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Randomized Response: A Survey Technique for Eliminating Evasive Answer BiasAuthor(s): Stanley L. WarnerSource: Journal of the American Statistical Association, Vol. 60, No. 309 (Mar., 1965), pp. 63-69Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2283137 .Accessed: 24/12/2013 08:44

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RANDOMIZEDRESPONSE:A SURVEYTECHNIQUEFOR ELIMINATING EVASIVEANSWERBIAS

STANLEY L.WARNERClaremont raduate chool

For various reasons ndividuals n a sample survey may prefer otto confide o the interviewer he correct nswers o certain questions.In such cases the individuals may elect not to reply t all or to replywith ncorrect nswers.The resulting vasive answer bias is ordinarilydifficult oassess. n this paper t is argued hat such bias is potentiallyremovable through allowing the interviewee to maintain privacythrough he deviceof randomizing isresponse.A randomized esponsemethod for estimating population proportion s presented s an ex-ample. Unbiasedmaximum ikelihood stimates re obtained nd theirmean square errors re compared with he mean square errors f con-ventional estimates under various assumptions bout the underlyingpopulation.

1. INTRODUCTION

FOR easons f modesty, ear f being hought igoted, r merely reluc-tance to confide ecrets o strangers, any ndividuals ttempt oevade

certain uestions ut to them by interviewers. n survey ernacular, hesepeoplebecome he non-cooperative }roup 5, pp. 235-72],either efusingoutright obe surveyed, r consenting o be surveyed ut purposely rovidingwrong nswers o the questions.n the onecasethere s the problem f refusalbias [1, pp. 355-61], [2, pp. 33-6], [5,pp. 261-9]; in the other ase there s theproblem f response ias [3,p. 89], [4, pp. 280-325].

The questions hat people end to evade are the questionswhich emandanswers hat are too revealing. nnocuous uestions rdinarily eceive oodresponse, ut questions equiring ersonal r controversial ssertions xciteresistance. hen esistancesencountered,heusualmodificationf he urveymethod ssimply n added ffort n the part f he nterviewero gain he on-fidence f he nterviewee. here s, however, natural eticence f he generalindividual oconfide ertain hings oanyone-let alone stranger-and hereis also a natural eluctanceo have confidentialtatements n a paper ontain-inghis name nd address. or somequestions t least,probably nly imitedgains re possible hrough rying opersuade he ntervieweehathesurrenderslittle y confiding o the nterviewer.

This paper suggests n alternate method or ncreasing ooperation. hemethod s built on the premise hat cooperation hould e naturally etter fthe questions llowanswerswhich eveal ess even to the nterviewer. ssen-tially he method nvolves hedevice hat-for certain uestions ot alreadyinnocuous-the nterviewee espondswith nswers hat furnish nformationonly n a probability asis.As an example, neapplicationmight nvolve heinterviewee'snlymaking true tatement ith given robability ess than1. In this ase,even he nterviewer ouldknow nly heprobability hat hegiven nswerwastrue. nasmuch s this ype f answer s lessrevealing hanan answer equired o be truthful ith probability , t is suggested hat this

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64 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1965

type of approach may ncourage reater ooperation or ertain urvey rob-lems.As anothermore etailed pplication f herandomized esponsemethod,the following ection utlines particular modelfor stimating populationproportion. he resulting stimates re then ompared ith onventional sti-mates nder arious ssumptions bout the cooperation f those nterviewed.

2. A RANDOM RESPONSE MODEL FOR PROPORTIONS

Suppose hat every person n a population elongs o either Group A orGroupB and t is required oestimate y survey heproportion elonging oGroupA. A simple andom ample fn people sdrawnwith eplacementromthe population nd provisions adefor achperson obe interviewed. eforethe nterviews, ach nterviewer s furnished ith n identical pinner ithfacemarked o that the spinner oints othe etter A with robability andto the etter with robability 1- p). Then, neach nterview, he ntervieweeis asked to spin the spinner nobserved y the nterviewer nd report nlywhether r not the pinner oints othe etter epresentinghegroup o whichthe nterviewee elongs. hat s, the nterviewees required nly osay yes orno according owhether r not he pinner oints othe correct roup; e doesnot report he group o which he spinner oints. Under he assumption hatthese es nd no reports re made ruthfully, aximumikelihoodstimates fthe rue opulation roportion re straightforward.

Let7r=the true probability f A in the population,p=the probability hat the spinner oints oA, and

xi= I if the th sample lement aysyes0l if the th sample lement aysno.

ThenP(Xi = 1) = rp +(-) (1 - p),P(Xj = 0) = (1 - T)p + (1 -p),

and arranging he indexing f the sample o that the first i report yeswhile hesecond n-n1) report no, the ikelihood f the sample s

L = [irp + (1 -p)]n-[(l - r)p + 7r(l - p)]n-n1 (1)The ogof the ikelihoods

logL = nilog [rp + (1 - r)(-p)] (2)+ (n-ni) log [(1 - r)p + r(l -p)],

and necessary onditions n 7r or maximum re(n - n1)(2p - 1) nl(2p - 1)

(1-X)p + r(1-p) - rp (-r)( -p)

or7rp + (1 - 7r) 1 - p) = n * (3)n


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RANDOMIZED RESPONSE 65Then, upposing 1/2, he maximum ikelihood stimate f 7r s

p-i ____e +n

(4)2p-1 (2p-1)n

The expected alue of the estimate s

1r- [p -1 + (1/n) ,EX,]

=- p_1 [p-1 + 7rp (1-lr)(1-p)] (5)

-7r,

and the variance f * isn Var Xi

(2p -1)In[rp + (1 -ir)(1-p)][( -7)p + 7r(l -p)]

(2p - 1)2n1/4 + (2p2 2p + 1/2)(-2ir2 + 27r 1/2)

(2p -1)2niF 1= 1 r 1 _ (X--1/2)21 (6)n - 16(p - 1/2)2 1

Expression5) shows is an unbiased stimate f he rue opulation ropor-tion r.1Moreover, ince r s a maximum ikelihood stimate nd any useful 'sare apt to be large, may be assumed ormally istributed bout r with hevariance ndicated n expression6). Thus all the usual confidence ntervalsare easily stablished. xpression 6) alsosetsout the separate ependence f

the variance f r upon he choice f p. In fact, dentifying-- (ir- 1/2)24_ _-_ (71-r)

n n

as the variance ue to sampling nd writing xpression6) as1 1 1

- (7r 1/2)2

Var =416(p -1/2)2 4

Var 7 = +'

(7)n nit s clear hat hevariance f can beexpresseds the um f hevariance ueto sampling lus hevariance ueto the random evice.

Twopractical uestions oncern heestimation ethod mplied y *. First,how ikely re people ocooperate nd tell the truth when sked o respond n

I The possibility f 79 akingvalues outside he0-1 range annot be ruled out, but this possibility s remote nlarge amples.


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66 AMERICAN STATISTICAL ASSOCIATIONJOURNAL, MARCH1965

the manner described? econd, how arge a sample s required o obtain variousdegrees f precision y this estimate s compared o the conventional stimate?

The first uestion is primarily n empirical question, but the rationale forexpecting etter ooperation s clear. The individual being nterviewed s askedfor ess. The matter of how much ess is summarized by the parameter p. Notefirst rom xpression 1) that if p = 1/2, the likelihood function oes not evendepend on ir. Thus, for p = 1/2, the interviewee would be furnishing o infor-mation at all. Then note that if p= 1, the entire procedure would reduce to theconventional procedure of requiring the individual to state unreservedlywhether r not he belonged o Group A. For p's between 1/2 and 1 (or between1/2 and 0) the person nterviewed rovides useful but not absolute informa-tion as to exactly which group he is in. In this context hep can be thought f asdescribing he nature of the cooperation etween he nterviewer nd the inter-viewee. As p goes from 1 to 1/2 the burden of cooperating passes from heinterviewee o the interviewer. t therefore eems reasonable to expect that forsome questions t least, p's less than 1should nduce greater ooperation n thepart of the person nterviewed.

The question of the sample size required for a given evel of precision lsodepends on the parameter . If a p close to 1 (or closeto 0) is adequate to insurecooperation, hen a smaller sample size is required than if a p close to 1/2 isrequired o insure ooperation. Values of p close to 1/2 convey ess informationfrom ach interview, hus hey lso imply ither larger variance of the estimateor a larger ample size. Substituting alues of p in expression 6) sets out theprecise relation. As an example, supposing a 7r .5 and a p halfway betweenthe zero and full nformation oints, .e., a p of 75, the variance shown by (6) is1/n. This would imply that the sample size should be about 400 in order tosecure a standard deviation of .05. By way of comparison, he conventionalestimation method equivalent to a p =1) would imply that a sample of onlyabout 100 would be sufficient or a standard deviation of .05-provided thatall the interviewees old the truth for he regular method.

The more pertinent omparisons re between he randomized stimates.andregular estimates under the assumption hat the regular estimates re handi-capped by less than 100 per cent truthfulness. uppose that in a regular urveyall consent to be surveyed, but members f Group A tell the truth only withprobability Ta and members f Group B tell the truth only with probabilityTb. Then, if Y,= 1 or 0 according s the ith member of the sample reports heis or is not in Group A, the conventional estimate of the true populationproportion is

n7r = - *(8)

n

The expected value, response bias [3, p. 89], and variance of this regular sti-mate are given by

EV = 7rTa + [(1 -7r)(1 - Tb)], (9)


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RANDOMIZED RESPONSE 67Bias' --E(7r-7r)

=r[Ta+ Tb-2] + [1-Tb], and (10)a [rTa + (1- ir)(1 - Tb)][O1 - rT4 -(1- 7r)(1 Tb)]

Var7r1

nTables 1 and 2 then compare the mean square errors the variance plus thesquare of the bias) of the randomized nd regular methods of estimation nderthe assumption hat the nterviewed ndividuals ell the truth n the randomizedmethod but only tell the truth n the non-random method with probabilitiesgiven by Ta and Tb. The left-hand wo columns of each table indicate various

TABLE 1. COMPARISON OFRANDOMIZED ANDREGULAR ESTIMATES

FOR TRUE PROBABILITY OFA=.6 ANDrn=1000Mean Square Error Randomized

Regular Estimates______ _ . Mean Square Error RegularProbability f Truth Bias

To To la p=.6 p=.7 p=.8 p =.9

.95 1.00 -.03 5.45 1.36 .60 .33

.90 1.00 -.06 1.62 .40 .18 .10

.70 1.00 -.18 .19 .05 .02 .01

.50 1.00 -.30 .07 .02 .01 .001.00 .95 .02 9.82 2.44 1.08 .601.00 .90 .04 3.41 .85 .37 .211.00 .70 .12 .43 .11 .05 .031.00 .50 .20 .16 .04 .02 .01

.95 .95 -.01 18.25 4.54 2.00 1.11

.90 .90 -.02 9.70 2.41 1.06 .59

.70 .70 -.06 1.62 .40 .18 .10

.50 .50 -.10 .61 .15 .07 .04

TABLE 2. COMPARISON OFRANDOMIZED AND REGULAR ESTIMATESFOR TRUE PROBABILITY OF A=.5 AND n=1000

Regular Estimates Mean Square Error Randomized

_____________________ - Mean Square Error RegularProbability f Truth Bas .Ta Tb P=a p.6 p=.7 p=.8 p-.9

.95 1.00 -.03 7.15 1.79 .79 .45

.90 1.00 -.05 2.28 .57 .25 .14.70 1.00 -.15 .28 .07 .03 .02

.50 1.00 -.25 .10 .03 .01 .01

.95 .95 .00 25.00 6.25 2.78 1.56

.90 .90 .00 25.00 6.25 2.78 1.56

.70 .70 .00 25.00 6.25 2.78 1.56

.50 .50 .00 25.00 6.25 2.78 1.56


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68 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1965paired alues or aand Tb.Thethird olumn hows he biasof henon-randommethod, nd the remaining olumns xhibit he atios f hemean quare rrorsof he randomizedstimates o the mean quare rrors f heregular stimatesfor arious alues f p. Tables1and2are

respectivelyppropriateor he

aseswhere hetrue probability f A is .6 and 5. The sample ize s set at 1000 neach ase.

3. CONCLUSIONS

Both ables re constructed nder he ssumption hat hep in each case slowenough o induce ull ooperationn the randomized pproach. hus theadvantages f he randomized ethod, hown ythose atios nthe ables hatare ess han 1, are n the nature f potential dvantages hat depend pon he

cooperationctually chieved y the randomized ethod. evertheless, hereis the clear uggestion hat the randomized ethod s apt to out-perform heregularmethod n a variety f situations. able 1 with Ta= 1 and Tb= .9, forexample,xhibits he ituation nwhichmembers f he minority opulationresent irectly onfidingotheir nterviewer heirminority tatus othe pointwhere enper cent of them ay A instead f B. The bias created s +.04, andthe ratio f mean quare rrors aries rom .41to 21,depending n the valueofp. Thepossiblemprovementhrough andomizationn this ase s evident.Anevengreater mprovementspossiblef t s the arger opulation hathesi-

tates to identify tself penly.This latter ase is exemplified y the row nwhich a.=9 and Tb=1.Moregenerallyt s to be observed hat-except or he caseswhere he bias

of he regular stimate s 0 or negligible-there ppear obe sizablepotentialgains hrough herandomized esponse.t should lso be kept nmind hat hepotential dvantages frandomizingre even arger or arger amples. or ex-ample, sample izeof 000would mply hat he ntry nTable1,column ,row2, would hange rom .62to .84. Thus the randomized method s to be pre-ferred n this nstance ven f p as low s .6 s required o assure ooperation.

The question s still openas to what methods f randomized esponse illprove hemost seful. ven with egard oestimating roportions, hemethodset out n Section is only ne of manypossibilities.t is interesting onoteinthis onnectionhat mathematicallyquivalentmodel o the oneof ection2 is furnished y simply equiring ach nterviewee omake statement hatis true with robability as to which f the two groups e is in. Thus n thismodel, he nterviewee, gain out of sight f the nterviewer, pins spinnerwhich oints o true with iven robability and to false with robability(1- p). Then he nterviewee akes statement hat s true r false ccording

to the waythe spinner ointed. sychologicallyhiswould ppear obe quitea different odel rom hat f ection ,but the tatistical roperties f he womodelsreequivalent.2hemaximumikelihoodstimate or he atter chemehasthe ame form nd the ame variance s the stimate fSection . There s

2Asbefore, p of furnishes o information, p of 1 furnishes ull nformation, nd other 's furnish nforma-tion depending n how far hey re from J. t is a feature f the dichotomous ature f the population hat tellingthe truth 2 of the time s equivalent o telling he truth 8 of the time.


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RANDOMIZEDRESPONSE 69thus question s to which f these r other quivalent andomized odels sto be preferred rom he tandpoinlt f ncreasing ooperation.

Finally, t should enoted hat t s easyto extend herandomized esponsetechnique o estimate istributions ther han that appropriate o a simpledichotomousariable.Asoneexample, hetechnique ouldbe applied oesti-mate five-classncome istribution hrough heobvious evice f stimatingthe proportionn eachclass eparately ythe method fSection . In this aseeach nterviewee ight e simply sked o make five eparate andomizede-sponses oncerning hether r not he wasin eachof the five eparate lasses.Just s with he proportion roblem,t s clear hat ther andomized esponsemethodsmay be imagined or his more eneral stimation roblem. nd ustas with he proportion roblem, he question f which pecific echniquewillprove uperior s a matter or mpiricalnvestigation.

4. ACKNOWLEDGMENTS

I am ndebted o the referee or elpful uggestions.REFERENCES

(1] Cochran, W. G., Sampling Techniques, econdEdition. New York: John Wiley andSons, nc., 1963.

12]Deming,W. E., SomeTheory f ampling.New York: John Wiley nd Sons, nc., 1950.[31Hansen, M. H., Hurwits, W. N., and Madow, W. G., Sample Survey Methods nd

Theory, olume . New York: John Wiley nd Sons, nc., 1953.[4] Hansen, M. H., Hurwitz, W. N., and Madow, W. G., Sample SurveyMethods ndTheory, olume I. New York: John Wiley and Sons, nc., 1953.

[5] Stephan, F. F., and McCarthy, . J., Sampling Opinions.New York: John Wiley andSons, nc., 1963.

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