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Alternative Estimators for Randomized ResponseTechniques in Multi-Character SurveysRaghunath Arnab aa Department of Statistics , University of Botswana , Gaborone , BotswanaPublished online: 18 Mar 2011.

To cite this article: Raghunath Arnab (2011) Alternative Estimators for Randomized Response Techniques in Multi-CharacterSurveys, Communications in Statistics - Theory and Methods, 40:10, 1839-1848, DOI: 10.1080/03610921003714188

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Communications in Statistics—Theory and Methods, 40: 1839–1848, 2011Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610921003714188

Alternative Estimators for Randomized ResponseTechniques inMulti-Character Surveys

RAGHUNATH ARNAB

Department of Statistics, University of Botswana, Gaborone, Botswana

In this article, alternative estimators for a multi-character survey based on either byprobability proportional to size with replacement (ppswr) or Rao–Hartley–Cochran(rhc, 1962) methods of sampling have been proposed. The study variables areconfidential in nature and randomized response technique is used for collection ofdata. It is further assumed that the study variable have a little correlation to thevariable for the selection of sample. The proposed alternative estimators fare betterthan the conventional estimators and their efficiencies are compared with the aid oflife data collected by CSO, government of Botswana.

Keywords Randomized response; Relative efficiency; Sampling design.

Mathematics Subject Classification 62D05.

1. Introduction

In socioeconomic and biometric research, we sometimes gather information relatingto highly sensitive issues such as induced abortion, drug addiction, HIV status,duration of suffering from AIDS, sufferings from amnesia due to narcotics uses,incidence of VDRL, tax evasion, and so on. In this situation, using the directmethod of interview (asking questions directly to the respondents), the respondentsvery often report untrue values or even refuse to respond because of the socialstigma or fear. In this situation, randomized response (RR) technique may be usedto collect more reliable data and better cooperation from the respondents.

RR technique was introduced by Warner (1965) for estimating �y, theproportion of population possessing a certain stigmatized character y (say).Warner’s (1965) technique was modified by Horvitz et al. (1967), Greenberg et al.(1969), Raghavrao (1978), Franklin (1989), Mangat and Singh (1990), Arnab (1996),Singh et al. (2010), Kim and Warde (2004, 2005a,b), and Singh (2008), among otherresearchers, for improving greater co-operation and efficiency.

Most of the surveys in practice are complex and multi-character surveys whereinformation of more than one character is collected at a time. Some of them areof a confidential nature while the others are not. In a complex survey sample

Received November 10, 2009; Accepted February 18, 2010Address correspondence to Raghunath Arnab, Department of Statistics, University of

Botswana, Private Bag UB 00705, Gaborone, Botswana; E-mail: arnabr@mopipi.ub.bw

1839

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1840 Arnab

is very often selected by varying probability sampling schemes using an auxiliaryvariable x as a measure of size. The resulting sampling design is expected to yieldefficient estimators for those characters (study variable) which are well-related tothe auxiliary variable x but may not provide efficient estimators for the characterswhich are poorly related to x.

In this article, we consider a multi-character survey where the sample is selectedeither by probability proportional to size with replacement (ppswr) or Rao–Hartley–Cochran (rhc, 1962) methods of sampling. We further assume that some of thecharacters are confidential in nature and RR techniques are used for collectionof data. The measure of size x for the selection of sample is poorly related tostudy variable y. As for an example, consider a household survey where informationon the sensitive character HIV status �y� along with non sensitive items such ashousehold income, expenditure on food, employment status, sources of income, andownership of cattle among others are collected. In this survey, household size maybe known in advance and may be used as size measure x, for the selection of sample.Obviously, the household size may have good correlation with household income orexpenditure but may have poor correlation with HIV status.

Rao (1966) first addressed the requirement of the adjustment of theconventional estimators in a multi-character survey where the study variable y isdirectly available from the respondents and the measure of size x for the selection ofsample is poorly related to the study variable y. The proposed alternative estimatorsof Rao (1966) found to fare better than the corresponding conventional estimatorsunder the following superpopulation model:

Model M1 � EM1�yi� = �� VM1�yi� = �2 and CM1�yi� yj� = 0 for i �= j� (1)

where �� �2�> 0� are unknown model parameters and EM1� VM1, and CM1 denote,respectively, the expectation, variance, and covariance with respect to the model M1.

In our present article, we consider the problem of estimation of the proportionof individual �y possessing certain sensitive characteristics A (say). In this situation,we denote yi = 1 if the ith individual possesses the character A and yi = 0 otherwise.Since y is an indicator variable, it cannot be proportional to the auxiliary variable xwhich is always positive, hence ppswr or rhc method of sampling may not produceefficient estimator for the population mean or total. In this situation the abovesuperpopulation model (1) is not appropriate.

In this article, we have proposed alternative estimators analogous to Rao(1966) for �y based on ppswr or rhc sampling schemes. The proposed alternativeestimators fare better than the conventional estimators when the measure of sizeof the selection of sample is poorly related to the study variable. The results arederived without assuming any superpopulation model. Finally, the performances ofthe proposed alternative estimators are judged with the live data.

1.1. Some Popular RR Techniques

In this section, we present a few popular RR techniques relevant to our discussion.Let U = �1� � I� � N be a finite population of N identifiable units and yi bethe value of the confidential character under study for the ith unit. The variable yitakes the value 1 if i ∈ A and it takes the value zero if i � A. Our aim is to estimate�y =

∑Ni=1 yi/N , the proportion of persons belong to the sensitive group using some

suitable RR devices described below.

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Randomized Response Techniques 1841

1.1.1. Warner (1965). From the population U� a sample of n units is selected bysimple random sampling with replacement (SRSWR) method. Each of the selectedunits has to perform a RR trial R1 (say) as follows. Respondent is to select acard at random from a pack of cards consisting of two types of cards withknown proportions. Card type 1, with proportion ���= 1/2� marked “I belong togroup A” and card type 2 with proportion 1− � marked “I do not belong toA”. Respondent will supply truthfully answer “yes” or “no”. The experiment isperformed in absence of the interviewer and hence the confidentiality is maintainedbecause the interviewer will not know which question the respondent is answered.Let zi denotes the randomized response obtained from ith respondent which takesthe value 1(0) if the response “yes” (“no”) was obtained. Denoting ER�VR� as theoperator of expectation (variance) for the randomized response, we get

ER�zi� = yi�+ �1− yi��1− �� and VR�zi� = ��1− ��

The revised randomized response ri = zi−�1−��

�2�−1� is an unbiased estimator of yi satisfies

ER�ri� = yi and VR�ri� =��1− ��

�2�− 1�2= �2

i �say� (2)

1.1.2. Horvitz et al. (1967). In this case, the RR trial R2 (say) consists of twoquestions—one related to the sensitive character (y) and other to a neutral character(q) such as (i) “I posses the character y” and (ii) “I posses the character q”. Therespondent should select either the question (i) with probability � or the question(ii) with probability 1− � by some suitable randomized device such as a spinner,random number table etc., and provide answer “Yes” or “No” to the interviewer.Let us define the randomized response obtained from the ith respondent as zi whichtakes the value one or zero corresponding to “yes” and “no” answer, respectively.Then we get

ER�zi� = yi�+ qi�1− �� and VR�zi� = ��1− ���yi + qi�

In this case, the revised RR is ri = zi−�q�1−��

�with

ER�ri� = yi and VR�ri� =��1− ���yi + qi�

�2= �2

i �say�� (3)

where �q =∑N

i=1 qi/N = proportion of persons posses non-sensitive character in thepopulation is assumed to be known.

1.1.3. Kuk (1990). Here two boxes each of which contain black and whitecards with known proportions �1� 1− �1 and �2� 1− �2, respectively (�1 �= �2).The respondent belongs to the group A will choose box 1 and others (group�A, say) choose box 2, respectively, and draw c cards at random and withreplacement. The respondent will report the number of black cards drawn ashis/her randomized response (z). In this situation, ER�zi� = yi�k�1�+ �1− yi��k�2� =

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k���1 − �2�yi + �2 and VR�zi� = yi�k�1�1− �1� + �1− yi��k�2�1− �2� . The revisedrandomized response ri = zi−k�2

k��1−�2�satisfies

ER�ri� = yi and VR�ri� =�k�1�1− �1� yi + �1− yi��k�2�1− �2�

�k��1 − �2� 2

= �2i �say� (4)

2. Alternate Estimators for ppswr and rhc Sampling Schemes

2.1. ppswr Sampling

Here, a sample of n units is selected by ppsw sampling scheme with normed sizemeasure pi = xi/X for the ith unit where xi is known positive size measure for theith unit with X =∑N

i=1 xi. The conventional unbiased estimator for the populationproportion �y =

∑Ni=1 yi/N for the direct response (DR) survey, where yi’s are

directly available from the respondent is

�y =1Nn

n∑i=1

yipi

Since yi’s are sensitive characters and not available from the respondents, a suitablerandomized response was obtained from the respondents in an independent mannersatisfying the model

ER�ri� = yi� VR�ri� = �2i = �yi + �wi + � = and Cov�ri� rj� = 0 for i �= j� (5)

where �� �, and � are non negative constants and wi is an indicator variable takingvalue 0 or 1. The class of RR models satisfy (5) will be denoted by �. For Warner(1965) model, � = � = 0 and � = ��1−��

�2�−1�2 ; Horvitz et al. (1967)’s model � = � = ��1−��/�2� � = 0� wi = qi; and for Kuk (1990)’s model � = k�1�1−�1�

k��1−�2�, � = k�2�1−�2�

k��1−�2�, � = 0,

wi = 1− yi, respectively.The conventional estimator �y for the RR survey reduces to

�yR = 1Nn

n∑i=1

ripi

Here, it should be noted that if the ith unit is selected in the sample ni times, ni

independent randomized responses are obtained from the ith unit.Clearly, the estimator �yR is unbiased for �y and has variance

Var��yR� = Vp�ER��yR� + Ep�VR��yR�

(Ep�Vp� denotes expectation (variance) with respect to the sampling design p)

= Vp

[1Nn

n∑i=1

yipi

]+ Ep

[1

�Nn�2

n∑i=1

�2i

p2i

]

= 1N 2n

[(N∑i=1

y2ipi

− N 2�2y

)+

N∑i=1

�2i

pi

] (6)

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Randomized Response Techniques 1843

Since yi = 1 if i ∈ A and yi = 0 for i � A we can write the above expression as

Var��yR� =1

N 2n

[(∑i∈A

1pi

− N 2�2y

)+

N∑i=1

�2i

pi

] (7)

Here, we propose an alternative estimator for �yR as

�yR�A� =1n

n∑i=1

ri =1Nn

n∑i=1

wi

pi

� where wi = Nripi (8)

Clearly, the proposed estimator �yR�A� is not an unbiased estimator of �y. The biasof �yR�A� is given by

B��yR�A� = E��yR�A� − �y =N∑i=1

yipi − �y

=N∑i=1

yi

(pi −

1N

)= 1

X

N∑i=1

yi

(xi −

X

N

)= 0

if y and x are uncorrelated. So, we get the following theorem:

Theorem 2.1. The proposed estimator �yR�A� is unbiased for �y if the study variable yis uncorrelated with the measure of size x.

Using the expression (6), we obtain the variance of �yR�A� as

Var��yR�A� =1

N 2n

[(∑i∈A

N 2p2i

pi

− N 2P2A

)+

N∑i=1

N 2p2i �

2i

pi

]

(where PA =

N∑i=1

yipi =∑i∈A

pi

)

= 1n

[(∑i∈A

pi − P2A

)+

N∑i=1

pi�2i

]

= PA�1− PA�

n+ 1

n

N∑i=1

pi�2i (9)

Theorem 2.2. For R ∈ �, Var��yR� ≥ Var��yR�A� if y is uncorrelated to w and x.

Proof. The expressions (8) and (9) yield

Var��yR�− Var��yR�A� = Q+G�

where

Q = 1n

(1N 2

∑i∈A

1pi

− �2y

)− PA�1− PA�

nand G = 1

N 2n

N∑i=1

�2i

pi

− 1n

N∑i=1

pi�2i (10)

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1844 Arnab

Now noting NA/∑

i∈A1pi≤∑

i∈A pi/NA, where NA =∑i∈A = number of

individuals belong to the sensitive group A = N�y, we get

Q ≥ 1n

(N 2

A

N 2PA

− �2y

)− PA�1− PA�

n= 1

n

(1− PA

PA

)��y + PA���A − PA� (11)

Now for uncorrelated x and y

�A − PA =N∑i=1

yi

(1N

− pi

)= − 1

X

N∑i=1

yi

(xi −

X

N

)= −N

XCov�x� y� = 0 (12)

Equations (11) and (12) yield

Q ≥ 0 (13)

Further the inequality∑N

i=1 yi/∑N

i=1 yipi ≤∑N

i=1 yipi/∑N

i=1 yi yields

N∑i=1

yi

(1pi

− N 2pi

)≥(∑N

i=1 yi)2∑N

i=1 yipi

− N 2N∑i=1

yipi

= N 2∑Ni=1 yipi

[( N∑i=1

yi

)2/N 2 −

( N∑i=1

yipi

)2]

= − N 2∑Ni=1 yipi

{(∑Ni=1 yi

)N

+( N∑

i=1

yipi

)}{( N∑i=1

yipi

)−(∑N

i=1 yi)

N

}

= − N 2∑Ni=1 yipi

{(∑Ni=1 yi

)N

+( N∑

i=1

yipi

)}NCov�yi� xi�

X (14)

Now putting �2i = �yi + �wi + � from (5), and using (14) we get

G = 1N 2n

N∑i=1

�2i

{1pi

− N 2pi

}

= 1N 2n

[�

N∑i=1

yi

(1pi

− N 2pi

)+ �

N∑i=1

wi

(1pi

− N 2pi

)+ �

N∑i=1

(1pi

− N 2pi

)]

≥ 1N 2n

[�

N∑i=1

yi�1pi

− N 2pi�+ �N∑i=1

wi

(1pi

− N 2pi

)]

Now using (14), we note that if the indicator variables y and w are uncorrelatedto x

G ≥ 0 (15)

The theorem follows from Eqs. (13) and (15).

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Randomized Response Techniques 1845

2.2. rhc Sampling

In rhc sampling scheme, at first, the population is divided at random into ngroups, each consisting of N/n (assuming integer) units. Then, from each group,one unit is selected with probability proportional to the normed size measure pi(>0�

∑Ni=1 pi = 1

). For the rhc sampling scheme conventional unbiased estimator for

�y is given by

�yR�rhc� =1N

n∑i=1

ripi

Pi� (16)

where Pi is the sum of pj values for the group containing the ith unit.The variance of �yR�rhc� is given by

V��yR�rhc� = Vp

[ER

{1N

n∑i=1

ripi

Pi

}]+ Ep

[VR

{1N

n∑i=1

ripi

Pi

}]

= V

(1N

n∑i=1

yipi

Pi

)+ E

[1N 2

n∑i=1

(Pi

pi

)2

�2i

] (17)

Now noting (i) V(∑n

i=1yipiPi

) = N−nn�N−1�

(∑Ni=1

y2ipi− Y 2

) = N−nn�N−1�

(∑i∈A

1pi− N 2�2

y

)and

(ii) E∑n

i=1

(Pi

pi

)2�2i =

∑Nj=1 aj�

2j with aj = 1

n�N−1� �N�n− 1�+ N−npj

} (Arnab, 2004),we get

V��yR�rhc� =1N 2

[N − n

n�N − 1�

(∑i∈A

1pi

− N 2�2y

)+

N∑j=1

aj�2j

] (18)

Following Rao (1966), we may take an alternative estimator for �y as

�∗yR�rhc� =

n∑i=1

riPi =1N

n∑i=1

qipi

Pi with qi = Nripi

Now using (18), the variance of �∗yR�rhc� comes out as

V��∗yR�rhc� = Vp

(1N

n∑i=1

qipi

Pi

)+ Ep

[1N 2

n∑i=1

(Pi

pi

)2

�2i

]where �2

i = �Npi�i�2

= 1N 2

[N − n

n�N − 1�

{N 2∑i∈A

pi − N 2�∑i∈A

pi�2

}+

N∑j=1

aj�2j

] (19)

The expressions (18) and (19) yield

V��yR�rhc� − V��∗yR�rhc� =

N − n

N − 1Q+D�

where Q is defined in (10) and D = 1N 2

∑Nj=1 aj

(�2j − �2

j

) = − 1nN�N−1�Cov�bj� pj� with

bj = �2j �1+ Npj��N�n− 1�+ �N − n�/pj.

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1846 Arnab

Now noting Q ≥ 0 and D ≥ 0 when correlation between each pair (yi, pi) and(bi, pi) becomes zero or negative. So, we get the following theorem.

Theorem 2.3. V��yR�rhc� ≥ V��∗yR�rhc� for R ∈ � when if yi’s are uncorrelated with

xi’s and b′is.

For Warner’s (1965) RR technique �2i = � = constant and hence bi’s have non

positive correlation with pi when bi is a decreasing function of pi which realizes if

npi <√n2�N − n�/�N 2�n− 1� (20)

For N > n+ 1, the condition (9) reduces to npi < 1 and it is likely to be hold for allpractical purposes specially for an inclusion probability proportional to size (IPPS)sampling scheme. Thus we have the following result.

Corollary 2.1. For Warner’s (1965) RR technique V��yR�rhc� ≥ V��∗yR�rhc� when if

yi’s are uncorrelated with xi and npi < 1.

3. Relative Efficiency

In this section, we compare performances of the proposed alternative estimatorswith the conventional estimators based on “Botswana AIDS impact survey” (BIAS)data collected by the CSO, govt. of Botswana. We consider one hundred householdsfrom BIAS data as a finite population. A household is defined as HIV negative(0) if none of the member of the household is HIV positive and a household isHIV positive (1) if at least one of the members is positive. The percentage relativeefficiency of the alternative estimator �yR�A� compared with the conventionalestimator �yR under ppwr sampling scheme and Warner’s RR is given by

E = V��yR�

V��yR�A��· 100 =

(1N 2

∑i∈A

1pi− �2

y

)+ �2

N 2

∑Ni=1

1pi

PA�1− PA�+ �2· 100 (21)

where �2 = ��1−��

�2�−1�2 .Similarly, the relative efficiency of �∗

yR�rhc�compared with the conventionalestimator �yR�rhc� for the rhc sampling scheme under Warner’s RR model isgiven by

E∗ = V��yR�rhc��

V��∗yR�rhc��

· 100 = KH1 + �2H2

KPA�1− PA�+ �2F2

· 100� (22)

where K = N−nn�N−1� , H1 = 1

N 2

∑i∈A

1pi− �2

y , H2 = 1n�N−1� ��n− 1�+ N−n

N 2

∑Nj=1

1pj , and

F2 = 1n�N−1� �N�n− 1�

∑Nj=1 p

2j + N − n .

Here, we note that the efficiency E is independent of the sample size n whileE∗ is not. The information on the household size and HIV status of a household isgiven in Table 1.

From the above table, we find X =∑Ni=1 xi = 288, �y = 49,

∑i∈A pi =

2344,∑100

i=11pi= 1746286,

∑100i=1 p

2j = 018397� and

∑i∈A

1pi= 7040914. The relative

efficiencies E and E∗ are given in Tables 2 and 3.

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Randomized Response Techniques 1847

Table 1HIV status for 100 households of Botswana

hh x y hh x y hh x y hh x y

10000 1 0 10035 1 0 10079 5 0 10107 3 110001 1 0 10036 6 1 10080 1 1 10108 2 110004 1 0 10038 1 1 10081 1 0 10109 3 010005 8 1 10039 2 0 10082 1 0 10110 5 110006 2 0 10041 7 0 10083 4 1 10111 1 110007 1 0 10042 2 1 10084 1 1 10112 5 110008 2 0 10043 2 0 10085 1 0 10113 5 110010 14 1 10044 9 0 10086 2 0 10115 1 110011 1 0 10045 2 0 10087 3 0 10116 1 110013 2 0 10046 4 1 10088 1 1 10117 1 010014 10 1 10047 2 0 10090 1 0 10118 3 110015 3 0 10048 2 1 10091 6 0 10119 8 110016 1 0 10049 1 0 10092 4 1 10121 1 010017 1 0 10051 2 0 10093 5 1 10122 2 110018 1 0 10052 1 0 10094 1 0 10125 15 110019 1 0 10053 4 0 10095 2 1 10128 3 010021 1 0 10054 4 1 10096 1 1 10129 5 110022 3 0 10055 5 0 10099 1 1 10130 1 010023 2 1 10056 5 0 10100 1 1 10131 1 110024 1 1 10058 1 1 10101 1 0 10132 1 110025 5 1 10063 1 0 10102 4 1 10133 2 110028 1 0 10065 1 0 10103 1 1 10134 6 110029 4 0 10070 2 0 10104 3 1 10135 4 110032 5 1 10074 1 0 10105 2 1 10136 1 010033 4 1 10076 1 0 10106 7 1 10137 3 1

hh = household identification, x = household size, y = HIV status.

Table 2Relative efficiency (E) of the alternative estimator under ppswr sampling

� 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9

E 189.2167 182.6875 178.1651 175.5061 175.5061 178.1650 182.6875 189.2167

Table 3Relative efficiency (E∗) of the alternative estimator under rhc sampling scheme

�

n 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9

10 180.5133 168.8916 161.4485 157.2826 157.2826 161.4485 168.8916 180.513315 175.4563 161.4427 152.8441 148.1533 148.1533 152.8440 161.4427 175.456325 164.8285 146.9847 136.9228 131.6907 131.6907 136.9227 146.9847 164.828540 147.4646 126.3341 115.8116 110.6830 110.6830 115.8115 126.3341 147.4646

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1848 Arnab

From Table 1, we conclude that the proposed alternative estimator providesenormous gain in efficiency over the conventional estimator. The gain in efficiencyincreases with the distance of � from .5. For the rhc sampling scheme, the gainin efficiencies of the alternative estimators are also observed. However, the gain inefficiency decreases with the increase of sample size as well as with the decrease ofthe distance of the � value from .5.

Acknowledgment

The author is thankful to the referee for his comments which greatly helped toimprove the manuscript.

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