A Modified Stratified Randomized Response Techniques

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1 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Abstract A Modified Stratified Radoized Resose Teciques F.B. Adebola, O.O. Joso,.A. Adegoke * Deartet of Statistics, Federal Uiversity of Tecology Akure, P.M.B. 704, igeria. * E-ail of te corresodig autor: urudee.adegoke@yaoo.co Tis aer resets a odificatio of Ki ad arde s (004) Stratified arer s Radoized Resose Tecique (SRRT) ad Ki ad Ela s (007) Stratified Urelated Questio Radoized Resose Tecique wit kow distributio (SURRT) usig otial allocatio. Our odels furter reduce te o-resose bias by itroducig te cocet of sub-sales of o-resodet develoed by Hase ad Hurwitz (946) to te above etioed odels. I tis aer we erfor a eirical ractice of our odel ad we also erfor te eirical coariso of our odels wit bot Ki ad arde (004) ad Ki ad Ela (007) odels. e discovered a ew ote i te eirical coarative study of our odels. Keywords: Close suervisio, Sesitive beavior, o-resodet, Sub-sale, Radoized resose teciques.. Itroductio I oter to reduce biasess of resodet i surveys, ost esecially i surveys tat te questios are so tat igt cause te resodet ot to eve aswer te questios or falsely aswer te questios; i tis regard we aeal to Radoized Resose Teciques. Radoized Resose Teciques is te etod of rotectig te resodet s idetity so as to collect reliable iforatio o caracteristics. Sesitive caracteristics ivolve; rae, abortio, Sexually Trasitted Ifectio/HIV etc. Radoized Resose Teciques el to reduce o-salig error; o-salig error ivolves resose bias ad o-resose bias arer (965). Resose bias is we te resodet gives a false aswer ad o-resose bias is we te resodet refuses to aswer te questio. Radoized Resose Tecique was itroduced by arer (965) to estiate te roortio of te eole i a give oulatio wo is ivolve i a articular social or legal stigatized beavior. arer uses a radoized device wit kow robability to ecourage te resodet for trutful aswer suc tat te researcer does ot kow wic of te two questios e or se aswers. Greeberg et al. (969) roosed te urelated questio RR odel tat is a variatio of arer's (965) RR odel i wic te resodet is reseted wit oe questio ad oe urelated questio (wic is ot related to te stigatized beavior). Hog et al. (994) reseted a stratified radoized resose tecique usig roortioal allocatio. Ki ad arde (004) reseted Stratified arer s Radoized Resose Model (SRRT) usig a otial allocatio. Ki ad Ela (007) reseted te Stratified Urelated Questio Radoized Resose Model (SURRT) usig a otial allocatio. Tis aer develos a Modified Stratified Radoized Resose Model usig otial allocatio. e develoed for arer (965) ad for Greeberg et al. (969). As stated earlier tat Radoized Resose Teciques els to reduce resose ad o-resose bias. Our odel furter reduces te o-resose bias by itroducig te cocet of sub-sales of o-resodet develoed by Hase ad Hurwitz (946). A coarative study of our two odels was doe as well so as to deterie te ost efficiet. Radoized Resose Teciques arer (965) gave a geius idea by usig radoized device to ecourage trutful aswer fro te resodet wit resect to a beavior. Te radoizig device, suc as a siig arrow, dice or cois is used to select oe of te two questios; suc as, I ave raed before (Preseted wit robability P) I ave ot raed before (Preseted wit robability -P) Te resodets ave te otios or o reseted to i or er. Te iterviewer does ot kow wic questio ay resodet as aswered but kows te robability P ad -P wit wic te two stateets are reseted. Here, wit a rado sale of resodets, te iterviewer records a bioial estiate 9

2 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 ˆ y of te roortio θ of aswers, were y is te uber of yes aswers. If te questios are aswered trutfully, te relatio betwee ad i te oulatio is give as: ere is te roortio of eole wit te stigatized or beavior i stratu usig arer s teciques ad is te robability of selectig te questio i stratu. ˆ (.) (.) Te Variace is give by V ˆ (.3) Horvitz et al. (967) roosed te urelated questio desig wic is a alterative to te arer s etod. Tey discussed tat cobiig oe questio ad oe urelated questio would eace ore cooeratio of te resodets. Te urelated questio desig was furter iroved by Greeberg et al. (969). Tey exaied te sae desig were te distributio of te o- questio is kow i advace. Te urelated questio desig wit a kow distributio is a silified versio of te origial urelated questio desig. To illustrate te atter of te desig; Questio : Do you ave ore ta oe girlfried? Questio : Did you watc te 9.00 cartoo yesterday? By resetig te secod questio i tis for gives te distributio of te urelated questio. If all resod trutfully, te oulatio roortio of yes aswers is give by K U (.4) ere K is te roortio i te saled oulatio wo watc te 9:00 cartoo yesterday i stratu wic is always assue to be kow. If K is kow, te obvious (ad axiu likeliood) estiate of U is ˆ U ˆ ˆ K (.5) ere U is te roortio of eole wit te stigatized or beavior i stratu usig Urelated desig teciques ad is te robability of selectig te questio i stratu. Te Variace is give by V ˆ Uk (.6) For a review, Hog et al. (994) suggested a Stratified Radoized Resose Teciques (RRT) uder te roortioal salig assutio. Uder Hog et al.'s (994) roortioal salig assutio, it ay be easy to derive te variace of te roosed estiator. However, it ay coe at a ig cost i ters of tie, effort, ad oey. 30

3 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 To roffer solutio to te deficiecy of Hog et al. (994), Ki ad arde (004) ad Ki ad Ela (005) bot reseted Stratified Radoized Resose Teciques (SRRT) usig a otial allocatio wic are ore efficiet ta te Hog et al. (994) Stratified Radoized Resose Teciques (SRRT) usig a roortioal allocatio. (See Ki ad arde (004)) Tey derived te robability of a "" aswer i stratu for tis rocedure as: If all resod trutfully, (.7) for =,,..., ere is te roortio of eole wit te trait i stratu. Te axiu likeliood estiate ˆ of a roortio is give by: ˆ (.8) ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ). Te iiu variace of te estiator ad Ela (005)): ˆ is give by (See Ki V ˆ ( ) ( ) ( ) (.9) Ki ad Ela (007) develo a Stratified Radoized Resose Teciques (SRRT) odel for Greeberg et al.'s (969) Urelated Questio Radoized Resose Teciques (URRT) odel usig a otial allocatio. Ki ad Ela (007) reseted Stratified Radoized Resose Teciques (SRRT) usig a otial allocatio wic are ore efficiet ta te Ki ad arde (004) ad Ki ad Ela (005) Stratified Radoized Resose Teciques (SRRT) (See Ki ad Ela (007)) Tey derived te robability of a "" aswer i stratu for tis rocedure as: If all resod trutfully, U K (.0) For =,,... ere K is te roortio i te saled oulatio wo watc te 0:00 ews yesterday ad it is kow. ere U is te roortio of eole wit te trait i stratu. Te axiu likeliood estiate ˆ of a roortio is give by: U U ˆ U U (.) ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ) Te iiu variace of te estiator U give K is give by (See Ki ad Ela (007)): 3

4 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 V ˆ U \ K w ( ) (.) Here, we develo a Modified Stratified arer s Radoized Resose Tecique (MSRRT) ad Modified Stratified Urelated Questio Radoized Resose Tecique wit kow distributio (MSURRT) usig otial allocatio 3. Our Model Hase ad Hurwitz (946) rovided a tecique for dealig wit te roble of te uit o-resose i ail surveys. Te tecique is as follows: Te oulatio cosidered is deoted by ad te oulatio is stratified ito ( >) strata. e ave te uber of uits i te oulatio i stratu. Stratificatio is te etod of usig auxiliary iforatio to icrease te recisio of te estiate of te oulatio caracteristics. ( See Okafor 00). A sale of size was deteried, te uber of sale i eac stratu was deteried usig te otial allocatio. Te we ave te uber of uit i eac stratu as suc tat. By Sile Rado Salig witout Relaceet; a questioaire is ailed to eac stratu ad ad is te uber of uit resoded ad ot resoded i stratu resectively. e ave 3. For te Resodet: Te iterviewer records a bioial estiate ˆ y of te roortio θ of aswers, were y is te uber of yes aswers. For arer (965): If te questios are aswered trutfully, te relatio betwee ere teciques ad ˆ (3.) ad i te oulatio is give as: is te roortio of eole wit te stigatized or beavior i stratu usig arer s is te robability of selectig te questio i stratu. ˆ Te axiu likeliood estiate ˆ (3.) of a roortio for te resodet is give by: ˆ (3.3) Te Variace for stratu is give by V ˆ (3.4) 3

5 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ) Te iiu variace of te estiator ˆ for te resodet is give by: V ˆ ( ( ) ) ( ) (3.5) ere For Greeberg et al. (969): If all resod trutfully, te oulatio roortio of yes aswers is give by K U (3.6) K is te roortio i te saled oulatio wo watc te 9:00 cartoo yesterday i stratu ere wic is always assue to be kow. If ˆ K is kow, te obvious (ad axiu likeliood) estiate of U is U ˆ ere U desig teciques ad ˆ K (3.7) is te roortio of eole wit te stigatized or beavior i stratu usig Urelated Te axiu likeliood estiate ˆU ˆ U U is te robability of selectig te questio i stratu. of a roortio U for te resodet is give by: (3.8) ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ). Te Variace for stratu is give by V ˆ U \ K (3.9) Te iiu variace of te estiator U give K is give by (See Ki ad Ela (007)): V ˆ \ ( U K ) (3.0) 33

6 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 ere 3. For te o-resodet: A subsale of, k uits is selected fro te o-resodets by SileRado k Salig witout Relaceet i stratu. Te desired iforatio is ow collected fro tese uits by close suervisio. Tis tie aroud it is assued tat all te Te iterviewer records a bioial estiate te uber of yes aswers ad For arer (965): ˆ y uits resod. of te roortio θ of aswers, were y is is te sub-sale of te o-resodet i stratu. If te questios are aswered trutfully, te relatio betwee ad i te oulatio is give as: (3.) ere is te roortio of eole wit te stigatized or beavior i stratu usig arer s teciques ad is te robability of selectig te questio i stratu. ˆ ˆ (3.) Te axiu likeliood estiate ˆ of a roortio for te resodet is give by: ˆ (3.3) Te Variace for stratu is give by V ˆ (3.4) ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ) 34

7 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Te iiu variace of te estiator ˆ for te resodet is give by: V ˆ ( ( ) ) ( ) (3.5) ere For Greeberg et al. (969): If all resod trutfully, te oulatio roortio of yes aswers is give by K U (3.6) ere K is te roortio i te saled oulatio wo watc te 9:00 cartoo yesterday i stratu wic is always assue to be kow. If K is kow, te obvious (ad axiu likeliood) estiate of U is ˆ U ˆ ˆ K (3.7) ere U is te roortio of eole wit te stigatized or beavior i stratu usig Urelated desig teciques ad is te robability of selectig te questio i stratu. Te axiu likeliood estiate ˆU of a roortio U for te resodet is give by: ˆ U U (3.8) ere for =,,, so tat ( is te uber of uit i te oulatio ad is te total uber of uits i stratu ) Te Variace for stratu is give by 35

8 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, \ ˆ K U V (3.9) Te iiu variace of te estiator U give K is give by (See Ki ad Ela (007)): ) ( \ ˆ K U V (3.0) ere 3.3 Te True Proortio of te Sesitive Beavior Te two ideedet estiates of te roortio of te beavior obtaied fro te resodet ad o-resodet strata are suitably weigted to for te estiate of te oulatio roortio of te beavior uder study (See Hase ad Hurwitz (946)). ow, extedig Hase ad Hurwitz (946) work to radoized resose teciques, Te we ave, Te ubiased estiator for roortio of te beavior is give by: ˆ w w (3.) is te estiate of te roortio of te beavior of te resodets at te first ailigs. is te estiate of te roortio of te beavior of te sub-saled o-resodets at te secod call. ere, w ad w Te ubiased variace of te beavior give by: ) ( ) ( ) ( ) ( ) ( ˆ ˆ k V ) ( ) ( ) ( k w (3.)

9 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Te ubiased variace of ˆ of is obtaied uo relacig ( ) Okafor (00). ; y is relaced by ; y by ad y T s is relaced by by Equatio (3.) ad (3.) ca be used for bot arer (965) ad Greeberg et al. (969) ( ), s by (See Hase ad Hurwitz (946)) ad 4 Our Study Ki ad arde (004) ad Ki ad Ela (007) ave reseted Stratified Radoized Resose Teciques (SRRT) ad as doe te efficiecy coariso (See Ki ad Ela (007)), but we still ave to coare our odel wit tis two odels. Here, we erfor a eirical ractice of our odel ad we also erfor te eirical coariso of our odels wit bot Ki ad arde (004) ad Ki ad Ela (007) odels. e aly our odel to sexual beavior aog studets i wic we adiistered five differet questios. Te questios ivolve:. Have you cotracted Sexually trasitted disease before?. Have you aborted for a lady or abort a regacy before? 3. Have you raed or beig raed before? Tese questios were cosidered because tey are social udesirable attributes suc tat eole do ot feel cofortable to disclose iforatio about it. For Greeberg et al (969), we aired tese questios wit a urelated questio wit kow distributio. 4 Study Desig Te oulatio cosidered i te survey was te udergraduate studets of te Federal Uiversity of Tecology, Akure, igeria. Te oulatio was stratified ito two i wic we ave te Male ad te Feale (tat is =, ). Stratificatio is te etod of usig auxiliary iforatio to icrease te recisio of te estiate of te oulatio caracteristics. ( See Okafor 00). Two questioaires were desig to evaluate te erforace of our odel; oe usig te arer (965) desig ad te oter usig te Greeberg et al (969) desig. e asked soe questios before askig te questios. Te total oulatio of studets i te uiversity ( = 05, (878 ales ad 369 feales)), we cose a sale size of =50 for eac desig, otial allocatio was used to deterie te uber of eole i eac stratu, we ave 8 ales ad 69 feales. Out of te 50 questioaire set out for te Stratified arer s Radoized Resose Teciques, oly 83 (44 ales ad 39 feales) was useable (resodet), 67 was recorded o-resodet ad out of te 50 questioaire set out for Stratified Urelated Questio Radoized Resose Teciques, oly 00 (55 ales ad 45 feales) was useable(resodet), 50 was recorded o-resodet. e erfor a sub-sale of o-resodet by coosig k =, tis coice of k is to get a large sub-sale. As te rocedure ilies, te sub-sale was doe wit close suervisio (tis does ot ea tat we actually kew wic questio te resodet aswered but we el to silify te istructio so as to get ore cooeratio of te resodet). All te k sub-sale resoses were ow useable. 37

10 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 4 Results Table. uber origially set uber uretured Te Resose ad te o-resose of te Survey *SRRT **SURRT MAE % FEMAE % MAE % FEMAE % 8 00% 69 00% 8 00% 69 00% 5 8.9% % 6.63% % uber retured % % % % Ivalid resose.5% % % % Useable resose % % % % o-resodet % % % % * Stratified arer s Radoized Resose Teciques (Ki ad arde (004)) ** Stratified Urelated Questio Radoized Resose Teciques (Ki ad Ela (007)) 4.. For Resodet Te table below resets te uber of yes resose ad te roortio of studet wit te beavior alogside wit te corresodig variaces. Table. Tye of Sexual beavio r For Male SRRT* Useable resos e Reso se Proortio wit te beavior Variace of te beavior SURRT** Usea ble reso se Res ose Proortio wit te beavior Variace of te beavior Relative Efficie cy: R.E Q % % % Q % % % Q % % % Table 3. Tye of Sexual beavio r For Feale SRRT* Useabl e resos e Reso se Proortio wit te beavior Variace of te beavior SURRT** Usea ble reso se Res ose Proortio wit te beavior Variace of te beavior Relative Efficie cy: R.E Q % % % Q % % % Q3 39.9% % % 38

11 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Table 4. Tye Sexual beavior of Proortio of te Sesitive Beavior (Male ad Feale) For Resodets Proortio wit te beavior SRRT* Variace of te beavior Proortio wit te beavior SURRT** Variace of te beavior Q 4.56% % % Q 0.6% % % Q3.54% % % Relative Efficiecy: R.E 4.. Sub-Sale of o-resodet Table 5. Te Table Below Sows te Result of te Survey: Total o-resose uber sub-saled uber retured Useable Resose SRRT* SURRT** MAE % FEMAE % MAE % FEMAE % % 5 00% 3 00% 00% 9 00% 5 00% 3 00% 00% 9 00% 5 00% 3 00% 00% Te table below resets te uber of yes resose ad te roortio of studet wit te beavior alogside wit te corresodig variaces for subsale of o-resose. Table 6. Tye of Sexual beavio r SRRT* Useab le reso se For Male Res ose Proortio wit te beavior * Variace of te beavior ** SURRT** Usea ble reso se Res ose Proortio wit te beavior* Variace of te beavior ** Relative Efficiec y: R.E *** Q % % % Q % % % Q % % % Table 7. For Feale SRRT SURRT Tye of Sexual beavior Usea ble reso se Res ose Proortio wit te beavior Variace of te beavior Usea ble reso se Res ose Proortio wit te beavior Variace of te beavior Relative Efficiec y: R.E Q % % % Q % % % Q % % % 39

12 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Table 8. Te Total Proortio of te Sesitive Beavior for te o-resodets Tye Sexual beavior of Proortio wit te beavior SRRT Variace of te beavior SURRT Proortio wit te beavior Variace of te beavior Q 3.8% % % Q 4.64% % % Q3 5.9% % 0.00 % Relative Efficiecy: R.E 4..3 Te true estiate (Cobied Estiate) Here we cobie te estiates fro te resodets ad te o-resodets usig te etod develoed by Hase ad Hurwitz (946). Table 9. Te True Proortio of te Sesitive Beavior for te Resodets ad o-resodets. arer (965) Greeberg et al (969) Tye of Sexual Proortio wit Variace of te Proortio Variace of Relative beavior te wit te te Efficiecy: beavior beavior beavior beavior Var RE Var A AU Q 6.87% % % Q 6.8% % % Q3.7% % % 4.3 Coarig te Efficiecy of our Model wit Ki ad arde (004) ad Ki ad Ela (007) Model Table 0. Te Variace of te Models ad te Relative Efficiecy QUESTIOS SRRT SURRT SRRT SURRT * R.E ** R.E ***R.E3 (004) (007) (OUR (OUR A = 00 C B = 00 D C = 00 D (A) (B) MODE) MODE) (C) (D) Q % 0% 40% Q % 08% 83% Q % 09% 8% *R.E is te relative efficiecy of Ki ad arde (004) ad our odel for arer (965). **R.E is te relative efficiecy of Ki ad Ela (007) odel ad our ode for Greeberg et al. (969). 40

13 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 ***R.E 3 is te relative efficiecy of our odel for arer (965) ad our odel for Greeberg et al. (969). Relative efficiecy greater ta 00% ilies tat deoiator is better (or ore efficiet), oterwise te uerator is better (or ore efficiet). Fro our results i te table above we ca see tat te Variaces i colu C ad D (wic are our odels) are lesser ta te variaces i colu A ad B (wic are te Ki ad arde (004) ad Ki ad Ela (007) odels). Fro R.E, our odel for arer (965) is ore efficiet ta te Ki ad arde (004) cosiderig te relative efficiecy. Fro R.E, our odel for Greeberg et al (969) is ore efficiet ta Ki ad Ela (007) cosiderig te relative efficiecy. e ca also deduce fro R.E 3 tat te coarative study sows tat our odel for arer (965) is ore efficiet ta our odel for Greeberg et al. (969). 5 Coclusio Fig. Te relative efficiecy of te variaces. e ave successfully reseted a odel for arer (965) usig otial allocatio ad for Greeberg et al. (969) usig otial allocatio. Te eirical coarative study as sow tat our odel is ore efficiet i ters of variace ta te Ki ad arde (004) ad Ki ad Ela (007) odels. ikewise, te coarative study of our two odels sows tat our odel for arer (965) is ore efficiet i ters of variace ta our odel for Greeberg et al. (969). As discussed earlier tat Greeberg et al. (969) ecourage ore cooeratio fro te resodet because of te urelated aired questio, wic ake it ore efficiet ta arer (965) (See Greeberg et al. (969)), but we ave bee able to cater for te deficiecy of arer (965) by itroducig subsale of o-resodet ad tis gave birt to a ew ote i te coarative study. Coclusively, as stated earlier tat Radoized Resose Teciques els to reduce resose ad o-resose bias, i additio, our odel furter reduces te o-resose bias. 4

14 Mateatical Teory ad Modelig ISS (Paer) ISS 5-05 (Olie) Vol.4, o.3, 04 Refereces Adebola, F.B. & Adegoke,.A. (03), A Survey of Exaiatio Malractices usig te Radoized Resose Tecique, Joural of te igeria Associatio of Mateatical Pysics 3, Greeberg, B.G., Abul-Ela, A.A., Sios,.R. & Horvitz, D.G. (969), Te Urelated Questio Radoized Resose: Teoretical Fraework, Joural of te Aerica Statistical Associatio 64, Hase, M.H. & Hurwitz,.. (946), Te Proble of o-resose i Sale Surveys, Joural of te Aerica Statistical Associatio 4, Hog, K., Yu, J. & ee, H. (994), A Stratified Radoized Resose Tecique, Te Korea Joural of Alied Statistics 7, Horvitz, D.G., Sa, B.U. & Sios,.R. (967), Te Urelated Questio Radoized Resose Model, Proceedigs of te Social Statistics Sectio. Joural of te Aerica Statistical Associatio 60, Ki, J-M. & arde,.d. (004), A Stratified arer's Radoized Resose Model, Joural of Statistical Plaig ad Iferece 0 (-), Ki, J-M. &Ela, M.E. (005), A Two-Stage Stratified arer's Radoized Resose Model Usig Otial Allocatio, Metrika 6, -7. Ki J-M. & Ela, M.E. (007), A Stratified Urelated Questio Radoized Resose Model, Statistical Paer, 48, Okafor, F.C. (00), Sale Survey Teory wit Alicatios, First editio, Afro-Orbis ublisers, sukka, igeria. arer, S.. (965), Radoized resose: A Survey Tecique for Eliiatig Evasive Aswer Bias, Joural of te Aerica Statistical Associatio 60,

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