A stratified randomized response model based on R. Singh, Singh, Mangat, and Tracy (1995) improved two-stage randomized response strategy is proposed. It has an optimal allocation and large gain in precision. Conditions are obtained under which the proposed model is more efficient than R. Singh et al. (1995) and H. P. Singh and Tarray (2015) models. Numerical illustrations are also given in support of the present study

Singh, Housila P.

Tarray, Tanveer A.

English

Digital Commons@Wayne State University

Journal of Modern Applied StatisticalMethodsVolume 17 | Issue 1 Article 2912-10-2018A Proficient Two-Stage Stratified RandomizedResponse StrategyTanveer A. TarrayIslamic University of Science and Technology, Awantipora, India, tanveerstat@gmail.comHousila P. SinghVikram University, Ujjain, India, hpsujn@gmail.comFollow this and additional works at: https://digitalcommons.wayne.edu/jmasmPart of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and theStatistical Theory CommonsThis Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted forinclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState.Recommended CitationTarray, Tanveer A. and Singh, Housila P. (2018) "A Proficient Two-Stage Stratified Randomized Response Strategy," Journal of ModernApplied Statistical Methods: Vol. 17 : Iss. 1 , Article 29.DOI: 10.22237/jmasm/1544453468Available at: https://digitalcommons.wayne.edu/jmasm/vol17/iss1/29Journal of Modern Applied Statistical Methods May 2018, Vol. 17, No. 1, eP2520 doi: 10.22237/jmasm/1544453468  Copyright © 2018 JMASM, Inc. ISSN 1538 − 9472    doi: 10.22237/jmasm/1544453468 | Accepted: April 12, 2017; Published: December 10, 2018. Correspondence: Tanveer A. Tarray, tanveerstat@gmail.com    2 A Proficient Two-Stage Stratified Randomized Response Strategy Tanveer A. Tarray Islamic University of Science and Technology Awantipora, India Housila P. Singh Vikram University Ujjain, India   A stratified randomized response model based on R. Singh, Singh, Mangat, and Tracy (1995) improved two-stage randomized response strategy is proposed. It has an optimal allocation and large gain in precision. Conditions are obtained under which the proposed model is more efficient than R. Singh et al. (1995) and H. P. Singh and Tarray (2015) models. Numerical illustrations are also given in support of the present study.  Keywords: Randomized response technique, stratified random sampling, dichotomous population, Sensitive attribute, estimation of proportion, optimum allocation  Introduction Warner (1965) introduced a randomized response (RR) model to estimate proportion for sensitive attributes including sexual orientation, criminal activity, child abuse, suicidal tendency in teenagers, all cases of AIDS, abortion, or drug addiction. Greenberg, Abul-Ela, Simmons, and Horvitz (1969) envisaged an unrelated question randomized response model using Warner’s sensitive question and unrelated question. This technique has generated much interest in the statistical literature since the publication of Warner’s RR model. The RR model has been studied by many authors, such as Chaudhuri and Muherjee (1988), Ryu, Hong, and Lee (1993), Tracy and Mangat (1996), S. Singh (2003), and Kim and Elam (2007). Hong, Yum, and Lee (1994) suggested a stratified RR technique using a proportional allocation. Kim and Warde (2004) presented a stratified RR technique based on Warner’s (1965) model that has an optimal allocation. Kim and Elam (2005) have applied Kim and Warde’s (2004) stratified Warner’s RR model to the two-stage model of Mangat and Singh (1990). Kim, Tebbs, and An (2006) proposed a Bayesian version of the Mangat (1994) model. Kim and Warde (2005) have TARRAY & SINGH 3 suggested a mixed randomized response model using simple random sampling which rectifies the privacy problem and extended the proposed model to stratified sampling. Lee, Uhm, and Kim (2011) have extended the work of Land, Singh, and Sedory (2011) in stratified sampling using a Poisson distribution. Recently, H. P. Singh and Tarray (2015) have applied a stratified RR model to the Tracy and Osahan (1999) two-stage model and derived the probability θi of “Yes” answer in stratum i for this procedure as:   ( )( )S S11 , for 1,2, ,2ii i i i i iPT T P i k  − = + − + =  K   (1)  where πSi is the proportion of people with the sensitive trait in a stratum i. The unbiased estimator STˆ  of a sensitive proportion estimate of S S1ki iiw ==  is given by   ( )( )( )ST S1 1ˆ 1 1 2ˆ ˆ1k ki i ii i ii i i i iT Pw wT P T = = − − −= =  + −      (2)  where wi = (Ni / N) for i = 1, 2,…, k, so that 11kiiw w== = ; N is the number of units in the entire population; Ni is the total number of units in the stratum i; and ˆi , the proportion of “Yes” answers obtained from the ni sampled respondents using simple random sampling with replacement (SRSWR) of stratum i, is a point estimate of θi in (1). The minimal variance of the estimator STˆ  under optimal allocation is given by   ( ) ( )( )( ) ( )( ) ( ) 21 2ST S S 211 1 2 1 11ˆV 14 1ki i i ii i iii i iT P T Pwn T P T  =  − − − − −  = − +  + −       (3)  In this paper, we have suggested a new stratified randomized response model based on the R. Singh et al. (1995) two-stage RR model. It is demonstrated that the estimator resulting from the proposed model is more efficient than those of its A PROFICIENT TWO-STAGE STRATIFIED RRS 4 competent models under the condition presented in the case of completely truthful reporting. Suggested Model The population is divided in to k strata and a sample is selected by SRSWR in each stratum. To get the full benefit from stratification, we assume that the number of units in each stratum is known. In the first stage of the survey interview, an individual respondent in the sample of stratum i is instructed to use the randomization device R1i, which consists of a sensitive question (S) card with probability Ti and a “Go to randomization device R2i in the second stage” direction card with probability (1 – Ti). The respondents in the second stage of the stratum i are instructed to use the randomization device R2i which uses three statements: (i) “I possess the sensitive attribute A;” (ii) “Yes;” and (iii) “No;” with probabilities Pi, (1 – Pi)αi, and (1 – Pi)(1 – αi), respectively, where αi ∈ [0, 1]. Let ni denote the number of units in the sample from stratum i and n denote the total number of units in sample from all stratum so that 1kiin n== . Thus, when respondents report truthfully, the probability of a "Yes" answer in stratum i for this procedure is given by   ( ) ( )S S1 1 , for 1,2, ,i i i i i i i iT T P P i k   = + − + − =   K   (4)  where θαi is the proportion of “Yes” responses and πSi is the proportion of respondents with the sensitive trait in the population from stratum i. If ˆi  denotes the estimate of the proportion of “Yes” answers in a stratum i, the relation (4) yields   ( )( )( )Sˆ 1 1ˆ , for 1,2, ,1i i i iii i iT Pi kT P T − − −= =+ −K   (5)  as an unbiased estimator of the proportion πSi of respondents with the sensitive trait in the population from stratum i. The variance of the estimator STˆ  is given by  TARRAY & SINGH 5  ( )( )( )( ) ( ) ( ) ( )( ) ( )S SS21ˆV1 1 1 1 2 1 1 11i iiii i Si i i i i i i i ii i i inT P T P T T Pn T T P    −=  − − + − − + − − −  ++ −     (6)  Because the selection in different strata are made independently, the estimator for individual strata can be added together to obtain an estimator for the whole population. The maximum likelihood estimator of πS, the proportion of respondents with the sensitive trait, is   ( )( )( )S S1 1ˆ 1 1ˆ ˆ1k ki i i ii i ii i i i iT Pw wT P T  = = − − −= =  + −      (7)  where N denotes the number of units in the entire population, Ni is the total number of units in the stratum i, and wi = (Ni / N) for i = 1, 2,…, k, so 11kiiw w== = . The proposed estimator Sˆ  is an unbiased estimator for the population proportion πS.  Theorem 1. The variance of the estimator Sˆ  is   ( )( )( )( ) ( ) ( ) ( )( ) ( )2S SS121ˆV1 1 1 1 2 1 1 11ki iii i ii i Si i i i i i i i ii i i iwn nT P T P T T Pn T T P    =−=  − − + − − + − − −  ++ −     (8)  Proof.  This follows from taking the variance of (7) and corollary 1 in section 5.5 of Cochran (1977). In practice, information on πSi is usually not known. But if prior information on πSi is available from past experience then it helps to establish the following optimal allocation formula:  Theorem 2. The optimal allocation of n to n1, n2,…, nk–1, and nk to derive the minimum variance of Sˆ  subject to 1kiin n==  is approximately given by A PROFICIENT TWO-STAGE STRATIFIED RRS 6  ( )( )( ) ( ) ( ) ( )( ) ( )( )( )( ) ( ) ( ) ( )( ) ( )S S122S S112211 1 1 1 2 1 1 1111 1 1 1 2 1 1 11ii i ii i Si i i i i i i i ii i i iki i iii i Si i i i i i i i ii i i innwT P T P T T Pn T T PwT P T P T T Pn T T P        ==− − − + − − + − − −  ++ −   − − − + − − + − − −  ++ −     (9)  Proof.  Follows from section 5.5 of Cochran (1977). The minimal variance of the estimator Sˆ  is given by   ( )( )( ) ( )( )( )( )( )( ) ( )( )( )S SS12212211ˆV1 1 2 1 111 1 1 1 11ki iii iSi i i i i i ii i i ii i i i i ii i i iwn nT P T T Pn T T PT P T Pn T T P   = −=   + − − − −++ −  − − − − − +  + −       (10)  By substituting (ni – 1) for ni in (8), the unbiased minimal variance of the estimator Sˆ  can be obtained.  Remark 1. The proposed procedure is equivalent to the H. P. Singh and Tarray (2015) procedure for α = 1 / 2.  Remark 2. For any given pair (Pi, Ti), the variance V( Sˆ i ) at (πSi, αi) and [(1 – πSi), (1 – αi)] has equal value. This means that V( Sˆ i ) is symmetric about the TARRAY & SINGH 7 point (1/2, 1/2) along any line in the (πSi, αi) plane passing through this point; see Corollary 2.3 in R. Singh et al. (1995, p. 268). Relative Efficiency An efficiency comparison of the proposed stratified randomized response technique and the two-stage randomized response techniques that were presented by R. Singh et al. (1995) will be conducted by comparing variances.  Theorem 3. Suppose there are two strata in the population, n = n1 + n2, P = P1 = P2 (P is the probability of selecting the sensitive question in the second stage), T = T1 = T2 (T is the probability of selecting the sensitive question in the first stage), and S 1 S1 2 S2ˆ ˆ ˆw w   = + . The proposed estimator Sˆ  is more efficient than the R. Singh et al. (1995) estimator SSMTˆ  (say), where πS1 ≠ πS2, under the following conditions:   ( ) ( ) ( ) 22 1S1 S2 1 2 1 21A A w w B − − + −  +    (11)  where   ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )( )( ) 1 S1 S1 S1 2 S2 S2 S2221 , 11 1 1 1 211 1 1 1 11A A B A A BT P T P TAT P TT P T PBT P T      = − + + = − + +      − − + − −=+ −− − − − −  =+ −   The following theorem exhibits that the proposed estimator Sˆ  is more efficient than the H. P. Singh and Tarray (2015) estimator STˆ :  Theorem 4. Assume that there are two strata in the population, n = n1 + n2, P = P1 = P2, and T = T1 = T2. The proposed estimator Sˆ  will be always more efficient than the H. P. Singh and Tarray (2015) estimator STˆ  (where πS1 ≠ πS2) under the following condition: A PROFICIENT TWO-STAGE STRATIFIED RRS 8  ( )( ) ( )( )1 1 1 1Either or1 1 2 1 1 2T P T P     −  −   − − − −     (12) Empirical Study To see the tangible idea about the performance of the proposed estimator Sˆ  over the H. P. Singh and Tarray (2015) estimator STˆ , compute the percent relative efficiency (PRE) of Sˆ  with respect to STˆ  for two strata, i.e. k = 2, P = P1 = P2, and T = T1 = T2, by using the formula   ( )( )( )( ) ( )( ) ( )STS STS21 1 2 221 1 1 2 2 2ˆVˆ ˆPRE , 100ˆV100w G H w G Hw G H w G H =  + + + =  + + +   (13)  where   ( ) ( )( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( ) ( )( ) ( )1 S1 S1 2 S2 S22S11 2S22 21 , 11 1 2 1 14 11 1 1 1 2 1 1 111 1 1 1 2 1 1 11i iG GT P T PHT P TT P T P T T PHT T PT P T P T T PHT T P         = − = −− − − − −=+ − − − + − − + − − − =+ −   − − + − − + − − − =+ −     The computed PRE( Sˆ , STˆ ) for n = 1000, α = 0 (0.1) 0.4, T = 0.1 (0.4) 0.9, P = 0.6,  0.8; and for πS1 = 0.28, πS2 = 0.33 and different values of w1 and w2 such that πS ≤ 0.5. Findings are shown in Table 1.   TARRAY & SINGH 9 Table 1. The relative efficiency of the proposed estimator πˆS with respect to the H. P. Singh and Tarray (2015) estimator πˆST, i.e. RE( πˆS, πˆST) = V( πˆST / πˆS), when n = 1000, T = 0.1, 0.4, 0.9, πS1 = 0.28, πS2 = 0.33, and different values of w1 and w2         P πS1 πS2 w1 w2 πS α T 0.30 0.33 0.36 0.39 0.42 0.28 0.33 0.9 0.1 0.285 0.0 0.6 145.05 141.75 138.58 135.60 132.80 0.28 0.33 0.9 0.1 0.285 0.0 0.7 128.99 127.10 125.29 123.55 121.88 0.28 0.33 0.9 0.1 0.285 0.0 0.8 118.54 115.59 114.66 113.76 112.89 0.28 0.33 0.8 0.2 0.290 0.1 0.6 140.19 137.56 134.56 131.56 129.53 0.28 0.33 0.8 0.2 0.290 0.1 0.7 125.53 123.89 122.32 120.82 119.37 0.28 0.33 0.8 0.2 0.290 0.1 0.8 114.28 113.47 112.67 111.90 111.50 0.28 0.33 0.7 0.3 0.295 0.4 0.6 129.61 127.61 125.70 123.90 122.18 0.28 0.33 0.7 0.3 0.295 0.4 0.7 117.43 116.34 115.29 114.29 113.32 0.28 0.33 0.7 0.3 0.295 0.4 0.8 108.64 108.14 107.65 107.18 106.72 0.28 0.33 0.6 0.4 0.300 0.0 0.6 144.20 140.91 137.82 134.90 132.15 0.28 0.33 0.6 0.4 0.300 0.0 0.7 128.39 126.54 124.76 123.05 121.41 0.28 0.33 0.6 0.4 0.300 0.0 0.8 116.16 115.23 114.33 113.44 112.59 0.28 0.33 0.5 0.5 0.305 0.1 0.6 139.46 136.61 133.91 131.37 128.96 0.28 0.33 0.5 0.5 0.305 0.1 0.7 125.02 123.41 121.87 120.38 118.96 0.28 0.33 0.5 0.5 0.305 0.1 0.8 113.97 113.17 112.39 111.63 110.90 0.28 0.33 0.4 0.6 0.310 0.4 0.6 129.11 127.13 125.25 123.47 121.78 0.28 0.33 0.4 0.6 0.310 0.4 0.7 117.10 116.03 115.00 114.01 113.05 0.28 0.33 0.4 0.6 0.310 0.4 0.8 108.46 107.96 107.49 107.02 106.13 0.28 0.33 0.3 0.7 0.315 0.0 0.6 143.37 140.13 137.09 134.22 131.51 0.28 0.33 0.3 0.7 0.315 0.0 0.7 127.81 125.99 124.24 122.56 120.95 0.28 0.33 0.3 0.7 0.315 0.0 0.8 115.80 114.89 114.00 113.14 112.30 0.28 0.33 0.2 0.8 0.320 0.1 0.6 138.74 135.93 133.28 130.77 128.40 0.28 0.33 0.2 0.8 0.320 0.1 0.7 124.52 122.94 121.42 119.96 118.56 0.28 0.33 0.2 0.8 0.320 0.1 0.8 113.66 112.87 112.11 111.37 110.65 0.28 0.33 0.1 0.9 0.325 0.4 0.6 128.62 126.66 124.81 123.05 121.38 0.28 0.33 0.1 0.9 0.325 0.4 0.7 116.78 115.72 114.71 113.73 112.79 0.28 0.33 0.1 0.9 0.325 0.4 0.8 108.28 107.80 107.32 106.87 106.43   Seven values of P and three different values of α were set to verify the relative efficiency of the suggested estimator Sˆ  with respect to the H. P. Singh and Tarray (2015) estimator STˆ . Shown in Table 1, the value of percent relative efficiency PRE( Sˆ , STˆ ) decreases as the values of P and α increase. The values of PRE( Sˆ , STˆ ) are greater than 100 for all values of πS1, πS2, w1, w2, P, α, and T considered here. The proposed estimator Sˆ  is more efficient than the H. P. Singh and Tarray (2015) estimator STˆ  in the case of the two strata in the population. It is further noted that, depending on the values of πS1, πS2, w1, and w2, A PROFICIENT TWO-STAGE STRATIFIED RRS 10 and keeping the cooperation from the respondents in view, the value of α as near to “Zero” or “Unity” as possible should be taken as an indication of a more efficient strategy. Acknowledgements The authors are thankful to the Editor-in-Chief and to the anonymous learned referee for their valuable suggestions regarding improvement of the paper. References Chaudhuri, A., & Mukerjee, R. (1988). Randomized response: Theory and techniques. New York, NY: Marcel-Dekker. Cochran, W. G. (1977). Sampling technique (3rd ed.). New York: John Wiley and Sons. Greenberg, B., Abul-Ela, A., Simmons, W. R., & Horvitz, D. G. (1969). The unrelated question randomized response: Theoretical framework. Journal of the American Statistical Association, 64(326),529-539. doi: 10.1080/01621459.1969.10500991 Hong, K. H., Yum, J. K., & Lee, H. Y. (1994). A stratified randomized response technique. Korean Journal of Applied Statistics, 7(1), 141-147. Retrieved from http://koreascience.or.kr/article/ArticleFullRecord.jsp?cn=GCGHDE_1994_v7n1_141 Kim, J.-M., & Elam, M. E. (2005). A two-stage stratified Warner’s randomized response model using optimal allocation. Metrika, 61(1), 1-7. doi: 10.1007/s001840400319 Kim, J.-M., & Elam, M. E. (2007). A stratified unrelated randomized response model. Statistical Papers, 48(2), 215-233. doi: 10.1007/s00362-006-0327-6 Kim, J.-M., Tebbs, J. M., & An, S.-W. (2006). Extensions of Mangat’s randomized-response model. Journal of Statistical Planning and Inference, 136(4), 1554-1567. doi: 10.1016/j.jspi.2004.10.005 Kim, J.-M., & Warde, W. D. (2004). A stratified Warner’s randomized response model. Journal of Statistical Planning and Inference, 120(1-2), 155-165. doi: 10.1016/s0378-3758(02)00500-1 TARRAY & SINGH 11 Kim, J.-M., & Warde, W. D. (2005). A mixed randomized response model. Journal of Statistical Planning and Inference, 133(1), 211-221. doi: 10.1016/j.jspi.2004.03.011 Land, M., Singh, S., & Sedory, S. A. (2012). Estimation of a rare attribute using Poisson distribution. Statistics, 46(3), 351-360. doi: 10.1080/02331888.2010.524300 Lee, G.-S., Uhm, D., & Kim, J.-M. (2011). Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution. Statistics, 47(3), 575-589. doi: 10.1080/02331888.2011.625503 Mangat, N. S. (1994). An improved randomized response strategy. Journal of the Royal Statistical Society. Series B (Methodological), 56(1), 93-95. Retrieved from https://www.jstor.org/stable/2346030 Mangat, N. S., & Singh, R. (1990). An alternative randomized procedure. Biometrika, 77(2), 439-442. doi: 10.1093/biomet/77.2.439 Ryu, J. B., Hong, K. H., & Lee, G. S. (1993). Randomized response model [Unpublished manuscript]. Seoul, Korea: Freedom Academy. Singh, H. P., & Tarray, T. A. (2015). A stratified Tracy and Osahan’s two-stage randomized response model. Communications in Statistics – Theory and Methods, 45(11), 3126-3137. doi: 10.1080/03610926.2014.895839 Singh, R., Singh, S., Mangat, N. S., & Tracy, D. S. (1995). An improved two stage randomized response strategy. Statistical Papers, 36(1), 265-271. doi: 10.1007/bf02926038 Singh, S. (2003). Advanced sampling theory with applications. Dordrecht, Netherlands: Kluwer Academic Publishers. Tracy, D. S., & Mangat, N. S. (1996). Some development in randomized response sampling during the last decade – A follow up of review by Chaudhuri and Mukherjee. Journal of Applied Statistical Science, 4(2/3), 147-158. Tracy, D. S., & Osahan, S. S. (1999). An improved randomized response technique. Pakistan Journal of Statistics, 15(1), 1-6. Warner, S. L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association, 60(309), 63-69. doi: 10.1080/01621459.1965.10480775 

A Proficient Two-Stage Stratified Randomized Response Strategy

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A Proficient Two-Stage Stratified Randomized Response Strategy

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