singh and Sharma¹⁸ estimator.

Table 9 Optimum Values and PRE of with Respect to , and Singh and Sharma18 for Population-II.

Scrambling Mechanism versus Direct Questioning

If no scrambled response technique has been used, then the estimator under the direct method is given as

(33)

Another estimator based on a sample of size m common to both occasions is a modified exponential-type estimator and structured as

(34)

The final estimator is a convex linear combination of two estimators and shown as

(35)

The minimum mean square error of the estimator up to first-order approximations is given as

(36)

where

The percent relative efficiencies have been computed using data in Estimators for Sensitive Population Mean at Current Occasion Under Proposed Scrambled Response Models for different choices of probabilities and are graphically presented in Figures 9 and 10.

Figure 9 Percent relative efficiency E1d for Model 1.

Figure 10 Percent relative efficiency E1d for Model 2.

Interpretation of Results

The following interpretations may be read from Tables 2–9 and Figures 1–10.

1. From Tables 2–5, it may be seen that the PREs of the proposed randomized response models with respect to the Bar-Lev et al³ model are always greater than 100, which indicates the dominating behaviors of the proposed models over the Bar–Lev et al³ model.
2. From Tables 2–5, it is clear that all the values of τ are greater than 0.5 and closer to 1. Hence, more privacy is protected and greater cooporation may be expected for the proposed models.
3. From Figures 1–8, it is visible that the values of unified measure of model quality for the proposed models are smaller than for the Bar-Lev et al³ model. Hence, the proposed models are better than the contemporary randomized response models given by Bar-Lev et al.³
4. From Tables 7–9, it may be seen that the suggested estimator under the proposed scrambled response models performs better than sample mean estimator, natural successive sampling estimator and Singh and Sharma¹⁸ estimators.
5. From Figures 9 and 10, it is visible that the proposed estimator under the scrambled response models when compared with the direct method performs better in terms of percent relative efficiency.

Conclusions and Recommendations

From the above results and interpretations we may conclude that the proposed models are uniformly dominating over the Bar-Lev et al³ model in terms of enhanced precision of estimates. It has been found that to deal with extremely sensitive questions, the proposed models are suitable with a high degree of privacy protection. The suggested estimator accomplishes good percent relative efficiency. Thus, the proposed randomized response models and proposed estimators may be recommended to survey practitioners encouragingly for use in the real life problems, whenever they intend to deal with the quantitative sensitive characteristics and for estimating the sensitive population mean when it changes frequently with the passage of time according to its nature.