# Partial Randomized Response Model for Simultaneous Estimation of Means of Two Sensitive Variables

Authors: Ghulam Narjis; Javid Shabbir; Ronald Onyango
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, a new partial randomized response model (RRM) has been proposed for estimating the population mean of two quantitative sensitive variables simultaneously. The utility of proposed model under stratification is also explored. The efficiency comparisons of the proposed model under simple and stratified random sampling are carried out numerically. A real data set was collected through direct questioning, proposed partial RRM and competitor randomized device from the students of statistics and animal sciences departments of Quaid-I-Azam University Islamabad, Pakistan. The performance of the proposed partial RRM is better than competitor RRM under simple and stratified random sampling.


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## Body

## 1. Introduction

The social survey is one of leading mechanisms to obtain reliable data on attitudes, behaviors, and opinions of the human population. Sometimes, the facts about the individuals are inaccessible to the investigators due to social stigma, such facts are considered as sensitive information. When asked directly, respondents may consciously or unconsciously provide incorrect information on stigmatizing characteristics. To reduce the bias and to procure reliable data, Warner [1] developed a randomized response model (RRM) to estimate the population proportion of a sensitive attribute. In Warner [1]’s model, a randomly selected proportion P of respondents are asked the sensitive question, and the remaining proportion 1−P of respondents are asked complement of the sensitive question. The researcher does not know whether the respondents answered the sensitive or insensitive question. Greenberg et al. [2] extended the Warner’s idea for mean estimation of quantitative sensitive variables. Some other developments for the estimation of mean are due to Eichhorn and Hayre [3]; Bar-Lev et al. [4]; Gupta et al. [5]; Gupta et al. [6] Hussain et al. [7]; Singh et al. [8]; Singh and Suman [9]; Lee and Hong [10]; Narjis and Shabbir [11, 12]; and Muneer et al. [13].Scrambled randomized response models are built on the idea of obtaining masked rather than actual responses. Masking can be achieved by adding to, subtracting from, or multiplying a random component to actual responses. Scrambled response models may be categorized as full, partial, and optional models. In full RRM, all respondents are requested to provide the scrambled response, whereas, in partial RRM, a randomly selected group of respondents are requested to provide the truthful response and remaining are requested to provide the scrambled response. In optional RRM, respondents are requested to provide the scrambled response if he/she considers the question sensitive, and truthful response if he/she considers the question to be non-sensitive. Mangat and Singh [14] and Gupta et al. [15] introduced the partial RRM and the optional RRM, respectively. The purpose of all RRMs are to protect privacy and increase cooperation.Researchers in the field of social, medical, and environmental sciences have well documented the situations where they may be interested to estimate the two dependent sensitive characters at the same time. For example, a researcher may have interest in estimating the proportion of population having income greater than a specific amount weighted according to whether they are tax evaders or not. Another example may be to estimate the proportion of gamblers who are also involved in robbery. Or one may estimate the proportion of induced abortion among females who have pre-marriage sexual relations. Christofides [16] introduced the RR model to estimate the proportions of two dependent sensitive attributes at the same time. Studies on estimation of two dependent sensitive attributes have been reported by Lee et al. [17]; Batool and Shabbir [18]; Ewemooje and Amahia [19]; Ewemooje [20]; and Ewemooje et al. [21].Similarly, the surveys related to household spending/expenditures that present households income and expenditure on different commodities comprises sensitive questions. For example, an economist may have interest in estimating the average difference between amount spent on alcoholic beverages and tobacco items and on food and non-food items. Or the interest may be in estimation of people’s actual income and the income reported in their tax return. However, the choice of RR models are very limited in aforementioned situations where one may need to estimate the mean or average of two quantitative sensitive variables at the same time. Recently, Ahmed et al. [22] introduced full scrambled RR model for simultaneous estimation of means of two quantitative sensitive variables, Hussain and Murtaza [23] had written corrigendum on Ahmed et al. [22] and provides the correct expression of ES13, ES23, σZA22, and σZA1ZA2, respectively.The notions and terminology are given in Sections1.1 and 1.2.


### 1.1. Notations and Terminology under Simple Random Sampling

Suppose a samples of size n is drawn under simple random sampling with replacement (SRSWR) from a finite population Ω=1,2,3,…….,N. Let Y1i and Y2i be two quantitative sensitive variables of interest with unknown mean and variance, which we wish to estimate. Assume S1 and S2 are two scrambling variables independent of both quantitative sensitive variables and with each other, the distribution of scrambling variables are known.LetES1=θ1=e10, ES2=θ2=e01, VS1=δ20, VS2=δ02, CovS1,S2=0, δab=ES1−θ1aS2−θ2b,EY1i=μy1, EY2i=μy2, VY1i=σy12, VY2i=σy22, and σy1y2=EY1i−μy1Y2i−μy2, where, a and b are non-negative integers.Lemma 1.
The moments of order four or less of scrambling variablesS1 and S2, the expression of ES1aS2b, a and b being non-negative integers with a+b≤4, are given by(1)ES12=θ12+δ20=e20,(2)ES13=δ30+3δ20θ1+θ13=e30,(3)ES14=δ40+4δ30θ1+6δ20θ12+θ14=e40,(4)ES22=θ22+δ02=e02,(5)ES23=δ03+3δ02θ2+θ23=e03,(6)ES24=δ04+4δ03θ2+6δ02θ22+θ24=e04(7)ES1S2=θ1θ2=e11,(8)ES12S2=θ12+δ20θ2=e21,(9)ES1S22=θ1θ22+δ02=e12,(10)ES12S22=θ12+δ20θ22+δ02=e22,(11)ES13S2=δ30+3δ20θ1+θ13θ2=e31,(12)ES1S23=θ1δ03+3δ02θ2+θ23=e13.


### 1.2. Notations and Terminology under Stratified Random Sampling

Consider a finite populationΩ=1,2,3,……,N which are partitioned into L homogeneous subgroups called strata, such that the hth stratum consists of Nh units, where h=1,2,…,L and ∑h=1LNh=N. A sample of size nh from Nh stratum is drawn by using SRSWR such that ∑h=1Lnh=n. Let Y1hi and Y2hi be the ith population values of two quantitative sensitive variables in the hth stratum, i=1,2,…,Nh and Wh=Nh/N is the known proportion of population units falling in the hth stratum. Similarly, S1h and S2h are two independent scrambling variables with known means and variances.LetES1h=θ1h=e10h, ES2h=θ2h=e01h, VS1h=δ20h, VS2h=δ02h, CovS1h,S2h=0, δabh=ES1h−θ1haS2h−θ2hb, EY1hi=μy1h, EY2hi=μy2h, VY1hi=σy1h2, VY2hi=σy2h2 and σy1hy2h=EY1hi−μy1hY2hi−μy2h, where, a and b are non-negative integers.Lemma 2.
The moments of order four or less of scrambling variablesS1h and S2h, the expression of ES1haS2hb, a and b being non-negative integers with a+b≤4, are given by(13)ES1h2=θ1h2+δ20h=e20h,(14)ES1h3=δ30h+3δ20hθ1h+θ1h3=e30h,(15)ES1h4=δ40h+4δ30hθ1h+6δ20hθ1h2+θ1h4=e40h,(16)ES2h2=θ2h2+δ02h=e02h,(17)ES2h3=δ03h+3δ02hθ2h+θ2h3=e03h,(18)ES2h4=δ04h+4δ03hθ2h+6δ02hθ2h2+θ2h4=e04h,(19)ES1hS2h=θ1hθ2h=e11h,(20)ES1h2S2h=θ1h2+δ20hθ2h=e21h,(21)ES1hS2h2=θ1hθ2h2+δ02h=e12h,(22)ES1h2S2h2=θ1h2+δ20hθ2h2+δ02h=e22h,(23)ES1h3S2h=δ30h+3δ20hθ1h+θ1h3θ2h=e31h,(24)ES1hS2h3=θ1hδ03h+3δ02hθ2h+θ2h3=e13h.The interest of researchers is always to investigate the true response from a population. To attain this desire, Mangat and Singh [14] proposed an ingenious partial RRT model by injecting an element of truthful responses into the Warner [1]’s model. Gupta and Thornton [24] described the partial RRT model for quantitative variables and many others also give the improvement in this area. It is important to note that almost all such types of RRT models can only estimate one quantitative sensitive variable at a time. So, in this article keeping in mind the desire of researchers, we propose an additive partial RRT model to estimate two quantitative sensitive variables simultaneously. The proposed model is an extension of Ahmed et al. [22]’s model under simple and stratified random sampling. The basic purpose of this study is to obtain truthful responses from some proportion of people and increase efficiency.This paper is organized as follows: In Section2, we give some existing RRT models. In Section 3, we introduce a partial randomized response model under SRSWR and numerically compare it with Ahmed et al. [22] model. In Section 4, we present a partial randomized response model under stratification and numerically compare it with stratified model of Ahmed et al. [22]. In Section 5, an application of real life data is given and comparison of proposed partial RRM is made with Ahmed et al. [22] model on basis of direct response technique. Finally, Section 6 provides a conclusion.


## 1.1. Notations and Terminology under Simple Random Sampling

Suppose a samples of size n is drawn under simple random sampling with replacement (SRSWR) from a finite population Ω=1,2,3,…….,N. Let Y1i and Y2i be two quantitative sensitive variables of interest with unknown mean and variance, which we wish to estimate. Assume S1 and S2 are two scrambling variables independent of both quantitative sensitive variables and with each other, the distribution of scrambling variables are known.LetES1=θ1=e10, ES2=θ2=e01, VS1=δ20, VS2=δ02, CovS1,S2=0, δab=ES1−θ1aS2−θ2b,EY1i=μy1, EY2i=μy2, VY1i=σy12, VY2i=σy22, and σy1y2=EY1i−μy1Y2i−μy2, where, a and b are non-negative integers.Lemma 1.
The moments of order four or less of scrambling variablesS1 and S2, the expression of ES1aS2b, a and b being non-negative integers with a+b≤4, are given by(1)ES12=θ12+δ20=e20,(2)ES13=δ30+3δ20θ1+θ13=e30,(3)ES14=δ40+4δ30θ1+6δ20θ12+θ14=e40,(4)ES22=θ22+δ02=e02,(5)ES23=δ03+3δ02θ2+θ23=e03,(6)ES24=δ04+4δ03θ2+6δ02θ22+θ24=e04(7)ES1S2=θ1θ2=e11,(8)ES12S2=θ12+δ20θ2=e21,(9)ES1S22=θ1θ22+δ02=e12,(10)ES12S22=θ12+δ20θ22+δ02=e22,(11)ES13S2=δ30+3δ20θ1+θ13θ2=e31,(12)ES1S23=θ1δ03+3δ02θ2+θ23=e13.


## 1.2. Notations and Terminology under Stratified Random Sampling

Consider a finite populationΩ=1,2,3,……,N which are partitioned into L homogeneous subgroups called strata, such that the hth stratum consists of Nh units, where h=1,2,…,L and ∑h=1LNh=N. A sample of size nh from Nh stratum is drawn by using SRSWR such that ∑h=1Lnh=n. Let Y1hi and Y2hi be the ith population values of two quantitative sensitive variables in the hth stratum, i=1,2,…,Nh and Wh=Nh/N is the known proportion of population units falling in the hth stratum. Similarly, S1h and S2h are two independent scrambling variables with known means and variances.LetES1h=θ1h=e10h, ES2h=θ2h=e01h, VS1h=δ20h, VS2h=δ02h, CovS1h,S2h=0, δabh=ES1h−θ1haS2h−θ2hb, EY1hi=μy1h, EY2hi=μy2h, VY1hi=σy1h2, VY2hi=σy2h2 and σy1hy2h=EY1hi−μy1hY2hi−μy2h, where, a and b are non-negative integers.Lemma 2.
The moments of order four or less of scrambling variablesS1h and S2h, the expression of ES1haS2hb, a and b being non-negative integers with a+b≤4, are given by(13)ES1h2=θ1h2+δ20h=e20h,(14)ES1h3=δ30h+3δ20hθ1h+θ1h3=e30h,(15)ES1h4=δ40h+4δ30hθ1h+6δ20hθ1h2+θ1h4=e40h,(16)ES2h2=θ2h2+δ02h=e02h,(17)ES2h3=δ03h+3δ02hθ2h+θ2h3=e03h,(18)ES2h4=δ04h+4δ03hθ2h+6δ02hθ2h2+θ2h4=e04h,(19)ES1hS2h=θ1hθ2h=e11h,(20)ES1h2S2h=θ1h2+δ20hθ2h=e21h,(21)ES1hS2h2=θ1hθ2h2+δ02h=e12h,(22)ES1h2S2h2=θ1h2+δ20hθ2h2+δ02h=e22h,(23)ES1h3S2h=δ30h+3δ20hθ1h+θ1h3θ2h=e31h,(24)ES1hS2h3=θ1hδ03h+3δ02hθ2h+θ2h3=e13h.The interest of researchers is always to investigate the true response from a population. To attain this desire, Mangat and Singh [14] proposed an ingenious partial RRT model by injecting an element of truthful responses into the Warner [1]’s model. Gupta and Thornton [24] described the partial RRT model for quantitative variables and many others also give the improvement in this area. It is important to note that almost all such types of RRT models can only estimate one quantitative sensitive variable at a time. So, in this article keeping in mind the desire of researchers, we propose an additive partial RRT model to estimate two quantitative sensitive variables simultaneously. The proposed model is an extension of Ahmed et al. [22]’s model under simple and stratified random sampling. The basic purpose of this study is to obtain truthful responses from some proportion of people and increase efficiency.This paper is organized as follows: In Section2, we give some existing RRT models. In Section 3, we introduce a partial randomized response model under SRSWR and numerically compare it with Ahmed et al. [22] model. In Section 4, we present a partial randomized response model under stratification and numerically compare it with stratified model of Ahmed et al. [22]. In Section 5, an application of real life data is given and comparison of proposed partial RRM is made with Ahmed et al. [22] model on basis of direct response technique. Finally, Section 6 provides a conclusion.


## 2. RRM in Literature

In this section, we consider the following existing RRMs.


### 2.1. Model under Simple Random Sampling

Ahmed et al. [22] proposed an additive and multiplicative model for estimation of mean of two quantitative sensitive variables simultaneously. Two responses are taken from each respondent, a scrambled response and fake response. The scrambled response from the ith respondent is obtained as(25)ZA1i=S1Y1i+S2Y2i.For the second response, each respondent is requested to rotate a spinner and respond accordingly as: the respondent is asked to report the value of scrambling variableS1 when the pointer lands in a shaded area, otherwise report the value of scrambling variable S2. Let P be the proportion of shaded area and 1−P be the proportion of non-shaded area of the spinner. Thus, the second response from the ith respondent, is given by(26)Zi=S1with probabilityP,S2with probability1−P.From equations (25) and (26), generate the response ZA2i as follows:(27)ZA2i=ZiZA1i=S12Y1i+S1S2Y2iwith robabilityP,S1S2Y1i+S22Y2iwith probability1−P.The unbiased estimators of the population meansμy1 and μy2, are given by(28)μ^yA1=Pe11+1−Pe02Z¯A1−e01Z¯A21−Pδ02e10−Pδ20e01,and(29)μ^yA2=e10Z¯A2−Pe20+1−Pe11Z¯A11−Pδ02e10−Pδ20e01.The variance of the proposed estimatorsμ^yA1 and μ^yA2, are given by(30)Vμ^yA1=Pe11+1−Pe022σZA12+e012σZA22−2Pe11+1−Pe02e01σZA1ZA2n1−Pδ02e10−Pδ20e012,and(31)Vμ^yA2=e102σZA22+Pe20+1−Pe112σZA12−2e10Pe20+1−Pe11σZA1ZA2n1−Pδ02e10−Pδ20e012,where (32)σZA12=μy12+σy12δ20+μy22+σy22δ02+σy12e102+σy22e012+2σy1y2e11,(33)σZA22=μy12+σy12Pe40+1−Pe22+μy22+σy22Pe22+1−Pe04+2μy1μy2+σy1y2Pe31+1−Pe13−Pe20+1−Pe11μy1+Pe11+1−Pe02μy22,and(34)σZA1ZA2=μy12+σy12Pe30+1−Pe21+μy22+σy22Pe12+1−Pe03+2μy1μy2+σy1y2Pe21+1−Pe12−μy1e10+μy2e01Pe20+1−Pe11μy1+Pe11+1−Pe02μy2.


### 2.2. Model under Stratified Random Sampling

From Ahmed et al. [22], in stratified random sampling, the scrambled response from the ith respondent of the hth stratum is obtained as(35)ZA1hi=S1hY1hi+S2hY2hi.The second response from theith respondent of the hth stratum is obtained as(36)Zhi=S1hwith probabilityPhS2hwith probability1−Ph.The unbiased estimators of the population meansμy1 and μy2 are given by(37)μ^yA1st=∑h=1LWhPhe11h+1−Phe02hZ¯A1h−e01hZ¯A2h1−Phδ02he10h−Pδ20he01h,and(38)μ^yA2st=∑h=1LWhe10hZ¯A2h−Phe20h+1−Phe11hZ¯A1h1−Phδ02he10h−Phδ20he01h.The variance of the proposed estimatorsμ^yA1st and μ^yA2st are given by(39)Vμ^yA1st=∑h=1LWh2σA1h2nh,and(40)Vμ^yA2st=∑h=1LWh2σA2h2nh,where(41)σA1h2=Phe11h+1−Phe02h2σZA1h2+e01h2σZA2h2−2Phe11h+1−Phe02he01hσZA1hZA2h1−Phδ02he10h−Pδ20he01h2,and(42)σA2h2=e10h2σZA2h2+Phe20h+1−Phe11h2σZA1h2−2e10hPhe20h+1−Phe11hσZA1hZA2h1−Phδ02he10h−Pδ20he01h2,(43)σZA1h2=μy1h2+σy1h2δ20h+μy2h2+σy2h2δ02h+σy1h2e10h2+σy2h2e01h2+2σy1hy2he11h,(44)σZA2h2=μy1h2+σy1h2Phe40h+1−Phe22h+μy2h2+σy2h2Phe22h+1−Phe04h+2μy1hμy2h+σy1hy2hPhe31h+1−Phe13h−Phe20h+1−Phe11hμy1h+Phe11h+1−Phe02hμy2h2,and(45)σZA1hZA2h=μy1h2+σy1h2Phe30h+1−Phe21h+μy2h2+σy2h2Phe12h+1−Phe03h+2μy1hμy2h+σy1hy2hPhe21h+1−Phe12h−μy1he10h+μy2he01hPhe20h+1−Phe11hμy1h+Phe11h+1−Phe02hμy2h.


## 2.1. Model under Simple Random Sampling

Ahmed et al. [22] proposed an additive and multiplicative model for estimation of mean of two quantitative sensitive variables simultaneously. Two responses are taken from each respondent, a scrambled response and fake response. The scrambled response from the ith respondent is obtained as(25)ZA1i=S1Y1i+S2Y2i.For the second response, each respondent is requested to rotate a spinner and respond accordingly as: the respondent is asked to report the value of scrambling variableS1 when the pointer lands in a shaded area, otherwise report the value of scrambling variable S2. Let P be the proportion of shaded area and 1−P be the proportion of non-shaded area of the spinner. Thus, the second response from the ith respondent, is given by(26)Zi=S1with probabilityP,S2with probability1−P.From equations (25) and (26), generate the response ZA2i as follows:(27)ZA2i=ZiZA1i=S12Y1i+S1S2Y2iwith robabilityP,S1S2Y1i+S22Y2iwith probability1−P.The unbiased estimators of the population meansμy1 and μy2, are given by(28)μ^yA1=Pe11+1−Pe02Z¯A1−e01Z¯A21−Pδ02e10−Pδ20e01,and(29)μ^yA2=e10Z¯A2−Pe20+1−Pe11Z¯A11−Pδ02e10−Pδ20e01.The variance of the proposed estimatorsμ^yA1 and μ^yA2, are given by(30)Vμ^yA1=Pe11+1−Pe022σZA12+e012σZA22−2Pe11+1−Pe02e01σZA1ZA2n1−Pδ02e10−Pδ20e012,and(31)Vμ^yA2=e102σZA22+Pe20+1−Pe112σZA12−2e10Pe20+1−Pe11σZA1ZA2n1−Pδ02e10−Pδ20e012,where (32)σZA12=μy12+σy12δ20+μy22+σy22δ02+σy12e102+σy22e012+2σy1y2e11,(33)σZA22=μy12+σy12Pe40+1−Pe22+μy22+σy22Pe22+1−Pe04+2μy1μy2+σy1y2Pe31+1−Pe13−Pe20+1−Pe11μy1+Pe11+1−Pe02μy22,and(34)σZA1ZA2=μy12+σy12Pe30+1−Pe21+μy22+σy22Pe12+1−Pe03+2μy1μy2+σy1y2Pe21+1−Pe12−μy1e10+μy2e01Pe20+1−Pe11μy1+Pe11+1−Pe02μy2.


## 2.2. Model under Stratified Random Sampling

From Ahmed et al. [22], in stratified random sampling, the scrambled response from the ith respondent of the hth stratum is obtained as(35)ZA1hi=S1hY1hi+S2hY2hi.The second response from theith respondent of the hth stratum is obtained as(36)Zhi=S1hwith probabilityPhS2hwith probability1−Ph.The unbiased estimators of the population meansμy1 and μy2 are given by(37)μ^yA1st=∑h=1LWhPhe11h+1−Phe02hZ¯A1h−e01hZ¯A2h1−Phδ02he10h−Pδ20he01h,and(38)μ^yA2st=∑h=1LWhe10hZ¯A2h−Phe20h+1−Phe11hZ¯A1h1−Phδ02he10h−Phδ20he01h.The variance of the proposed estimatorsμ^yA1st and μ^yA2st are given by(39)Vμ^yA1st=∑h=1LWh2σA1h2nh,and(40)Vμ^yA2st=∑h=1LWh2σA2h2nh,where(41)σA1h2=Phe11h+1−Phe02h2σZA1h2+e01h2σZA2h2−2Phe11h+1−Phe02he01hσZA1hZA2h1−Phδ02he10h−Pδ20he01h2,and(42)σA2h2=e10h2σZA2h2+Phe20h+1−Phe11h2σZA1h2−2e10hPhe20h+1−Phe11hσZA1hZA2h1−Phδ02he10h−Pδ20he01h2,(43)σZA1h2=μy1h2+σy1h2δ20h+μy2h2+σy2h2δ02h+σy1h2e10h2+σy2h2e01h2+2σy1hy2he11h,(44)σZA2h2=μy1h2+σy1h2Phe40h+1−Phe22h+μy2h2+σy2h2Phe22h+1−Phe04h+2μy1hμy2h+σy1hy2hPhe31h+1−Phe13h−Phe20h+1−Phe11hμy1h+Phe11h+1−Phe02hμy2h2,and(45)σZA1hZA2h=μy1h2+σy1h2Phe30h+1−Phe21h+μy2h2+σy2h2Phe12h+1−Phe03h+2μy1hμy2h+σy1hy2hPhe21h+1−Phe12h−μy1he10h+μy2he01hPhe20h+1−Phe11hμy1h+Phe11h+1−Phe02hμy2h.


## 3. Proposed Partial RRM under Simple Random Sampling

In this section, we propose a partial randomized response model for simultaneous estimation of means of two quantitative sensitive variables. In the proposed partial RRM, each respondent selected in theith sample is requested for two responses by using two randomized response (RR) devices. The RR Device I provides the scramble and true response of sensitive variables, whereas the RR Device II provides the fake response, that is free from the sensitive variables.The RR Device I, bearing two types of statements:(i)
Report the additive true value of both sensitive variables, sayY1i+Y2i with probability T and(ii)
Report the scrambled response as,S1Y1i+S2Y2i with probability 1−T. Mathematically, each respondent is requested to report the response Z1i as(46)Z1i=Y1i+Y2iwith probabilityTS1Y1i+S2Y2iwith probability1−T.The partial randomized responseZ1i in the ith sample is given by(47)Z1i=βiY1i+Y2i+1−βiS1Y1i+S2Y2i.For the second response, each respondent is requested to use RR Device 2 which is same as equation (26); thus, the fake response Zi from equation (26) in the ith sample is given by(48)Zi=γiS1i+1−γiS2i,where γi and βi are Bernoulli random variables with means P and T, respectively, which are the known parameters.From equations (47) and (48), we generate response Z2i as follows:(49)Z2i=ZiZ1i=S1Y1i+S1Y2iwith probabilityPTS12Y1i+S1S2Y2iwith probability1−TPS2Y1i+S2Y2iwith probabilityT1−PS1S2Y1i+S22Y2iwith probability1−P1−T.The generated responseZ2i in the ith sample is given by(50)Z2i=γiβiS1Y1i+S1Y2i+1−βiγiS12Y1i+S1S2Y2i+1−γiβiS2Y1i+S2Y2i+1−γi1−βiS1S2Y1i+S22Y2i.Taking expected values on both sides of equations (47) and (50), we have(51)EZ1i=Tμy1+μy2+1−Te10μy1+e01μy2,and(52)EZ2i=PTe10μy1+e10μy2+1−TPe20μy1+e11μy2+1−PTe01μy1+e01μy2+1−P1−Te11μy1+e02μy2.From equations (51) and (52), by the method of moments, we have(53)A11μ^y1+A12μ^y2=Z¯1,and(54)A21μ^y1+A22μ^y2=Z¯2,whereA11=T+1−Te10A12=T+1−Te01A21=PTe10+1−TPe20+1−PTe01+1−P1−Te11A22=PTe10+1−TPe11+1−PTe01+1−P1−Te02.Solving equations (53) and (54) by using Cramer’s rule, we have unbiased estimators of μy1 and μy2, respectively, given by(55)μ^y1=A22Z¯1−A12Z¯21−T1−Pδ02A11−Pδ20A12,and(56)μ^y2=A11Z¯2−A21Z¯11−T1−Pδ02A11−Pδ20A12.Theorem 1.
The variance of proposed estimatorsμ^y1 and μ^y2 is, respectively, given by(57)Vμ^y1=A222σZ12+A122σZ22−2A12A22σZ1Z2n1−T21−Pδ02A11−Pδ20A122,and(58)Vμ^y2=A112σZ22+A212σZ12−2A11A21σZ1Z2n1−T21−Pδ02A11−Pδ20A122,where (59)σZ12=μy12+σy12T+1−Te20+μy22+σy22T+1−Te02+2μy1μy2+σy1y2T+1−Te11−A11μy1+A12μy22,(60)σZ22=μy12+σy12PTe20+1−TPe40+1−PTe02+1−P1−Te22+μy22+σy22PTe20+1−TPe22+1−PTe02+1−P1−Te04+2μy1μy2+σy1y2PTe20+1−TPe31+1−PTe02+1−P1−Te13−A21μy1+A22μy22,and(61)σZ1Z2=μy12+σy12PTe10+1−PTe01+1−TPe30+1−P1−Te21+μy22+σy22PTe10+1−PTe01+1−TPe12+1−P1−Te03+2μy1μy2+σy1y2PTe10+1−PTe01+1−TPe21+1−P1−Te12−A11μy1+A12μy2A21μy1+A22μy2.Proof.
Note that the variance expressions for two estimatorsμ^y1 and μ^y2 can be obtained through the formula VarA±B=VarA+VarB±2CovA,B.
Now, the varianceσZ12 is given by(62)σZ12=EZ1i2−EZ1i2=TEY1i+Y2i2+1−TES1Y1i+S2Y2i2−EβiY1i+Y2i+1−βiS1Y1i+S2Y2i2,or(63)σZ12=μy12+σy12T+1−TES12+μy22+σy22T+1−TES22+2μy1μy2+σy1y2T+1−TES1S2−A11μy1+A12μy22.
On substituting the values ofES12, ES22, and ES1S2 in equation (63), we have equation (59).
The varianceσZ22 is given by(64)σZ22=EZ2i2−EZ2i2=PTES1Y1i+S1Y2i2+1−TPES12Y1i+S1S2Y2i2+1−PTES2Y1i+S2Y2i2+1−P1−TES1S2Y1i+S22Y2i2−EγiβiS1Y1i+S1Y2i+1−βiγiS12Y1i+S1S2Y2i+1−γiβiS2Y1i+S2Y2i+1−γi1−βiS1S2Y1i+S22Y2i2,or(65)σZ22=μy12+σy12PTES12+1−TPES14+1−PTES22+1−P1−TES12S22+μy22+σy22PTES12+1−TPES12S22+1−PTES22+1−P1−TES24+2μy1μy2+σy1y2PTES12+1−TPES13S2+1−PTES22+1−P1−TES1S23−A21μy1+A22μy22.
On substituting the values ofES12, ES22, ES14, ES24, ES1S2, ES12S22, ES13S2, and ES1S23 in equation (65), we have equation (60).
The covarianceσZ1Z2 between Z1i and Z2i is given by(66)σZ1Z2=CovZ1i,Z2i=EZ1iZ2i−EZ1iEZ2i=EγiβiβiY1i+Y2iS1Y1i+S1Y2i+1−βiS1Y1i+S2Y2iS1Y1i+S1Y2i+1−βiγiβiY1i+Y2iS12Y1i+S1S2Y2i+1−βiS1Y1i+S2Y2iS12Y1i+S1S2Y2i+1−γiβiβiY1i+Y2iS2Y1i+S2Y2i+1−βiS1Y1i+S2Y2iS2Y1i+S2Y2i+1−γi1−βiβiY1i+Y2iS1S2Y1i+S22Y2i+1−βiS1Y1i+S2Y2iS1S2Y1i+S22Y2i−EβiY1i+Y2i+1−βiS1Y1i+S2Y2iEγiβiS1Yi+S1Y2i+1−γiβiS12Yi+S1S2Y2i+1−γiβiS2Yi+S2Y2i+1−γi1−βiES1S2Yi+S22Y2i,or(67)σZ1Z2=μy12+σy12PTES1+1−PTES2+1−TPES13+1−P1−TES12S2+μy22+σy22PTES1+1−PTES2+1−TPES1S22+1−P1−TES23+2μy1μy2+σy1y2PTES1+1−PTES2+1−TPES12S2+1−P1−TES1S22−A11μy1+A12μy2A21μy1A22μy2.
On substituting the values ofES1, ES2, ES12, ES22, ES13, ES23, ES1S2, ES12S2, and ES1S22 in equation (67), we have equation (61).Corollary 1.
The unbiased estimators forVμ^y1 and Vμ^y2 are, respectively, given by(68)V^μ^y1=A222sZ12+A122sZ22−2A12A22sZ1Z2n1−T21−Pδ02A11−Pδ20A122,and(69)V^μ^y2=A112sZ22+A212sZ12−2A11A21sZ1Z2n1−T21−Pδ02A11−Pδ20A122,where sZ12=n−1−1∑i=1nZ1i−Z¯12, sZ22=n−1−1∑i=1nZ2i−Z¯22 and
sZ1Z2=n−1−1∑i=1nZ1i−Z¯1Z2i−Z¯2 are unbiased estimator of σZ12, σZ22, and σZ1Z2, respectively.Remark 1.
WhenT=0, the proposed partial RRM reduces to Ahmed et al. [22] model.


### 3.1. Percent Relative Efficiency under SRSWR

In this section, we compute the percent relative efficiency (PRE) of proposed estimatorsμ^y1 and μ^y2 over the estimators μ^yA1 and μ^yA2, respectively, as(70)PRE1=Vμ^yA1Vμ^y1×100,and(71)PRE2=Vμ^yA2Vμ^y2×100.We performed a simulation study to verify the superiority of the proposed partial RRM through FORTRAN program and showed the situation where the proposed methods might be more efficient than the method of Ahmed et al. [22]. The simulation results give large number of situations where PREi, i=1,2 values of proposed partial RRM are more than 100. However, we presented only few values of PRE(1) and PRE(2) in Table 1, for different parameter values, P=0.5, various values of T, μy1=20, μy2=30, two values of σy1 and σy2, θ1=4, θ2=7.2, δ20=2, δ02=4, δ30=−0.17, δ03=−1.65, δ40=12.13, and δ04=52.29. The efficiency comparison on the values of scrambling variables that were earlier used in Ahmed et al. [22] model is also carried out, but our proposed model is less efficient on those values. Thus, we conclude that the efficiency of proposed partial RRM model can be increased or decreased by using different scrambling variables.Table 1
PRE of proposed partial RRM over Ahmed et al. [22] model.
ρy1y2σy1σy2T=0.1T=0.3T=0.5T=0.7T=0.9PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)−0.922112.87110.15145.35133.00189.95157.43246.90172.20239.90114.7026112.87110.15145.37133.02190.01157.47247.05172.31240.38114.9362112.87110.15145.34133.00189.92157.40246.81172.14239.61114.5666112.87110.15145.36133.02190.00157.46247.02172.29240.27114.88−0.722112.87110.15145.35133.00189.95157.43246.90172.20239.89114.6926112.87110.15145.36133.02190.00157.47247.04172.30240.36114.9262112.87110.15145.34132.99189.92157.40246.80172.13239.58114.5566112.87110.15145.36133.02189.99157.46246.99172.27240.19114.84−0.522112.87110.15145.35133.00189.95157.43246.89172.20239.88114.6926112.87110.15145.36133.02190.00157.47247.03172.30240.33114.9062112.87110.15145.34132.99189.92157.40246.79172.13239.56114.5466112.87110.15145.36133.01189.98157.45246.97172.25240.10114.80−0.322112.87110.15145.35133.00189.95157.43246.89172.19239.87114.6826112.87110.15145.36133.02190.00157.46247.03172.29240.30114.8962112.87110.15145.34132.99189.91157.40246.78172.12239.53114.5266112.87110.15145.35133.01189.97157.44246.94172.23240.02114.76−0.122112.87110.15145.35133.00189.95157.42246.89172.19239.86114.6826112.87110.15145.36133.02189.99157.46247.02172.28240.27114.8862112.87110.15145.33132.99189.91157.39246.78172.11239.50114.5166112.87110.15145.35133.01189.96157.43246.91172.21239.94114.720.122112.87110.15145.35133.00189.95157.42246.88172.19239.85114.6726112.87110.15145.36133.02189.99157.46247.01172.28240.24114.8662112.87110.15145.33132.99189.91157.39246.77172.11239.48114.5066112.87110.15145.35133.00189.95157.42246.89172.19239.86114.680.322112.87110.15145.35133.00189.95157.42246.88172.19239.84114.6726112.87110.15145.36133.02189.99157.46247.00172.27240.22114.8562112.87110.15145.33132.99189.91157.39246.76172.10239.45114.4866112.87110.15145.34133.00189.94157.42246.86172.17239.78114.640.522112.87110.15145.35133.00189.95157.42246.88172.19239.83114.6726112.87110.15145.36133.01189.99157.45246.99172.26240.19114.8462112.87110.14145.33132.99189.90157.39246.75172.10239.42114.4766112.87110.15145.34133.00189.93157.41246.84172.16239.69114.600.722112.87110.15145.35133.00189.95157.42246.88172.18239.82114.6626112.87110.15145.36133.01189.98157.45246.98172.26240.16114.8262112.87110.14145.33132.99189.90157.38246.74172.09239.39114.4666112.87110.15145.34133.00189.92157.40246.81172.14239.61114.560.922112.87110.15145.35133.00189.94157.42246.87172.18239.81114.6626112.87110.15145.36133.01189.98157.45246.97172.25240.13114.8162112.87110.14145.33132.99189.90157.38246.73172.08239.37114.4466112.87110.15145.34132.99189.92157.39246.79172.12239.54114.52


## 3.1. Percent Relative Efficiency under SRSWR

In this section, we compute the percent relative efficiency (PRE) of proposed estimatorsμ^y1 and μ^y2 over the estimators μ^yA1 and μ^yA2, respectively, as(70)PRE1=Vμ^yA1Vμ^y1×100,and(71)PRE2=Vμ^yA2Vμ^y2×100.We performed a simulation study to verify the superiority of the proposed partial RRM through FORTRAN program and showed the situation where the proposed methods might be more efficient than the method of Ahmed et al. [22]. The simulation results give large number of situations where PREi, i=1,2 values of proposed partial RRM are more than 100. However, we presented only few values of PRE(1) and PRE(2) in Table 1, for different parameter values, P=0.5, various values of T, μy1=20, μy2=30, two values of σy1 and σy2, θ1=4, θ2=7.2, δ20=2, δ02=4, δ30=−0.17, δ03=−1.65, δ40=12.13, and δ04=52.29. The efficiency comparison on the values of scrambling variables that were earlier used in Ahmed et al. [22] model is also carried out, but our proposed model is less efficient on those values. Thus, we conclude that the efficiency of proposed partial RRM model can be increased or decreased by using different scrambling variables.Table 1
PRE of proposed partial RRM over Ahmed et al. [22] model.
ρy1y2σy1σy2T=0.1T=0.3T=0.5T=0.7T=0.9PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)−0.922112.87110.15145.35133.00189.95157.43246.90172.20239.90114.7026112.87110.15145.37133.02190.01157.47247.05172.31240.38114.9362112.87110.15145.34133.00189.92157.40246.81172.14239.61114.5666112.87110.15145.36133.02190.00157.46247.02172.29240.27114.88−0.722112.87110.15145.35133.00189.95157.43246.90172.20239.89114.6926112.87110.15145.36133.02190.00157.47247.04172.30240.36114.9262112.87110.15145.34132.99189.92157.40246.80172.13239.58114.5566112.87110.15145.36133.02189.99157.46246.99172.27240.19114.84−0.522112.87110.15145.35133.00189.95157.43246.89172.20239.88114.6926112.87110.15145.36133.02190.00157.47247.03172.30240.33114.9062112.87110.15145.34132.99189.92157.40246.79172.13239.56114.5466112.87110.15145.36133.01189.98157.45246.97172.25240.10114.80−0.322112.87110.15145.35133.00189.95157.43246.89172.19239.87114.6826112.87110.15145.36133.02190.00157.46247.03172.29240.30114.8962112.87110.15145.34132.99189.91157.40246.78172.12239.53114.5266112.87110.15145.35133.01189.97157.44246.94172.23240.02114.76−0.122112.87110.15145.35133.00189.95157.42246.89172.19239.86114.6826112.87110.15145.36133.02189.99157.46247.02172.28240.27114.8862112.87110.15145.33132.99189.91157.39246.78172.11239.50114.5166112.87110.15145.35133.01189.96157.43246.91172.21239.94114.720.122112.87110.15145.35133.00189.95157.42246.88172.19239.85114.6726112.87110.15145.36133.02189.99157.46247.01172.28240.24114.8662112.87110.15145.33132.99189.91157.39246.77172.11239.48114.5066112.87110.15145.35133.00189.95157.42246.89172.19239.86114.680.322112.87110.15145.35133.00189.95157.42246.88172.19239.84114.6726112.87110.15145.36133.02189.99157.46247.00172.27240.22114.8562112.87110.15145.33132.99189.91157.39246.76172.10239.45114.4866112.87110.15145.34133.00189.94157.42246.86172.17239.78114.640.522112.87110.15145.35133.00189.95157.42246.88172.19239.83114.6726112.87110.15145.36133.01189.99157.45246.99172.26240.19114.8462112.87110.14145.33132.99189.90157.39246.75172.10239.42114.4766112.87110.15145.34133.00189.93157.41246.84172.16239.69114.600.722112.87110.15145.35133.00189.95157.42246.88172.18239.82114.6626112.87110.15145.36133.01189.98157.45246.98172.26240.16114.8262112.87110.14145.33132.99189.90157.38246.74172.09239.39114.4666112.87110.15145.34133.00189.92157.40246.81172.14239.61114.560.922112.87110.15145.35133.00189.94157.42246.87172.18239.81114.6626112.87110.15145.36133.01189.98157.45246.97172.25240.13114.8162112.87110.14145.33132.99189.90157.38246.73172.08239.37114.4466112.87110.15145.34132.99189.92157.39246.79172.12239.54114.52


## 4. Proposed Partial RRM under Stratified Random Sampling

In this section, we present a partial randomized response model under stratification; a subsample in each stratum is drawn using a SRSWR sampling. Each sampled respondent in thehth stratum is requested for two responses by using two randomized response (RR) devices. The RRh Device I provides the scramble and true response of sensitive variables, whereas the RRh Device II provides the fake response, that is free from the sensitive variables.TheRRh Device I, bearing two types of statements: (i) Report the additive true value of both sensitive variables, say Y1hi+Y2hi with probability Th, and (ii) Report the scrambled response as, S1hY1hi+S2hY2hi with probability 1−Th. Mathematically, each ith respondent of the hth stratum is requested to report the response Z1hi as:(72)Z1hi=Y1hi+Y2hiwith probabilityThS1hY1hi+S2hY2hiwith probability1−Th.The partial randomized responseZ1hi in the ith sample of the hth stratum is given by(73)Z1hi=βhiY1hi+Y2hi+1−βhiS1hY1hi+S2hY2hi,and for the second response, each respondent is requested to use RRh Device II which is same as equation (36); thus, the fake response Zhi from in the ith sample of the hth stratum is given by(74)Zhi=γhiS1h+1−γhiS2h,where γhi and βhi are Bernoulli random variables with mean Ph and Th, respectively, which are the known parameters.From equations (73) and (74), we generate response Z2hi as follows:(75)Z2hi=ZhiZ1hi=S1hY1hi+S1hY2hiwith probabilityPhThS1h2Y1hi+S1hS2hY2hiwith probability1−ThPhS2hY1hi+S2hY2hiwith probabilityTh1−PhS1hS2hY1hi+S2h2Y2hiwith probability1−Ph1−Th.The generated responseZ2hi in the ith sample of the hth stratum is given by(76)Z2hi=γhiβhiS1hY1hi+S1hY2hi+1−βhiγhiS1h2Y1hi+S1hS2hY2hi+1−γhiβhiS2hY1hi+S2hY2h+1−γhi1−βhiS1hS2hY1hi+S2hi2Y2hi.Taking expected values on both sides of equations (73) and (76), we have(77)EZ1hi=Thμy1h+μy2h+1−The10hμy1h+e01hμy2h,and(78)EZ2h=PhThe10hμy1h+e10hμy2h+1−ThPhe20hμy1h+e11hμy2h+1−PhThe01hμy1h+e01hμy2h+1−Ph1−The11hμy1h+e02hμy2h.From equations (77) and (78), by the method of moments, we have(79)A11hμ^y1h+A12hμ^y2h=Z¯1h,and(80)A21hμ^y1h+A22hμ^y2h=Z¯2h,whereA11h=Th+1−The10hA12h=Th+1−The01hA21h=PhThe10h+1−ThPhe20h+1−PhThe01h+1−Ph1−The11hA22h=PhThe10h+1−ThPhe11h+1−PhThe01h+1−Ph1−The02h.Solving equations (79) and (80) by using Cramer’s rule, we have unbiased estimators of μy1 and μy2, respectively, given by(81)μ^y1st=∑h=1LWhA22hZ¯1h−A12hZ¯2h1−Th1−Phδ02hA11h−Phδ20hA12h,and(82)μ^y2st=∑h=1LWhA11hZ¯2h−A21hZ¯1h1−Th1−Phδ02hA11h−Phδ20hA12h.Theorem 2.
The variance of proposed estimatorsμ^y1st and μ^y2st is, respectively, given by(83)Vμ^y1st=∑h=1LWh2σ1h2nh,and(84)Vμ^y2st=∑h=1LWh2σ2h2nh,where(85)σ1h2=A22h2σZ1h2+A12h2σZ2h2−2A12hA22hσZ1hZ2h1−Th21−Phδ02hA11h−Phδ20hA12h2,and(86)σ2h2=A11h2σZ2h2+A21h2σZ1h2−2A11hA21hσZ1hZ2h1−Th21−Phδ02hA11h−Phδ20hA12h2,(87)σZ1h2=μy1h2+σy1h2Th+1−The20h+μy2h2+σy2h2Th+1−The02h+2μy1hμy2h+σy1hy2hTh+1−The11h−A11hμy1h+A12hμy2h2,(88)σZ2h2=μy1h2+σy1h2PhThe20h+1−ThPhe40h+1−PhThe02h+1−Ph1−The22h+μy2h2+σy2h2PhThe20h+1−ThPhe22h+1−PhThe02h+1−Ph1−The04h+2μy1hμy2h+σy1hy2hPhThe20h+1−ThPhe31h+1−PhThe02h+1−Ph1−The13h−A21hμy1h+A22hμy2h2,and(89)σZ1hZ2h=μy1h2+σy1h2PhThe10h+1−PhThe01h+1−ThPhe30h+1−Ph1−The21h+μy2h2+σy2h2PhThe10h+1−PhThe01h+1−ThPhe12h+1−Ph1−The03h+2μy1hμy2h+σy1hy2hPhThe10h+1−PhThe01h+1−ThPhe21h+1−Ph1−The12h−A11hμy1h+A12hμy2hA21hμy1h+A22hμy2h.
Proof is simple, so omitted.Corollary 2.
The variance of proposed estimatorsμ^y1st and μ^y2st under different methods of sample allocation are as:(i)
Equal Allocation:(90)Vμ^y1stEA=Ln∑h=1LWh2σ1h2,and(91)Vμ^y2stEA=Ln∑h=1LWh2σ2h2.(ii)
Proportional allocation:(92)Vμ^y1stPA=1n∑h=1LWhσ1h2,and(93)Vμ^y2stPA=1n∑h=1LWhσ2h2.(iii)
Optimum Allocation:(94)Vμ^y2stOA=1n∑h=1LWhσ2h2ησ1h2+1−ησ2h2/Ch∑h=1LWhησ1h2+1−ησ2h2Ch,and(95)Vμ^y1stOA=1n∑h=1LWhσ1h2ησ1h2+1−ησ2h2/Ch∑h=1LWhησ1h2+1−ησ2h2Ch,wherenh=nWhησ1h2+1−ησ2h2/Ch/∑h=1LWhησ1h2+1−ησ2h2/Ch,which are estimated by using linear cost function such asC=C0+∑h=1LnhCh (where C0 is fixed cost, and Ch is the variable cost in each stratum) and the compromised variance as:Vcomp=∑h=1LWh2ησ1h2+1−ησ2h2/nh, whereas η is known constant.Corollary 3.
The unbiased estimators forVμ^y1st and Vμ^y2st are, respectively, given by(96)V^μ^y1st=∑h=1LWh2s1h2nh,and(97)V^μ^y2st=∑h=1LWh2s2h2nh,where(98)s1h2=A22h2sZ1h2+A12h2sZ2h2−2A12hA22hsZ1hZ2h1−Th21−Phδ02hA11h−Phδ20hA12h2,and(99)s2h2=A11h2sZ2h2+A21h2sZ1h2−2A11hA21hsZ1hZ2h1−Th21−Phδ02hA11h−Phδ20hA12h2,
sZ1h2=nh−1−1∑i=1nhZ1hi−Z¯1h2,sZ2h2=nh−1−1∑i=1nhZ2hi−Z¯2h2and
sZ1hZ2h=nh−1−1∑i=1nhZ1hi−Z¯1hZ2hi−Z¯2h are unbiased estimator of σZ1h2, σZ2h2 and σZ1hZ2h, respectively.


### 4.1. Percent Relative Efficiency under Stratification

In this section, we compute the percent relative efficiency (PRE) of proposed partial RRM over Ahmed et al. [22] model under stratification using proportional allocation method. For numerical comparison, we use real data set that is taken from Rosner [25]; the childhood respiratory disease study of Boston. We consider Y1=AGE of a child and Y2=FEV (forced expiratory volume) both as sensitive variables as earlier used by Ahmed et al. [22]. The population is subdivided into two strata on the basis of gender. The PREs of proposed estimators μ^y1st and μ^y2st with respect to Ahmed et al. [22] estimators μ^yS1st and μ^yS2st, respectively, are defined as:(100)PRE1=Vμ^yA1stPAVμ^y1stPA×100=∑h=1LWhσA1h2∑h=1LWhσh12×100,and(101)PRE2=Vμ^yA2stPAVμ^y2stPA×100=∑h=1LWhσA2h2∑h=1LWhσh22×100.The results are presented in Table2, the scrambling variable are same as in Ahmed et al. [22] for both strata, that is, θ1h=2.6, θ2h=4.5, δ20h=2.3, δ02h=1.2, δ30h=2.2, δ03h=2.4, δ40h=3.2, and δ04h=3.2. The PRE1 and PRE2 are free from the sample size. The numerical comparison shows that the efficiency of proposed partial RRM may be increased by choosing appropriate values of design parameters Ph,Th. We also observe that when Th=0, the proposed model reduces to Ahmed et al. [22] model.Table 2
PRE of proposed partial RRM with respect to Ahmed et al. [22] model under stratification.
T1,T2P1,P2=0.2P1,P2=0.4P1,P2=0.6P1,P2=0.8PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)0.01001001001001001001001000.11301261041011071041081050.21721601081001131061151090.3229203110981191071231120.4307257109921241061301120.5413323105831251011341100.65523969469122911331020.7713458765010974122860.88264584828814895590.964229016836184624We also compute thePRE of proposed partial RRM under SRSWR over proposed partial RRM under stratification to observe the gain in efficiency due to stratification. For both estimators, the PRE is almost 100 at different values of design parameters. This is because the variation between strata is almost same, and the randomization devices are also identical.In the next section, we consider an application of a real data set.


## 4.1. Percent Relative Efficiency under Stratification

In this section, we compute the percent relative efficiency (PRE) of proposed partial RRM over Ahmed et al. [22] model under stratification using proportional allocation method. For numerical comparison, we use real data set that is taken from Rosner [25]; the childhood respiratory disease study of Boston. We consider Y1=AGE of a child and Y2=FEV (forced expiratory volume) both as sensitive variables as earlier used by Ahmed et al. [22]. The population is subdivided into two strata on the basis of gender. The PREs of proposed estimators μ^y1st and μ^y2st with respect to Ahmed et al. [22] estimators μ^yS1st and μ^yS2st, respectively, are defined as:(100)PRE1=Vμ^yA1stPAVμ^y1stPA×100=∑h=1LWhσA1h2∑h=1LWhσh12×100,and(101)PRE2=Vμ^yA2stPAVμ^y2stPA×100=∑h=1LWhσA2h2∑h=1LWhσh22×100.The results are presented in Table2, the scrambling variable are same as in Ahmed et al. [22] for both strata, that is, θ1h=2.6, θ2h=4.5, δ20h=2.3, δ02h=1.2, δ30h=2.2, δ03h=2.4, δ40h=3.2, and δ04h=3.2. The PRE1 and PRE2 are free from the sample size. The numerical comparison shows that the efficiency of proposed partial RRM may be increased by choosing appropriate values of design parameters Ph,Th. We also observe that when Th=0, the proposed model reduces to Ahmed et al. [22] model.Table 2
PRE of proposed partial RRM with respect to Ahmed et al. [22] model under stratification.
T1,T2P1,P2=0.2P1,P2=0.4P1,P2=0.6P1,P2=0.8PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)PRE(1)PRE(2)0.01001001001001001001001000.11301261041011071041081050.21721601081001131061151090.3229203110981191071231120.4307257109921241061301120.5413323105831251011341100.65523969469122911331020.7713458765010974122860.88264584828814895590.964229016836184624We also compute thePRE of proposed partial RRM under SRSWR over proposed partial RRM under stratification to observe the gain in efficiency due to stratification. For both estimators, the PRE is almost 100 at different values of design parameters. This is because the variation between strata is almost same, and the randomization devices are also identical.In the next section, we consider an application of a real data set.


## 5. Application of Real Data Set

Hussain et al. [26] estimated the average total number of classes that were missed by the students and Gjestvang and Singh [27] considered the problem of estimating the average GPA of students. In this study, we simultaneously estimate the average total number of classes that were missed by the students and average GPA of the students, by using the proposed partial RRM under SRSWR.We took a sample of 80 students from Stat 317, Stat 629, and Zoo 203 classes to estimate the average GPA and average total number of missed classes by the students, at Quaid-i-Azam University, Islamabad. We generated 20 random numbers ofS1 and S2 separately from Poisson distribution with means 5 and 2, respectively. To collect the data through proposed RRM, two decks of cards were used: Deck I, a deck of yellow cards and Deck II, a deck of blue cards. A deck of yellow cards bearing two different types of statements by setting T=0.5 to get the scrambled and true responses, whereas deck of blue cards bearing values of two scramble variables by setting P=0.7 to get fake responses.In Deck I or deck of yellow cards,20 cards, on 10 cards, we wrote the statements:(102)PleaseReport:GPAscored inlast semester+Total classesmissed duringlast semester,and, on remaining 10 cards, we wrote the statements:(103)PleaseReport:GPAscored inlast semesterS1randomnumber+Total classesmissed duringlast semesterS2randomnumber.In this process,20 values of Si, i=1,2 were written on each card.The Deck II or deck of blue cards, consists of20 cards, out of which 14 cards had the values of S1 and the remaining six cards had values of S2, with statement “kindly report one selected random number.”Before starting the data collection process, we highlighted the importance of randomized response methods and explained the proposed partial RRM and Ahmed et al. [22] model to the students. Each student was requested to draw one card from the Deck I and report the response as requested on the card, on yellow color paper. Similarly, repeat the process for Deck II and report the response as requested on the card, on blue color paper. Then, each student was provided a pink color card written on Ahmed et al. [22] model which was similar to equation (103) and requested to provide the scramble response on pink color paper. After marking the three responses on three different color papers, the students were advised to staple these papers together and put them into the box lying on the table. Finally, all students wrote the actual GPA and true total number of classes which they had missed during last semester on white color paper without disclosing their identity. Table 3 presents the responses obtained from the students.Table 3
Data obtained through Ahmed et al. [22]; proposed partial RRM and direct responses.
Data obtained through RR device I of Ahmed et al. [22] model45, 38.5, 28.6, 7.2, 31.1, 51.2, 42.9, 30.6, 8.4, 38, 29.2, 21.5, 32.2, 15.6, 15, 18.9, 27.7, 32, 36.5, 16.5, 35.4, 28, 24.2, 5.6, 16.6, 24.5, 16.4, 39.3, 34.8, 13.8, 20.6, 25, 26.8, 38.7, 36.4, 10.2, 27, 21.5, 25, 28.2, 43, 28.8, 34.7, 32, 31.4, 8.8, 63.8, 9.2, 15.8, 61.5, 27.4, 30.5, 29.9, 22.5, 14.2, 25, 14.6, 28.1, 18.8, 28.4, 14.3, 27.5, 38.5, 30.6, 9, 34.7, 13.4, 19.5, 14.5, 33.6, 54, 25, 17.6, 6.3, 41, 28.4, 37, 10.4, 27.5, 22.6Data obtained through RR device I of proposed partial RRM by settingT=0.510.5, 38.5, 28.6, 7.2, 31.1, 9.8, 10.9, 30.6, 6.7, 38, 7.6, 8.3, 8.4, 11.8, 15, 10.3, 27.7, 7.2, 36.5, 16.5, 35.4, 6.4, 24.2, 5.6, 10.2, 24.5, 7.1, 39.3, 34.8, 13.8, 12.2, 25, 11.2, 38.7, 36.4, 7.6, 9, 21.5, 7, 6.7, 43, 28.8, 34.7, 32, 8.3, 8.8, 11.1, 9.2, 15.8, 61.5, 8.2, 6.7, 7.7, 10.1, 10.1, 8.6, 14.6, 28.1, 18.8, 28.4, 14.3, 27.5, 8.9, 9.1, 9, 34.7, 7.8, 19.5, 14.5, 7.6, 54, 7.5, 17.6, 6.3, 41, 28.4, 5.5, 7.2, 27.5, 22.6Data obtained through RR device II for fake responses by settingP=0.76, 5, 6, 1, 9, 4, 9, 4, 2, 5, 7, 1, 1, 2, 5, 3, 7, 4, 7, 5, 3, 5, 6, 1, 3, 2, 1, 3, 3, 1, 3, 2, 1, 9, 4, 1, 2, 3, 7, 6, 5, 6, 1, 4, 1, 2, 9, 4, 2, 5, 7, 5, 1, 5, 2, 5, 3, 7, 3, 7, 3, 5, 5, 6, 1, 9, 3, 5, 1, 6, 6, 8, 1, 1, 1, 6, 1, 2, 5, 4GPA on last semester from direct response3.4, 2.5, 3.2, 3.33, 3.4, 3.6, 3.05, 2.5, 3, 4.2, 4.1, 3.25, 3.3, 3, 3, 3.5, 2.9, 2.6, 3.1, 2.9, 3.9, 2.1, 3.5, 3.9, 3.4, 3, 2.5, 2.6, 3, 2.5, 2.6, 3, 3, 3, 3.7, 4, 4, 3.2, 3.5, 3.8, 4, 3, 3.8, 3.4, 2.8, 2.7, 3.5, 2.02, 3.6, 3.2, 3.2, 3.2, 1.9, 4, 3.5, 2.5, 3.65, 2.15, 3, 3.9, 4, 3.7, 3.3, 3.86, 3, 3.4, 2.9, 3.6, 3.63, 3.4, 2.8, 3, 2.94, 2.7, 2.6, 3.7, 3.5, 2.6, 2.3, 3Total classes missed during last semester from direct response6, 7, 2, 9, 4, 10, 5, 3, 6, 0, 0, 5, 12, 6, 0, 8, 4, 2, 5, 3, 3, 4, 3, 3, 2, 12, 12, 6, 6, 7, 2, 4, 5, 4, 0, 6, 10, 4, 3, 4, 3, 9, 4, 6, 5, 12, 2, 5, 5, 5, 6, 6, 3.3, 8, 5, 8, 4, 0, 6, 0, 0, 7, 3, 5, 10, 4, 7, 9, 5, 9, 6, 3, 10, 0, 5, 2, 6, 5, 6Table4 presents results of the survey. The estimates of the means μy1 and μy2 of proposed partial RRM are close to the estimates based on true responses as compared to the estimates obtained from Ahmed et al. [22] model. We noted that the standard error values are large due to small sample sizes; thus, we suggest that a large scale sample survey should be conducted in future to reach more realistic outcomes. The estimates of randomized response models reflect that the students are more reluctant to admit the total number of missed classes to do something unrelated to university study through direct questioning, whereas GPA is a less sensitive question for students. In conclusion, one can see that the proposed method of collecting scrambled data on sensitive issues can be used safely and securely.Table 4
Results of the survey.
ModelEstimated meanStandard error95% confidence intervalμ^y1μ^y2μ^y1μ^y2μ^y1μ^y2Proposed partial RRM3.186.170.571.12[2.06, 4.30][3.97, 8.37]Ahmed et al. [22]2.337.530.471.15[1.41, 3.25][5.26, 9.80]Direct questioning3.195.050.060.34[3.07, 3.31][4.38, 5.72]


## 6. Conclusion

In the present paper, we have suggested a partial randomized response model for estimating two population means simultaneously. Through simulation study and real life data application, it is observed that the proposed partial RRM performs better than the Ahmed et al. [22] model. The superiority of suggested partial RRM under stratification revealed through numerical comparison and it is observed that the proposed partial RRM under stratified random sampling performs better as compared to stratified model of Ahmed et al. [22]. Moreover, we also observed that design parameters play an important role in increasing or decreasing the efficiency of suggested models. The main advantage of proposed partial RRM is that it enables researchers to collect truthful responses at least from some proportion of people. Thus, the proposed partial randomized response model are therefore recommended for its use in practice as an alternative to Ahmed et al. [22] randomized response model.



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*Source: 6083646-2022-08-23.xml*