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Warner's Randomized Response Model: A Bayesian Approach

Author(s): Robert L. Winkler and Leroy A. FranklinSource: Journal of the American Statistical Association, Vol. 74, No. 365 (Mar., 1979), pp. 207-214Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286752.

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Warner'sRandomized esponseModel:A BayesianApproachROBERT . WINKLERnd LEROYA. FRANKLIN*

In randomizedesponseampling, riornformations of particularvaluebecausethe randomizationffectivelyeduces heamount fsample nformation. arner'smodelwith beta prior istributionfor heproportionf nterest ields osterior istributionshataremixturesfbetadistributions.hese mixturesrereadilynterpret-able, but approximationshat provide eta posterior istributionssimplifyhe analysis.Anexamplendicates hat he approximationsare quiteaccurate, rovidesome nsightnto he effectfdifferentpriordistributions,nd demonstrateshe practicalvalue of priorinformationn the attempt o obtainprecise stimateswhenran-domized esponsemethods reused.KEY WORDS: Randomizedesponse; ayesiannference; arner'srandomizedesponsemodel;Mixturesfbeta distributions;pproxi-mateposterioristributions.

1. INTRODUCTIONIn survey sampling,problemsoften arise when re-spondentsare asked sensitive questions (e.g., questionsabout highly ersonalor controversialmatters).To avoidproviding he requestedinformation,ome respondentsmay refuseto answer questions or may intentionallyfalsify heir answers. Thus, it is difficulto make in-ferencesbout sensitive opicsbased on survey amplingin which ensitive uestionsare asked directly.An attempt o avoid suchdifficultiess represented ythe development f randomized esponse amplingplans,which nject a probabilistic lement nto the questioningprocedure.A chance device is used by the respondent oselectthequestionthatis to be answered.As long as theoutcomeofthe chancedevice is not knownto the inter-viewer, herespondent an answerhonestlywithout ullyrevealing nformationegardinghe sensitive ssue.Sincethe propertiesof the chance device are knownto theexperimenter, he answer provides some informationabout thesensitive opic,albeitnotas much nformationas an honest answer to a directquestion. The loss of

informationecause of the randomizationmay be a muchless seriousproblem hanthe oss of nformationecauseofnonresponse r lyingwithout herandomization.Several randomized responseplans have been devel-oped, and a comprehensiveurveyof such plans is pro-vided in Horvitz,Greenberg, nd Abernathy 1976). Inthisarticle,we focuson theoriginal andomized esponsemodel proposed by Warner (1965). The study of in-*Robert . Winklers ProfessorndLeRoyA. FranklinsVisitingAssistantrofessor,oth ntheDepartmentfQuantitativeusinessAnalysis, raduate choolofBusiness,ndianaUniversity,loom-ington,N 47401. The authors re gratefulo the refereesnd as-sociate ditor or heir elpfulommentsn an earlier ersionf hisarticle.n particular,he ssociate ditoruggested pproximationinSection .

ferences ased on Warner'smodel (and other andomizedresponse models as well) has been limitedto classicalprocedures. ome specific pplications, ncluding studyof organized crime and a study of the incidence ofabortions, are discussed in Horvitz, Greenberg,andAbernathy 1976).The objective of this article s to present Bayesianapproach to Warner's model. The Bayesian frameworkprovides natural way to studyand interpretituationssuch as randomized response sampling, where onlypartial nformations available. Furthermore,hepracti-cal value ofthe Bayesian approachis particularly reatwhen sample informations relatively imited,and oneeffect of randomizationis effectively o reduce theamountofsampleinformation.Warner's randomized responsemodel is discussed inSection 2, and beta prior distributionsfor Warner'smodel areused inSection3. In Section4, approximationsthat facilitate heapplicationof Bayesian procedures orthe Warnermodel are presented.To illustrate he pro-cedures,randomizedresponsedata fromLiu and Chow(1976) concerninghe incidenceof abortions n Taiwanare analyzed in Section 5 using the methodologyofSections3 and 4. Section6 contains brief ummary nddiscussion.

2. WARNER'S ANDOMIZED ESPONSEMODELIn Warner's model, a respondent nswers "Yes" or"No" to eitherthe sensitivequestionof interest r thecomplementaryuestion.For example, suppose thatweare interestedn whether personbelongsto GroupA.The respondent ses a chancedeviceto selectQuestion1,"Do you belongto GroupA?," or Question2, "Do you

belongto GroupAC?,"whereAc is thecomplement f A.Thus, iftherespondent ays "Yes," the interviewer oesnot know whetherthe "Yes" refers o Question 1 orQuestion2.We assume that simple randomsampling s used togeneratethe sample and that the parameterof interestis 'r,the proportion fthe sampledpopulationbelongingto GroupA. However,the data bear moredirectly ponthe parameter X, the probability that a randomlyselectedmember fthepopulationwillanswer"Yes." If? Journal f theAmericanStatisticalAssociationMarch1979,Volume74,Number365

Theory nd Methods Section207

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208 Journal ftheAmericantatisticalAssociation, arch 979p denotesthe probability hat the chance device selectsQuestion1, then r and X are relatedas follows:

X= (2p-1)r + (1-p) , (2.1)or r= [X-(l-p)]/(2p-1) (2.2)Here p is chosenby the experimenternd is thereforeknown, and we assume without oss of generality hatp > 0.5.Suppose that a sample of size n withreplacement orwithoutreplacement rom population arge enoughtobe consideredeffectivelynfinite)results in r "Yes"answers.The samplingprocess s Bernoulli n X,and thelikelihood unctions4(r,n X) = Xr(1 - X)n-r for 1 - p < X < p (2.3)In terms f ir,4(r,n r) = [(2p - 1)ir (1 - p)]r[p - (2p - 1)7r]n-rfor O 0. In this article, f and Frepresent ensityfunctions nd cumulativedistributionfunctions, espectively.Also, parameters re treated asrandom variables in the Bayesian approach, but thedistinctionbetween a randomvariable and its valuesshould be clearfrom he context, o for implification eomitany distinction n termsofsymbols e.g., no tildesorupper-case etters re used in thissense).The likelihood unction or sample ofr "Yes" answersand n - r "No" answers is given by (2.4), and theposterior ensity s thereforef r Ir, n) oc ia'-'(1 - r)''-1[(2p - 1) r

+ (1 - p) ]r[p - (2p - 1) r]n-r (3.2)for 0 < r < 1. Expanding the expressions n squarebrackets, implifying,nd normalizing, e get

nf r Ir, n) = wt Qr Ia' + t, /' + n - t) (3.3)t=OforO < r < 1,where

nWt= Wt*/E w8* (3.4)8=0and

W n\ B(a' +t, /3'+ n - t)tJ B (a', /3')min(r, t) t\In -t\* ( j). r-j pn-t-r+2j(l - p)t+r-2i . (3.5)j=o \j r- j

The posteriordistribution f xr, hen, s a mixture fbeta distributions. oreover, heweightswo, . , wnhavean intuitivelyppealing interpretation.f we define asthe actual number frespondentsn Group A, thevalueoft s neverobserved, ut before hesample s taken,thepredictive istributionft sa beta-binomial istribution:P (tjn a', /3')= (n\ B (' + t /3'+ n - t)kt B(' i3')

for t =O,1,...,n . (3.6)Also, givent,the distribution fr,thenumberof "Yes"answers,can be foundby separatelyconsidering i, thenumber f"Yes" answers mongthet nGroupA and r2,the numberof "Yes" answersamong the n - t not inGroupA. Givent,riand r2 rebinomial, nd thedistribu-tion of r = ri + r2 is simplya convolutionof the twobinomial distributions:

minr, ) t\n -t\P(r in,t) = , (t)(n t) pn-t-r+2j(1 - p)t+r-2jj=O \g/ \r- j (3.7)for r = 0, 1, .. ., n. From (3.4)-(3.7), we have:Wt = P(tl n,ae',/3')P(rj , t)/E P(s In, e', 3')P(r n, )

8 =0= P (tj|n,r) . (3.8)

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Winklernd Franklin: andomizedesponse: Bayesian pproach 209Hence,each term n the sumin (3.3) is theproductof

(a) the posterior robability hat exactlytof thenrespondentsctuallybelongto GroupA; and(b) the posteriordistribution f 7r given that ex-actly t of the n respondents ctuallybelongtoGroupA.Because of the randomized response plan, we neverobservet, and the uncertaintybout t is what causes theposteriordistribution f 7r to be a mixtureof n + 1differenteta distributionsnsteadof just a singlebetadistribution.With non-randomized-responseampling,t = r is observed,so that w= 1 and wt= 0 fort $ r,which eads to a singlebeta distributionn (3.3).

4. APPROXIMATIONSThe beta priordistributionor rsuggestednSection3leads to a posterior istributionhat is easy to explain nterms ftherandomized esponseprocedure utmaynot

be easy to workwith n practice.The computations n-volved in determininghe weights n (3.4) are cumber-some, and once the weights are determined,findingprobabilitiesfromthe posteriordistributionmay bedifficult. or instance,to findthe posteriorprobabilityassociated with an intervalof values of irnecessitatesfindinghe probability f that intervalforn + 1 differ-ent beta distributions.As n increases,the difficulties,both n terms f the weights nd interms f workingwiththe posterior istribution,re intensified.Since natural-conjugateprior distributions re usedwidely in Bayesian inferencebecause of their tract-ability,a natural-conjugate pproximation o the betaprior distribution or ir mightbe viewed as a usefulalternative. t is thereforef nterest o showbrieflywhynatural-conjugate istributionsrenot easytoworkwithin the randomized response situation. From (2.3),Warner'smodel behaves like a Bernoulliprocess in Xwith the restriction hat 1 -p < X < p. Thus, betadistributions runcated at 1 -p and p are natural-conjugateforX, nd the implieddistributionor r isf r) = K(a, b) 2p -) f,s[(2p - 1)7r (1 -p) la, b]for O < r < 1, (4.1)where K(a, b) = [F#(p a, b) - Fp 1 - pla, b)j', witha > 0 and b > 0. This distribution s a "stretched"truncatedbeta obtained by taking the truncatedbetaf(X) defined n [1 - p, p] and stretchingt out to cover[O, 1]. The mean and varianceof ir reE(ir) = {[aK(a, b)/(a + b)K(a + 1,b)] - (1 -p)}(2p - 1) (4.2)and

V(7r = 2p 1)a(a + 1) [ K(a, b((a + b) a.+ b + 1) K(a + 2, b)J_(a + -)[K b)( 3

If the priordistribution f 7r is given by (4.1) and asample results n r "Yes" answersand n - r "No" an-swers, hen theposterior istribution f7r s of the sameform s (4.1) witha and b replacedby a + r and b + n- r,respectively. hus,Bayes' theorem s easy to apply,but terms uch as K (a, b) make it difficulto workwiththe prior nd posterior istributions.

In the remainder f thissection,we return o thecaseofbeta priordistributions or r nd we consider wo ap-proximationshat seem to providereasonable fitswhilebeing relatively asy to use. The approximations epre-senttwo differentpproaches,one using an approximatepriordistribution nd an exact likelihoodfunction,ndthe otherusing an exact priordistributionnd an ap-proximateikelihoodfunction.Approximation involvesbeta approximations or heprior nd posterior istribu-tions of X, and Approximation involves a Bernoulliapproximation o (2.4), the likelihoodfunctionfor 7r.First,we discussApproximation .The beta distribution or7r givenby (3.1) impliesadistribution orX hat s a betadistributionn [1- p,p],f(X) = [B (a', d')]-I(X - 1 + p) a'-l(p - ),0'-1-2p-1)1-a'-.8' for 1 - p < X< p X (4.4)which snot hesame as a beta distribution n [0, 1] thatis truncated t 1 - p and p. Thus, f X) in (4.4) doesnotcombine ractablywith he ikelihood unctionn (2.3). Iff(X) can be approximatedby a beta distribution, ow-ever, heapplicationofBayes' theorem s simple,yieldinga posterior eta distribution orX.The approximation anthen be used in reverse o find n approximate osteriordistributionorXwhich mplies beta distributionorv.To make the approximation s easy to deal with as pos-sible,we ignore he imitation fXto [1 - p, p] inorderto work with untruncatedrather than truncatedbetadistributions orX. When the approximation s used inreverse fter he inclusionofthe sampleevidence, dis-tributiondefined n [1 - p, p] for X is found, nd thecorresponding istribution orir is properlydefinedon[O, 1].We generate beta approximationo f(X)by equatingmeans andvariances.Let a* and,8* enote heparametersofthebeta approximation,whichmeansthat

a= E2(X)[1 - E(X)][V(X)]-' - E(X) (4.5)andA8* E(X)[1 - E(X)]2[V(X)]-' - [1 - E(X)], (4.6)where (X) = (2p - 1)E(7r) + (1 -p),

V(X) = (2p - 1)2V(7r) , E(Xr) a'/ a' +,/')and V(r) = ax'/'/(a' + /'8)2(a' + Of+ 1). Givenr "Yes"answers n n trials,the posterior istribution orXcor-responding o the priorbeta approximation s anotherbeta distributionwith parameters ax**- a* + r and,8*= /3* n -r, and we want to approximatethisbeta distributionwith a distribution f the form 4.4)withparameters " and /3". Equating means and vran-

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210 Journal ftheAmericantatistical ssociation, arch1979ances onceagain,we getae" { E X r,n) - 1 + p]2[p - E(X r,n)]- [E(xJr,n) - 1 +p]V(XIr,n)}(2p - 1)-'E[V(x ,n) -I (4.7)and/8" {[E(xlr, n) - 1 + p]Ep - E(XIr, n)]2- [p - E(X r, n)JV(Xr, n) I 2p -1)-1[V(xI ,n) -I , (4.8)where E(XItr, ) = a**/at**+ A3**)and

V(XIr,n) = a**/3**/(a** 3**)2((a** + 3** + 1)The approximateposteriordistribution or 7r is then abeta distribution ithparameters " and ,B".Next,we considerApproximation , which focusesonthe likelihoodfunctionnstead ofthe priordistribution.A likelihoodfunction ftheform

t*(r*, *7) = r*(1 - )n*-r* (4.9)combines ractablywith hebeta priordistributionivenby (3.1). Thus,wewantto find *and n*to make (4.9) agood fit to (2.4). Provided that 1 - p < r/n< p,equating modes and the curvature at the modes oflog 4(r,n I r) nd logt* r*,n* r)yields

r*/n*= [(r/n) - (1 - p)]/(2p - 1) = *, (4.10)the maximum ikelihoodestimateof ir from 2.6), and

* A (1C2) [ 2log t(r, nlIr)1

n(2p - 1)2* 1 -) (4.11)-( A) (.1

whereS=r/n. The approximateposteriordistributionfor r s then a beta distribution ithparameters " = a'+ r* and d" = 3' + n* - r*.The sample results an bethought of as approximately equivalent to a non-randomized-responseamplewithr* "Yes" answers ndn*- r*"No" answers.The examples in Section 5 shed some light on the'goodness" of the two approximations hat have beenconsideredn thissection. t is useful o notethat in thelimit,as p-* 1, p- 0.5, n-*0, or n-* oo, both ap-proximationsgreeperfectly ith heexact results.Whenp -*1, a" -* a' + r and /" --*1' + n - r forboth ap-proximations s the data-generating rocess approachesa Bernoulli process in 7r.As p -* 0.5, a" -- a' and"1 ' in both cases, which means that in the limit,rand n - r provideno informationbout 7r. f n - O,a" -* a' and ,B -* /', which makes sense because theamount of sample information iminishes o nothing.Finally,as n * oc, thepriordistributions overwhelmedby the sample evidence and the central imittheoremimplies hat withprobability ne,the exactposterior is-tribution nd both approximations pproach a normal

distribution ithmean* and variance (1- *)/n* (seeLindley1965, p. 129).5. AN EXAMPLE: BORTIONSN TAIWAN

To illustratethe Bayesian procedures. eveloped inSections 3 and 4, we considerdata reported n Liu andChow (1976) concerning he incidenceof inducedabor-tions nTaichung,Taiwan. Liu andChow used a multiple-trial version fWarner'smodelinwhichtherandomizedresponseprocedurewas repeated forthe same questionthreetimesper respondent n orderto obtain more in-formation nd increase heefficiencyfestimation. incewe are concerned nlywith hesingle-trialWarnermodelinthisarticle,we use only thedata from hefirst rialforeach respondent.The population of interestn the Liu and Chow studywas marriedwomen of age 20 to 44, and GroupA con-sisted of those women who had experienced nducedabortions. With p = 0.3, the surveyyielded 90 "Yes"answers nd 60 "No" answers.The procedures evelopedhere assumethatp > 0.5, so we convert he data to theequivalent = 0.7,r = 60,andn - r = 90. The maxi-mum ikelihood stimates f Xand ir re, from 2.5) and(2.6), X = 0.40 and * = 0.25, and theestimatedvarianceof * is 0.01.We considerfivedifferenteta priordistributions or7r: a' = 1, d' = 1; a' = 2, B' = 4; a' = 10, d' = 20;a' = 2, /' = 8; and a' = 10, d' = 40. For each of thesedistributions,he exact posteriordistribution or r (asgiven in Section3) was foundnumerically nd two ap-proximatebeta posterior istributions or7r as given nSection5) werealso found.The required ntegrals, ereand for other numericalresults in this section, wereevaluatedvia RombergQuadrature.The five priordensitiesand the correspondingxactand approximate osterior ensities romApproximation1 are shown in the figure.The posteriormeans andstandard deviations are presented n the table (underr = 60, n = 150), as are the maximumdifferencese-tween the exact and approximateposterior umulativedistributionunctions,

D = max Fexact(7rr, n) - Fapproximate(7r,n) , (5.1)for each case. Furthermore, our other sample sizes(n = 15, 75, 300, and 450) are considered n the table,withr/nheld constant t 0.4.The inferentialstatement of primary interest inBayesian analysis s an entireposterior istribution,utpoint nd interval stimates an,ofcourse,be determinedfromthat distribution.The posteriormeans of ir pre-sentedin the table can be considered s pointestimatesof 7r nd can be comparedwith themaximum ikelihoodestimateof7r, .25, to providean indication ftheeffectofdifferentriordistributionsn theestimate.The priordistribution auses a shift n the estimatetoward thepriormean of r, ' /(a' + /'), and this hift ppears tobesubstantial s a proportionf a'/a' (+ /3')] -* unlessn

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Winklernd Franklin: andomizedesponse: Bayesian pproach 211ExactandApproximate osteriorMeans (,i") and StandardDeviations o-") nd Values ofDfor elected Beta PriorDistributions or T and Selected Sample Outcomes

r =6, n =15 r=30, n =75 r =60, n =150 r = 120, n =300 r = 180, n =450Ap- Ap- Ap- Ap- Ap- Ap- Ap- Ap- Ap- Ap-Ex- prox. prox. Ex- prox. prox. Ex- prox. prox. Ex- prox. prox. Ex- prox. prox.a/ B, act 1 2 act 1 2 act 1 2 act 1 2 act 1 2

1,1 Itt" .361 .385 .379 .266 .298 .294 .254 .276 .274 .252 .264 .263 .251 .259 .259O' .226 .214 .220 .130 .127 .130 .098 .095 .096 .070 .069 .069 .058 .057 .057D .054 .034 .086 .070 .073 .062 .054 .045 .043 .0372,4 A" .305 .313 .313 .268 .282 .283 .257 .270 .270 .253 .261 .261 .252 .258 .258' .155 .155 .156 .110 .111 .111 .088 .087 .088 .066 .066 .066 .055 .055 .055D .017 .018 .040 .040 .043 .043 .037 .038 .033 .03310,20 u" .327 .328 .328 .310 .312 .313 .297 .299 .301 .283 .285 .287 .275 .277 .279' .081 .082 .082 .071 .073 .074 .063 .065 .065 .053 .054 .055 .047 .048 .048D .003 .005 .009 .018 .012 .024 .013 .027 .012 .0272,8 Itt" .201 .207 .208 .209 .221 .224 .219 .230 .233 .230 .237 .239 .236 .241 .242' .114 .113 .113 .096 .092 .092 .082 .077 .077 .064 .061 .061 .054 .052 .052D .024 .028 .058 .067 .059 .068 .046 .054 .037 .04510,40 u" .201 .202 .202 .206 .207 .208 .211 .212 .214 .218 .220 .221 .223 .225 .226

Gi .055 .055 .055 .053 .052 .052 .050 .049 .049 .046 .044 .044 .042 .040 .040D .003 .004 .012 .017 .018 .026 .021 .032 .021 .034

is very large relative to a' + d'. For instance,whena' 10, /' = 40, r = 60, and n = 150, the posteriormean (0.211) is much closer to the prior mean (0.20)than to themaximum ikelihood stimate.The posteriordistributions re all positively skewed, so that theirmodes, whichcan also be considered s point estimates,are slightly owerthan the correspondingmeans. As forprecision, he posterior tandard deviation is less thanthe standarderror f the maximum ikelihood stimate,which is 0.10, because of the additional informationprovidedby the priordistribution. s a result,Bayesiancredible intervals tend to be narrowerthan the cor-responding onfidence ntervals,with the differencenwidth beingreasonably arge unless a' and /' are quitesmall comparedto n* = (al" + /3") - (a' + o').The shifting f point estimates oward the priormeanand the reduction n width of interval estimates is atypicalfeature f Bayesian procedures s comparedwithclassical procedures.Such effects eem especially pro-nounced in the above analysis. The prior distributionseems to assume a moresignificantole than mightbeexpected n view of the samplesize,and thisreflectshefactthat randomization ffectivelyeducesthe amountof ample nformation,s noted nSection1.The approxi-mationsused to generateposterior istributionsrovidean indication f the amountof nformationhat s "lost."If we think fn* = (c" + ,/") - (a' + ,8')as a measureof"equivalent sample size," then wemightviewn*/n asa roughmeasure of the proportion f information e-tained by the randomized response procedure. WithApproximation , n*/n can be determined rom 4.11):

n*/n = *(1 - *)/( +c)(c+ 1-), (5.2)where c = (1 - p)/(2p - 1). For the cases consideredin the table, 7r 0.25, c = 0.75, and n*/n = 0.125,which means that the effect f randomization s effec-

tivelyto reduce the samplesize by a factor f eight.Theposteriormean of ir is a weighted verage of the priormean and *?, where the weights are proportionaltoa' + d' and n/8, respectively.This demonstrates hepractical value of prior informationn attemptingtoobtain precise estimates when randomized responsemethods re used.With respectto the two approximations iscussedinSection 4, the means, variances,and values of D pre-sented in the table indicate that for these cases, bothapproximations eem to providegood fits to the exactposterior istributions. ome additionalnumerical esultsobtainedforothercases supportthiscontention;detailsare omittedhereto save space. For bothapproximations,the highestvalues ofD in the table occurwhena' = /'= 1, in which case the priordistributionf ir s uniformon [0, 1]. Even thesevalues ofD do not seem toobad,and the situationquickly mproves s a' and d' increase.With Approximation , no beta distribution n [0, 1]provides goodfit o f(X),which s uniform n [1 - p, p].As a' and d' increase with 1 - p < a'/(a' + /') < p,the effects f truncation re reduced. With Approxima-tion2, any differencesetween heexact and approximatelikelihoodfunctions re most noticeablewhen there islittle prior information. hese differencesend to be"washed out" as the amount of prior informationn-creases and the posteriordistributions less dependenton the sample evidence.For example,the reductions nD from a ,' = 2, /' 4 to a' = 10, /' = 20 or from' = 2, /' = 8 to ' = 10, /' = 40 are considerable, ascan be seenfrom he table.The fit achieved by the approximations ppears toimprove s n -* 0 orn -* oo, as would be expectedfromthe imiting esults iven nSection4. In fact, hesamplesize from he Liu and Chow study (n = 150) providespoor fitrelativeto most of the othersample sizes con-

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Winkler nd Franklin:RandomizedResponse:A BayesianApproach 213DENSITY

8.007.507.006.506.005.505.004.50 ------------ PRIOR

EXACT OSTERIOR4.00 a'=-10, -40-5 | l---------------APPROXIMATE POSTERIOR3.503.00 i2.502.001.501.00.50.00 .8.20 .40 .60 .80 1.00

provide beta posterior distributionsfor 7r certainlysimplify heanalysis. The examplepresentednSection 5indicates hatthese approximations eem to providegoodfitsto the exact posteriordistribution.n addition toillustrating he Bayesian procedures, he example givessome nsight nto the effect fdifferentriordistributionson inferences bout r and demonstrates he practicalvalue of prior informationn the attempt to obtainprecise stimateswhenrandomized esponsemethods reused.To apply in practice the methodsdeveloped in thisarticle, user mustassess a beta priordistributionor7r.The assessmentof prior distributions or proportions sdiscussed nWinkler 1967). Various fractiles fthe priordistribution an be assessed, and tables (e.g., Pearson1968) can be used to fit a beta distribution o thesefractiles.fmoments re assessed orcalculated,formulasfor hebeta mean and variance can be solved for ' andp3'.Alternatively,he usermight ssess a beta distribu-tion directlyby interpretinghe prior information sbeing equivalentto a sample (nota randomized esponsesample)with ' individuals nGroupA and,3' ndividualsnot in GroupA. Finally,whenthe prior nformationsvery sparse, a diffuse eta priordistribution e.g., animproper distributionwith a' = ,B'= 0 or a uniformdistribution itha' = ,3' = 1) mightbe considered.

Once a' and ,3'have been chosen nd thesample results(r "Yes" answers n n trials) obtained, one of the ap-proximationsn Section 4 can be used to determine napproximate osterior istribution.WithApproximation1, calculate E X) = [(2p - 1)a'/ (a' + d') ] + 1 - p andV(X) = (2p -1 )2a'/3'/(C' + /'t)2(aC + /' + 1) and use(4.5) and (4.6) to find a* and 3*. Next, findE(X Ir,n)= (a* + r)/((a* + /3* + n) andV(X r, n) = (a* + r) 3* + n-r)/

(a* +,8* + n)2(oa* +,8* + n + )and use (4.7) and (4.8) to determine " and /3". WithApproximation , solve (4.10) and (4.11) forr* and n*,then ompute " = a' + r*and 3" = /3'+ n -r*.The approximate beta posterior distribution canbe summarized in terms of moments such as themean, a"/(a" + 3"), and the variance, ac"3B"/(a"/ + #/3)2(a"f+ /3" + 1). Point estimates therthanthe mean are the mode (a" - 1)/(a" + 8" - 2) (as-suming hat a" > 1 and /3"> 1), or the median, which(along with certain other fractiles)can be foundfromtables (e.g., Pearson 1968). Credible ntervals re usefulsummary measures of posteriordistributions.Credibleintervals uttingoff qual probabilities n the two tailsof the distribution an be found by determining ppro-priate fractiles rom ables,or highest ensity egions or

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