Linearization Variance Estimators for Longitudinal Survey Data

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1 Suary Lnearzaton Varance Estators for Lontunal Survey Data A. Denat an J.. K. Rao A. Denat, Socal Survey Methos Dvson, Statstcs Canaa, Ottawa, Ontaro, Canaa, KA 06 J.. K. Rao, School of Matheatcs an Statstcs, Carleton Unversty, Ottawa, Ontaro, Canaa, KS 5B6 In survey sapln, aylor lnearzaton s often use to obtan varance estators for nonlnear fnte populaton paraeters, such as ratos, reresson an correlaton coeffcents, whch can be expresse as sooth functons of totals. aylor lnearzaton s enerally applcable to any sapln esn, but t can lea to ultple varance estators that are asyptotcally esn unbase uner repeate sapln. he choce aon the varance estators requres other conseratons such as ( approxate unbaseness for the oel varance of the estator uner an assue oel, ( valty uner a contonal repeate sapln fraewor. Denat an Rao (00 propose a new approach to ervn aylor lnearzaton varance estators that leas rectly to a unque varance estator that satsfes the above conseratons. In ths paper, we extene the wor of Denat an Rao (00 to eal wth lontunal surveys whch lea to epenent observatons an to ultple wehts on the sae unt. We conser a varety of lontunal sapln esns, covern panel surveys, househol panel surveys as well as rotatn surveys. Key Wors: Multple wehts; Repeate surveys; aylor lnearzaton.. Introucton aylor lnearzaton s a popular etho of varance estaton for coplex statstcs such as rato an reresson estators an lostc reresson coeffcent estators. It s enerally applcable to any sapln esn that perts unbase varance estaton for lnear estators, an t s coputatonally spler than a resapln etho such as the jacnfe. However, t can lea to ultple varance estators that are asyptotcally esn unbase uner repeate sapln. he choce aon the varance estators, therefore, requres other conseratons such as ( approxate unbaseness for the oel varance of the estator uner an assue oel, ( valty uner a contonal repeate sapln fraewor. For exaple, n the context of sple rano sapln an the rato estator, Y ˆ R = ( y / x X, of the populaton total Y, Royall an Cuberlan (98 showe that a coonly use lnearzaton varance estator ϑ = n s oes not trac the contonal varance of Y ˆ ven x, unle the jacnfe varance estator L ( z ϑ J. Here y an x are the saple eans, X s the nown populaton total of an auxlary varable x, varance of the resuals z = y ( y / x an ( n, J x jacnfe varance estator, ϑ, we obtan a fferent lnearzaton varance estator, tracs the contonal varance as well as the uncontonal varance, where or ϑ J ay be preferre over stuy to eonstrate that both ϑ L. Vallant (993 obtane ϑ J an JL Särnal, Swensson an Wretan (989 showe that R s z s the saple enote the saple an populaton szes. By lnearzn the ϑ = ( X / x ϑ, whch also X = X / s the ean of x. As a result, ϑ JL ϑ JL for the post-stratfe estator an conucte a sulaton ϑ possess oo contonal propertes ven the estate post-strata counts. ϑ JL s both asyptotcally esn unbase an asyptotcally oel ˆ unbase n the sense of E ϑ JL Var (YˆR, where E enotes oel expectaton an Var s the oel varance ( = of YˆR uner a Αrato oel : E ( y = β x ; =,..., an the y s are nepenent wth oel varance Var (y = σ x, σ > 0. hus, ϑ JL s a oo choce fro ether the esn-base or the oel-base perspectve. Denat an Rao (00 propose a new approach to varance estaton that s theoretcally justfable an at the sae te leas rectly to a ϑ -type varance estator for eneral esns. Afterwar, Denat an Rao (00 extene ther JL JL ( Y R L Pae 38

2 etho to the case of ssn responses when ajustent for coplete nonresponse an putaton base on sooth functons of observe values, n partcular rato putaton, are use. Whle several ethos have been propose to correctly estate the varance of an estator uner crosssectonal ata, ethos of varance estaton fro lontunal ata are lte, espte an ncrease n recent years n the nuber of lontunal surveys. he ynac nature of lontunal populatons such as brths an eaths, the ltaton of lontunal saples to tae nto account such features of lontunal populatons, an ultple wehts are one of the ffcultes nherent to lontunal ata n aton to the custoary cross-sectonal ffcultes such as coplex esns, ssn ata an so on. hs paper s a frst attept to exten the Denat an Rao (00,00 etho to the case of lontunal survey ata. Secton ves a bref account of the etho for the case of cross-sectonal ata, an secton 3 presents the extenson to lontunal survey ata.. Cross-Sectonal Data We ve a bref account of the Denat an Rao (00 etho uner full response. Suppose an estator θˆ of a paraeter θ can be expresse as a fferentable functon (Yˆ of estate totals Y ˆ = ( Y ˆ Yˆ,...,, where Y ˆ y s an estator of the populaton Y j, j =,..,, where = 0 f the unt s not n the saple s, j = U j U s the set of populaton unts, an θ = (Y wth Y = ( Y,...,Y. We ay wrte θˆ as ˆ θ = f ( s, A an θ = f (, A y th, where Ay s an atrx wth j colun y j = ( y j,..., y j, j =,...,, an s the -vector of 's. For exaple, f θˆ enotes the rato estator Y ˆR = X ( U y /( U x, then =, y y, y = x an f (, reuces to the total Y, notn that X ( Y / X = Y. ote that Y ˆ s a functon of = A y ( s = (,..., s, y, x an the nown total X, but we roppe X for splcty an wrte Y ˆ f (s, y, x. If the Horvtz- hopson wehts are use, then =/ π for s, where π s the probablty of selectn unt n the saple s. Let f ( b, Ay = f ( b for arbtrary real nubers b = ( b,...,. Denat an Rao (00 showe that the aylor lnearzaton of ˆ θ θ, naely s equvalent to ˆ θ θ = ( Yˆ ( Y ˆ θ θ = z = b ( ( a / a ( Yˆ Y, a= Y ( f ( b / b ( ( s, where an ( a / a = ( ( a / a,..., ( a / a z = ( z,..., z wth z = f ( b / b. It follows fro (. that b= a varance estator of θˆ s approxately ven by the varance estator of the estate total ˆ( z = Y z, ; that s, var( ˆ θ ϑ( z,, where ϑ ( y, enotes the varance estator of Y ˆ = Yˆ( y, n operator notaton usn the vector of esn wehts lnearzaton varance estator s. ow we replace z by z b= = f ( b / b b=s R = R y ( (., snce z =s are unnown, to et a ϑ ( ˆ θ = ϑ( z,. (. L ote that ϑ L (θˆ ven by (. s sply obtane fro the forula ϑ ( y, for Y ˆ Yˆ( y, by replacn y by = for s. ote that we o not frst evaluate the partal ervatves f (b / b at b = to et z an then substtute estates for the unnown coponents of z. Our etho, therefore, s slar n sprt to Bner(996 s approach. he varance estator (θˆ s val because s a consstent estator of ϑ L z z. z Pae 39

3 Suppose θˆ s the rato estator Y ˆ R = X ( y /( x, where enotes suaton over U. hen f (b = X ( b y /( b x XYˆ( y, b / Yˆ( x, b an = ( y Rˆ x X z = f ( b / b =. b=s Xˆ For sple rano sapln, ˆ ϑ L ( Y R = ϑ( z, arees wth ϑjl = ( X / x ϑl. he above ervaton s sple an natural. On the other han, n the stanar lnearzaton etho, θˆ s frst expresse n ters of eleentary coponents Y ˆ,...,Ŷ as (Yˆ an the partal ervatves (a / a j are then evaluate at a = Y. It s nterestn to note that all the coponents Y ˆ use the sae wehts (s an our approach always taes j partal ervatves of f (b wth respect to b at b = s. It s not necessary to frst express θˆ n ters of eleentary coponents. Denat an Rao (00 apple the etho to a varety of probles, covern reresson calbraton estators of a total Y an other estators efne ether explctly or plctly as solutons of estatn equatons. hey obtane a new varance estator for a eneral class of calbraton estators that nclues eneralze ran rato an eneralze reresson estators. hey also extene the etho to two-phase sapln an obtane a varance estator that aes fuller use of the frst phase saple ata copare to tratonal lnearzaton varance estators. 3. Lontunal Data As n the usual fnte populaton stuaton, we entfy a populaton of sze urn a fxe pero of te by a set of nces P = {,..., }. Let P enote the cross-sectonal populaton at te t an let enote the cross-sectonal populaton ebershp ncator for unt, J =,...,,.e. = f P an = 0 f P wth J + = J = the sze of the cross-sectonal populaton P, where enotes suaton over the populaton unts. In ths secton, we conser varance estaton of a fferentable functon of estate cross-sectonal totals uner a seres of lontunal sapln esns assun full response an no post-stratfcaton. It s shown that a varance estator can be obtane throuh varance estaton of a esn-wehte estator of total of a synthetc populaton. he case of estators obtane as soluton of estatn equatons can be obtane alon the lnes of Denat an Rao (00, an the case of estators obtane fro survey ata wth ssn responses can be obtane alon the lnes of Denat an Rao (00. In ths verson, etals of these extensons are otte for space reason. o scuss ultple wehts nherent n repeate surveys, suppose an estator θˆ of a paraeter θ can be expresse as a fferentable functon ˆ θ = ( Yˆ of estate cross-sectonal totals Y ˆ Y ˆ ( ( Y ˆ =,...,, where ˆ (t Y s an estator of the populaton total Y = J y at te t, t =,...,, an θ = (Y wth Y = (Y,..., Y. We ay wrte Yˆ as ˆ t t = = A t t y Y s y t wth specfe constants, where A = (,...,, are vectors of cross-sectonal wehts at te t, (t s the -vector of s an y = (y,..., y. In ths secton we show that the estator θˆ can be wrtten as ˆ θ = f ( uner a we class of lontunal survey esns. herefore, t follows fro Denat an Rao (00 that a lnearzaton varance estator of θˆ s ven by ϑ L ( ˆ θ = ϑ( z,, (3. where = vech( A, (3. ( ( ( A,..., A A =, A = (,...,, are vectors of esn wehts, vech( A enotes the vector obtane by stacn stnct colun of ( t ( t J J A unerneath each other, n orer fro left to rht, an ( ( 3 Pae 40

4 z = f ( b. (3.3 b b= 3. Inepenent Saples where Uner nepenent saples we have =, Y ˆ ( t = Yˆ( y, = y, an = (,..., = vech( ( ( (,..., s the vector esn wehts at te, (3.4 t. If the Horvtz-hopson wehts are use then = J a / π, = f unt belons to the cross-sectonal saple s an = 0 f not, π an = Pr( s / J =. 3. Panel Saple a Uner panel saple stuaton, where observatons are collecte over te on the sae saple unts, we have =, Y ˆ (0 = Yˆ( y, an = a, (3.5 where, =,...,, are the basc survey wehts of the unque saple selecte at the ntal wave (say at te t = Househol Panel Saple Uner househol panel saple stuaton, where observatons are collecte over te not only on the saple unts selecte at the ntal wave but also on non-saple unts who jon househols contann at least one saple unt, we have =, ˆ Y = y, an where f the unt t s = j J j I j j s α j j j = lves n the sae househol as unt at te t, an s obtane throuh the ultplcty approach, Sren ( Multple Saples, (3.6, α = α = / J I a, I s an ncator varable ncatn a = j E( a. he value for ew entrants who have zero probablty to jon the ntal populaton unts are not covere, unless a suppleentary saple s taen. hat s why ultple saples are use at the estaton stae at te t uner househol panel surveys. s Uner ultple saples stuaton each saple, =,..., can be vewe as a saple fro the sae cross-sectonal F populaton usn a fferent sapln frae. he fraes are probably ncoplete, but the unon of the fraes covers the entre cross-sectonal populaton. he fraes overlap. Let = F J α j J be the contonal frae ebershp ncator varable,.e., J f = an J = 0 otherwse. When cobnn the saples, the wehts have to be ajuste to account for the ultplcty of the sapln fraes: s s α = ( = / J +, where t =, (3.7 where α are constants satsfyn the constrants J α an are the cross-sectonal wehts resultn fro frae F. A frst choce for α s α t J + s the ultplcty of unt at te t,.e., the total nuber of sapln fraes reportn unt at te t, (Sren, 970. A secon choce for α s 4 Pae 4

5 = J f / J f α t A y n whch case one wants to tae nto account the sapln fracton of each sapln frae (Kalton an Anerson, 986. Uner ultple fraes, the estate total Y can be expresse as before as splcty. 3.5 Rotatn Saples U U. Hence, estaton of the varance can be one alon the lne of sub-secton Detals are otte for Rotatn saples are use as a coprose between nepenent saples an panel surveys. Rotatn saples control overlap between successve te peros an prove unbase cross-sectonal estates whle tan avantae of the correlaton between two te peros. For exaple, n labor-force surveys whch are conucte onthly n any countres, overlap s controlle by re-ntervewn a hh fracton of prevous selecte househols (say rotaton roups whle new househols are selecte for a frst ntervew(say one rotaton roup for a total of rotaton roups or panel saples. Each househol reans n the sae rotaton roup for a preeterne nuber of onths. Unle ultple saples, rotatn roups o not overlap. A sple exaple of a rotatn roup s ven by the follown two steps: (a Splt the populaton at rano nto roups U ( =,..., each of sze /. (b Splt each roup at rano nto / n saples each of sze n /. he follown table llustrates a rotaton pattern wth =, = 0, n = 4 an 50% overlap. able : Exaple of Rotatn Schee roup Saple s ˆ f e 3 s s 5 s s s 5 Cross-sectonal sa ple ( ( (3 s s s s s s s = s = = s Each cross-sectonal saple s copose of saples. he saple s appears n the frst cross-sectonal saple (te t =, the saples s an s appear n two te peros(te an 3, whle the saple s appears only n the thr cross-sectonal saple (te t = 3. he frst step conssts on selectn non-overlap saples fro the populaton wth a coplete coverae of the unon of the saples. Let a ( U be the roup U, =,...,, ebershp n cator varables wth a ( U =, a ( U =, an a ( U ah ( U h = ( = h. he secon step s a repetton of the frst step wthn each s s on al saple s s ebershp ncator varables wth a s s = ns, ˆ ˆ s s, an a s s a q q = = q. We have =, Y = Y ( y, /, an roup. Let a be the c ton a = s ( ( ( ( ( ( s = vech(,...,,...,,...,, (3.8 ( ( wth = (,...,, = ( U, ( U = J a ( U / E ( a ( U, ( 5 Pae 4

6 a = a / E (, an a = a s s f s s s the saple use at te t n roup. VarE( Yˆ where Uner rotatn sa ples, the varance of = Var( Y = 0, so an estator of Var (θˆ s ven by an ϑ (. enotes a contonal varance estator of a total. Coposte estaton ϑ ˆ θ = Yˆ t L ( ϑ ( z, = vech( ( ˆ t ( ˆ by Var( Y = EVar Y + VarE( Yˆ ( (. ow (3.9, (3.0 Uner rotatn saples, t s avantaeous to use a coposte estator. Suppose the follown sple coposte estator ˆ ˆ ˆ ˆ ( ˆ ( t ˆ ( t Y = Y ˆ α ( Yu Y β Y Y. (3. ˆ u ˆ t s ven,,..., where Y s an estator of the populaton total usn the unatche saple, Y an Y are estators of the populaton totals usn atche saples, an α an ˆβ are assue to be for splcty reresson paraeters estators as n the RE approach. he case of optal reresson s covere n Denat an Rao (00. A varance estator of the coposte estator, θ = ( Y, can be obtane usn (3.9 wth ( ( vech(,..., he coposte estator (3. has three coponents as n the AK coposte estator, but t s not recursve an estates for fferent oans a up unle n the case of AK coposte estator. Conclun Rears We have presente a new approach to varance estaton uner lontunal survey ata. A val varance estator s ven uner a varety of lontunal sapln esns. Extenson to estators efne as solutons to survey wehte estatn equatons s currently uner nvestaton. References Bner, D. (996, ΑLnearzaton Methos for Snle Phase an wo-phase Saples: A Cooboo Approach, Survey Methooloy,, 7-. Denat, A. an Rao, J.. K. (00,Α Lnearzaton Varance Est ators for Survey Data, Methooloy Branch Worn Paper, SSMD-00-00Ean Rao, J.. K. (00, ΑLnearzaton Varance Estators for Survey Data Wth Mssn Responses, n Statstcs Canaa. Denat, A. Proceen of the Secton Survey Research Methos, Aercan Statstcal Assocaton. Kalton,. an Anerson, D. (986, Sapln Rare Populatons, Journal of the Royal Statstcal Socety, Seres A, 49, Part, pp Royall, R. M., an Cuberlan, W.. (98, ΑAn Eprcal Stuy of the Rato Estator an Estators of ts Varance, Journal of the Aercan Statstcal Assocaton, 76, Särnal, C.-E., Swensson, B., an Wretan, J.H. (989,Αhe Wehte Resual echnque for Estatn the Varance of the eneral Reresson Estator of the Fnte Populaton otal, Boetra, 76, Sren, M.. (970, Househol Surveys wth Multplcty, Journal of the Aercan Statstcal Assocaton, 65, Vallant, R. (993,ΑPostsratfcaton an Contonal Varance Estaton, Journal of the Aercan Statstcal Assocaton, 88, ( ( ˆ ˆ ( t =. (3. 6 Pae 43