Small Area Estimation for Business Surveys

Transcription

1 ASA Secton on Survey Research Methods Small Area Estmaton for Busness Surveys Hukum Chandra Southampton Statstcal Scences Research Insttute, Unversty of Southampton Hghfeld, Southampton-SO17 1BJ, U.K. (emal: Abstract In busness surveys, data typcally are skewed and the standard approach for small area estmaton (SAE) based on lnear mxed models lead to neffcent estmates. In ths paper, we dscuss SAE technques for skewed data that are lnear followng a sutable transformaton. In ths context, mplementaton of the emprcal best lnear unbased predcton (EBLUP) approach under transformaton to a lnear mxed model s complcated. However, ths s not the case wth the model-based drect (MBD) approach (Chambers and Chandra, 006), whch s based on weghted lnear estmators. We extend the MBD approach to skewed data usng sample weghts derved va model calbraton based on a log transform model wth random area effects. Our results show ths estmator s both effcent and robust wth respect to the dstrbuton of these random effects. An applcaton to real data demonstrates the satsfactory performance of the method. Keywords: Small areas, Skewed data, MBD, Model Calbraton, Expected value model. 1. Introducton The standard methods for SAE assume a lnear mxed model can be used to characterze the small areas of nterest. However, t happens (typcally for skewed data) that the varable of nterest Y s lnear on some transformed scale (e.g. n busness surveys, often varables are lnear on log scale). In ths context, estmaton based on lnear model for Y leads to neffcent estmates. In such stuaton, an approprate technque for SAE should essentally be based on a lnear mxed model for a transformed varable. In ths paper we explore transform varable based estmaton n context of SAE for skewed data, focussng on the wdely used log transformaton. In ths paper we extend the MBD approach of Chambers and Chandra (006) to SAE for skewed data. In partcular, we consder the use of sample weghts derved va model calbraton (Wu and Stter, 001) based on a log transform model wth random area effects. In the followng secton we summarze the model calbraton approach for estmaton of populaton quanttes. In secton 3 we then dscuss the expected value model derved from a transform lnear mxed model for SAE of skewed data. Secton 4 ntroduces the survey weghts based on expected value model derved from a transform lnear mxed model and descrbes the MBD estmator for SAE n ths case. In secton 5 we provde llustratve emprcal results. Fnally, n secton 6 some concludng remarks are made.. Model Calbraton for Populaton Estmaton In ths secton we brefly revew model calbraton for estmaton of populaton level quanttes. To start, we fx our notaton. Let Y denote an N-vector of populaton values of a characterstc of nterest, and suppose that our prmary am s estmaton of the total T y of the values n Y (or ther mean). In order to assst us n ths obectve, we shall assume that we have access to X, an N p matrx of values of p auxlary varables that are related, n some sense, to the values n Y. In partcular, we assume that the ndvdual sample values n X are known. The non-sample values n X may not be ndvdually known, but are assumed known at some aggregate level. At a mnmum, we know the populaton totals T x of the columns of X. Gven ths set up, Devlle and Särndal (199) ntroduced the notaton of a calbraton estmator of populaton total of Y as T y = w s y, where the calbraton weghts w s are chosen to mnmse ther average dstance, from the basc desgn weghts, subect to the calbraton N constrant wx = x s 1 T = = x. There s an mplct underlyng assumpton that Y and X are lnearly related that makes ths a vald argument. If the underlyng model s non-lnear then the calbrated estmator derved under a lnearty assumpton cannot be very effcent. Let us assume the relatonshp between Y and X can be descrbed by a super populaton model E ( Y X) = h( X; η) and V ( Y X) =Ω (1) where η typcally vector-valued model parameter, and the mean functon hxη ( ; ) s a known functon of X and η, the varance Ω s a functon of X and hxη ( ; ). Here E andv denotes the expectaton and varance wth respect to model. The model (1) s qute general and ncludes lnear, non-lnear, and generalzed lnear models as specal cases. In ths context, Wu and Stter, (001) proposed the use of sample weghts derved va model calbraton. They defned the calbraton estmator for populaton 803

2 ASA Secton on Survey Research Methods 1 mean of Y as Yc = N w s y wth weghts sought to mnmze the dstance measure under the constrants: w N s = and N wh s = h = 1, where η s a desgn consstent estmator forη. Provded the model (1) s a reasonable one, y s then (at least approxmately) a lnear functon of ts ftted values hx ( ; η ) under ths model. The basc dea of ths approach s then we can carry out lnear estmaton usng these ftted or expected values as auxlary varables. That s calbraton s performed wth respect to the populaton mean of the ftted values h = h( x ; η) of hx ( ; η ). The above dscusson represents what mght be referred to the desgn-based nterpretaton of model calbraton. A model-based perspectve on model calbraton can be descrbed as follows. We assume that Y and hxη ( ; ) are related by the lnear model of the form Y = αj + ε () where J denotes the desgn matrx for the lnear model (3) lnkng Y and hxη ( ; ), α = ( α0, α 1) s a vector of unknown parameters, ε denotes a N- vector of random varables wth E ( ε ) = 0 and V ( ε) =Ω= [ ω k ]. We called model () the expected value or ftted value model defned by (1). For α 0 = 0 n model () we refer as rato specfcaton of ths model, otherwse regresson specfcaton. The model () can have ether rato or regresson specfcaton. Wthout loss of generalty, we arrange the vector Y so that the frst n elements correspond to the sample unts, and partton Y, J and Ω accordng to sample and non-sample unts. Where J s denoted the n 1 vector of ftted values of the auxlary varables and Ω ss s the n n covarance matrx assocated wth the n sample unts that make up the n 1 sample vector Y s. A subscrpt of r s used to denote correspondng quanttes defned by the N n non-sample unts, wth Ω rs denotng the ( N n) n matrx defned by Cov( Yr, Y s). In what follows we denote 1 N, 1 n and 1 r as vectors of 1 s and I N and N, I n and I r as dentty matrces of order N, n n respectvely. In practce the varance components that defne covarance matrx Ω are unknown and so need to be estmated from the sample data. We use a hat to denote such an estmate. Further, throughout ths paper we assume that samplng s unnformatve, so the sample data also follow the populaton model. Gven ths notaton, the sample weghts that defne the BLUP for populaton total of Y under a general lnear ftted value model () are h 1 w = 1 + H ( J 1 J 1 ) + ( I H J ) Ω Ω 1 (3) BLUP n h N s n n h s ss sr r where Hh = ( J sωss Js) J sω ss. See Royall (1976). The sample weghts (3) derved va model calbraton are calbrated on J. The weghts (3) are based on a model approprate for estmaton of populaton as a whole and usng these weghts for SAE wll be neffcent. The most commonly used class of models for small area estmaton model s essentally a mxed model. The next secton descrbes the models sutable for SAE. 3. Small Area Models under Transformaton Let Y be the N 1 vector of values of varable of nterest n small area ( = 1,..., m) and let X be the N p matrx of values of the auxlary varables assocated wth Y. We assume that Y and X are not related by a lnear model on themselves, but they are lnearly related on logarthm (natural) transform model. We consder the followng lnear mxed model specfcaton for the dstrbuton of l = log( Y) gven Z : l = Z β + Gu + e (4) where Z = (1 N,log( X )) s the N ( 1) p+ matrx of values of the auxlary varables n area, β s a ( p + 1) 1 vector of fxed effects, G s a N q matrx of known covarates charactersng dfferences between small areas, N s the number of populaton unts n the small area, 1 N s a vector of 1 s of order N, u s a random area effect assocated wth the th small area and e s a N 1 vector of ndvdual level random errors. The two random varables u and e are assumed to be ndependently normally dstrbuted, wth zero means and wth varances Vu ( ) = Σ and Ve ( ) = σ e I N respectvely. The varance-covarance matrx of l s V = GΣ ( θ) G + σein, wth v = GΣ ( θ ) G + Var( e ) and v = G Σ ( θ ) G,, k = 1,..., N. k k By groupng the area-specfc models (4) over the populaton, we are led to the populaton level model: l = Zβ + Gu+ e (5) where l = ( l 1,..., l m ), Z = ( Z 1,..., Z m ), G = dag( G ;1 m), u = ( u 1,..., u m ) and e= ( e 1,..., e m ). The varance matrx of l s V = dag( V ;1 m). We assume that Z has full column rank. In practce the varance components of the model that defne the covarance matrx V are unknown and we estmate them from the sample data under the model (5). The estmated varance matrx of l s V = dag( V ;1 m) wth V = σ I + GΣ G. We consder the partton of l, Z, e N G and V nto sample and non-sample components as mentoned before (3). We use smlar notaton at the 804

3 ASA Secton on Survey Research Methods small area level by ntroducng an extra subscrpt to denote small area. Wth ths notaton, and assumng (5) holds, the 1-1 m m Z V Z Z V β 1 l EBLUE of β s = ( 1 s ss s ) ( = = 1 s ss s ) wth E ( β) β = = 1 s ss s. We denote φ = Zβ wth E ( φ ) = Z β and = and m 1 V ( β ) ( Z V Z ) -1 V ( ) ( φ = ZV β ) Z, where a ( k = ZV β ) Z k 0 as n. We denote by a = ( a11,..., an N ) and v = ( v11,..., vn N ), the vectors of dagonal elements of the covarance matrces V ( φ ) and V ( l ) respectvely. In order to use the Chambers and Chandra (006) MBD method to get estmates for small areas we requre sample weghts. For skewed data that follows a lnear mxed model on the log scale (5), the sample weghts can be derved va model calbraton, so frst we need to evaluate expected value model (Secton ). In other words, we need to evaluate the frst and second moments under the model (4) to derve the sample weghts (3). We can use parameter estmates derved under model (4) to obtan the predcted values of the transform varable and then back-transform to get predcted values of Y. These lead to the naïve-lognormal predctor. However, ths predctor s based. Bas corrected frst and second order moments that defne the expected value model are expressed below. Let us consder Zβ + v / E ( ) ( Y = e E Y ) (6) Thus, we need to adust ths bas. Usng two-step Taylor seres approxmaton, a second order bas corrected estmate of E ( Y ) s defned as 1 ( ; ) exp( v Y = h Z η = k Zβ + ), = 1,..., N (7) / so that ( Zβ + v E Y ) e = E ( Y ). Here ( ) k = 1 + a + Var ( v )/4 s the bas correcton and Var( v ) s the asymptotc covarance matrx of v gven by nverse of the relevant nformaton matrx (Rao, 003). Under normalty of the random errors u and e, covarance between Y and Y k n small area s ω 1 ( Z ) ( v v ) + Zk β + kk v k [ ( 1)] k = Zβ v v [ ( 1)] = e e e f k e e e f k (8) We group the bas corrected predctor (7) and the covarance (8) at the small area level as 1 ( ; ) exp( v Y = h Z η = k Zβ + ), (9) Var ( Y ) =Ω = [ ω ] = A A (10) k ( Z β );1 where A { dag e N } = and s N N postve defnte matrx wth ( k, ) th elements as ( v )/ { + v v δ kk (e k k = e 1)}. The area-specfc approxmately bas corrected estmator (9) and covarance matrx (10), grouped at populaton level defne the populaton level verson of expected value model E ( Y h) = α01 N + α1h= αj and V ( Y h) =Ω (11) where h = ( h 1,..., h m ) and Ω = dag( Ω;1 m). 4. SAE under the Expected Value Model (11) Wth approprate sample and non-sample partton of Y, J and Ω, as n secton, the EBLUP verson of sample weghts (3) under the model (11) are h 1 1 ( 1 1 ) ( ) EBLUP n h N s n n h s ss sr1r w = + H J J + I H J Ω Ω (1) H ( J J ) J where h = sωss s sω ss. We note that the weghts (1) depend on random area effects of the mxed model (4) va the covarance structure of model (11) and are thus sutable for small area estmaton. We now use the MBD approach of Chambers and Chandra (006) to defne estmator for small areas. They only consder the Háek form of the MBD estmator for small areas usng sample weghts derved under a lnear mxed model. However, the weghts (1) are derved va model calbraton under the expected value model (11) where estmator s defned as the HT form (Secton ). Thus, we consder both forms of MBD estmators. The sample weghts (1) assocated wth the sample unts n the small area can be used to defne the followng MBD estmators for the th small area meany : The Háek form of the weghted sample for area Háek s s Y w y w = (13) The Horvtz-Thompson form of the weghted sample for area HT Y = w y N (14) s Both estmators (13) and (14) also depend on how the model calbraton weghts (1) are specfed. In partcular, we consder two dfferent specfcatons for the expected value model (11), the rato and the regresson specfcaton (see below equaton ()). Ths leads to four dfferent MBD estmators that are set out below. Estmator Estmator type Model specfcaton TrMBD1 Háek type Rato TrMBD H-T type Rato TrMBD3 Háek type Regresson TrMBD4 H-T type Regresson 805

4 ASA Secton on Survey Research Methods Estmaton of MSE of (13) and (14) follows the approach of Chambers and Chandra (006), and treats these expressons as smple weghted doman mean estmates under the populaton level model (3). Under ths approach, the sample weghts derved from (1) are treated as fxed and the predcton varance of (13) and (14) s estmated usng a standard robust varance estmator. See Royall and Cumberland (1978). A plug-n estmate of the squared bas of (13) and (14) under ths model s added to ths estmated predcton varance to fnally defne a smple estmate of the MSE. Note that under ths approach the EBLUP weghts underlyng (13) and (14) borrow strength va the assumed small area model (11), but ths model s not used n nference. In partcular, we treat the expected value model (11) as a vehcle for generatng estmaton weghts, but base nference on the model (), thus ensurng consstency wth the way mean squared errors are estmated at populaton level. 5. Smulaton Study In ths secton we llustrate the performance of seven dfferent small area estmators. These are the proposed MBD estmators (TrMBD1-TrMBD4) for skewed data (Secton 4), the Háek type (MBD1), and HT type (MBD) MBD estmators based on sample weghts derved under a lnear mxed model and the EBLUP under a lnear mxed model. We consder two types of smulaton studes. The frst type of study uses model-based smulaton to generate artfcal populaton and sample data. These data are then used to contrast the performance of dfferent estmators. The second type of smulaton study was carred out usng real data and desgnbased smulatons to test these estmators n the context of a real populaton and realstc samplng methods. Three measures of estmaton performance were computed usng the estmates generated n the smulaton study. These were the relatve mean error and the relatve root mean squared error (RMSE), both expressed as percentages, of regonal mean estmates and the coverage rate (CR) of nomnal 95 per cent confdence ntervals for regonal means. 5.1 The Model Based Smulaton Study In model-based smulatons, we consder a populaton sze N=1500 and generated randomly the small area populaton szes N (=1,...,m=30) so that = N. Further, we consder n=600 and N generated n = N( n/ N) wth n = n. These were fxed for all smulatons. We generated the populaton values y from a multplcatve β model y = 5.0x ue. The generated populaton s skewed on the raw scale and lnear on the log transform scale. The random errors e were ndependently generated from a lognormal (LN) dstrbuton wth LN (0, σ e ). The random area effects u and auxlary varables x were generated from LN (0, σ u ) and LN (6, σ x ) respectvely. The values of parameter σ e and σ u were chosen so that ntra-area correlaton vares between We used sx dfferent sets of parameter to brng dfferent level of varaton n generated data as shown below: Parameter β σ u σ e σ x Par Par Par Par Par Par Usng ths generated data we estmated the parameters usng the lme functon n R, and then calculated the estmates for small areas (Secton 4). We replcated 1000 smulaton runs. The results from ths smulaton study are reported n Table The Desgn Based Smulaton Study In desgn-based smulatons, our basc data come from the same sample of 165 Australan broadacre farms (AAGIS) that were used n the smulaton study reported n Chandra and Chambers (005). In partcular, we use the same target populaton of 8198 farms (obtaned by samplng wth replacement from the orgnal sample of 165 farms wth probabltes proportonal to ther sample weghts). The same 1000 ndependent stratfed random samples as used n Chandra and Chambers (005) were then drawn from ths (fxed) populaton, wth total sample sze n each draw equal to the orgnal sample sze (165) and wth the small areas of nterest defned by the 9 Australan agrcultural regons represented n ths populaton. Sample szes wthn these regons were fxed to be the same as n the orgnal sample. Note that these vared from a low of 6 to a hgh of 117, allowng an evaluaton of the performance of the dfferent methods consdered across a range of realstc small area sample szes. Here, our am s to estmate average annual farm costs (A$) n these regons wth farm sze (hectares) as auxlary varable. We used random ntercept model specfcaton of the mxed model. Detals of ths smulated populaton are descrbed n Chandra and Chambers (005). Table set out the results from ths smulaton study. 5.3 Results of the Smulaton Studes Results from Table 1 show that the average relatve mean errors (RMEs) and the average relatve RMSEs for Haek type of estmators (TrMBD1 and 806

5 ASA Secton on Survey Research Methods TrMBD3) under expected value model (11) are sgnfcantly large. Further, hgh coverage rates under these methods (TrMBD1 and TrMBD3) are the consequence of large bases. The HT type estmators (TrMBD and TrMBD4) derved under rato and regresson specfcatons for the expected value model are almost dentcal. Among conventonal calbraton weghtng based MBD estmators, both Haek type (MBD1) and HT type (MBD) estmators are dentcal. Therefore, n further dscusson we drop the Haek type of estmator under model calbraton and HT type estmator under classcal calbraton. Further, Table 1 shows that the average RMEs and the average relatve RMSEs for TrMBD are consstently lower than both MBD1 and EBLUP. However, wth same order of RMEs, the relatve RMSEs of EBLUP s lower order than MBD1. The average coverage rates for TrMBD are relatvely hgher wth smaller wdth as compare to MBD1 and EBLUP. Wth almost same coverage rates, EBLUP has smaller average wdths than MBD1. We notced that both the RME and the relatve RMSEs of TrMBD are smaller than MBD1 and EBLUP n all regons. Further, the RMEs and the relatve RMSE of MBD1 and EBLUP ncrease proportonate to non-lnearty (Par1 to Par6) n the data. The coverage rate ncreases and the wdth decreases, hence accuracy ncreases n transformaton-based methods. Further, the relatve nterval wdth under TrMBD reduced more rapdly as non-lnearty n data ncreases. The results ndcate a sgnfcant gan due to transformaton based method of small area estmaton for skewed data. Ths gan s proportonate to non-lnearty n the data. Between MBD1 and EBLUP methods, the EBLUP appears to perform better. The results from the desgn-based smulaton usng the real data (AAGIS) show that the average RME of TrMBD s smaller than EBLUP and but larger than MBD1. The relatve RMSE of TrMBD s margnally larger and the average coverage rate hgher (Table ). However, Fgure 1 ndcates that the hgh RME and relatve RMSE of TrMBD s due to an outler n regon 1. The TrMBD s more affected by ths outlyng pont. If we dscard the outler contamnated estmates and examne the average based on 8 regons then TrMBD seems to be performng better. Overall transform varable based SAE methods for AAGIS data appears to provde effcent set of estmates. The TrMBD method provdes sgnfcant gan under lnearty on transform model. The gan may not be sgnfcant f lnearty does not hold. However, t s safer to use TrMBD method even though transform model s approxmately lnear. For AAGIS data, ftted model on log scale s not exactly lnear. Consequently, TrMBD method of SAE performs margnally better overall. 6. Conclusons and Further Research Our results show that transformed varable based method for SAE of skewed data performs well. We note that the gan n effcency by accountng nonlnearty n data va log-transform lnear model s qute sgnfcant. Further, even though assumed model devates slghtly from lnearty on transform scale, the proposed method stll works well wth margnal gan. These results are based on normalty assumpton of random errors. However, we also nvestgated under gamma dstrbuton for the random errors and notced that method s robust wth respect to dstrbuton of random errors. The applcaton of proposed SAE technques to real data from AAGIS provdes a satsfactory performance. In applcaton of ths method, dentfcaton of an approprate transform model s crucal, otherwse results can be msleadng. Acknowledgement The research presented n ths paper s part of my PhD work and supervsed by Professor Ray Chambers. Hs valuable gudance s gratefully acknowledged. The author also gratefully acknowledges the fnancal support provded by a PhD scholarshp from the U.K. Commonwealth Scholarshp Commsson. References Chambers, R.L. and Chandra, H. (006). Improved Drect Estmators for Small Areas. Submtted. Chandra, H. and Chambers, R. (005). Comparng EBLUP and C-EBLUP for Small Area Estmaton. Statstcs n Transton, 7, Devlle, J.C. and Särndal, C.E. (199). Calbraton Estmators n Survey Samplng. Journal of the Amercan Statstcal Assocaton, 87, Prasad, N.G.N. and Rao, J.N.K. (1990). The Estmaton of the Mean Squared Error of Small Area Estmators. Journal of the Amercan Statstcal Assocaton, 85, Rao, J.N.K. (003). Small Area Estmaton. New York: Wley. Royall, R.M. (1976). The Lnear Least-Squares Predcton Approach to Two-Stage Samplng. Journal of the Amercan Statstcal Assocaton, 71, Royall, R.M. and Cumberland, W.G. (1978). Varance Estmaton n Fnte Populaton samplng. Journal of the Amercan Statstcal Assocaton, 73, Wu, C. and Stter, R.R. (001). A Model Calbraton Approach to Usng Complete Auxlary Informaton from Survey Data. Journal of the Amercan Statstcal Assocaton, 96,

6 ASA Secton on Survey Research Methods Table 1 Average relatve mean error (ARME), average relatve RMSE (ARRMSE), average coverage rate (ACR) and average -sgma confdence nterval wdth (AW) for model based smulatons. All averages are expressed as percentages. Crteron Estmator Par1 Par Par3 Par4 Par5 Par6 ARME TrMBD TrMBD TrMBD TrMBD MBD MBD EBLUP ARRMSE TrMBD TrMBD TrMBD TrMBD MBD MBD EBLUP ACR TrMBD TrMBD TrMBD TrMBD MBD MBD EBLUP AW TrMBD x x x10 6 TrMBD x x x10 6 TrMBD x x x10 6 TrMBD x x x10 6 MBD x x x10 6 MBD x x x10 6 EBLUP x10 4 0x x10 6 Table Average relatve mean error (ARME), average relatve RMSE (ARRMSE) and average coverage rate (ACR) for AAGIS data. All averages are expressed as percentages. Crteron Estmator Average of 9 areas Average of 8 areas ARME TrMBD MBD EBLUP ARRMSE TrMBD MBD EBLUP ACR TrMBD MBD EBLUP

7 ASA Secton on Survey Research Methods Fgure 1 Regonal performance of TrMBD (sold lne), MBD1 (dashed lne) and EBLUP (thn lne) for AAGIS data Percentage Relatve Bas Regon(ordered by populaton sze) Percenatge Relatve RM SE Regon(ordered by populaton sze) Coverage Rate Regon(ordered by populaton sze) 809