An Improved Randomized Response Strategy
Author(s): N. S. Mangat
Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 56, No. 1
(1994), pp. 93-95
Published by: Wiley for the Royal Statistical Society
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J.R. Statist.Soc. B (1994)
56, No. 1,pp. 93-95
An ImprovedRandomizedResponse Strategy
By N. S. MANGATt
PunjabAgricultural
University,
Ludhiana,India
[ReceivedJuly1991. Final revisionAugust 1992]
SUMMARY
A randomizedresponseprocedureis proposedwhichhas thebenefitof simplicity
overthat
of Mangat and Singh.Conditionsare obtainedunderwhichtheproposedstrategyis better
thanthoseof Warneror Mangat and Singh.
Keywords: EQUAL PROBABILITIES WITH REPLACEMENT; ESTIMATION OF PROPORTION;
RANDOMIZED RESPONSE TECHNIQUE
1. INTRODUCTION
To procure
dataforestimating
ofa population
trustworthy
ir,theproportion
havinga
sensitiveattributeA say, Warner(1965) introduceda procedurecalled the
randomizedresponse(RR) technique.Subsequently,
severalotherworkershave
RR strategies
variousalternative
suggested
(see Hedayatand Sinha (1991) fora
review).Recently,
Mangatand Singh(1990)proposeda two-stage
RR procedure.
Theirmethod,requiring
theuse of tworandomized
devices,makestheinterview
Thismotivated
a littlecumbersome.
procedure
theauthorto considera relatively
Thesituation
wheretherespondents
simpleRR strategy.
arenotcompletely
truthful
intheiranswershas also beenconsidered.
2. METHOD PROPOSED
Each of n respondents,
assumedto be selectedby equal probabilities
with
isinstructed
tosay'yes'ifheorshehastheattribute
replacement
sampling,
A. Ifheor
she does not have attribute
is requiredto use the Warner
A, the respondent
deviceconsisting
oftwostatements:
randomization
A' and
(a) 'I belongto attribute
(b) 'I do nothaveattribute
A',
withprobabilitiesp
andI -p respectively.
represented
Thenheorsheistoreport'yes'
to theoutcomeofthisrandomization
or 'no' according
deviceandtheactualstatus
toattribute
thatheorshehaswithrespect
A. Thewholeprocedure
iscompleted
bythe
unobserved
The probability
respondent
of a yesanswerforthis
bytheinterviewer.
is givenby
procedure
a = Xr+ (l-ir)(l-p).
Ludhiana 141004,
tAddressforcorrespondence:Departmentof Plant Breeding,Punjab Agricultural
University,
India.
i 1994 RoyalStatisticalSociety
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All use subject to JSTOR Terms and Conditions
0035-9246/94/56093
94
[No. 1,
MANGAT
inbothgroupA andgroupnot-A,
Sincetheyesanswermaycomefromrespondents
yeswillnotbe violated.However,the
of thepersonreporting
theconfidentiality
'no' willalwayscomefromgroupnot-A.Sinceitisassumedthat
reporting
individuals
are
stigma,therespondents
to groupnot-Adoesnotcarryanyperceived
belonging
assumednotto objectto theprocedure.
canbe easily
of 'rfortheproposedprocedure
likelihood
estimator
Themaximum
seento be
7rm=
(1+p)/p,
of yesanswersobtainedfromthen sampled
where&^is theobservedproportion
individuals.
for
*m is unbiased
B(n, a), theestimator
Since(xfollowsthebinomialdistribution
to thefollowing
result.
7r.Thisleadssimply
v(*m) arerespectively
Theorem1. ThevarianceV(rm)anditsunbiasedestimator
givenby
V(jm)
=
r(1-x)/n + (1-wr)(1-p)/np
(2.1)
and
V(Wm)=
(I
- a!)l(n-I)p
2.1. EfficiencyComparison
On usingequation(2.3) of Mangatand Singh(1990)and equation(2.1) of this
theorem.
paper,we arriveat a resultstatedinthefollowing
*m willbe moreefficient
thantheMangat
Theorem
2. The proposedestimator
if
and Singh(1990)estimator
7 > I - p(l T){1 -(1 -T)(l)} P))
ii> 1
{12p-1+ 2T(I1-p)}2
9
P 9 0.5,
'O5
(2.2)
22
intheirfirst
thesensitive
deviceandp
attribute
whereTis theprobability
representing
hasdecidedthevaluesofp and T,itiseasytoverify
isas above.Oncetheinvestigator
(2.2) byusinga priorguessaboutthevalueof 7rfroman earlierstudyor
inequality
T= 0, forwhichMangatandSingh's(1990)resultreducesto
On putting
pilotsurvey.
(2.2), weobtainthefollowing.
Warner's(1965)result,in inequality
ismoreefficient
thanthatofWarner(1965)if
3. Theproposedstrategy
Theorem
X > 1 - { p/(2p _ 1)}2,
whichalwaysholdsforp>
T3
3. INCOMPLETELY TRUTHFUL REPORTING
thattherespondents
tothesensitive
classreport
belonging
Let T3betheprobability
inthenon-sensitive
Therespondents
grouphaveno reasontotella lie.The
thetruth.
thusbecomes
ofyesanswersfortheproposedprocedure
probability
, = wT3+ (l-7r)(1-p).
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1994]
RANDOMIZED RESPONSE
95
Here,theestimator
*m is biasedwith
magnitude
ofbiasgivenby
B(*m) = 1r(T3-l)/p.
Thenwehavethefollowing
theorem
forwhichtheproofis straightforward.
Theorem4. Themean-square
erroroftheestimator
*m is givenby
MSE(*m) ['rT3(1-XT3) + (1 -ir)(1 -p){(1 - (1 - w)(1 -p) - 2xT3}
+
{(T3
_
1)}2]
/p2.
(3.1)
Thisexpression
can be comparedwithexpressions
(1.3) and (3.3) of Mangatand
Singh(1990)foran efficiency
comparison
basedontheminimum
mean-square
error.
ACKNOWLEDGEMENTS
The authoris grateful
to therefereeand ProfessorRavindraSingh,fortheir
valuablecomments
and suggestions.
REFERENCES
Hedayat,A. S. andSinha,B. K. (1991)DesignandInference
inFinitePopulation
Sampling.
NewYork:
Wiley.
Mangat,N. S. and Singh,R. (1990)An alternative
randomized
response
procedure.
Biometrika,
77,
439-442.
Warner,S. L. (1965)Randomized
response:a survey
foreliminating
technique
evasiveanswerbias.
J.Am. Statist.Ass., 60,63-69.
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