An Improved Warner s Randomized Response Model

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1 Iteratioal Joural of Statistics ad Applicatios 05, 5(6: DOI: 0.593/j.statistics A Improved Warer s Radomized Respose Model F. B. Adebola, O. O. Johso * Departmet of Statistics, Federal Uiversit of Techolog, Aure, Nigeria Abstract This paper presets a modificatio of Warer s [8] Radomized Respose model. Accordig to O Muircheartaigh et al [7], o-respose is ievitable i a surve; i view of this, our model further reduces the o-respose bias b further samplig for the o-respodets. I this paper we performed a empirical practice of our model ad we also performed the empirical compariso of our model with Warer [8] model. We discovered that our model is more efficiet tha the Warer [8] model. Kewords Close supervisio, Sesitive behavior, No-respodet, Sub-sample, Radomized respose techiques. Itroductio Warer [8] proposed the radomized respose techique as a surve techique to reduce potetial bias due to o-respose ad social desirabilit whe asig questios about sesitive behaviors (see Warer [8], for a comprehesive review. The method ass respodets to use a radomizatio device, such as a coi, dec of cards, spiers whose outcome is ot ow b the eumerator. The outcome of the radomized device determies which of the two questios the respodet aswers. A lot of improvemets have bee doe to Warer s radomized respose model, to metio few, Greeberg et al. [4], Gupta ad Shabbir [5], Adebola ad Adegoe [], Adepetu ad Adebola [3], Adebola et al. []. I this paper, we develop a Modificatio of Warer s Radomized Respose Techiques b itroducig the cocept of sub-sample of o-respodet. Radomized Respose Techiques helps to reduce respose ad o-respose bias while our model further reduces the o-respose bias. Item o-respose occurs whe the respodet refuses to aswer the sesitive part of the questio which is the major cocer of the iterviewer. I sectios that follow, we preset the Warer s [8] Radomized Respose Model, propose Radomized Respose Model ad thereafter its relative efficiec over the existig oe.. Warer s Radomized Respose Model * Correspodig author: teeja@gmail.com (O. O. Johso Published olie at Copright 05 Scietific & Academic Publishig. All Rights Reserved Warer [7] gave a geius idea b usig radomized device to ecourage truthful aswer from the respodet with respect to a sesitive behavior. The radomizig device, such as a spiig arrow, dice or cois is used to select oe of the two questios; such as, I am HI positive (class A, preseted with probabilit P I am HI egative (class B, preseted with probabilit -P The respodets have the optios Yes or No preseted to him or her. The iterviewer does ot ow which questio a respodet has aswered but ows the probabilit P ad -P with which the two statemets are preseted. Here, with a radom sample of respodets, the iterviewer records a biomial estimate θˆ of the proportio θ of Yes aswers, where is the umber of es aswers. If the questios are aswered truthfull, the relatio betwee θ ad π i the populatio is give as: πp + ( ( π ( p + ( θ π (. Where π is the proportio of people with the stigmatized or sesitive behavior usig Warer s techiques ad p is the probabilit of selectig the sesitive questio. θ (, p (. ( p Proof From equatio (., we have θ πp + ( ( π Maig π the subject of the relatio, we have θ ( p (

2 64 F. B. Adebola et al.: A Improved Warer s Radomized Respose Model The ubiased estimator πˆ is give b: The ariace is give b ( π of a sesitive proportio π ( ( p ˆ θ ˆ (.3 ( ( p ˆ θ ( ( p ˆ θ ( π ( p Recall that ( c 0 The, Thus, ( ( p, where c is a costat. 0 ( π ˆ θ ( p ( θ ( p θ ( π (.4 Where θˆ follows a biomial distributio, Now to fid the ubiased estimator of the variace ( p + ( ( p + p θ π θ π ( π ( ( π ( ( p π p + p p + p 4pπ + 4p π 4p π 4pπ π + π + p p p( ( p ( 4p 4p+ π ( 4p 4p+ π p p p( ( π π ( p p p The ariace is give b ( π p( ( p ( π π The ubiased variace estimator ˆ ( πˆ proportio πˆ is give b: ˆ ( ˆ π p( ( p of a sesitive ( ˆ π ˆ π (.5 The secod term i ˆ( πˆ is the variace that ˆ ( πˆ would have if all respodets aswered truthfull a direct questio about class A membership. Except b chose ππ AA ear 0.5 ad p > 0.85, the first term is greater tha the secod, ofte much greater. The method is thus quite imprecise i geeral. This might be expected sice the iterviewer does ot ow whether a es aswer implies membership i a class A or the opposite. However, Warer s method ma give a smaller mea square error (MSE tha a direct sesitive questio would, if the latter produced umerous refusals or false aswers. 3. Our Model Several radomized respose techiques has bee developed, the models developed do ot tae ito cosideratio of item o-respose (refusal to aswer the sesitive part of the questio. No-respose is a importat source of o-samplig error i surve samplig, it occurs whe some but ot all the required iformatio is collected from the sample uit. The most damagig is uit o-respose where a samplig uit refuses to aswer the sesitive part of the radomized respose techiques desiged questioaire. I view of this, we proposed a improved Warer s radomized respose model that is based o sub-sample of o-respodet so as to iduce a better estimate of the proportio of people with the stigmatized or sesitive behavior. Questioaires were set out ad the umber of useable resposes were recorded (useable resposes at the first iterview give as while the remaiig were referred to o-respose give as. I order to further reduce the o-respose bias the a surve o sub-sample of o-respodet come to place. It is assumed that the whole of the sub-sample respoded to the surve ad are useable resposes which would be achieved b close supervisio. Close supervisio i this cotext does t mea the iterviewer ows the questio aswered b the respodet but it meas the respodet is properl istructed ad moitored o a oe to oe basis. Let be the umber es respose from the respodet at the first iterview. Let m be the umber of es aswer from the sub-sample of o-respodet. Let be the sample size of the Surve. Let m be the sub-sample size. Let be the uit

3 Iteratioal Joural of Statistics ad Applicatios 05, 5(6: which is used to tae the sub-sample. The proportio of es respose from our model is give b: m θ (3. + B simplifig, we have (see theorem for proof: ˆ θ ˆ θ + ˆ θ (3. Where θˆ be the proportio of es aswer, ˆ θ be the proportio of es aswer form the respodet at the first iterview ad ˆ θ be the proportio of es aswer from the sub-sample of o-respodet. From the Warer s radomized respose techiques, the proportio of es respose from our model is give b: θ π p+ ( ( π + π p+ ( ( π ( + π p+ ( ( π (3.3 Let B +, the we have: θ B π p+ ( ( π Solvig for π, we have, θ B( B( p The we have, θ ( + ( + p ( ( Recall that θ θ + θ θ+ θ + p + p ( ( ( ( Hece, the ubiased estimator of πˆ is give b: θ ( ( ( + ( p θ+ + p ˆ The variace of the estimator is give b: v( ˆ θ + v( ˆ θ ( ˆ π + p ( ( θ( θ θ ( θ + ( ˆ π ( + ( p B simplifig, we have (3.4 (3.5 ( ˆ π p( ( ( ( ( π p ( π ( π ( π ( + + p p p + B further simplificatio, The ubiased variace estimator ˆ ( πˆ of a sesitive proportio π is give b: ( ( p ( π p ˆ π ( ˆ π + + (3.6 The, ˆ ( ˆ π ( π w + (3.7 Where ( π w is the variace for a Warer s model. Theorem : The proportio of es respose, θ is give b: ˆ θ ˆ θ + ˆ θ Proof: From the Hase ad Hurwitz [6], which itroduces the cocept of subsample of o-respodet, we have θ θ + w w θ m Where w, w, θ, θ m. m The, we have θ + (3.8 m Recall that m B substitutig for m i equatio (3.8, we have θ + The ubiased estimator of θ is give b ˆ θ ˆ θ + ˆ θ Where ˆ θ ad ˆ θ. Theorem : Give that is a iteger value ad > Show that < +

4 66 F. B. Adebola et al.: A Improved Warer s Radomized Respose Model Proof: Give that, That is, B re-arragig, Thus whe ( Divide through b ( + Multipl through b The, ( ( ( ( + ( ( + + From equatio (, we have > ( ( + + ( + ( ( + ( Divide through b (, we have + Subtract from both sides, we have Divide through b, we have The, From equatio ( ad (3, usig trasitivit law + 4. Comparative Stud of our Model Here we performed the comparative stud of our model; this ca be achieved mathematicall ad empiricall. Mathematicall, it follows that the proposed model is more efficiet tha the Warer s radomized respose model if we have; (3 Relative efficiec (RE ariace of proposed mod el ariace of Warer ' s mod el p( ( p( + ( p < ( π π + + p RE < π ( π B simplifig, we have RE < + Sice theorem holds, the the variace of our proposed model is less tha the variace of Warer s RRT. Empiricall, to also validate our coclusio o the proposed model we preset the tables below. Table. Table showig the relative efficiec whe 50, 0., p0.7 N π P K Warer s ariace Proposed ariace Relative Efficiec (% % % % % % Table. Table showig the relative efficiec whe 500, 0., p0.7 N π P K Warer s ariace Proposed ariace Relative Efficiec (% % % % % % We ca deduce from the empirical compariso that the choice of plas a major role i the comparative stud. It ca be derived from the table that gave the miimum variace i the proposed model; coclusivel, the smaller the choice of, the more efficiet the proposed model is over the covetioal Warer s model. 5. Coclusios This paper preseted a improved Warer s radomized respose model; the proposed strateg further reduces the o-respose bias b itroducig the cocept of sub-samples of o-respodet. The proposed model is liel to iduce better estimate with a reduced variace. Moreover, the

5 Iteratioal Joural of Statistics ad Applicatios 05, 5(6: proposed model is more efficiet tha the Warer s model. Lastl, we are able to coclude that the smaller the choice of (the uit which is used to divide the o-respodet so as to have the sub-sample size, the higher the gai i efficiec of the proposed model over the covetioal Warer s model. REFERENCES [] Adebola, F.B. ad Adegoe, N.A. (03: A Surve of Examiatio Malpractices usig the Radomized Respose Techique. Joural of the Nigeria Associatio of Mathematical Phsics, 3, [] Adebola, F. B., Johso, O. O., & Adegoe, N. A. (04: A Modified Stratified Radomized Respose Techiques. Mathematical Theor ad Modelig, 4(3, 9-4. [3] Adepetu, A.O. ad Adebola, F.B. (04: O the Relative Efficiec of the Proposed Reparametized Radomized Respose Model. Iteratioal Joural of Mathematical Theor ad Modelig, 4, [4] Greeberg, B.G., Abul-Ela, A.A., Simmos, W.R. ad Horvitz, D.G. (969: The Urelated Questio Radomized Respose: Theoretical Framewor. Joural of the America Statistical Associatio 64, [5] Gupta, S.N. ad Shabbir, J. (006: A Alterative to Warer s Radomized Respose Model. Joural of Moder Applied Statistical Methods, 5, [6] Hase, M.H. ad Hurwitz, W.N. (946: The Problem of No-Respose i Sample Surves, Joural of the America Statistical Associatio 4, [7] O Muircheartaigh, C. ad Campaelli, P. (999: A multilevel exploratio of the role of iterviewers i surve o-respose. Joural of the Roal Statistical Societ, Series A 6, [8] Warer, S.L. (965: Radomized respose: A Surve Techique for Elimiatig Evasive Aswer Bias, Joural of the America Statistical Associatio 60,