1 Introduction

Let Y = {y₁, y₂, …, y_M} be a finite population of which estimating proportion of individuals having sensitive attribute is the prime interest. To procure such data, a respondent may be reluctant to disclose truth or provide incorrect information due to his/her privacy concerns. Therefore, such data collected through conventional (direct) ways are likely to include response-bias. In this connection [1], proposed the randomized response method with objective to collect trustworthy data by protecting privacy of the interviewee (respondent). This method, based on a randomizing device, requires interviewee to report ‘yes’ or ‘no’ as per statement pointed out by the randomizing device without revealing it to the interviewer. This technique is quite encouraging as respondent cannot be identified i.e., which question he/she has answered. After development of this pioneer work in simple random sampling (SRS) framework, different researchers have proposed a verity of methods which protect respondent’s privacy by different ways and (or) give precise estimate under SRS scheme. For example, see [2–5] and references therein.

McIntyre [6] devised the RSS scheme as oppose to the conventional SRS scheme for investigation of population characteristics. This method considers ranking of the small sets of units by eyes or any other method that involves minimal cost before acquiring final sample. To choose a sample of size m under RSS scheme, the experimenter first needs to identify m sets of units each of size m from the target population and rank the units within each set. Now, for i = 1, 2, …, m, select ith smallest unit from the ith set. The same process can be repeated r times, if required, to obtain the sample of size mr. After development of RSS scheme [7], administrated its mathematical setup. To cope with the situation when direct ranking of main variable is troublesome [8], suggested the idea of indirect ranking using a concomitant variable that is highly correlated with the variable of interest. In this method, ranking is done on an auxiliary variable to choose corresponding units of the main variable. For instance, in a forest survey study, we wish to estimate height of trees; but ranking of trees may be problematic. On the other hand, the measurement on diameter of tree, which is also highly correlated with height of the tree, is easy to determine.

To select m(≥ 2) units, the RSS scheme that based on an auxiliary variable X can be described as: (i) construct m sets of units each of size m from bivariate population (X,Y). (ii) In each set, obtain the exact measurement on X, and arrange Y with respect to X. (iii) Select Y values corresponding to ith (i = 1, 2, …, m) ordered unit of X in the ith set. The steps (i)-(iii) can be cycled for r times, if desired, to get a final sample of size N = mr. For more literature and application of RSS design, the interested reader is referred to the studies [9–13].

Terpstra [14] have investigated usual (insensitive) population proportion using RSS scheme and then compared with SRS rival. Following [14, 15] suggested a new proportion estimator in RSS scheme and claimed that it works better than that of [14]. Later on [16], has pointed out some bottlenecks associated with the estimator in [15] and suggested new improved estimators in RSS schemes.

Recently [17], have introduced a new sensitive proportion estimator in RSS scheme that beats SRS-based competitor suggested by [1]. In this study, keeping in view different perceptions of sensitive inquiry, we suggest a mixed (partial randomized response) method for acquiring reliable data using RSS scheme, wherein ranking is done via a continuous auxiliary variable. We discuss its properties and compare with SRS and RSS competitors given in [1, 17] respectively. The rest parts of this study are arranged as: A review of existing sensitive proportion estimators is given in Section 2. The proposed estimator, along with its properties, is developed in Section 3. Application to AIDS data is considered in Section 4. Cost analysis is done in Section 5, and final remarks are included in Section 6.

2 A review of existing methods

Let Y_1j, Y_2j, …, Y_mj denotes a simple random sample with replacement sampling (SRSWR) of size m in jth cycle, for j = 1, 2, …, r. To collect true response, each interviewee is provided with a randomizing device, say a spinner, to choose one of the following two assertion, without revealing it to the interviewer:

1. I have the sensitive attribute A
2. I do not have the sensitive attribute A

with, respective, pre-determined selection probabilities p ≠ 0.5 and 1 − p. Each interviewee uses the said device and reports ‘yes’ or ‘no’ as per selected statement and his/her correct status. Then ‘yes’ response of ith (i = 1, 2, …, m) respondent at jth cycle can be written as Let m₁ be the total number of ‘yes’ responses from the sample size N = mr. [1] established the maximum likelihood (ML) estimate of π as given by , and showed that is unbiased and its variance is given by (2.1) Let Y be a Bernoulli variate and X is a continuous insensitive auxiliary variable having cumulative distribution function (cdf) F_X(x). Let Y given X = x also follows Bernoulli distribution i.e., , where g(x) is a function with range (0, 1). From [17], the marginal distribution of Y is with parameter π = E[g(X)], ‘E’ stands for expected value. In this study we assume that π = E[g(β₀ + β₁X)], β₀, β₁ ∈ ℜ, g(⋅) is a inverse logit or probit link function, and X follows normal distribution with mean 2 and variance 1 or standard uniform distribution.

Let {(Y_[i]j, X_(i)j): i = 1, 2, …, m;j = 1, 2, …, r} be a RSS of size N, where Y_[i]j denotes ith imperfect ranked unit in jth cycle, and X_(i)j serves ith perfect ranked unit emerged in jth cycle. From [14], Y_[i] is with mean π_[i] = E[g(β₀ + β₁X_(i))] and variance . [14] introduced insensitive proportion () in RSS and is given by (2.2) Let the respondents under RSS scheme are directed to select one of the two aforesaid statements (a) and (b) by using randomizing device suggested in [1]. The respondent reports ‘yes’ (‘no’) according to the outcomes of the device and his/her actual status. Let Y_[i]j = 1 if ith ranked unit reports ‘yes’. Then Let are independent and identically distributed (i.i.d) Bernoulli randomized responses with parameter pπ_[i] + (1 − p)(1 − π_[i]), then ML estimate of π_[i] for the given data is , where ; is the total number of successes observed under ith ranked unit. Now, overall measure of π, using the relation , see the reference [17], is given by (2.3) and its associated variance is (2.4)

3 The proposed method

In this section, we propose a partial randomized response model to gather reliable data for estimation of population sensitive proportion under RSS scheme. This technique is applicable in the situation in which some respondents prefer ‘direct response’ to ‘randomized response’ for a sensitive inquiry. Suppose, taking into account cost and time, the experimenter decides to collect k ≥ 2 ‘direct responses’ by the technique introduced by [14] and the remaining m − k ≥ 2 units through randomized response technique suggested by [17]. Assuming that k² + (m − k)² identifiable units are available and following the lines of Eqs (2.2) and (2.3), we propose a mixed (partial randomized response) model as given by (3.1) N′ = (N − rk). It is obvious that Y_[i]j and are Bernoulli variates with, respective, parameter π_[i] and λ_[g] = (2p − 1)π_[g] + (1 − p). It is noteworthy that , and are special cases of . For k = 0, reduces to . Similarly, for k = m, becomes ; when k = 0 and m = 1, simplifies to .

Lemma: The proposed estimator possesses the following properties:

1. (i). It is an unbiased estimator i.e., E() = π
2. (ii). It is more precise than and i.e.,

Proof:

(i) From Eq (3.1), we have This completes proof (i).

To prove (ii), again using Eq (3.1), we have (3.2) Since the term is always greater than or equal to zero, it is easy to observe that . Moreover [17], have showed that . This completes the proof (ii).

The relative precision (RP) of with respect to can be evaluated by the formula (3.3) From [17], the limiting distribution of the proposed estimator, when m is fixed and r → ∞, is given by (3.4) Some interesting results can be obtained from Eq (3.4). For k = 0, it gives the result that presented in [17]. Furthermore, for k = m or p = 0(1), it becomes [14] under direct inquiry about the attribute of interest. Moreover, in the situation when both p and m are equal to 1, we get conventional method of direct interaction with the respondent under SRS. Note that these are not frequent occurrences; but can happen. Finally, the variance given in Eq (3.4) can be estimated by substituting π_[i] by . In this way, the estimated variance of ith ranked unit becomes independent of g(⋅). This validates asymptotic inference from the suggested method.

4 Application

We have first collected a real age data set of 50 AIDS patients from a local government hospital, Rawalpindi, Pakistan. Then each respondent was approached and requested to answer the question-whether he/she has had a sexual relation with any sex-worker or otherwise. The respondent was given option to either respond directly using SRS or via [1] randomizing device with p = 0.2 and 1 − p = 0.8 respectively. All interviewees were also assured of their identity will never be disclosed, and made it clear that this information could help in their treatments. They were convinced enough that almost half-25 patients gave consent to disclose true information directly. After converting ‘yes’ and ‘no’ responses into binary response, we have computed a true sensitive proportion π = 0.473. We have also computed mean and variance of X and its correlation with Y as given by and ρ_X,Y = 0.417. The main purpose of this survey was to make known of the true proportion, and compare performance of the above discussed estimators.

5 Cost analysis

In survey sampling, several factors such as cost, time and accuracy or precision are taken into account to choose an appropriate sampling design. Generally, cost is main focus in almost all sampling surveys. However, this important factor was neglected in the previous Sections by assuming that there is no cost associated with the ranking of sampling units.

Following [18], we develop a cost model in RSS to assess performance of the estimators. Let c_s denotes cost of stratification attached with each measured unit in ranked set sampling. In common practice, this is the cost of drawing m − 1 units and accomplishing judgment ordering of the m units of a set. Similarly, c_q serves the cost of drawing and quantifying a unit without classification or ranking.

Now, the relative efficiency (RE) is defined as the ratio of the variance of the estimator under SRS and RSS with assumption that the total cost, say C, is same for both sampling schemes. Again, using the reference [18], the RE of with respect to is given by (5.1) It is clear from Eq (5.1) that for fixed c_q when c_s varies, the magnitude of RE decreases. Moreover, maximum RE value can be gained when c_s = 0. To graphical present behavior of Eq (5.1), we assume different combinations of (c_q, c_s) measured in dollar as {(20, 2), (20, 4)}. Note that, the values of , h = a, k, w are same as obtained in Subsection 4.2.

In S7–S10 Figs, we have presented RE values of vs for different combinations of (c_q, c_s) at m = 4. Likewise, the RE values of vs for different (c_q, c_s) at m = 5 are displayed in S11–S14 Figs.

As expected, RE rapidly decreases as c_s increases and vice-versa. However, the proposed model still remains superior. Hence, it is appropriate to estimate sensitive proportion using partial randomized response model when there is negligible (minimum) cost involved for ranking of units.

6 Conclusion

This study has proposed a partial randomized response model for efficiently estimating sensitive proportion under RSS scheme, wherein ranking is done via a continuous auxiliary variable. Both mathematical and numerical results supported the suggested model relative to the existing ordinary models. Moreover, the limiting distribution of the new model is also discussed and then derived some interesting results from it. The graphical representation of the numerical results revealed that the RP values between and are symmetric about p = 0.5, and became larger when m(k) or magnitude of correlation between X and Y increases. On the other hand, RP values between and show similar trend, except, also work efficiently even at low correlation (ρ < 0.5). Finally, the cost analysis has also been done which also advocated supremacy of the new model without compromising privacy of the respondents. Moreover, the proposed model provide the options of both direct and indirect (randomized response), therefore, it is flexible. Furthermore, the cost of direct response is far less than indirect query. Hence, proposed model is highly recommended for sensitive proportion estimation−being flexible, economical and efficient.

Supporting information

Acknowledgments

The authors are grateful to an Academic Editor and two anonymous reviewers for providing useful comments that substantially improved the previous version of this study.