On Covariance Estimation when Nonrespondents are Subsampled

Transcription

1 Metodološk zvezk, Vol. 5, No., 008, 95- On Covarance Estmaton when Nonrespondents are Sbsampled Wocech Gamrot Abstract The phenomenon of nonresponse n a sample srve redces the precson of parameter estmates ntrodces the bas. Several procedres have been developed to compensate for these effects. An mportant technqe s the two-phase (or doble) samplng scheme whch reles on sbsamplng the nonrespondents re-approachng them n order to obtan the mssng data. Ths paper focses on the applcaton of doble samplng to estmate the fnte poplaton covarance. Two covarance estmators sng combned data from the ntal sample the sbsample are consdered. Ther propertes are derved. Two specal cases of the general procedre are dscssed. Introdcton The estmaton of covarances between ndvdal poplaton characterstcs of the covarance matr as a whole n the nonresponse staton has enoed vvd nterest for ears. The prncple of mamm lkelhood (ML) plas a promnent role n constrctng estmators for covarance matrces as well as for ndvdal covarances. Fnkbener (979) apples Fletcher-Powell optmm-seekng algorthm for obtanng ML estmates of covarance matr for factor analss. Lee(986) proposes the estmaton procedre nder whch mssng vales are vewed as latent varables estmators are obtaned nder normalt assmptons va generalzed least sqares ML sng Fsher scorng (teratvel reweghted Gass-Newton algorthm). A coherent theor on the se of mamm lkelhood prncple to estmate the covarance matr nder some model descrbng the dstrbton of poplaton vales s gven b Lttle Rbn (987). The Epectaton-Mamzaton (EM) algorthm b Dempster, Lard Rbn (977) s contemplated as a stard method for dealng wth nonresponse n a wde range of cases wth specal emphass on the normal dstrbton. Ths approach s frther developed n sbseqent papers. Woodrff (990) consders Wocech Gamrot, Department of Statstcs, nverst of Economcs, Bogccka 4, 40-6 Katowce, Pol; gamrot@ae.katowce.pl.

2 96 Wocech Gamrot alternatve estmator based on the EM algorthm wth addtonal restrctons epressed b the regresson sperpoplaton model. Schneder (00) proposes a reglarzed EM algorthm tlzng rdge regresson, for the case nmber of varables eceeds the sample sze. Bentler Lang (003) dscss the applcaton of an EM-tpe gradent algorthm for mamm lkelhood estmaton n the contet of two-level strctral eqaton modellng. Jamshdan (997) apples EM for confrmator factor analss. Jamshdan Bentler (999) eplore the possbltes of sng several complete-data algorthms (EM, Fletcher- Powell Fsher s scorng) to provde mamm lkelhood estmates of the covarance matr wth mssng data. nder a competng approach Van Praag, Dkstra VanVelzen (985) constrct an asmptotcall dstrbton-free (ADF) estmator of the covarance matr based on lnear regresson. Armnger Sobel (990) abon normalt constrants se psedo-mamm lkelhood (PML) method b Armnger Schoenberg (989) to constrct estmators of covarance matrces whle Yan Bentler (995) provde theoretcal stfcaton for the se of PML n the contet of non-normal dstrbtons. Gold, Bentler Km (003) compare ADF estmators wth mamm lkelhood estmators n the contet of strctral eqaton modellng. Sgnfcant attenton s also devoted to the se of varos mptaton methods n covarance estmaton. Brown (994) compares estmators for lstwse deleton, parwse deleton mean mptaton. Klne (998) compares the propertes of estmators compted nder mean mptaton, regresson mptaton pattern matchng. Schafer Olsen (998) emplo a method of mltple mptaton nvented b Rbn (987). The develop data agmentaton algorthm provde stfcaton for ts se on the gronds of Baesan theor. Data agmentaton s also consdered n the paper of Graham Hofer (000) compared wth the EM method. The comparson of estmators for lstwse parwse deleton, mean mptaton fll-nformaton mamm lkelhood s gven b Wothke (000). In ths paper the problem of estmatng ndvdal covarance between two poplaton characterstcs s dscssed nder the qas-romzaton approach (Oh Scheren (983)). It s assmed that the nonresponse s a rom phenomenon bt poplaton vales are fed the are not sbect to modelng as opposed to most of the papers referenced above. Let s consder some characterstcs X Y n fnte poplaton of sze N. Fed vales of these characterstcs are respectvel denoted b,..., N,..., N. The am of the srve s to estmate some fnctons of these vales called poplaton parameters. Let the poplaton be srveed accordng to the followng general two-phase samplng procedre. In the frst phase a rom sample s of sze n s drawn from accordng to some arbtrar samplng desgn p(s) determnng s ndvdal nclson probabltes of the frst order = p(s) of the s, second order = p(s), for. We assme that nonresponse appears n the srve, conseqentl some nts respond whle others do not. The sample s

3 On Covarance Estmaton when Nonrespondents are Sbsampled 97 ma conseqentl be dvded nto two dsont sbsets s s sch that nts from s respond nts from s do not. Followng Cassel et al. (983) we assme that nonresponse s a rom event governed b some probablt dstrbton q(s s) sall referred to as response dstrbton (see e.g. Särndal et al. 99). In general, t s defned condtonall wth respect to s n order to reflect the nteractons between sampled nts. The response dstrbton determnes ndvdal response probablt of an -th nt = ρ (s s) ont response probablt of -th -th nt s s q ρ s s, = q(s s) for. The dstrbton q(s s) also determnes the behavor of the rom set s whch ma be epressed as a fncton of s s, so for an s =s-s we have (s s) = q(s s) q(s, s s). The second phase of q = the srve s then carred ot n ths second phase a sbsample s of sze n s drawn from s accordng to the samplng desgn p (s s,s ) whch s characterzed b another set of nclson probabltes of the frst order = p'(s' s,s ) s,s s' second order = p'(s' s,s ). We assme that necessar efforts s,s s', are ndertaken n the second phase that garantee obtanng complete responses from all sbsampled nts. In the setp descrbed above three sorces of sample romness were defned, each of them respectvel assocated wth probablt dstrbton p(s), q(s s) p (s s,s ). All epectatons wll be compted ontl wth respect to these three probablt dstrbtons nless otherwse stated. Estmaton of the poplaton total Let s consder the poplaton total of X (the same ma be defned for Y or an other characterstc): t = (.) nder complete response t s nbasedl estmated b the Horvtz-Thompson (95) statstc: tˆ = (.) s In partclar, b pttng = for we obtan an nbased estmator of the poplaton sze N n the form: s Nˆ = (.3)

4 98 Wocech Gamrot Both estmators above are generall based nder nonresponse. As an eample consder determnstc nonresponse model, accordng to whch the poplaton s dvded nto two strata:, of szes N N sch that ρ = for ρ =0 otherwse. If the sample s drawn sng smple rom samplng wthot replacement (SRSWOR), hence nclson probabltes of the frst second order are respectvel eqal to =n/n =n(n-)/(n(n-)) for then the estmator trns ot to be based ts bas s eqal to: t = B ) (tˆ = t t = (.4). The bas does not depend on the sample sze hence t does not tend to zero wth growng n. However, sng the data from both phases we can constrct an nbased estmator of t n the form (Särndal et al 99): tˆ = s s' s, s (.5) Pttng = for we agan obtan an nbased estmator of N: Nˆ = s s' s, s (.6) These estmators wll be sed frther as bldng blocks for more complcated estmaton procedres. 3 Estmaton of the poplaton covarance Let s consder the poplaton covarance between X Y defned b the epresson: or b the eqvalent formla: t C (X, Y) = (3.) N N N C (X, Y) = t t t (3.) N N(N ) =, t = t =. nder complete response the covarance s often estmated b respectve statstcs: Ĉ = tˆ (3.3) N N(N ) (X, Y) tˆ tˆ

5 On Covarance Estmaton when Nonrespondents are Sbsampled 99 Ĉ = tˆ (3.4) Nˆ Nˆ (Nˆ ) (X, Y) tˆ tˆ nknown poplaton totals are replaced wth correspondng Horvtz- Thompson estmators. As ndcated b Särndal et al (99), the latter s sall preferred to the former de to better varance propertes. These covarance estmators are however based nder nonresponse, As a specal case let s consder SRSWOR determnstc nonresponse. For large sample sze appromate bases of both estmators respectvel take the form: AB(Ĉ ) = C C (3.5) C AB(Ĉ C ) = C C (3.6) # = t t t (3.7) N N(N ) = t t t (3.8) N N (N ) # t = whle, t = t =. Hence the bas does not tend to zero when n grows n an apparent analog to the epresson (.4). In order to correct for the bas we propose to replace Horvtz-Thompson estmators wth ther nbased doble samplng conterparts. Ths leads to two alternatve estmators of the poplaton covarance: Ĉ (X, Y) = tˆ tˆ tˆ (3.9) N N(N ) Ĉ (X, Y) = tˆ tˆ tˆ (3.0) Nˆ Nˆ (Nˆ ) sng Talor lnearzaton we obtan the appromate varance of Ĉ (X, Y) : s,s AV(Ĉ (X, Y)) = E pq (3.) (N ),, s s,s s, s = ( X)( Y) XY (3.) whle form: X = t / N Y = t / N. The appromate bas ma be epressed n the

6 00 Wocech Gamrot s,s AB(Ĉ (X, Y)) = E pq N (N ),, s s,s s, s (3.3) It s worth notng, that ths appromaton of the bas s obtaned b epng the estmator n Talor seres ncldng terms p to the second order, as opposed to more crde appromatons based onl on frst-order terms. The smbols AB AV are sed to dstngsh lnearzaton-based appromatons from the eact bas varance. sng the same method we obtan the appromate varance of Ĉ (X, Y) n the form: AV(Ĉ * * * * s,s = (X, Y)) E pq (N ),, s s,s s, s * (3.4) = ( X)( Y) C (X, Y) (3.5) ts second-order bas: * * s,s AB (Ĉ ) = E pq (3.6) N (N ),, s s,s s, s (( )( ) C (X, Y) ) * = N ( ) N(N ) (3.7) The smbol ( ) appearng n epressons above represents the epectaton wth E pq respect to frst-phase samplng desgn p(s) to the response dstrbton q(s s). On assmed level of generalt t s mpossble to elmnate them from AV formlas bt we ma acheve ths b makng addtonal assmptons concernng the second phase samplng desgn response dstrbton. The eample s shown n the net secton. On the other h, the appromate varances ma be estmated, wthot an assmptons, b emplong the approach of Särndal et al (99) whch leads to respectve statstcs: û û ûû s,s Vˆ (Ĉ = (X, Y) (N ), s s', s' s,s s,s s, s (3.8)

7 On Covarance Estmaton when Nonrespondents are Sbsampled 0 Vˆ (Ĉ * * * * û û û û s,s = (X, Y)) (N ), s s', s' s,s s,s s, s û (3.9) t t t t = N (3.0) N N t t = Ĉ (X, Y) Nˆ Nˆ (3.) * û * = s,s s,s s,s for for for for, s s s, s, s, s (3.) If the statstcs û * û estmated constants * wthot error, then varance estmators above wold respectvel be nbased for AV(Ĉ (X, Y)) AV(Ĉ (X, Y)). Obvosl the do not some bas appears, bt we ma hope that t remans modest tends to zero for large samples. We wll now present two mportant specal cases of the general procedre presented above. 4 Eqal probablt samplng Let s assme as n papers of Srnath (97) Rao (986) that SRSWOR s sed n both phases. Hence nclson probabltes of the frst second order are respectvel eqal to =n/n =n(n-)/(n(n-)) for n the frst phase whle s,s = n' / n s,s = n'(n' ) /(n (n )) for n the second phase. We also assme that the sbsample sze s a lnear fncton of the nonrespondent sbset sze accordng to formla: n ' = cn 0<c< s a constant fed n advance. Frthermore, we assme determnstc nonresponse model descrbed earler. nder these assmptons we have Nˆ N. Conseqentl both estmators Ĉ (X, Y) Ĉ are mtall eqvalent take the common form: Ĉ = tˆ tˆ tˆ (4.) N N(N )

8 0 Wocech Gamrot tˆ tˆ N = n s c s' tˆ N = n s c s' N = n s c s' (4.) (4.3) (4.4) The appromate varance of Ĉ (X, Y) ma be epressed as: N N n W c AV(Ĉ ) = S () S () (4.5) (N ) Nn n c S () = (4.6) N N S () = (4.7) N N The second-order appromate bas s N n W c AB(Ĉ ) = C C N (N ) Nn n c (4.8) C = N N N (4.9) W =N /N. Both appromate bas appromate varance decrease when ntal sample sze n grows. From (3.8) we also obtan the varance estmator: N N n Vˆ (Ĉ (X, Y)) = (n )S ( ) s (N ) n(n ) N(n ) cn (n ) n nn Ss' ( ) (s s' c(n n) n ) (4.0)

9 On Covarance Estmaton when Nonrespondents are Sbsampled 03 s = û (4.) n s s' = û (4.) n' s' S (4.3) s ( ) = (û s ) n s û S (4.4) s '( ) = (û s' ) n' s' t t t t = N. (4.5) N N 5 neqal probablt samplng We wll now focs on another specal case of the general two-phase procedre. The determnstc nonresponse model assmng that response probabltes are ether eqal to zero or eqal to nt s seldom realstc. In practce the are more lkel to take an vale from the <0,> nterval depend on the alar varables as well as the varable nder std. In partclar t ma be more reasonable to assme that response probabltes are descrbed b logstc model gven b epresson: ρ = ( ep( β )) (5.) s some vector of alar varables correspondng to the -th poplaton nt whle β s the parameter vector. Also, more sophstcated samplng desgns that make se of alar nformaton to mprove the effcenc of parameter estmates are often preferred to the SRSWOR. One of sch desgns, known as Pareto samplng (Rosén 997), allows to draw a fed-sze sample wth frst order nclson probabltes appromatel proportonal to the vales z,...,z N of the alar characterstc Z. Accordng to ths procedre the sample s of the sze n s drawn n followng two steps (Särndal Lndström 005): ) For an a realzaton of a rom varable havng nform dstrbton on the <0,> nterval s generated the followng epresson s evalated:

10 04 Wocech Gamrot q ( ) = (5.) ( ) the desred nclson probablt s gven b epresson z = n (5.3) z If >0 then -th nt s atomatcall nclded n the sample nclson probabltes for remanng nts are recompted accordngl. ) The poplaton sbset consstng of n nts havng the largest vales of q k s nclded n the sample. Proposed covarance estmators are not eqvalent now. Comptaton of ther propertes n sch a staton sng general formlas for appromate varance appromate bas derved above s possble when some reasonable assmptons are made abot response dstrbton. However, to evalate eact nclson probabltes t s necessar to se nmercal procedres developed b Ares (000). Ths makes the analtcal comparson of covarance estmators dffclt. Hence, a smlaton std was carred ot n order to compare both covarance estmators. Drng smlaton eperments, the poplaton nder std was represented b the data obtaned from the Polsh 996 agrcltral censs representng 40 farms n three boroghs (Bolesław, Gręboszów, Radgoszcz) of the Dąbrowa Tarnowska dstrct. Three varables were sed ncldng farm sales (X), farm cattle stock (Y) farm area (Z). The covarance between X Y was the estmated parameter. A logstc nonresponse model was arbtrarl assmed statng that poplaton nts respond ndependentl wth response probabltes respectvel eqal to ρ=(ep(β 0 β β )) - for wth the parameter vector β=[β 0,β,β ] chosen arbtrarl. Two smlaton eperments were respectvel eected for β=[0,0,0] (ρ ndependent on X,Y) β=[0,,] (ρ dependent on X,Y). The smlatons were ndependentl repeated for Pareto samplng tlzng Z as the alar varable n both phases, for smple rom samplng wthot replacement n both phases. The same sample sze n=00, 50,..., 600 was alwas assmed for both desgns. For each sample sze the drawng of sample-sbsample pars was smlated, the propertes of estmators were assessed on the bass of ther emprcal dstrbtons. Hence, each pont on followng graphs corresponds to estmates. Estmators Ĉ (X, Y), Ĉ Ĉ (X, Y) were compared. In the graphs the are respectvel denoted b abbrevatons (Cov, Cov Cov_srs). The mean sqare error (MSE), the bas the share of bas n MSE observed n the frst eperment are respectvel shown on Fgres, 3.

11 On Covarance Estmaton when Nonrespondents are Sbsampled 05 MSE 4,0E07 3,5E07 3,0E07,5E07,0E07,5E07,0E07 5,0E06 0,0E n Cov Cov Cov_srs Fgre : MSE as a fncton of n for β=[0,0,0], BIAS 0,0E00-5,0E0 -,0E0 -,5E0 -,0E0 -,5E0-3,0E0-3,5E0-4,0E n Cov Cov Cov_srs Fgre : Bas as a fncton of n for β=[0,0,0]. The MSE of all three estmators decreases wth growng sample sze. Ths observaton s consstent wth asmptotc reslts obtaned for the SRSWOR case. For an vale of n the estmator ) Ĉ (X,Y s mch less accrate than others. The

12 06 Wocech Gamrot lowest mean sqare error s observed for Ĉ, whch s sgnfcantl better than other two n terms of MSE. The bas of all three estmators s negatve ts absolte vale decreases wth growng n. Ths s also consstent wth asmptotc reslts for SRSWOR. The share of bas n the MSE s appromatel constant for each estmator. It does not eceed,4% for Ĉ 0,4% for Ĉ Ĉ. Hence, each estmator ma be treated as appromatel nbased. BB/MSE,6%,4%,%,0% 0,8% 0,6% 0,4% 0,% 0,0% n Cov Cov Cov_srs Fgre 3: Share of bas n MSE as a fncton of n for β=[0,0,0]. The mean sqare error (MSE), the bas the share of bas n MSE observed n the second eperment are respectvel shown on Fgres 4, 5 6. Agan, the MSE of all three estmators tends to decrease wth growng ntal sample sze. For an vale of n observed MSE of the estmator Ĉ s the lowest observed MSE of the estmator Ĉ s the hghest. The bas s negatve for Ĉ Ĉ whle oscllatng arond zero for Ĉ. Its absolte vale s lowest for Ĉ hghest for Ĉ. The observed share of bas n the MSE s agan neglgble, not eceedng 0,05% for ) Ĉ (X,Y 0,5% for the other two estmators.

13 On Covarance Estmaton when Nonrespondents are Sbsampled 07 MSE 4,5E07 4,0E07 3,5E07 3,0E07,5E07,0E07,5E07,0E07 5,0E06 0,0E n cov cov cov_srs Fgre 4: MSE as a fncton of n for β=[0,,]. 00 BIAS cov cov cov_srs n Fgre 5: Bas as a fncton of n for β=[0,,].

14 08 Wocech Gamrot 0,6% 0,5% BB/MSE 0,4% 0,3% 0,% 0,% cov_ cov_ cov_srs 0,0% n Fgre 6: Share of bas n MSE as a fncton of n for β=[0,,]. 6 Conclsons In ths paper two nonresponse-corrected estmators of the fnte poplaton covarance are consdered nder qas-romzaton approach. The are compted sng the data obtaned sng a general two-phase samplng procedre nvolvng arbtrar samplng desgns n both phases. Ther appromate varances bases are derved nder stochastc nonresponse characterzed b arbtrar response dstrbton. No model assmptons on poplaton dstrbton are made for ther dervaton, whch allevates the rsk of model msspecfcaton. For the mportant specal case of smple rom samplng wthot replacement determnstc nonresponse derved formlas sggest that proposed estmators are nearl nbased. The reslts of smlaton eperments carred ot for another specal case of stochastc nonresponse Pareto samplng also seem to spport ths hpothess for estmators Ĉ Ĉ. It s worth notng that the propertes of proposed estmators have been derved smlatons were eected nder the assmpton of complete response n the second phase. Frther research s needed on ther propertes f ths assmpton s not satsfed. Acknowledgements The research has been spported b the grant No: H0B 0 30 from the Polsh Mnstr of Edcaton Scence.

15 On Covarance Estmaton when Nonrespondents are Sbsampled 09 References [] Ares, N. (000): Technqes to Calclate Eact Inclson Probabltes for Condtonal Posson Samplng Pareto ps Samplng Desgns, PhD Thess, Chalmers, Göteborg nverst. Göteborg. [] Armnger, G. Schoenberg, R. (989): Psedo-mamm lkelhood estmaton a test for msspecfcaton n mean covarance strctre models. Pschometrka, 54, [3] Armnger, G. Sobel, M.E. (990): Psedo-mamm lkelhood estmaton of mean covarance strctres wth mssng data. Jornal of the Amercan Statstcal Assocaton, 85, [4] Bentler, P.M. Lang, J. (003): Two-Level mean covarance strctres: Mamm Lkelhood va the EM algorthm. In Rese, S.P., Dan, N. (Eds.): Mltlevel Modellng Methodologcal Advances, Isses Applcatons. London: Pscholog Press. [5] Brown, R.L. (994): Effcac of the ndrect approach for estmatng strctral eqaton models wth mssng data: A comparson of fve methods. Strctral Eqaton Modellng,, [6] Cassel, C.M., Särdal, C.E., Wretman, J. (983): Some ses of statstcal models n connecton wth the nonresponse problem. In Madow, W.G. Olkn, I. (Eds.): Incomplete Data n Sample Srves. Vol II, New York: Academc Press,. [7] Dempster, A.P., Lard, N.M., Rbn, D. (977): Mamm lkelhood from ncomplete data va the EM algorthm. Jornal of the Roal Statstcal Socet, 39, -38. [8] Fnkbener, C. (979): Estmaton for the mltple factor model when data are mssng. Pschometrka, 44, [9] Gold, M.S., Bentler, P.M., Km, K.H. (003): A comparson of mamm lkelhood asmptotcall dstrbton free methods of treatng ncomplete nonnormal data. Strctral Eqaton Modellng, 0, [0] Graham, J.W. Hofer, S.M. (000): Mltple mptaton n mltvarate research. In Lttle, T.D., Schnabel, K.., Bamert, J. (Eds.): Modellng Longtdnal Mltlevel Data: Practcal Isses, Appled Approaches Specfc Eamples. Mahwah, NJ: Lawrence Erlbam. [] Horvtz, D.G. Thompson, D.J. (95): A generalzaton of samplng wthot replacement from a fnte nverse. Jornal of the Amercan Statstcal Assocaton, 47, [] Jamshdan, M. (997): An EM algorthm for ML factor analss wth mssng data. In Berkane, M. (Ed.): Latent Varable Modellng Applcatons to Casalt. New York: Sprnger,

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17 On Covarance Estmaton when Nonrespondents are Sbsampled Longtdnal Mltlevel Data: Practcal Isses, Appled Approaches Specfc Eamples. Mahwah, NJ: Lawrence Erlbam. [30] Yan, K.-H. Bentler, P.M. (995): Mean covarance strctre analss wth mssng data. CLA Statstcal Seres Report No. 93, Los Angeles: nverst of Calforna.