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1 Unversty of Wollongong Research Onlne Centre for Statstcal & Survey Methodology Workng Paper Seres Faculty of Engneerng and Informaton Scences 2009 Borrowng strength over space n small area estmaton: Comparng parametrc, semparametrc and non-parametrc random effects and M-quantle small area models R. Chambers Unversty of Wollongong, ray@uow.edu.au N. Tzavds Unversty of Manchester N. Salvat Unversty of Psa Recommended Ctaton Chambers, R.; Tzavds, N.; and Salvat, N., Borrowng strength over space n small area estmaton: Comparng parametrc, semparametrc and non-parametrc random effects and M-quantle small area models, Centre for Statstcal and Survey Methodology, Unversty of Wollongong, Workng Paper 12-09, 2009, 13p. Research Onlne s the open access nsttutonal repostory for the Unversty of Wollongong. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au

2 Centre for Statstcal and Survey Methodology The Unversty of Wollongong Workng Paper Borrowng strength over space n small area estmaton: Comparng parametrc, sem-parametrc and non-parametrc random effects and M- quantle small area models Ray Chambers, Nkos Tzavds, Ncola Salvat Copyrght 2008 by the Centre for Statstcal & Survey Methodology, UOW. Work n progress, no part of ths paper may be reproduced wthout permsson from the Centre. Centre for Statstcal & Survey Methodology, Unversty of Wollongong, Wollongong NSW Phone , Fax Emal: anca@uow.edu.au

3 Borrowng strength over space n small area estmaton: Comparng parametrc, sem-parametrc and non-parametrc random effects and M-quantle small area models Ray Chambers 1, Nkos Tzavds 2, Ncola Salvat 3 Centre for Statstcal and Survey Methodology School of Mathematcs and Appled Statstcs, Unversty of Wollongong 1 Centre for Census and Survey Research, Unversty of Manchester 2 Dpartmento d Statstca e Matematca Applcata all Economa, Unverstà d Psa, Italy 3 Abstract In recent years there have been sgnfcant developments n model-based small area methods that ncorporate spatal nformaton n an attempt to mprove the effcency of small area estmates by borrowng strength over space. A popular approach parametrcally models spatal correlaton n area effects usng Smultaneous Autoregressve (SAR) random effects models. An alternatve approach ncorporates the spatal nformaton va M-quantle Geographcally Weghted Regresson (GWR), whch fts a local model to the M-quantles of the condtonal dstrbuton of the outcome varable gven the covarates. A further approach uses splne approxmatons to ft nonparametrc unt level nested error regresson and M-quantle regresson models that reflect spatal varaton n the data and then uses these nonparametrc models for small area estmaton. In ths presentaton we contrast the performance of these alternatve small area models usng data wth geographcal nformaton. We also examne how these models perform when estmaton s for out of sample areas.e. areas wth zero sample, and dscuss ssues related to estmaton of mean squared error of the resultng small area estmators. Our analyss s llustrated usng smulatons based on data from the U.S. Envronmental Protecton Agency s Envronmental Montorng and Assessment Program. Keywords: Spatal nformaton; Robust regresson; Iteratvely Reweghted Least Squares; Nonparametrc smoothng. 1 Introducton In recent years there have been sgnfcant developments n model-based small area estmaton. The most popular approach to small area estmaton employs random effects models for estmatng doman specfc parameters (Rao, 2003). An alternatve approach to small area estmaton that relaxes the parametrc assumptons of random effects models by employng M-quantle models was recently proposed by Chambers & Tzavds (2006). Typcally, random effects models assume ndependence of the random area effects. Ths ndependence assumpton s also mplct n M-quantle small area models. In economc, envronmental and epdemologcal applcatons, however, observatons that are spatally close may be more related than observatons that are further apart. Ths spatal correlaton can be accounted for by extendng the random effects model to allow for spatally correlated area effects usng, for example, a Smultaneous Autoregressve (SAR) model (Anseln, 1988; Cresse, 1993). The applcaton of Smultaneous Autoregressve models to small area estmaton enables researchers to borrow strength over space and hence mprove the precson of small area estmates. In ths context, Sngh et al. (2005) and Prates & Salvat (2008) proposed the use of the Spatal Emprcal Best Lnear Unbased Predctor (SEBLUP). SAR models allow for spatal correlaton n the error structure. An alternatve approach to ncorporatng the spatal nformaton n the regresson model s by assumng that the regresson coeffcents vary spatally across the geography of nterest. Geographcally Weghted Regresson (GWR) (Brundson et al., 1996; Fotherngham et al., 1997, 2002; Yu & Wu, 2004) extends the tradtonal regresson model by allowng local rather than global parameters to be estmated. That s, GWR drectly models spatally non-statonarty n the mean structure of the model. In a recent paper Salvat et al. (2008) proposed an M-quantle GWR small area model. In dong so, the authors frst proposed an extenson to the GWR model, M-quantle GWR model,.e. a locally robust model for the M-quantles of the condtonal dstrbuton of the outcome varable gven the covarates. Ths model was then used to defne a predctor of the small area characterstc of nterest that accounts for spatal structure of the data. The M-quantle GWR small area model ntegrates the concepts of robust small area estmaton and borrowng strength over space wthn a unfed modelng framework. When the functonal form of the relatonshp between the response varable and the covarates s unknown or has a complcated functonal form, an approach based on use of a nonparametrc regresson model usng penalzed splnes can offer sgnfcant advantages compared wth one based on a lnear model. By expressng the splne coeffcents n the model as random effects, Ruppert et al. (2003) show how fttng a p-splne model s equvalent to fttng a lnear mxed model. On the bass of ths property, Opsomer et al. (2008) have recently proposed a new approach to SAE that extends the unt level nested error regresson model (Battese et al., 1988) by combnng small area random effects wth a p-splne regresson model. Prates et al. (2008) have extended ths approach to the M-quantle method for the estmaton of the small area parameters usng a nonparametrc specfcaton of the condtonal M-quantle of the response varable gven the covarates. The use of bvarate p-splne approxmatons to ft nonparametrc unt level nested error and M-quantle regresson models allows for reflectng spatal varaton n the data and then uses these nonparametrc models for small area estmaton. 1

4 In ths paper we contrast SAR, unt level nested error p-splne regresson, M-quantle GWR and M-quantle splne models n terms of ther performance usng data wth geographcal nformaton. We also examne whether estmaton for out of sample areas.e. areas wth zero sample szes can be mproved usng models that borrow strength over space. The structure of the paper s as follows. In secton 2 we revew unt level mxed models wth ndependent and spatally correlated random area effects for small area estmaton. In secton 3 we present M-quantle and M-quantle GWR models for small area estmaton. In secton 4 we descrbe the nonparametrc unt level nested error and M-quantle regresson models. Estmaton of the mean squared error of the resultng small area predctors under modelng approaches s dscussed. For explorng the research questons of ths paper we employ a real survey datasets: data from the U.S. Envronmental Protecton Agency s Envronmental Montorng and Assessment Program (EMAP). The dataset contans geo-referenced and n secton 5 we use desgn-based smulaton experments for assessng the performance of the dfferent small area predctors and ther assocated estmates of mean squared error consdered n ths paper. Fnally, n secton 6 we summarze our man fndngs. 2 Unt level mxed models for small area estmaton Let x j denote a vector of p auxlary varables for each populaton unt j n small area and assume that nformaton for the varable of nterest y s avalable only from the sample. The target s to use the data to estmate varous area-specfc quanttes. A popular approach for ths purpose s to use mxed effects models wth random area effects. A lnear mxed effects model s y j = x T jβ + z j γ + ɛ j, j = 1..., n = 1,..., d (1) where β s the p 1 vector of regresson coeffcents, γ denotes a random area effect that characterzes dfferences n the condtonal dstrbuton of y gven x between the d small areas, z j s a constant whose value s known for all unts n the populaton and ɛ j s the error term assocated wth the j-th unt wthn the -th area. Conventonally, γ and ɛ j are assumed to be ndependent and normally dstrbuted wth mean zero and varances σγ 2 and σɛ 2 respectvely. The Emprcal Best Lnear Unbased Predctor (EBLUP) of the mean for small area (Battese et al., 1988; Rao, 2003) s then ˆm MX [ y j + j s j r ŷ j where ŷ j = x T j ˆβ + z j ˆγ, s denotes the n sampled unts n area, r denotes the remanng N n unts n the area and ˆβ and ˆγ are obtaned by substtutng an optmal estmate of the covarance matrx of the random effects n (1) nto the best lnear unbased estmator of β and the best lnear unbased predctor of γ respectvely. Model (1) can be extended to allow for correlated random area effects. Let the devatons v from the fxed part of the model X T β be the result of an autoregressve process wth parameter ρ and proxmty matrx W (Anseln, 1988; Cresse, 1993), then ] v = ρwv + γ v = (I ρw) 1 γ (3) where I s a d d dentty matrx. Combnng (1) and (3), wth ε ndependent of v, the model wth spatally correlated errors can be expressed as y = X T β + Z(I ρw) 1 γ + ε. (4) The error term v then has a d d Smultaneously Autoregressve (SAR) dsperson matrx gven by G = σ 2 γ [ (I ρw T )(I ρw)] 1. (5) The W matrx descrbes the neghbourhood structure of the small areas and ρ defnes the strength of the spatal relatonshp between the random effects of neghbourng areas. Under (4), the Spatal Best Lnear Unbased Predctor (Spatal BLUP) of the small area mean and ts emprcal verson (SEBLUP) are obtaned followng Henderson (1975). In partcular, the SEBLUP of the small area mean, m, s ˆm MX/SAR [ y j + j s j r ŷ j 2.1 MSE estmaton for small area estmates under the SAR mxed model An expresson for the mean squared error (MSE) under the SEBLUP model and ts estmator are obtaned followng the results of Kackar & Harvlle (1984), Prasad & Rao (1990) and Datta & Lahr (2000). More specfcally, the MSE estmator conssts of three components denoted by g 1, g 2 and g 3 : MSE[ ˆm MX/SAR ] = g 1 (σγ, 2 σε, 2 ρ) + g 2 (σγ, 2 σε, 2 ρ) + g 3 (σγ, 2 σε, 2 ρ). ] (2) (6) 2

5 The components of the MSE are due to the varablty assocated wth the estmaton of the random effects (g 1 ), the estmaton of β (g 2 ) and the estmaton of (σγ, 2 σε, 2 ρ) (g 3 ). Note that due to the ntroducton of the addtonal parameter ρ, component g 3 of the MSE s not the same as n the case of the EBLUP (Prasad & Rao, 1990). In practcal applcatons the predctor ˆm MX/SAR has to be assocated wth an estmator of MSE[ ˆm MX/SAR ]. Followng the results of Harvlle & Jeske (1992) and Zmmerman & Cresse (1992) an approxmately unbased mean squared error estmator of s gven by mse[ ˆm MX/SAR ] g 1 (ˆσ 2 γ, ˆσ 2 ε, ˆρ) + g 2 (ˆσ 2 γ, ˆσ 2 ε, ˆρ) + 2g 3, (ˆσ 2 γ, ˆσ 2 ε, ˆρ) (7) when ˆσ 2 γ, ˆσ 2 ε, ˆρ are REML estmators. See Sngh et al. (2005) and Prates & Salvat (2008) for detals. 3 M-quantle models for small area estmaton A recently proposed approach to small area estmaton s based on the use of M-quantle models (Chambers & Tzavds, 2006; Tzavds et al., 2008). A lnear M-quantle regresson model s one where the q th M-quantle Q q (x; ψ) of the condtonal dstrbuton of y gven x satsfes Q q (X; ψ) = X T β ψ (q). (8) Here ψ denotes the nfluence functon assocated wth the M-quantle. For specfed q and contnuous ψ, an estmate ˆβ ψ (q) of β ψ (q) s obtaned va teratve weghted least squares. Followng Chambers & Tzavds (2006) an alternatve to random effects for characterzng the varablty across the populaton s to use the M-quantle coeffcents of the populaton unts. For unt j wth values y j and x j, ths coeffcent s the value θ j such that Q θj (x j ; ψ) = y j. These authors observed that f a herarchcal structure does explan part of the varablty n the populaton data, unts wthn clusters (areas) defned by ths herarchy are expected to have smlar M-quantle coeffcents. When the condtonal M-quantles are assumed to follow a lnear model, wth β ψ (q) a suffcently smooth functon of q, ths suggests a predctor of m of the form ˆm MQ [ y j + j s {x T ˆβ ] j ψ (ˆθ )} j r where ˆθ s an estmate of the average value of the M-quantle coeffcents of the unts n area. Typcally ths s the average of estmates of these coeffcents for sample unts n the area. These unt level coeffcents are estmated by solvng ˆQ qj (x j ; ψ) = y j for q j wth ˆQ q denotng the estmated value of (8) at q. When there s no sample n area then ˆθ = 0.5. Tzavds et al. (2008) refer to (9) as the nave M-quantle predctor and note that ths can be based. To rectfy ths problem these authors propose a bas adjusted M-quantle predctor of m that s derved as the mean functonal of the Chambers & Dunstan (1986) (CD hereafter) estmator of the dstrbuton functon and s gven by ˆm MQ/CD = + td ˆF [ (t) y j + j s {x T ˆβ j ψ (ˆθ )} + N n n j r 3.1 M-quantle geographcally weghted model for small area estmaton (9) {y j x T ˆβ ] j ψ (ˆθ )}. (10) j s SAR mxed models are global models.e. wth such models we assume that the relatonshp we are modelng holds everywhere n the study area and we allow for spatal correlaton at dfferent herarchcal levels n the error structure. One way of ncorporatng the spatal structure of the data n the M-quantle small area model s va an M-quantle GWR model (Salvat et al., 2008). Unlke SAR mxed models, M-quantle GWR are local models that allow for a spatally non-statonary process n the mean structure of the model. Gven n observatons at a set of L locatons {u l ; l = 1,..., L; L n} wth n l data values {(y jl, x jl ); j = 1,..., n l } observed at locaton u l, an M-quantle GWR model s defned as Q q (X; ψ, u) = X T β ψ (u; q) (11) where now β ψ (u; q) vares wth u as well as wth q. The M-quantle GWR s a local model for the entre condtonal dstrbuton -not just the mean- of y gven x. Estmates of β ψ (u; q) n (11) can be obtaned by solvng L n l w(u l, u) ψ q {y jl x T jlβ ψ (u; q)}x jl = 0 (12) l=1 j=1 3

6 where ψ q (t) = 2ψ(s 1 t){qi(t > 0)+(1 q)i(t 0)} and s s a sutable robust estmate of scale such as the medan absolute devaton (MAD) estmate s = medan y jl x T jl β ψ(u; q) / It s also customary to assume a Huber type nfluence functon although other nfluence functons are also possble ψ(t) = ti( c t c) + sgn(t)i( t > c). Provded c s bounded away from zero, an teratvely re-weghted least squares algorthm can then be used to solve (12), leadng to estmates of the form { 1X ˆβ ψ (u; q) = X T W (u; q)x} T W (u; q)y. (13) In (13) y s the vector of n sample y values and X s the correspondng desgn matrx of order n p of sample x values. The matrx W (u; q) s a dagonal matrx of order n wth entres correspondng to a partcular sample observaton and equal to the product of ths observaton s spatal weght, whch depends on ts dstance from locaton u, wth the weght that ths observaton has when the sample data are used to calculate the spatally statonary M-quantle estmate ˆβ ψ (q). At ths pont we should menton that the spatal weght s derved from a spatal weghtng functon whose value depends on the dstance from sample locaton u l to u such that sample observatons wth locatons close to u receve more weght than those further away. One popular approach to defnng such a weghtng functon s to use { w(u l, u) = exp 0.5(d (ul,u)/b) 2}, where d (ul,u) denotes the Eucldean dstance between u l and u and b s the bandwdth, whch can be optmally defned usng a least squares crteron (Fotherngham et al., 2002). It should be noted, however, that alternatve weghtng functons, for example the b-square functon, can also be used. Salvat et al. (2008) also proposed a reduced M-quantle GWR that combnes local ntercepts wth global slopes and s defned as Q q (X; ψ, u) = X T β ψ (q) + δ ψ (u; q). (14) Ths s ftted n two steps. At the frst step we gnore the spatal structure n the data and estmate β ψ (q) drectly va the teratve re-weghted least squares algorthm used to ft the standard lnear M-quantle regresson model (8). At the second step geographc weghtng s appled to estmate δ ψ (u; q) usng ˆδ ψ (u; q) = n 1 L l=1 n l w(u l, u) ψ q {y jl x T ˆβ jl ψ (q)}. Hereafter we refer to (11) and (14) as the MQGWR and MQGWR-LI (Local Intercepts) models respectvely. The prmary am of ths paper s to employ the MQGWR and MQGWR-LI models for estmatng the area mean m of y. Followng Chambers & Tzavds (2006) ths can be done by frst estmatng the M-quantle GWR coeffcents {q j ; j s} of the sampled populaton unts wthout reference to the small areas of nterest. A grd-based nterpolaton procedure for dong ths under (8) s descrbed n Chambers & Tzavds (2006) and can be drectly used wth the M-quantle GWR models (11) and (14). In partcular, we adapt ths approach wth M-quantle GWR models by frst defnng a fne grd of q values over the nterval (0, 1) and then use the sample data to ft (11) or (14) for each dstnct value of q on ths grd and at each sample locaton. The M-quantle GWR coeffcent for unt j wth values y j and x j at locaton u j s computed by nterpolatng over ths grd to fnd the value q j such that Q qj (x j ; ψ, u j ) = y j. Provded there are sample observatons n area, an area-specfc M-quantle GWR coeffcent, ˆθ, can be defned as the average value of the sample M-quantle GWR coeffcents n area. Followng Salvat et al. (2008), the bas-adjusted M- quantle GWR predctor of the mean n small area s MQGW R/CD ˆm [ y j + j s j r j=1 ˆQˆθ (x j ; ψ, u j ) + N n n where ˆQˆθ (x j ; ψ, u j ) s defned ether va the MQGWR model (11) or va the MQGWR-LI (14). 3.2 MSE estmaton for small area estmates under the M-quantle GWR model ] {y j ˆQˆθ (x j ; ψ, u j )}. (15) j s Mean squared error estmaton for the M-quantle GWR small area estmates s based on the Chambers et al. (2008) estmator and s also descrbed n Salvat et al. (2008). To start wth we note that (15) can be expressed as a weghted sum of the sample y-values MQGW R/CD ˆm w T y (16) 4

7 where w = N 1 + H T n jx j N n H T n jx j. (17) j r j s Here 1 s the n-vector wth j th component equal to one whenever the correspondng sample unt s n area and s zero otherwse and { 1X H j = X T W (u j ; ˆθ )X} T W (u j ; ˆθ ). Gven the lnear representaton (16), an estmator of a frst order approxmaton to the mean squared error of ths predctor can be computed followng standard methods of robust mean squared error estmaton for lnear predctors of populaton quanttes (Royall & Cumberland, 1978). Put w = (w j ) ths estmator s of the form where λ jk = v( ˆm MQGW R/CD ) = N 2 λ jk k:n k >0 j s k { } (w j 1) 2 + (n 1) 1 (N n ) I(k = ) + wjk 2 I(k ). { y j ˆQˆθk (x j ; ψ, u j )} 2 (18) 4 Nonparametrc small area models Although very useful n many stuatons, lnear mxed models depend on dstrbutonal assumptons for the random part of the model and do not easly allow for outler robust nference. In addton, the fxed part of the model may not be flexble enough to handle cases n whch the relatonshp between the varable of nterest and the covarates s more complex than that assumed by a lnear model. Opsomer et al. (2008) extend model (1) to the case n whch the small area random effects can be combned wth a smooth, non-parametrcally specfed trend. In partcular, n the smplest case y j = m(x 1j ) + z j γ + ɛ j, j = 1..., n = 1,..., d, (19) where m( ) s an unknown smooth functon of the varable x 1. The estmator of the small area mean s [ ] NP MX ˆm y j + ŷ j j s j r (20) as n (2) where ŷ j = ˆm(x 1j ) + z j ˆγ. By usng penalzed splnes as the representaton for the non-parametrc trend, Opsomer et al. (2008) express the non-parametrc small area estmaton model as a random effects model wth ˆm(x 1j ) = ˆβ 0 + ˆβ 1 x 1j + a j δ. Then the ŷ j value ncludes the splne functon, whch s treated as a random effect, and the small area random effect. The latter can be easly extended to handle bvarate smoothng and addtve modelng. These authors proposed and studed the theoretcal propertes of the mean squared error of (20). They extended the results of Prasad & Rao (1990) and Das et al. (2008) to the case of a splne-based random effect. Opsomer et al. (2008) also proposed a bootstrap estmator for the MSE, whch performs reasonably well, but s computatonally ntensve. Prates et al. (2008) have extended ths approach to the M-quantle method for the estmaton of the small area parameters usng a nonparametrc specfcaton for the condtonal M-quantles of the response varable gven the covarates. When the functonal form of the relatonshp between the q-th M-quantle and the covarates devates from the assumed one, the lnear M-quantle regresson model can lead to based estmators of the small area parameters. When the relatonshp between the q-th M-quantle and the covarates s not lnear, a p-splnes M-quantle regresson model may have sgnfcant advantages compared to the lnear M-quantle model. The small area estmator of the mean may be taken as n (9) where the unobserved value for populaton unt r j s predcted usng ŷ j = x j ˆβψ (ˆθ ) + a j ˆν ψ (ˆθ ), where ˆβ ψ (ˆθ ) and ˆν ψ (ˆθ ) are the coeffcent vectors of the parametrc and splne components, respectvely, of the ftted p- splnes M-quantle regresson functon at ˆθ. In case of p-splnes M-quantle regresson models the bas-adjusted estmator for the mean s gven by NP MQ/CD ˆm = 1 { ŷ j + N } (y j ŷ j ), (21) N n j U s where ŷ j denotes the predcted values for the populaton unts n s and n U. NP MQ/CD Followng the approach descrbed n Chambers et al. (2008), for fxed q, the ˆm n (21) can be wrtten as lnear combnaton of the observed y j. The derved weghts are treated as fxed and a plug n estmator of the mean squared error of estmator has been obtaned by usng standard methods for robust estmaton of the varance of unbased weghted lnear 5

8 estmators (Royall & Cumberland, 1978) and by followng the results due to Chambers et al. (2008). See Salvat et al. (2008a) for detals. The use of bvarate p-splne approxmatons to ft nonparametrc unt level nested error and M-quantle regresson models allows for reflectng the spatal varaton n the data and then uses these nonparametrc models for small area estmaton. In partcular, for M-quantle models, as we have just dealt wth flexble smoothng of quantles n scatterplots, we can now handle the way n whch two contnuous varables affect the quantles of the response wthout any structural assumptons: Q q (x 1, x 2, ψ) = m ψ,q (x 1, x 2 ),.e. we can deal wth bvarate smoothng. It s of central nterest n a number of applcaton areas as envronment and publc health. It has partcular relevance when geographcally referenced responses need to be converted to maps. As seen earler, p-splnes rely on a set of bass functons to handle nonlnear structures n the data. Bvarate smoothng requres bvarate bass functons; Ruppert et al. (2003) advocate the use of radal bass functons to derve Low-rank thn plate splnes. In partcular, the followng model s assumed at quantle q for unt : m ψ,q [x 1, x 2 ; β ψ (q), γ ψ (q)] = β 0ψ (q) + β 1ψ (q)x 1 + β 2ψ (q)x 2 + a ν ψ (q). (22) Here z s the -th row of the followng n K matrx A = [C( x κ k )] 1 n [C(κ k κ k )] 1/2 1 k,k 1 k K K, (23) where C(t) = t 2 log t, x = (x 1, x 2 ) and κ k, k = 1,..., K are knots. See Prates et al. (2008) for detals on ths. The choce of knots n two dmensons s more challengng than n one. One approach could be that of layng down a rectangular lattce of knots, but ths has a tendency to waste a lot of knots when the doman defned by x 1 and x 2 has an rregular shape. In one dmenson a soluton to ths ssue s that of usng quantles. However, the extenson of the noton of quantles to more than one dmenson s not straghtforward. Two solutons suggested n lterature that provde a subset of observatons ncely scattered to cover the doman are space fllng desgns (Nychka & Saltzman, 1998) and the clara algorthm (Kaufman & Rousseeuw, 1990). The frst one s based on the maxmal separaton prncple of K ponts among the unque x and s mplemented n the felds package of the R language. The second one s based on clusterng and selects K representatve objects out of n; t s mplemented n the package cluster of R. 4.1 A note on small area estmaton for out of sample areas In some stuatons we are nterested n estmatng small area characterstcs for domans (areas) wth no sample observatons. The conventonal approach to estmatng a small area characterstc, say the mean, n ths case s synthetc estmaton. Under the mxed model (1) or the SAR mxed model (4) the synthetc mean predctor for out of sample area s MX/SY NT H ˆm where U = s r. Under M-quantle model (8) the synthetc mean predctor for out of sample area s MQ/SY NT H ˆm j U x T j ˆβ, (24) j U x T j ˆβ ψ (0.5). (25) We note that wth synthetc estmaton all varaton n the area-specfc predctons comes from the area-specfc auxlary nformaton. One way of potentally mprovng the conventonal synthetc estmaton for out of sample areas s by usng a model that borrows strength over space such as an M-quantle GWR model and nonparametrc unt level nested error and M-quantle regresson models. In ths case a synthetc-type mean predctors for out of sample area are defned by MQGW R/SY NT H ˆm NP MQ/SY NT H ˆm NP MX/SY NT H ˆm j U ˆQ0.5 (x j ; ψ, u j ) (26) j U ˆm(x 1j, u j ) (27) j U [x j ˆβψ (0.5) + z j ˆν ψ (0.5, u j )]. (28) Emprcal results that address the ssue of out of sample area estmaton are set out n sectons 5. 6

9 5 Desgn-based smulaton study In ths secton we present results from a smulaton study that was used to examne the performance of the small area predctors dscussed n the prevous sectons. We present a desgn-based smulaton usng data from the Envronmental Montorng and Assessment Program (EMAP) that forms part of the Space Tme Aquatc Resources Modellng and Analyss Program (STARMAP) at Colorado State Unversty. The survey data used n ths desgn-based smulaton comes from the U.S. Envronmental Protecton Agency s Envronmental Montorng and Assessment Program (EMAP) Northeast lakes survey (Larsen et al., 1997). Between 1991 and 1995, researchers from the U.S. Envronmental Protecton Agency (EPA) conducted an envronmental health study of the lakes n the north-eastern states of the U.S. For ths study, a sample of 334 lakes was selected from the populaton of 21, 026 lakes n these states. The lakes formng ths populaton were grouped accordng to dgt Hydrologc Unt Codes (HUCs) of whch 64 contaned less than 5 observatons and 27 dd not have any observatons. The varable of nterest was Acd Neutralzng Capacty (ANC), an ndcator of the acdfcaton rsk of water bodes. Snce some lakes were vsted several tmes durng the study perod and some of these were measured at more than one ste, the total number of observed stes was 349 wth a total of 551 measurements. In addton to ANC values, the EMAP data set also contaned the elevaton and geographcal coordnates of the centrod of each lake n the target area (HUC). For sampled locatons we know the exact spatal coordnates of the correspondng locaton. For non-sampled locatons the centrod of the lake s known. Hence, detaled nformaton on the spatal coordnates for non-sampled locatons exsts as the geography defned by the lakes s below the geography of nterest defned by the HUCs. The am of ths desgn-based smulaton was (a) to compare the performance of the dfferent small area predctors of the mean of ANC n each HUC and (b) to evaluate the performance of the dfferent predctors for estmatng the mean ANC for out of sample HUCs. In order to do ths, we frst created a populaton of ANC values wth smlar spatal characterstcs to that of the lakes sampled by EMAP. A total of 200 ndependent random samples were then taken from each HUC, that had been sampled by EMAP, wth sample szes set equal to the max(5, n ) where n s the sample sze of each HUC n the orgnal EMAP dataset. No observatons were taken from HUCs that had not been sampled by EMAP. Ths process led to a total sample sze of 652 ANC values from 86 HUCs. In order to generate a populaton dataset that had smlar spatal structure to that of the EMAP sample data, we allocated ANC values to the non-sampled lakes as follows: (1) we frst randomly ordered the non-sampled locatons n order to avod lst order bas and gave each sampled locaton a donor weght equal to the nteger component of ts survey weght mnus 1; (2) takng each non-sample locaton n turn, we chose a sample locaton as a donor for the j th non-sample locaton by selectng one of the ANC values of the EMAP sample locatons wth probablty proportonal to w(u j, u) = exp{ 0.5(d (uj,u)/b) 2 }. Here d (uj,u) s the Eucldean dstance from the j th non-sample locaton u j to the locaton u of a sampled locaton and b s the GWR bandwdth estmated from the EMAP data; and (3) we reduced the donor weght of the selected donor locaton by 1. We compare the followng small area predctors (a) EBLUP (2), (b) M-quantle CD (10) (MQ), (c) M-quantle GWR CD (15) under model (11) (MQGWR), (d) M-quantle GWR CD (15) under the local ntercepts model (14) (MQGWR-LI), (e) SEBLUP (6) (f) nonparametrc EBLUP (20) (NPEBLUP) and (g) nonparametrc M-quantle CD (21) (NPMQ). For the M-quantle GWR predctors we use the centrod of the lake. The relatve bas (RB) and the relatve root mean squared error (RRMSE) of estmates of the mean value of ANC n each HUC were computed. Before presentng the results from ths smulaton study we would lke to show some model and spatal dagnostcs. Fgure 1 shows normal probablty plots of level 1 and level 2 resduals obtaned by fttng a two-level (level 1 s the lake and level 2 s the HUC) mxed model to the synthetc populaton data. The normal probablty plots ndcate that the Gaussan assumptons of the mxed model are not met. Hence, the use of a model that relaxes these assumptons, such as the M-quantle model, can be justfed n ths case. In order to detect whether there s spatal autocorrelaton n the EMAP data we computed the Moran s I coeffcent. The standardzed Moran s I s analogous to the correlaton coeffcent, and ts values range from 1 (strong postve spatal autocorrelaton) to -1 (strong negatve spatal autocorrelaton). For the EMAP data Moran s I = 0.61 ndcatng a postve spatal correlaton. There s also evdence of a non-statonary process. In partcular, usng an ANOVA test proposed by Brundson et al. (1999) we rejected the null hypothess of statonarty of the model parameters. Based on the spatal dagnostcs we expect that ncorporatng the spatal nformaton n small area estmaton may lead to gans n effcency. The results set out n Tables 1 and 2 show the across areas and smulatons dstrbuton of relatve bas and relatve root mean squared error for n sample and out of sample areas respectvely. Focusng frst on Table 1 we note that all small area predctors based on the dfferent varants of the M-quantle GWR model have sgnfcantly lower relatve bas than the EBLUP SEBLUP, and NPEBLUP predctors wth the MQGWR predctor performng best. Examnng the performance n terms of relatve root mean squared error we note that the small area predctors that account for the spatal structure of the data have on average smaller root mean squared errors wth the NPEBLUP, SEBLUP and MQGWR predctors performng best. The ncreased relatve root mean squared error of the MQGWR predctors can be explaned by the bas varance trade off assocated wth the use of robust methods. That s, although by usng the M-quantle GWR model we reduce the bas of the pont estmates, the MQGWR predctors have hgher varablty. One way of potentally tacklng ths problem s by makng the M-quantle GWR model less robust for example by settng n the Huber nfluence functon c > These results also show that 7

10 Sample Quantles Sample Quantles Theoretcal Quantles Theoretcal Quantles Fgure 1: Normal probablty plots of level 2 (left) and level 1 resduals (rght) derved by fttng a two level lnear mxed model to the synthetc populaton data. there s a substantal number of n-sample HUCs where the MQGWR predctor has lower RRMSE than the NPEBLUP and SEBLUP predctors. Focusng now on Table 2 we note that for out of sample areas NPMQ and MQGWR-based small area predctors have lower relatve bases and lower root mean squared errors than the EBLUP, NPEBLUP and SEBLUP predctors. Ths supports our orgnal hypothess that the M-quantle GWR model offers a straghtforward approach for mprovng synthetc estmaton for out of sample areas. The performance of the SEBLUP predctor n ths case may be surprsng. However, we should bear n mnd that for out of sample areas we use a synthetc SEBLUP. A more elaborate method for out of sample areas under the SAR model has been proposed by Sae & Chambers (2005). Fgures 2 and 3 show how the dfferent mean squared estmators track the true mean squared error of the dfferent predctors. Here we see that mean squared estmator descrbed n Tzavds et al. (2008), and ts verson (18) under the M-quantle GWR model, perform well n terms of trackng the true mean squared error of the M-quantle small area predctors. Fnally, we see that the Prasad-Rao type MSE estmators of the EBLUP, NPEBLUP and SEBLUP perform poorly n ths applcaton as far as trackng the area-specfc mean squared error s concerned. Ths phenomenon has been also reported n other desgn-based studes (Chambers et al., 2008). As the model dagnostcs have already demonstrated, for ths data the Gaussan assumptons of the mxed model are not satsfed. Ths provdes a further explanaton for the performance of the Prasad-Rao type mean squared estmators n ths case. 6 Conclusons In ths paper we contrasted dfferent approaches for borrowng strength over space n small area estmaton usng parametrc, semparametrc and nonparametrc small area models. Our results show that ncorporatng the spatal nformaton n small area estmaton can lead to sgnfcant gans n the effcency of the small area estmates. The penalzed splnes model appears to be a useful tool when the functonal form of the relatonshp between the varable of nterest and the covarates s left unspecfed and the data are characterzed by complex patterns of spatal dependency. An advantage of the M-quantle GWR-based estmators s that they perform better than the SEBLUP, NPEBLUP estmator for estmatng parameters for out of sample areas. One approach for potentally mprovng the performance of the SEBLUP estmator for out of sample areas s to use the Sae & Chambers (2005) SAR mxed model. A further advantage of the M-quantle GWR model s that t allows for outler robust nference. As we llustrated n secton 5, the volaton of the assumptons of the mxed model may lead to substantal bas n the small area estmates derved wth the EBLUP SEBLUP and NPEBLUP predctors. The volaton of the assumptons of the mxed model also affects the performance of the Prasad-Rao type mean squared error estmators. On the other hand, the use of a robust approach to small area estmaton tackles the problem of bas though at the expense of hgher varablty. Approaches to balancng ths bas varance trade off were brefly descrbed n secton 5. In a recent paper Snha & Rao (2008) proposed the use of a robust mxed model for small area estmaton. The robust mxed model can be drectly compared to the M-quantle 8

11 Table 1: Desgn-based smulaton results usng the EMAP data. Results show the dstrbuton of Relatve Bas (RB) and Relatve Root Mean Squared Error (RRMSE) over areas and smulatons for the 86 sampled HUCs. Percentle of across area dstrbuton Predctor Mean Relatve Bas (%) EBLUP SEBLUP MQ MQGWR MQGWR-LI NPEBLUP NPMQ RRMSE (%) EBLUP SEBLUP MQ MQGWR MQGWR-LI NPEBLUP NPMQ Table 2: Desgn-based smulaton results usng the EMAP data. Results show the dstrbuton of Relatve Bas (RB) and Relatve Root Mean Squared Error (RRMSE) over areas and smulatons for the 27 out of sample HUCs. Percentle of across area dstrbuton Predctor Mean Relatve Bas (%) EBLUP SEBLUP MQ MQGWR MQGWR-LI NPEBLUP NPMQ RRMSE (%) EBLUP SEBLUP MQ MQGWR MQGWR-LI NPEBLUP NPMQ

12 RMSE RMSE HUC HUC RMSE RMSE HUC HUC Fgure 2: HUC-specfc values of actual desgn-based RMSE (sold lne) and average estmated RMSE (dashed lne). Top left s the approxmaton to the RMSE of the MQ predctor. Top rght s the approxmaton to the RMSE of the MQGWR predctor. Bottom left s the approxmaton to the RMSE of the MQGWR-LI predctor and bottom rght s the approxmaton to the RMSE of the NPMQ (ag.sp) predctor. Estmates of the RMSE under the dfferent models are obtaned usng the mean squared error estmator suggested by Chambers et al. (2008). 10

13 RMSE RMSE HUC HUC RMSE HUC Fgure 3: HUC-specfc values of actual desgn-based RMSE (sold lne) and average estmated RMSE (dashed lne). Top left s the Prasad & Rao (1990) approxmaton to the RMSE of the EBLUP predctor. Top rght s the approxmaton to the RMSE of the SEBLUP predctor usng the RMSE estmator of secton 2.1. Bottom left s the approxmaton to the RMSE of the NPEBLUPusng the RMSE estmator suggested by Opsomer et al. (2008). 11

14 small area model and we currently workng on comparng the two models. Extendng further the outler robust mxed model nto an outler robust SAR mxed model wll further mprove the collecton of small area estmaton tools. References ANSELIN, L. (1988). Spatal Econometrcs. Methods and Models. Boston: Kluwer Academc Publshers. BANERJEE, S., CARLIN, B. & GELFAND, A. (2004). Herarchcal Modelng and Analyss for Spatal Data. New York: Chapman and Hall. BATTESE, G. E., HARTER, R.M. & FULLER, W.A.(1988). An error-components model for predcton of county crop areas usng survey and satellte data. Journal of the Amercan Statstcal Assocaton 83, BRUNDSON, C., FOTHERINGHAM, A.S. & CHARLTON, M.(1996). Geographcally weghted regresson: a method for explorng spatal nonstatonarty. Geographcal Analyss 28, BRUNDSON, C., FOTHERINGHAM, A.S. & CHARLTON, M.(1999). Some notes on parametrc sgnfcance tests for geographcally weghted regresson. Journal of Regonal Scence 39, CHAMBERS, R. & DUNSTAN, R. (1986). Estmatng dstrbuton functon from survey data. Bometrka 73, CHAMBERS, R. & TZAVIDIS, N. (2006). M-quantle Models for Small Area Estmaton. Bometrka 93, CHAMBERS, R., TZAVIDIS, N. & CHANDRA, H. (2008). On robust mean squared error estmaton for lnear predctors for domans. [paper submtted for publcaton, avalable upon request] CRESSIE, N. (1993). Statstcs for Spatal Data. New York: John Wley & Sons. DAS, K., JIANG, J. & RAO, J.N.K. (2004). Mean squared error of emprcal predctor. Ann. Statst. 32, DATTA, G.S. & LAHIRI, P. (2000). A Unfed Measure of Uncertanty of Estmates for Best Lnear Unbased Predctors n Small Area Estmaton Problem. Statstca Snca 10, FOTHERINGHAM, A.S., BRUNDSON, C. & CHARLTON, M.(1996). Two technques for explorng non-statonarty n geographcal data. Geographcal Systems 4, FOTHERINGHAM, A.S., BRUNDSON, C. & CHARLTON, M.(1996). Geographcally Weghted Regresson West Sussex: John Wley & Sons. KACKAR, R. & HARVILLE, D. (1984). Approxmatons for standard errors of estmators for fxed and random effects n mxed models. Journal of the Amercan Statstcal Assocaton 79, KAUFMAN, L. & ROUSSEEUW, P. (1990). Fndng Groups n Data: An Introducton to Cluster Analyss. New York: Wley. HENDERSON, C. (1975). Best lnear unbased estmaton and predcton under a selecton model. Bometrcs 31, HARVILLE, D.A. & JESKE, D.R. (1992). Mean squared error of estmaton or predcton under a general lnear model. Journal of the Amercan Statstcal Assocaton 87, LARSEN, D. P., KINCAID, T. M., JACOBS, S. E. & URQUHART, N. S.(2001). Desgns for evaluatng local and regonal scale trends. Boscence 51, LONGFORD, N.T. (2007). On standard errors of model-based small area estmators. Survey Methodology 33, NYCHKA, D. & SALTZMAN, N. (1998). Desgn of ar qualty montorng networks. In Nychka, Douglas, Pegorsch, Walter W. and Cox, Lawrence H. (eds), Case studes n envronmental statstcs OPSOMER, J. D. CLAESKENS, G., RANALLI, M. G., KAUERMANN, G. & BREIDT, F. J.(2008). Nonparametrc small area estmaton usng penalzed splne regresson. Royal Statstcal Socety, Seres B 70, PETRUCCI, A. & SALVATI, N. (2006). Small area estmaton for spatal correlaton n watershed eroson assessment. Journal of Agrcultural, Bologcal and Envronmental Statstcs 11, PRASAD, N. & RAO, J. (1990). The estmaton of the mean squared error of small-area estmators. Journal of the Amercan Statstcal Assocaton 85,

15 PRATESI, M. & SALVATI, N. (2007). Small area estmaton: the EBLUP estmator based on spatally correlated random area effects. Statstcal Methods & Applcatons 17, PRATESI, M., RANALLI, M. G.& SALVATI, N. (2008). Semparametrc M-quantle regresson for estmatng the proporton of acdc lakes n 8-dgt HUCs of the Northeastern US. Envronmetrcs 19, RAO, J.N.K., KOVAR, J.G. & MANTEL, H.J. (1990). On Estmatng Dstrbuton Functons and Quantles from Survey Data Usng Auxlary Informaton. Bometrka 77, RAO, J. N. K. (2003). Small Area Estmaton. London: Wley. ROYALL, R.M. & CUMBERLAND, W.G. (1978). Varance estmaton n fnte populaton samplng. Journal of the Amercan Statstcal Assocaton 73, RUPPERT, D., WAND, M. P.& CARROLL, R.(2003). Semparametrc Regresson. Cambrdge: Cam- brdge Unversty Press. SAEI, A. & CHAMBERS, R. (2005). Emprcal Best Lnear Unbased Predcton for Out of Sample Areas. Workng Paper M05/03, Southampton Statstcal Scences Research Insttute, Unversty of Southampton. SALVATI, N., TZAVIDIS, N., PRATESI, M. & CHAMBERS, R. (2008). Small Area Estmaton Va M-quantle Geographcally Weghted Regresson. [paper submtted for publcaton, avalable upon request] SALVATI, N., RANALLI, M.G. & PRATESI, M. (2008a). Nonparametrc M-quantle Regresson usng Penalzed Splnes n Small Area Estmaton [paper submtted for publcaton, avalable upon request] SINGH, B., SHUKLA, G. & KUNDU, D. (2005). Spato-temporal models n small area estmaton. Survey Methodology 31, SINHA, S.K. & RAO, J.N.K. (2008). Robust Small Area Estmaton. [paper submtted for publcaton] TZAVIDIS, N., MARCHETTI, S. & CHAMBERS, R. (2008). Robust predcton of small area means and dstrbutons. [paper submtted for publcaton, avalable upon request] YU, D.L. & WU, C. (2004). Understandng populaton segregaton from Landsat ETM+magery: a geographcally weghted regresson approach. GISence and Remote Sensng 41, ZIMMERMAN, D.L. & CRESSIE, N. (1992). Mean squared predcton error n the spatal lnear model wth estmated covarance parameters. Ann. Inst. Stat. Math. 44,