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On a qualitative comparison of the proposed randomized

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On a qualitative comparison of the proposed randomized

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 On a Qualitative Comparison of the Proposed Randomized Response Technique with Hussain and Shabbir (2007) F.B. Adebola1 & A.O. Adepetun2*Department of Mathematical Sciences, Federal University of Technology ,PMB 704,Akure, Ondo State, Nigeria. Email: akinolaoladiran@yahoo.comAbstractRandomized Response Techniques (RRTs) were developed for the purpose of improving response rate byprotecting surveyee’s privacy and avoiding answer bias.In this paper, we succinctly investigated the trend of theresponse rate (π) with both the proposed Randomized Response Technique (RRT) and the conventional one. Wefound that the proposed (RRT) is better than the conventional one at each response rate since it gives smallervariance.Keywords: Response rate, Randomized Response Techniques (RRTs), error variance, unpleasant character.1.IntroductionObtaining information directly about a unpleasant character such as shoplifting, cheating in an examination,tax evasion etc. in a human population survey is a difficult activity. A survey may receive untruthfully answersfrom the survey respondents when he/she uses direct questioning approach. Because of many reasons,information about incidence of unpleasant characters in the population is necessary. Warner (1965) proposedthe initial randomized-response techniques which is an effective survey method to find such estimates while atthe same time protecting individual’s anonymity. To date, a large number of developments and variants ofWarner’s Randomized Response Technique (RRT) have been put forward by several researchers. Greenberg et al.(1969), Mangat and Singh (1990), Mangat (1994), Christofides (2003), Kim and Warde (2004), Adebola andAdepetun (2011) are some of the many to be noted. In the next section, we present conventional technique,Proposed Randomized Response Technique and subsequently its efficiency over the conventional one in terms ofthe response rate and the variances respectively.2. The Conventional TechniqueHussain and Shabbir (2007) proposed a Randomized Response Technique (RRT) based on the random use of oneof the two randomization devices R1 and R2. In design, the two randomization devices R1 and R2 are the same asthat of Warner’s (1965) device but with different probabilities of selecting the sensitive question. The ideabehind this suggestion is to decrease the suspicion among the respondents by providing them choice to randomlychoose the randomization device itself. As a result, respondents may divulge their true status. A simple randomsample with replacement (SRSWR) sampling is assumed to select a sample of size n. Let ∝ and β be any two αpositive real numbers chosen such that q ൌ , ሺα ് βሻ is the probability of using R1, where R1 consists of the ∝ାβ βtwo statements of Warner’s device but with preset probabilities P1 and 1 െ P1 and 1 െ q ൌ is the ∝ାβprobability of using R2 ,where R2 consists of the two statements of Warner’s device also with preset probabilitiesPଶ and 1 െ P2 respectively. For the ith respondent, the probability of a “yes” response is given by ஑ ஒPሺyesሻ ൌ ∅ ൌ ሾ P1 π ൅ ሺ1 െ P1 ሻሺ1 െ πሻሿ ൅ ሾ P2 π ൅ ሺ1 െ P2 ሻሺ1 െ πሻሿ (2.1) ஑ାஒ ஑ାஒTo provide the equal privacy protection in both the randomization devices R1 and R2, we put P1 ൌ 1 െ P2 intoequation (2.1), obtained: 61
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 πሾሺߙ െ βሻሺ2P1 െ 1ሻሿ ൅ P1 β ൅ P2 α∅ൌ ሺ2.2ሻ ߙ൅ߚ Hence, ∅ሺα ൅ βሻ െ P1 β െ P2 απൌ , P1 ് 1ൗ2 , ∝് β ሺ2.3ሻ ሺ2P1 െ 1ሻሺα െ βሻThe unbiased moment estimator of true probability of yes response (response rate) π was given by ෡ ∅ሺα ൅ βሻ െ P1 β െ P2 απൌෝ ሺ2.4ሻ ሺ2P1 െ 1ሻሺα െ βሻ ෡Where ∅ ൌ y and y is the number of respondents reporting a “yes” answer when P1 ൌ 1 െ P2 . The variance of nthe estimator was given then by πሺ1 െ πሻ ሺPଶ α ൅ Pଵ βሻሺPଵ α ൅ Pଶ βሻ ܸሺߨ ௖௢௡௩ ൌ ොሻ ൅ ሺ2.5ሻ n nሺ2Pଵ െ 1ሻଶ ሺα െ βሻଶ ሺα ൅ βሻଶ3.The Proposed TechniqueIt was quite obvious that despite the successful attempts by several authors in developing an efficientRandomized Response Techniques (RRTs), the developed techniques only considered a two-option of “yes” and“no” response. As a result of which we propose a new Randomized Response Technique (RRT) that will bebased on the random use of one of the three randomization devices, Rଵ , R ଶ and R ଷ . In design, the threerandomization devices Rଵ , R ଶ and R ଷ are similar to that of Warner’s device but with different probabilities ofselection. In addition to α and β proposed earlier by Hussain and Shabbir, we introduce δ, a positive real αnumber such that q ൌ , α ് β ് δ is the probability of using Rଵ , where Rଵ consists of the two αାβାδstatements of Warner’s device and the new introduce device also with preset probabilities P1, Pଶ and Pଷrespectively. By adopting Hussain and Shabbir’s probability of a “yes” response for the ith respondent, theprobability of a “yes” response when the third option “undecided” is included is given by ஑ ஒ ஔ ܳሺ‫ݏ݁ݕ‬ሻ ൌ ߮ ൌ [ P1 π ൅ ሺ1 െ P1 ሻሺ1 െ πሻሿ ൅ ሾP2 π ൅ ሺ1 െ P2 ሻሺ1 െ πሻሿ ൅ ሾP3 π ൅ ሺ1 െ ஑ାஒାஔ ஑ାஒାஔ ஑ାஒାஔ P3 ሻሺ1 െ πሻሿ (3.1)In order to provide the equal privacy protection in the three randomization devices R1, R2, and R ଷ , we putP1 ൌ 1 െ P2 െ P3 into equation (3.1), obtained: ߮ሺߙ ൅ ߚ ൅ ߜሻ െ ሾሺߙ ൅ ߚ ൅ ߜሻ െ P1 α െ P2 β െ P3 δሿߨൌ ሺ3.2ሻ 2P1 α ൅ 2P2 β ൅ 2P3 δ െ α െ β െ δHence, the unbiased sample estimate of ߨ is given as ߮ ൅ ߚ ൅ ߜሻ െ ሾሺߙ ൅ ߚ ൅ ߜሻ െ P1 α െ P2 β െ P3 δሿ ොሺߙߨൌො ሺ3.3ሻ 2P1 α ൅ 2P2 β ൅ 2P3 δ െ α െ β െ δ xWhere ߮ ൌ ො and x is the number of respondents reporting a “yes” answer when P1 ൌ 1 െ P2 െ P3 . The nvariance of the estimator is given then by πሺ1 െ πሻ ሺPଵ α ൅ Pଶ β ൅ Pଷ δሻሺPଷ α ൅ Pଶ β ൅ Pଵ δሻܸሺߨ ൌ ොሻ ൅ ሺ3.4ሻ n ݊ሾ2Pଵ ሺα െ δሻ ൅ 2Pଶ ሺβ െ δሻ െ ሺα ൅ β െ δሻሿଶ ሺߙ ൅ ߚ ൅ ߜሻଶ4.Proposed Technique versus Conventional TechniqueWe show that the new RRT is better than the existing ones by comparing it with the conventional one at varyingresponse rate keeping the sample size constant. Hence, the proposed tripartite Randomized Response Technique(RRT) will be better than the conventional Randomized Response Technique (RRT) if we have 62
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012ܸሺߨ ௖௢௡௩ െ Vሺπሻ୮୰୭୮ ൐ 0 ොሻ ෝ (4.1)Or if஠ሺଵି஠ሻ ሺ୔మ ஑ା୔భ ஒሻሺ୔భ ஑ା୔మ ஒሻ πሺଵିπሻ ሺ୔భ ஑ା୔మ ஒା୔య ஔሻሺ୔య ஑ା୔మ ஒା୔భ ஔሻ ൅ െ െ ൐ 0 ሺ4.2ሻ ୬ ୬ሺଶ୔భ ିଵሻమ ሺ஑ିஒሻమ ሺ஑ାஒሻమ ୬ ௡ሾଶ୔భ ሺ஑ିஔሻାଶ୔మ ሺஒିஔሻିሺ஑ାஒିஔሻሿమ ሺఈାఉାఋሻమOr if ሺPଶ α ൅ Pଵ βሻሺPଵ α ൅ Pଶ βሻ ሺPଵ α ൅ Pଶ β ൅ Pଷ δሻሺPଷ α ൅ Pଶ β ൅ Pଵ δሻ ଶ ሺα െ βሻଶ ሺα ൅ βሻଶ െ ൐ 0 ሺ4.3ሻnሺ2Pଵ െ 1ሻ ݊ሾ2Pଵ ሺα െ δሻ ൅ 2Pଶ ሺβ െ δሻ െ ሺα ൅ β െ δሻሿଶ ሺߙ ൅ ߚ ൅ ߜሻଶThe condition given in (4.3) is true, for Pଵ , Pଶ , Pଷ and π ranging from 0.1 to 0.9 if α and β,α and δ, β and δ differ from each other by at least 9 where ߙ, ߚ, and δ are any suitable positive real numbers.Table 4.1: Comparison between conventional RRT and proposed RRT when Pଵ ൌ 0.6, Pଶ ൌ 0.3, Pଷ ൌ 0.1, α ൌ 20, β ൌ 11, δ ൌ 2, n=50, for varying of response rate (π) ߨ ܲ૚ ܲ૛ ܲ૜ α ߚ Δ Conventional Proposed Variance Variance0.1 0.6 0.3 0.1 20 11 2 0.00304 0.002260.2 0.6 0.3 0.1 20 11 2 0.00444 0.003660.3 0.6 0.3 0.1 20 11 2 0.00544 0.004660.4 0.6 0.3 0.1 20 11 2 0.00604 0.005260.5 0.6 0.3 0.1 20 11 2 0.00624 0.00546 63
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 64
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012Figure 4.1: Graph showing comparison between conventional RRT and proposed RRT whenP_1=0.6,P_2=0.3,P_3=0.1, α=20,β=11,δ=2, n=50, for varying of response rate (π) 0.007 0.006 0.005 0.004 Var(π) 0.003 Conventional variance 0.002 Proposed variance 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 πFigure 4.2: Graph showing comparison between conventional RRT and proposed RRT when Pଵ ൌ 0.4, Pଶ ൌ0.4, Pଷ ൌ 0.2, α ൌ 20, β ൌ 11, δ ൌ 2, n=50, for varying of response rate (π) 0.006 0.005 0.004 Var(π) 0.003 Conventional variance Proposed variance 0.002 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 π 65
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012Figure 4.3: Graph showing comparison between conventional RRT and proposed RRT when Pଵ ൌ 0.2, Pଶ ൌ0.5, Pଷ ൌ 0.3, α ൌ 20, β ൌ 11, δ ൌ 2, n=50, for varying of response rate (π) 0.007 0.006 0.005 0.004 Var(π) 0.003 Conventional variance Proposed variance 0.002 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 πFigure 4.4: Graph showing comparison between conventional RRT and proposed RRT when Pଵ ൌ 0.15, Pଶ ൌ0.6, Pଷ ൌ 0.25, α ൌ 20, β ൌ 11, δ ൌ 2, n=50, for varying of response rate (π) 0.006 0.005 0.004 Var(π) 0.003 Conventional variance 0.002 Proposed variance 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 π5.ConclusionIn this study, we reviewed the conventional Randomized Response Technique (RRT). 66
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012The efficiency of our proposed Randomized Response Technique over that of the conventional one was alsoverified when both techniques were compared using the response rate approach. It was obvious in the results onTables and Figures 4.1,4.2,4.3 and 4.4 above that the proposed technique is indeed better than the conventionalone.References1. Adebola,F.B. and Adepetun, A.O. (2011):A new Tripartite Randomized Response Technique. Journal of theNigerian Association of Mathematical Physics, Volume 19: pp 119-122.2. Christofides TC (2003): A generalized Randomized Response Technique. Metrika 57: 195-2003. Greenberg, B., Abul-Ela, A; Simmons, W. and Horvitz, D. (1969): The Unrelated question randomizedresponse: Thereotical framework. Journal of American Statistical Association, 64, 529 – 539.4. Hussain, Z. and Shabbir, J. (2007): Randomized use of Warner’s Randomized Response Model. Interstat:April #7.http://interstat.statjournals.net/INDEX/Apro7.html5. Kim J. and Warde W.D., (2004): A mixed Randomized Response Model. J., Statist.Plann.Inference, 110, 1-11.6. Mangat, N. S. and Singh, R. (1990): An alternative randomized response procedure. Biometrika 77,439-442.7. Singh, S. (2002): Randomized response model. Metrika, 56, 131-142.8. Warner SL (1965) : Randomized Response: A survey Technique for Eliminating Evasive answer Bias.Journal of the American Statistical Association 60: 63 – 69. 67
  8. 8. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. Prospective authors ofIISTE journals can find the submission instruction on the following page:http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar

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