Iteratioal Joural of Busiess ad Social Sciece ol No ; October 00 A New Mixed adomized espose Model Aesha Nazuk NUST Busiess School Islamabad, Paksta E-mail: Aeshaazuk@bsedupk Phoe: 009-5-9085-367 Abstract Javid Shabbir Departmet of Statistics Quaid-i-Azam Uiversit Islamabad, Pakista I this stud we preset a modificatio of Kim ad Warde (004 model to estimate the proportio of a qualitative sesitive variable It has bee umericall show that the proposed model performs better tha the model of Kim ad Warde (004 ad Moors (97 Kewords: adomized respose techique (T, simple radom samplig with replacemet, sesitive variable, ad iocuous variable Itroductio Sesitive stud variables are ofte dealt i surve research such as proportio of adulterated milk packs of a compa, proportio of illicit drugs usage, etc Warer (965 itroduced a techique to estimate the true proportio of qualitative sesitive variable Greeberg et al (97 preseted a revised versio of Warer (965 techique for qualitative variables As far as qualitative sesitive variables are cocered, ma researchers have modified the Warer (965 model, some of them iclude Magat et al (997, Sigh et al (000, Chag ad Huag (00, Chag et al (004, ad Gupta et al (006 Kim ad Warde (004 preseted a ew radomized respose model usig simple radom sample with replacemet samplig scheme which improves the privac of respodets I this paper we have modified the Kim ad Warde (004 model to estimate the proportio of qualitative sesitive variable Mai aim of modificatio has bee to reduce the variace of estimator, for proportio of qualitative sesitive variable, ad to improve the privac protectio of the respodets Proposed Model Let a radom sample of size be selected usig simple radom samplig with replacemet Each respodet i the sample is istructed to aswer a iocuous questio I possess the iocuous characteristic Y If the aswer to the iitial direct questio is Yes the the respodet is istructed to go to radomizatio device otherwise to Where cosists of two statemets (i I belog to sesitive group ad (ii I belog to the iocuous group, with respective probabilit p ad ( p While cosists of the same pair of statemets as i but with respective probabilit p ad ( p I order to offer privac to the respodets the are ot required to tell that which radomizig device the have used Let ad be the umber of respodets usig ad + = respectivel such that ( 86
Cetre for Promotig Ideas, USA wwwijbssetcom Note that the respodets comig to have reported a Yes to the iitial direct questio therefore = i Deote b the probabilit of Yes from the respodets usig The p p = + ( p ( p = + ( A ubiased estimator for the true proportio of the sesitive trait is as follows ( p prop( =, ( p where is the sample proportio of Yes respose from the radomizig device The variace of prop ( is give b ( p + p ( = = (3 p prop ( p ( ( Note that the respodets usig have reported a No to the iitial direct questio therefore = 0 i Deote b the probabilit of Yes from the respodets usig, which is give b ( = p + p = p (4 Proceedig similarl as we have doe for, we get ( p prop( =, (5 p where is the sample proportio of Yes respose from the radomizig device The variace of prop ( is give b, ( ( p ( prop ( = = (6 p p Now we shall pool the two estimators usig weights, to form a estimator for prop prop ( pro p ( = + (7 Now the variace of prop is give b, ( p + ( p ( p prop = + where p p = p (see Lake (975 (8 Testig the Hpothesis of Truthful eportig arious researchers icludig Chag ad Huag (00 ad Chag et al (004 icorporated the possibilit of less tha completel reportig, eve with T The proposed model has bee equipped with the probabilities of truthful resposes We rewrite the proportio of Yes from the two devic ad, icorporatig the probabilit of truthful reportig Let probabilit of truthful reportig is deoted b T, where 0 T ad it is assumed that i Let Ti, i =,, be the probabilit of truthful respose i the first ad secod devices respectivel ii espodets do ot lie about the iocuous trait; the ma lie for 87
Iteratioal Joural of Busiess ad Social Sciece ol No ; October 00 Note that the respodets comig to have reported a Yes to the iitial direct questio therefore = i Deote b the probabilit of Yes from the respodets usig, which is give b = p T + ( p = p T + ( p (9 A ubiased estimator for the true proportio of the sesitive trait is as follows ( p = (0 t (, pt where is the sample proportio of Yes respose from the radomized respose The variace of t ( is give b ( t ( ( T p ( T = = ( ( ( T p pt Proceedig similarl as we have doe for the first device, we get, = p T + ( p = pt ( t ( = (3 p T The variace of t ( is give b ( t ( p T = (4 ( pt Now we formulate the weighted estimator for t t ( t(, = + The variace of t is give b ( t ( T p ( T = + pt ( p T T p = / p, (see Lake (975 ( where ( (5 (6 We ca test that whether the probabilit of truthful reportig is oe or less tha oe i first radomizig device, b testig H 0 : T = vs H : T < 0 The associated critical regio is z α / ( where z α / is the αth quitile poit of the stadard ormal distributio Similar aalsis ca be doe for 3 Estimatig the True Probabilit of Truthful eportig Oe ma be iterested i estimatig the true probabilities of truthful reportig i the populatio belogig to the two radom devices 88
Cetre for Promotig Ideas, USA wwwijbssetcom From (9 we costruct a ubiased estimator of T as ( p T = (7 p where is the sample proportio of Yes respose from the radomizig device The variace of T is give b ( ( ( ( T p T T = = (8 p p Similarl the estimator for T is obtaied from ( as follows T = p The variace of T is give b ( T ( pt T = (9 p 3 Efficiec Compariso A efficiec compariso of the proposed model, uder completel truthful respodig case, has bee doe with the Kim ad Warde (004 model From Kim ad Warde (004, we have ( ( p λp ( + ( λ ( KW = +, (3 p The PE of the prop with respect to KW, is as follows, ( i PE = 00 ( prop ( ( p λp + ( λ + p PE = 00 (3 ( p + ( p ( p + p I Tables ad, oe ma easil see that the proposed estimator prop is alwas, at least umericall, efficiet tha the Kim ad Warde s (004 estimator KW 5 Coclusio I this paper we have modified the Kim ad Warde (004 model We have also worked i the less tha completel truthful reportig but it has bee assumed that the respodets ma lie about the sesitive stud variable but ot the iocuous variable Ackowledgemet The authors are especiall appreciative for Dr Zawar Hussai of Quaid e Uiversit, Islamabad, Pakista 89
Iteratioal Joural of Busiess ad Social Sciece ol No ; October 00 efereces Chag, HJ ad Huag, KC (00: Estimatio of proportio ad sesitivit of a qualitative character Metrika, ol 53, pp 69-80 Chag, HJ, Wag, C,L ad Huag, KC (004: O estimatig the proportio of a qualitative sesitive character usig radomized respose samplig, Qualit ad Quatit, ol 38, pp 675-680 Chag, HJ, Wag, C,L ad Huag, KC (004: Usig radomized respose to estimate the proportio ad truthful reportig probabilit i a dichotomous fiite populatio, Joural of Applied Statistics, ol 3, pp 565-573 Gupta, S, Shabbir, J ad Iembo, (006: Modificatios to Warer s model usig blak cards, America Joural of Mathematical ad Maagemet Scieces, ol 6, pp 85-96 Greeberg, BG, Kubler,, Aberath, J ad Horvitz, DG (97: Applicatios of the techique i obtaiig quatitative data, Joural of the America Statistical Associatio, ol 66, pp 43-50 Kim, J ad Warde, WD (004: A mixed radomized respose model Joural of Statistical Plaig ad Iferece, ol 33, pp - Warer, SL (965: adomized respose: A surve techique for elimiatig evasive aswer bias Joural of the America Statistical Associatio, ol 60, pp 63-69 Magat, NS, Sigh, ad Sigh, S (997: iolatio of respodet s privac i Moors model its rectificatio through a radom group strateg respose model Commuicatios i Statistics Theor ad Methods, ol 6 pp 43 55 Sigh, S, Sigh, ad Magat NS (000: Some alterative strategies to Moor s model i radomized respose samplig- a surve techique for elimiatig evasive aswer bias Joural of Statistical Plaig ad Iferece, ol 83, pp 43-55 Table = PE p = 0 p = 03 p = 05 p = 07 p = 09 0 90 83486 00 880000 436835 93679 50 50 749 47990 7566 97739 393 70 30 566096 4657 7959 47337 088 0 90 785796 55430 9400 30080 498 50 50 6649 487 334 76397 4 70 30 6866 5878 055 39634 3599 0 90 755678 30 5004 4643 3436 50 50 38369 434364 45669 64474 7707 70 30 676 6096 74496 35050 053 0 90 740760 677 437500 8545 657 50 50 53653 4479 4065 5748 45 70 30 75453 7494 75000 3384 0838 0 90 739366 08068 400000 0360 996 50 50 743 46839 40000 5348 30 70 30 85997 9084 77777 36 075 0 90 7538 0309 3788 98 936 50 50 03636 50333 44067 5988 0857 70 30 0848 36546 83606 33 06584 00 0 0 03 04 05 06 90