Statistical matching using fractional imputation

Transcription

1 Sascs Publcaons Sascs 6-06 Sascal achng usng fraconal puaon Jae Kwang K Iowa Sae Unvers, jk@asae.edu El J. Berg Iowa Sae Unvers, elb@asae.edu Taesung Park Seoul Naonal Unvers Follow hs and addonal works a: hp://lb.dr.asae.edu/sa_las_pubs Par of he Desgn of Eperens and Saple Surves Coons, Sascal Mehodolog Coons, and he Sascal Models Coons The coplee bblographc nforaon for hs e can be found a hp://lb.dr.asae.edu/ sa_las_pubs/. For nforaon on how o ce hs e, please vs hp://lb.dr.asae.edu/ howoce.hl. Ths Arcle s brough o ou for free and open access b he Sascs a Iowa Sae Unvers Dgal Reposor. I has been acceped for ncluson n Sascs Publcaons b an auhorzed adnsraor of Iowa Sae Unvers Dgal Reposor. For ore nforaon, please conac dgrep@asae.edu.

2 Surve Mehodolog, June 06 9 Vol. 4, No., pp Sascs Canada, Caalogue No. -00-X Sascal achng usng fraconal puaon Jae Kwang K, El Berg and Taesung Park Absrac Sascal achng s a echnque for negrang wo or ore daa ses when nforaon avalable for achng records for ndvdual parcpans across daa ses s ncoplee. Sascal achng can be vewed as a ssng daa proble where a researcher wans o perfor a jon analss of varables ha are never jonl observed. A condonal ndependence assupon s ofen used o creae pued daa for sascal achng. We consder a general approach o sascal achng usng paraerc fraconal puaon of K (0) o creae pued daa under he assupon ha he specfed odel s full denfed. The proposed ehod does no have a convergen epecaon-asaon (EM) sequence f he odel s no denfed. We also presen varance esaors approprae for he puaon procedure. We eplan how he ehod apples drecl o he analss of daa fro spl quesonnare desgns and easureen error odels. Ke Words: Daa cobnaon; Daa fuson; Ho deck puaon; Spl quesonnare desgn; Measureen error odel. Inroducon Surve saplng s a scenfc ool for akng nference abou he arge populaon. However, we ofen do no collec all he necessar nforaon n a sngle surve, due o e and cos consrans. In hs case, we wsh o eplo, as uch as possble, nforaon alread avalable fro dfferen daa sources fro he sae arge populaon. Sascal achng, soees called daa fuson (Baker, Harrs and O Bren 989) or daa cobnaon (Rdder and Moff 007), as o negrae wo or ore daa ses when nforaon avalable for achng records for ndvdual parcpans across daa ses s ncoplee. D Orazo, Zo and Scanu (006) and Leulescu and Agafe (03) provde coprehensve overvews of he sascal achng echnques n surve saplng. Sascal achng can be vewed as a ssng daa proble where a researcher wans o perfor a jon analss of varables ha are never jonl observed. Morar and Scheuren (00) provde a heorecal fraework for sascal achng under a ulvarae noral assupon. Rässler (00) develops ulple puaon echnques for sascal achng wh pre-specfed paraeer values for nondenfable paraeers. Lahr and Larsen (005) address regresson analss wh lnked daa. Rdder and Moff (007) provde a rgorous reaen of he assupons and approaches for sascal achng n he cone of econoercs. Sascal achng as o consruc full augened daa fles o perfor sascall vald jon analses. To splf he seup, suppose ha wo surves, Surve A and Surve B, conan paral nforaon abou he populaon. Suppose ha we observe and fro he Surve A saple and observe and fro he Surve B saple. Table. llusraes a sple daa srucure for achng. If he Surve B saple (Saple B) s a subse of he Surve A saple (Saple A), hen we can appl record lnkage echnques (Herzog, Scheuren and Wnkler 007) o oban values of for he surve B saple. However, n an cases, such perfec achng s no possble (for nsance, because he saples a conan. Jae Kwang K, Deparen of Sascs, Iowa Sae Unvers, Aes, IA 500, U.S.A. E-al: jk@asae.edu; El Berg, Deparen of Sascs, Iowa Sae Unvers, Aes, Iowa, U.S.A. E-al: elb@asae.edu; Taesung Park, Deparen of Sascs, Seoul Naonal Unvers, Seoul, Korea. E-al: aesungp@gal.co.

3 0 K, Berg and Park: Sascal achng usng fraconal puaon non-overlappng subses), and we a rel on a probablsc wa of denfng he sascal wns fro he oher saple. Tha s, we wan o creae for each eleen n saple B b fndng he neares neghbor fro Saple A. Neares neghbor puaon has been dscussed b an auhors, ncludng Chen and Shao (00) and Beauon and Bocc (009), n he cone of ssng surve es. Table. A sple daa srucure for achng X Y Y Saple A o o Saple B o o Fndng he neares neghbor s ofen based on how close he are n ers of s onl. Thus, n an cases, sascal achng s based on he assupon ha and are ndependen, condonal on. Tha s,. (.) Assupon (.) s ofen referred o as he condonal ndependence (CI) assupon and s heavl used n pracce. In hs paper, we consder an alernave approach ha does no rel on he CI assupon. Afer we dscuss he assupons n Secon, we presen he proposed ehods n Secon 3. Furherore, we consder wo eensons, one o spl quesonnare desgns (n Secon 4) and he oher o easureen error odels (n Secon 5). Resuls fro wo sulaon sudes are presened n Secon 6. Secon 7 concludes he paper. Basc seup For splc of he presenaon, we consder he seup of wo ndependen surves fro he sae arge populaon conssng of N eleens. As dscussed n Secon, suppose ha Saple A collecs nforaon onl on and and Saple B collecs nforaon onl on and. To llusrae he dea, suppose for now ha,, are generaed fro a noral dsrbuon such ha N,. Clearl, under he daa srucure n Table., he paraeer s no esable fro he saples. The condonal ndependence assupon n (.) ples ha = and =. Tha s, s copleel deerned fro oher paraeers, raher han esaed drecl fro he realzed saples. Sascs Canada, Caalogue No. -00-X

4 Surve Mehodolog, June 06 Snhec daa puaon under he condonal ndependence assupon n hs case can be pleened n wo seps: [Sep ] Esae f fro Saple A, and denoe he esae b f ˆ. [Sep ] For each eleen n Saple B, use he f ˆ. a a value o generae pued value(s) of fro Snce values are never observed n Saple B, snhec values of are creaed for all eleens n Saple B, leadng o snhec puaon. Hazza (009) provdes a nce revew of leraure on puaon ehodolog. K and Rao (0) presen a odel-asssed approach o snhec puaon when onl s avalable n Saple B. Such snhec puaon copleel gnores he observed nforaon n fro Saple B. Sascal achng based on condonal ndependence assues ha Cov, = 0. Thus, he regresson of on and usng he pued daa fro he above snhec puaon wll esae a zero regresson coeffcen for. Tha s, he esae ˆ for ˆ = ˆ ˆ ˆ, 0 wll esae zero. Such analses can be sleadng f CI does no hold. To eplan wh, we consder an oed varable regresson proble: = 0 = 0 z e z e where ze,, e are ndependen and are no observed. Unless = = 0, he laen varable z s an unobservable confoundng facor ha eplans wh Cov, 0. Thus, he coeffcen on n he populaon regresson of on and s no zero. Noe ha he CI assupon s an assupon for odel denfcaon. Anoher denfng assupon s he nsruenal varable (IV) assupon, as descrbed n he followng reark. Reark. We presen a foral descrpon of he IV assupon. Frs, assue ha we can decopose as =, such ha () f,, = f, () f, = a f, b for soe a b. Thus, s condonall ndependen of gven and bu s correlaed wh gven. Noe ha a be null or have a degenerae dsrbuon, such as an nercep. The varable sasfng he above wo condons s ofen called an nsruenal varable (IV) for. The dreced acclc graph n Fgure. llusraes he dependence srucure of a odel wh an nsruenal varable. Rdder and Moff (007) used ecluson resrcons o descrbe he nsruenal varable assupon. One eaple where he nsruenal varable assupon s reasonable s repeaed surves. In he repeaed surve, suppose ha s he sud varable a ear and sasfes Markov proper Sascs Canada, Caalogue No. -00-X

5 K, Berg and Park: Sascal achng usng fraconal puaon P,, = P, where P denoes a cuulave dsrbuon funcon. In hs case, s an nsruenal varable for In fac, an las observaon of s s he nsruenal varable for.. s Fgure. Graphcal llusraon of he dependence srucure for a odel n whch s an nsruenal varable for and s an addonal covarae n he odels for and. Under he nsruenal varable assupon, one can use wo-sep regresson o esae he regresson paraeers of a lnear odel. The followng eaple presens he basc deas. Eaple. Consder he wo saple daa srucure n Table.. We assue he followng lnear regresson odel: where e 0, e = e, (.) 0 and e s ndependen of j, j, j for all, j. In hs case, a conssen esaor of =,, can be obaned b he wo-sage leas squares (SLS) ehod as follows: 0. Fro Saple A, f he followng workng odel for 0 u = u, u 0, (.) o oban a conssen esaor of =,, defned b 0 =,, = X X XY ˆ ˆ ˆ ˆ 0 where X = X0, X, X s a ar whose h row s,, and Y s a vecor wh beng he h coponen.. A conssen esaor of = 0,, s obaned b he leas squares ehod for he regresson of on, ˆ, where ˆ = ˆ 0 ˆ ˆ. Sascs Canada, Caalogue No. -00-X

6 Surve Mehodolog, June 06 3 Aspoc unbasedness of he SLS esaor under he nsruenal varable assupon s dscussed n Append A. The SLS ehod s no drecl applcable f he regresson odel (.) s nonlnear. Also, whle he SLS ehod gves esaes of he regresson paraeers, SLS does no provde conssen esaors for ore general paraeers such as =Pr < <3. Sochasc puaon can provde a soluon for esang a ore general class of paraeers. We eplan how o odf paraerc fraconal puaon of K (0) o address general purpose esaon n sascal achng probles. 3 Fraconal puaon We now descrbe he fraconal puaon ehods for sascal achng whou usng he CI assupon. The use of fraconal puaon for sascal achng was orgnall presened n Chaper 9 of K and Shao (03) under he IV assupon. In hs paper, we presen he ehodolog whou requrng he IV assupon. We onl assue ha he specfed odel s full denfed. The denfabl of he specfed odel can be easl checked n he copuaon of he proposed procedure. To eplan he dea, noe ha s ssng n Saple B and our goal s o generae fro he condonal dsrbuon of gven he observaons. Tha s, we wsh o generae fro f, f, f. (3.) To generae fro (3.), we can consder he followng wo-sep puaon:. Generae. Accep fro f ˆ. f, a f s suffcenl large. Noe ha he frs sep s he usual ehod under he CI assupon. The second sep ncorporaes he nforaon n. The deernaon of wheher f, s suffcenl large requred for Sep s ofen ade b applng a Markov Chan Mone Carlo (MCMC) ehod such as he Meropols-Hasngs algorh (Chb and Greenberg 995). Tha s, le be he curren value of n he Markov Chan. Then, we accep wh probabl f, R, = n,. f, Such algorhs can be copuaonall cubersoe because of slow convergence of he MCMC algorh. Paraerc fraconal puaon of K (0) enables generang pued values n (3.) whou requrng MCMC. The followng EM algorh b fraconal puaon can be used:. For each B, generae pued values of, denoed b,,, fro f ˆ, a where fˆa denoes he esaed dens for he condonal dsrbuon of gven obaned fro Saple A. Sascs Canada, Caalogue No. -00-X

7 4 K, Berg and Park: Sascal achng usng fraconal puaon. Le ˆ be he curren paraeer value of n f,. For he assgn he fraconal wegh such ha j= w j =. j w f ˆ j, ; 3. Solve he fraconall pued score equaon for h j pued value j, j b j j= w w S ;,, = 0 (3.) B o oban ˆ, where S ;,, = log f, ;, and w b s he saplng wegh of un n Saple B. 4. Go o Sep and connue unl convergence. When he odel s denfed, he EM sequence obaned fro he above PFI ehod wll converge. If he specfed odel s no denfable hen here s no unque soluon o azng he observed lkelhood and he above EM sequence does no converge. In (3.), noe ha, for suffcenl large, ˆ ˆ ˆ j S ;,, f, ; fa d j wj S ;,, j ˆ ˆ j= f, ; fa d = E S;, Y,, ;. If s caegorcal, hen he fraconal wegh can be consruced b he condonal probabl correspondng o he realzed pued value (Ibrah 990). Sep s used o ncorporae observed nforaon of n Saple B. Noe ha Sep s no repeaed for each eraon. Onl Sep and Sep 3 are eraed unl convergence. Because Sep s no eraed, convergence s guaraneed and he observed lkelhood ncreases, as long as he odel s denfable. See Theore of K (0). Reark 3. In Secon, we nroduce IV onl because hs s wha s pcall done n he leraure o ensure denfabl. The proposed ehod self does no rel on hs assupon. To llusrae a suaon where we can denf he odel whou nroducng he IV assupon, suppose ha he odel s wh e N0, and e e N = e = e 0 0 0,. Then f = f, f d s also a noral dsrbuon wh ean 0 0 and varance. Under he daa srucure n Table., such a odel s denfed whou assung he IV assupon. The Sascs Canada, Caalogue No. -00-X

8 Surve Mehodolog, June 06 5 assupon of no neracon beween and n he odel for s ke o ensurng he odel s denfable. j Insead of generang fro fˆ a, we can consder a ho-deck fraconal puaon (HDFI) ehod, where all he observed values of n Saple A are used as pued values. In hs case, he fraconal weghs n Sep are gven b where ˆ j 0 w w f, ; ˆ, j j w j0 fˆ a j =. w fˆ ka ka a j k (3.3) The nal fraconal wegh w j0 n (3.3) s copued b applng porance weghng wh = fˆ fˆ fˆ d w fˆ a j a j a a a j A as he proposal dens for. j The M-sep s he sae as for paraerc fraconal puaon. See K and Yang (04) for ore deals on HDFI. In pracce, we a use a sngle pued value for each un. In hs case, he fraconal weghs can be used as he selecon probabl n Probabl-Proporonal-o-Sze (PPS) saplng of sze =. For varance esaon, we can eher use a lnearzaon ehod or a resaplng ehod. We frs consder varance esaon for he au lkelhood esaor (MLE) of. If we use a paraerc odel f = f ; and f, ;, he MLE of =, s obaned b solvng S, S, = 0,0, (3.4) where S = was, S = log f ; s he score funcon of, A S, = ES X, Y ;,, S = wbs, and S B = log f, ; s he score funcon of. Noe ha we can wre S, = wbes, ;. Thus, and B S f ; f, ; d S = wb B f ; f, ; d = w ES S, ; B B b w ES, ; ES, ; b Sascs Canada, Caalogue No. -00-X

9 6 K, Berg and Park: Sascal achng usng fraconal puaon S f ; f, ; d S = wb B f ; f, ; d = wbe S, ; B w ES S, ; B B b w ES, ; ES, ;. b Now, S can be conssenl esaed b ˆ B = w w S S S, ˆ ˆ ˆ b j j j (3.5) B j= where ˆ ˆ j S j = S ;,, ˆ ˆ j S j = S ;,,, and ˆ ˆ j S wjs Also, S can be conssenl esaed b where = ;,. ˆ ˆ b j j (3.6) Iˆ = w w S B B j= ˆ B = w w S S S, b j j j B j= ˆ ˆ ˆ S = S ;, j, and S = w S. j Usng a Talor epanson wh respec o, and we can wre Wrng j= j j S S E S E S S = S KS, ˆ,, ˆ V E S VS KSE S. S = w s, b B j= Sascs Canada, Caalogue No. -00-X

10 Surve Mehodolog, June 06 7 wh s = ES, ;, a conssen esaor of VS can be obaned b applng a desgn-conssen varance esaor o w s b ˆ B wh sˆ ˆ = w. j= js j Under sple rando saplng for Saple B, we have Also, VKS s conssenl esaed b ˆ VS = n sˆ sˆ. B B V ˆ = KV ˆ ˆ S K ˆ, where Kˆ = Bˆ ˆ I, ˆB s defned n (3.5), and Iˆ = S evaluaed a ˆ =. Snce he wo ers S and S are ndependen, he varance can be esaed b ˆ ˆ ˆ ˆ ˆ ˆ V I V S VI, where Î s defned n (3.6). More generall, one a consder esaon of a paraeer defned as a roo of he census esang N equaon U ;,, = 0. = Varance esaon of he FI esaor of copued fro j w w U ;,, = 0 b j s dscussed n Append B. B j= 4 Spl quesonnare surve desgn In Secon 3, we consder he suaon where Saple A and Saple B are wo ndependen saples fro he sae arge populaon. We now consder anoher suaon of a spl quesonnare desgn where he orgnal saple S s seleced fro a arge populaon and hen Saple A and Saple B are randol chosen such ha A B = S and A B =. We observe, fro Saple A and observe, fro Saple B. We are neresed n creang full augened daa wh observaon,, n S. Such spl quesonnare surve desgns are ganng popular because he reduce response burden (Raghunahan and Grzzle 995; Chpperfeld and Seel 009). Spl quesonnare desgns have been nvesgaed, for eaple, for he Consuer Ependure surve (Gonzalez and Elnge 008) and he Naonal Assessen of Educaonal Progress (NAEP) surve n he US. In applcaons of splquesonnare desgns, analss a be neresed n ulple paraeers such as he ean of and he ean of, n addon o he coeffcen n he regresson of on. We consder a desgn where he orgnal Saple S s paroned no wo subsaples: A and B. We assue ha s observed for S, s colleced for A and s colleced for B. The probabl of selecon no A or B a depend on bu does no depend on or. As a consequence, he desgn used o selec subsaple A or B s non-nforave for he specfed odel (Fuller 009, Chaper 6). We le w denoe he saplng wegh assocaed wh he full saple S. We assue a procedure s avalable for esang he varance of an esaor of he for Yˆ = w, and we denoe S Vˆ w. he varance esaor b s S Sascs Canada, Caalogue No. -00-X

11 8 K, Berg and Park: Sascal achng usng fraconal puaon A procedure for obanng a full pued daa se s as follows. Frs, use he procedure of Secon 3 o j oban pued values : B, j =,, and an esae, ˆ, of he paraeer n he dsrbuon f, ;. The esae ˆ s obaned b solvng j j j= w w S ;,, = 0, (4.) B where S ;,, = log f, ;. Gven ˆ, generae pued values j f, ; ˆ, for A and j =,,. Under he assupon ha he odel s denfed, he paraeer esaor ˆ generaed b solvng (4.) s full effcen n he sense ha he pued value of for Saple A leads o no effcenc gan. To see hs, noe ha he score equaon usng he pued value of s copued b Because,, j j w S w wjs A j= B j= ;,, ;,, = 0. (4.) are generaed fro f ˆ, ;, j l A j= A p w S ;,, = w E S ;,, Y, ; ˆ. Thus, b he proper of score funcon, he frs er of (4.) evaluaed a = ˆ s close o zero and he soluon o (4.) s essenall he sae as he soluon o (4.). Tha s, here s no effcenc gan n usng he pued value of n copung he MLE for n f, ;. However, he pued values of can prove he effcenc of nferences for paraeers n he jon dsrbuon of,. As a sple eaple, consder esaon of, he argnal ean of. Under sple rando saplng, he pued esaor of = E Y s where,, j ˆ I, =, n A j= B are generaed fro ˆ (4.3) f, ;. For suffcenl large, we can wre ˆ = ˆ = ˆ E, ;. n A B I, n A B Under he seup of Eaple., we can epress ˆ = ˆ ˆ ˆ 0 B where ˆ 0, ˆ, ˆ ˆ ˆ ˆ ˆ =0 0 sasfes Sascs Canada, Caalogue No. -00-X

12 Surve Mehodolog, June 06 9 and ˆ = ˆ 0 ˆ ˆ wh ˆ 0, ˆ, ˆ sasfng ˆ 0 ˆ ˆ =0. Thus, A gnorng he saller order ers, we have V V V ˆ = ˆ I, n nb n whch s saller han he varance of he drec esaor ˆ = n. b b B 5 Measureen error odels We now consder an applcaon of sascal achng o he proble of easureen error odels. Suppose ha we are neresed n he paraeer n he condonal dsrbuon f ;. In he orgnal saple, nsead of observng,, we observe,, where s a conanaed verson of. Because nference for based on, a be based, addonal nforaon s needed. One coon wa o oban addonal nforaon s o collec, n an eernal calbraon sud. In hs case, we observe, n Saple A and, n Saple B, where Saple A s he calbraon saple, and Saple B s he an saple. Guo and Lle (0) dscuss an applcaon of eernal calbraon. The eernal calbraon fraework can be epressed as a sascal achng proble. Table 5. akes he connecon beween sascal achng and eernal calbraon eplc. An nsruenal varable assupon pers nference for based on daa wh he srucure of Table.. In he noaon of he easureen error odel, he nsruenal varable assupon s f, = f and f = a f = b, (5.) for soe a b. The nsruenal varable assupon a be judged reasonable n applcaons relaed o error n covaraes because he subjec-aer odel of neres s f, and s a conanaed verson of ha conans no addonal nforaon abou gven. Table 5. Daa srucure for easureen error odel Surve A (calbraon sud) o o Surve B (an sud) o o For full paraerc f and f, one can use paraerc fraconal puaon o eecue he EM algorh. Ths ehod requres evaluang he condonal epecaon of he coplee-daa score funcon gven he observed values. To evaluae he condonal epecaon usng fraconal puaon, we frs epress he condonal dsrbuon of gven, as, f, f f. (5.) Sascs Canada, Caalogue No. -00-X

13 30 K, Berg and Park: Sascal achng usng fraconal puaon We le an esaor fˆa of f be avalable fro he calbraon saple (Saple A). Ipleenaon of he EM algorh va fraconal puaon nvolves he followng seps:. For each B, generae fro f ˆ, for j =,,. j. Copue he fraconal weghs wh j= w j =. 3. Updae b solvng j a w f ˆ j ; B j b j j= w w S ;, = 0, where S ;, = log f ;. 4. Go o Sep unl convergence. The ehod above requres generang daa fro f. For soe nonlnear odels or odels wh non-consan varances, sulang fro he condonal dsrbuon of gven a requre Mone Carlo ehods such as accep-rejec or Meropols Hasngs. The sulaon of Secon 6. eeplfes a sulaon n whch he condonal dsrbuon of has no closed for epresson. In hs case, we a consder an alernave approach, whch a be copuaonall spler. To descrbe hs approach, le h be he workng condonal dsrbuon, such as he noral dsrbuon, fro whch saples are easl generaed. We assue ha esaes fˆa and hˆa of f and h, respecvel, are avalable fro Saple A. Ipleenaon of he EM algorh va fraconal puaon hen nvolves he followng seps:. For each B, generae fro h ˆ, for j =,,. j. Copue he fraconal weghs wh j= w j j a j ˆ ˆ j ; ˆ j a a w f f h (5.3) =. 3. Updae b solvng B j b j j= 4. Go o Sep unl convergence. w w S ;, = 0. Varance esaon s a sraghforward applcaon of he lnearzaon ehod n Secon 3. The hodeck fraconal puaon ehod descrbed n Secon 3 wh fraconal weghs defned n (3.3) also eends readl o he easureen error seng. Sascs Canada, Caalogue No. -00-X

14 Surve Mehodolog, June Sulaon sud To es our heor, we presen wo led sulaon sudes. The frs sulaon sud consders he seup of cobnng wo ndependen surves of paral observaon o oban jon analss. The second sulaon sud consders he seup of easureen error odels wh eernal calbraon. 6. Sulaon one To copare he proposed ehods wh he esng ehods, we generae 5,000 Mone Carlo saples of,, wh sze n = 400, where 0,, and N,, = e, (6.) 0 e N =,, =,,. Noe ha, n hs seup, we have f, = f and so he varable plas he role of he nsruenal varable for. Insead of observng,, jonl, we assue ha onl, are observed n Saple A and onl, are observed n Saple B, where Saple A s obaned b akng he frs n a =400 eleens and Saple B s obaned b akng he reanng n b = 400 eleens fro he orgnal saple. We are neresed n esang four paraeers: hree regresson paraeers 0,, and = P <, <3, he proporon of < and <3. Four ehods are consdered n esang he paraeers:. Full saple esaon (Full): Uses he coplee observaon of, n Saple B.. Sochasc regresson puaon (SRI): Use he regresson of on fro Saple A o oban ˆ 0, ˆ, ˆ, where he regresson odel s = 0 e wh e 0,. For each j j B, =0 pued values are generaed b = ˆ 0 ˆ e where j e N 0,. ˆ 3. Paraerc fraconal puaon (PFI) wh = 0 usng he nsruenal varable assupon. 4. Ho-deck fraconal puaon (HDFI) wh =0 usng he nsruenal varable assupon. Table 6. presens Mone Carlo eans and Mone Carlo varances of he pon esaors of he four paraeers of neres. SRI shows large bases for all paraeers consdered because s based on he condonal ndependence assupon. Boh PFI and HDFI provde nearl unbased esaors for all paraeers. Esaors fro HDFI are slghl ore effcen han hose fro PFI because he wo-sep procedure n HDFI uses he full se of respondens n he frs sep. The heorecal aspoc varance of ˆ copued fro PFI s ˆ 0.7 V Sascs Canada, Caalogue No. -00-X

15 3 K, Berg and Park: Sascal achng usng fraconal puaon whch s conssen wh he sulaon resul n Table 6.. In addon o pon esaon, we also copue varance esaors for PFI and HDFI ehods. Varance esaors show sall relave bases (less han 5% n absolue values) for all paraeers. Varance esaon resuls are no presened here for brev. Table 6. Mone Carlo eans and varances of pon esaors fro Sulaon One. (SRI, sochasc regresson puaon; PFI, paraerc fraconal puaon; HDFI; ho-deck fraconal puaon) Paraeer Mehod Mean Varance 0 Full SRI PFI HDFI Full SRI PFI HDFI Full SRI PFI HDFI Full SRI PFI HDFI The proposed ehod s based on he nsruenal varable assupon. To sud he sensv of he proposed fraconal puaon ehod o volaons of he nsruenal varable assupon, we perfored an addonal sulaon sud. Now, nsead of generang fro (6.), we use =0.5 3 e, (6.) where e N0, and can ake non-zero values. We use hree values of, 0,0.,0., n he sensv analss and appl he sae PFI and HDFI procedure ha s based on he assupon ha s an nsruenal varable for. Such assupon s sasfed for = 0, bu s weakl volaed for = 0. or = 0.. Usng he fraconall pued daa n saple B, we esaed hree paraeers, = E Y, s he slope for he sple regresson of on, and 3 = P <, < 3, he proporon of < and <3. Table 6. presens Mone Carlo eans and varances of he pon esaors for hree paraeers under hree dfferen odels. In Table 6., he absolue values of he dfference beween he fraconall pued esaor and he full saple esaor ncrease as he value of ncreases, whch s epeced as he nsruenal varable assupon s ore severel volaed for larger values of, bu he dfferences are relavel sall for all cases. In parcular, he esaor of s no affeced b he deparure fro he nsruenal varable assupon. Ths s because he puaon esaor under he ncorrec puaon odel sll provdes an unbased esaor for he populaon ean as long as he regresson puaon odel conans an nercep er (K and Rao 0). Thus, hs led sensv analss Sascs Canada, Caalogue No. -00-X

16 Surve Mehodolog, June suggess ha he proposed ehod sees o provde coparable esaes when he nsruenal varable assupon s weakl volaed. Table 6. Mone Carlo eans and Mone Carlo varances of he wo pon esaors for sensv analss n Sulaon One (Full, full saple esaor; PFI, paraerc fraconal puaon; HDFI; ho-deck fraconal puaon) Model Paraeer Mehod Mean Varance =0 Full PFI HDFI Full PFI HDFI Full PFI HDFI =0. Full PFI HDFI Full PFI HDFI Full PFI HDFI =0. Full PFI HDFI Full PFI HDFI Full PFI HDFI Sulaon wo In he second sulaon sud, we consder a bnar response varable, where and N Bernoull p, (6.3) log p = 0,,. In he an saple, denoed b B, nsead of observng,, we observe,, where 0 = u, (6.4) Sascs Canada, Caalogue No. -00-X

17 34 K, Berg and Park: Sascal achng usng fraconal puaon and u N 0,. We observe,, =,, na n a calbraon saple, denoed b A. For he sulaon, na = n B = 800, =, =, =0, =, 0 0 =0.5, =0.4, =0, and =. Prar neres s n esaon of and esng he null hpohess ha =. The Mone Carlo (MC) saple sze s,000. We copare he PFI esaors of o hree oher esaors. Because he condonal dsrbuon of gven s non-sandard, we use he weghs of (5.3) o pleen PFI, where he proposal dsrbuon hˆa s an esae of he argnal dsrbuon of based on he daa fro Saple A. We consder he followng four esaors: j. PFI: For PFI, he proposal dsrbuon for generang s a noral dsrbuon wh ean ˆ and varance ˆ, where ˆ and ˆ are he au lkelhood esaes of and, respecvel, based on Saple A. The fraconal weghs defned n (5.3) has he for ˆ ˆ j j ˆj j a w p p f, (6.5) j where pˆ j = ep ˆ 0 ˆ and fˆa s he esae of f based on au lkelhood esaon wh he Saple A daa. The puaon sze = Nave: A nave esaor s he esaor of he slope n he logsc regresson of on for B. 3. Baes: We use he approach of Guo and Lle (0) o defne a Baes esaor. The odel for hs sulaon dffers fro he odel of Guo and Lle (0) n ha he response of neres s bnar. We pleen GIBBS saplng wh JAGS (Pluer 003), specfng dffuse proper pror dsrbuons for he paraeers of he odel. Leng 6 we assue a pror ha N I 7 = log,log,,,,,, 0 0 0,0, where I 7 s a 7 7 den ar, and he noaon N 0, V denoes a noral dsrbuon wh ean 0 and covarance ar V. The pror dsrbuon for he power s unfor on he nerval 5,5. To evaluae convergence, we eane race plos and poenal scale reducon facors defned n Gelan, Carln, Sern and Rubn (003) for 0 prelnar sulaed daa ses. We nae hree MCMC chans, each of lengh,500 fro rando sarng values and dscard he frs 500 eraons as burn-n. The poenal scale reducon facors across he 0 sulaed daa ses range fro.0 o., and he race plos ndcae ha he chans well. To reduce copung e, we use,000 eraons of a sngle chan for he an sulaon, afer dscardng he frs 500 for burn-n. 4. A Weghed Regresson Calbraon (WRC) esaor. The WRC esaor s a odfcaon of he weghed regresson calbraon esaor defned n Guo and Lle (0) for a bnar response varable. The copuaon for he weghed regresson calbraon esaor nvolves he followng seps: () Usng ordnar leas squares (OLS), regress on for he calbraon saple. Sascs Canada, Caalogue No. -00-X

18 Surve Mehodolog, June () Regress he logarh of he squared resduals fro sep () on he logarh of calbraon saple. Le ˆ denoe he esaed slope fro he regresson. ˆ, for he () Usng weghed leas squares (WLS) wh wegh regress on for he calbraon saple. Le 0 ˆ and ˆ be he esaed nercep and slope, respecvel, fro he WLS regresson. (v) For each un n he an saple, le ˆ = ˆ 0 ˆ. (v) The esae of 0, s obaned fro he logsc regresson of on ˆ n he an saple. Table 6.3 conans he MC bas, varance, and MSE of he four esaors of. The nave esaor has a negave bas because s a conanaed verson of. The PFI esaor s superor o he Baes and WRC esaors. We copue an esae of he varance of he PFI esaors of usng he varance epresson based on he lnear approaon. We defne he MC relave bas as he rao of he dfference beween he MC ean of he varance esaor and he MC varance of he esaor o he MC varance of he esaor. The MC relave bas of he varance esaors for PFI s neglgble (less han % n absolue values). Table 6.3 Mone Carlo (MC) eans, varances, and ean squared errors (MSE) of pon esaors of fro Sulaon Two. (PFI, paraerc fraconal puaon; WRC, weghed regresson calbraon; MC, Mone Carlo; MSE, ean squared error) Mehod MC Bas MC Varance MC MSE PFI Nave Baes WRC Concludng rearks We approach sascal achng as a ssng daa proble and propose he PFI ehod o oban conssen esaors and correspondng varance esaors. Under he assupon ha he specfed odel s full denfed, he proposed ehod provdes he pseudo au lkelhood esaors of he paraeers n he odel. A suffcen condon for odel denfabl s he esence of an nsruenal varable n he odel. The easureen error fraework of Secon 5 and Secon 6., where eernal calbraon provdes an ndependen easureen of he rue covarae of neres, s a suaon n whch he sud desgn a be judged o suppor he nsruenal varable assupon. The proposed ehodolog s applcable whou he nsruenal varable assupon, as long as he odel s denfed. If he odel s no denfable, hen he EM algorh for he proposed PFI ehod does no necessarl converge. In pracce, one can rea he specfed odel as denfed f he EM sequence converges. Tha s, as long as he EM sequence converges, Sascs Canada, Caalogue No. -00-X

19 36 K, Berg and Park: Sascal achng usng fraconal puaon he resulng analss s conssen under he specfed odel. Ths s one of he an advanages of usng he frequens approach over Baesan. In he Baesan approach, s possble o oban he poseror values even under non-denfed odels and he resulng analss can be sleadng. Tesng wheher he IV assupon holds n he daa a hand s uch ore dffcul, perhaps possble, under he daa srucure n Table.. Insead, gven he specfed odel, we can onl check wheher he odel paraeers are full esable. On he oher hand, wheher he specfed odel s a good odel for he daa a hand s a dfferen sor. Model dagnoscs and odel selecon aong dfferen denfable odels are ceranl poran fuure research opcs. Sascal achng can also be used o evaluae effecs of ulple reaens n observaonal sudes. B properl applng sascal achng echnques, we can creae an augened daa fle of poenal oucoes so ha causal nference can be nvesgaed wh he augened daa fle (Morgan and Wnshp 007). Such eensons wll be presened elsewhere. Acknowledgeens We hank Professor Yanuan Ma, an anonous referee and he Asssan Edor (AE) for ver consrucve coens. The research of he frs auhor was parall suppored b Bran Pool progra (3S ) fro Korean Federaon of Scence and Technolog Soce and b a gran fro NSF (MMS-339). The research of he second auhor was suppored b a Cooperave Agreeen beween he US Deparen of Agrculure Naural Resources Conservaon Servce and Iowa Sae Unvers. The work of he hrd auhor was suppored b he Bo-Snerg Research Projec (03M3A9C407858) of he Mnsr of Scence, ICT and Fuure Plannng hrough he Naonal Research Foundaon n Korea. Append A. Aspoc unbasedness of SLS esaor Assue ha we observe, n Saple A and observe, n Saple B. To be ore rgorous, we can wre, a a o denoe he observaon, n Saple A. Also, we can wre, b b o denoe he observaons n Saple B. In hs case, he odel can be wren as = e = e a 0 a a a a b 0 b b b b wh Ee a a =0 and Eeb b, b = 0. Noe ha b s no observed fro he saple. Insead, we use ˆ b usng he OLS esae obaned fro Saple A. Wrng X a =, a a and X b =, b b, we have ˆ ˆ b = Xb X axa X aa = Xba. The SLS esaor of =,, s hen 0 Sascs Canada, Caalogue No. -00-X

20 Surve Mehodolog, June where Z =,,. Thus, we have b b ˆb b ˆ = SLS ZZ Z b b b b We a wre where Ee b b =0. Snce we have ˆ = ZZ Z Z SLS b b b b b = ZZ Z ˆ e. b b b b b b = e = X e b 0 b b b b b ˆ = X XX X b b a a a a = X X X X X e b a a a a a = X X XX Xe, b b a a a a ˆ = e X X X Xe b b b b a a a a (A.) and (A.) becoes ˆ =. (A.) SLS ZZ b b Zb eb Xb XaXa Xe a a eb Assue ha he wo saples are ndependen. Thus, Eeb a, b, a = 0. Also, E ZZ Z e,,, = 0. Thus, and b b b b a b a b E ˆ,, = E ZZ ZX XX Xe,, SLS a b a b b b b a a a a a b a ZZ ˆ b b ZX b b a ZZ ZX XX Xe = ZZ Z X XX X X b b b b a a a a b b b b a a a a a =. Ths er has zero epecaon aspocall because nb Z bzb and nb Z bxb are bounded n probabl and ˆa converges o zero. B. Varance esaon N N Le he paraeer of neres be defned b he soluon o UN = U ;, = 0. = We assue ha U =0. Thus, paraeer s pror ndependen of whch s he paraeer n he daagenerang dsrbuon of,,. Under he seup of Secon 3, le ˆ = ˆ, ˆ le ˆ be he soluon o U ˆ =0 where be he MLE of =, obaned b solvng (3.4). Also, j U = w w U ;,, B j= b j Sascs Canada, Caalogue No. -00-X

21 38 K, Berg and Park: Sascal achng usng fraconal puaon and j ˆ j ˆ j j ; ; w f f h wh w =. j= j Here, h s he proposal dsrbuon of generang pued values of n he paraerc fraconal puaon. B nroducng he proposal dsrbuon h, we can safel gnore he j dependence of pued values on he esaed paraeer value ˆ. B Talor lnearzaon, ˆ ˆ ˆ U U U U Noe ha where I = S. Also, ˆ I S ˆ S S where j S = ww S ;,. j B j= Thus, we can esablsh U ˆ U K S K S, where K = DI and = I = E S, D = EU S and D = EU S, we have where = EU, K D I wh I ES ˆ = V U V V V = V w u KS, B =, u = ˆ;, ˆ EU ;, V = V K ws. A conssen esaor of each coponen can A be developed slarl o Secon 3. and References Baker, K.H., Harrs, P. and O Bren, J. (989). Daa fuson: An apprasal and eperenal evaluaon. Journal of he Marke Research Soce, 3, 5-. Sascs Canada, Caalogue No. -00-X

22 Surve Mehodolog, June Beauon, J.-F., and Bocc, C. (009). Varance esaon when donor puaon s used o fll n ssng values. The Canadan Journal of Sascs/La Revue Canadenne de Sasque, 37, 3, Chen, J., and Shao, J. (00). Jackknfe varance esaon for neares neghbor puaon. Journal of he Aercan Sascal Assocaon, 96, 453, Chb, S., and Greenberg, E. (995). Jackknfe varance esaon for neares neghbor puaon. The Aercan Sascan, 46, Chpperfeld, J.O., and Seel, D.G. (009). Desgn and esaon for spl quesonnare surves. Journal of Offcal Sascs, 5,, D Orazo, M., Zo, M.D. and Scanu, M. (006). Sascal Machng: Theor and Pracce. Chcheser, UK: Wle. Fuller, W.A. (009). Saplng Sascs, Hoboken, NJ: John Wle & Sons, Inc. Gelan, A., Carln, J.B., Sern, H.S. and Rubn, D.B. (003). Baesan Daa Analss, Chapan and Hall Tes n Sascal Scence. Chapan and Hall/CRC, second edon. Gonzalez, J., and Elnge, J. (008). Adapve ar saplng for he consuer ependure quarerl nervew surve. In Proceedngs of he Surve Research Mehods Secon, Aercan Sascal Assocaon, Guo, Y., and Lle, R.J. (0). Regresson analss wh covaraes ha have heeroskedasc easureen error. Sascs Medcne, 30, 8, Hazza, D. (009). Ipuaon and nference n he presence of ssng daa. In Handbook of Sascs, Volue 9, Saple Surves: Theor Mehods and Inference, (Eds., C.R. Rao and D. Pfefferann), Herzog, T.N., Scheuren, F.J. and Wnkler, W.E. (007). Daa Qual and Record Lnkage Technques. New York: Sprnger. Ibrah, J.G. (990). Incoplee daa n generalzed lnear odels. Journal of he Aercan Sascal Assocaon, 85, K, J.K. (0). Paraerc fraconal puaon for ssng daa analss. Boerka, 98, 9-3. K, J.K., and Rao, J.N.K. (0). Cobnng daa fro wo ndependen surves: A odel-asssed approach. Boerka, 99, K, J.K., and Shao, J. (03). Sascal Mehods n Handlng Incoplee Daa, Chapan and Hall/CRC. K, J.K., and Yang, S. (04). Fraconal ho deck puaon for robus nference under e nonresponse n surve saplng. Surve Mehodolog, 40,, -30. Lahr, P., and Larsen, M.D. (005). Regresson analss wh lnked daa. Journal of he Aercan Sascal Assocaon, 00, Leulescu, A., and Agafe, M. (03). Sascal achng: A odel based approach for daa negraon. Eurosa Mehodologes and Workng Papers. Morgan, S.L., and Wnshp, C. (007). Counerfacuals and Causal Inference: Mehods and Prncples for Socal Research. New York, USA: Cabrdge Unvers Press. Morar, C., and Scheuren, F. (00). Sascal achng: A paradg for assessng he unceran n he procedure. Journal of Offcal Sascs, 7, Sascs Canada, Caalogue No. -00-X

23 40 K, Berg and Park: Sascal achng usng fraconal puaon Pluer, M. (003). JAGS: A progra for analss of Baesan graphcal odels usng Gbbs saplng. In Proceedngs of he 3 rd Inernaonal Workshop on Dsrbued Sascal Copung. Raghunahan, T.E., and Grzzle, J.E. (995). A spl quesonnare desgn. Journal of he Aercan Sascal Assocaon, 90, Rässler, S. (00). Sascal Machng: A Frequens Theor, Praccal Applcaons, and Alernave Baesan Approaches. New York: Sprnger-Verlag. Rdder, S., and Moff, R. (007). The econoercs of daa cobnaon. Handbook of Econoercs, Sascs Canada, Caalogue No. -00-X