Small Area Estimation via M-Quantile Geographically Weighted Regression

Transcription

1 Small Area Estmaton va M-Quantle Geographcally Weghted Regresson CCSR Workng Paper Ncola Salvat, Nkos Tzavds, Monca Prates & Ray Chambers One popular approach to small area estmaton when data are spatally correlated s to employ Smultaneous Autoregressve Regresson (SAR) random effects models to defne an extenson to the Emprcal Best Lnear Unbased Predctor namely, the Spatal Emprcal Best Lnear Unbased Predctor (SEBLUP) (Sngh et al., 2005 and Prates and Salvat, 2007). SAR models allow for spatal correlaton n the error structure. An alternatve approach for ncorporatng the spatal nformaton n the regresson model s va Geographcally Weghted Regresson (GWR) (Brunsdon et al., 1996; Fotherngham et al., 1997). GWR extends the tradtonal regresson model by allowng local rather than global parameters to be estmated. In ths paper we nvestgate the use of GWR n small area estmaton based on the M-quantle modellng approach (Chambers and Tzavds, 2006). In dong so we frst propose an M-quantle GWR model that s a local model for the M-quantles of the condtonal dstrbuton of the outcome varable gven the covarates

2 Small Area Estmaton Va M-quantle Geographcally Weghted Regresson Ncola Salvat (1), Nkos Tzavds (2), Monca Prates (1) and Ray Chambers (3) ABSTRACT One popular approach to small area estmaton when data are spatally correlated s to employ Smultaneous Autoregressve Regresson (SAR) random effects models to defne an extenson to the Emprcal Best Lnear Unbased Predctor namely, the Spatal Emprcal Best Lnear Unbased Predctor (SEBLUP) (Sngh et al., 2005 and Prates and Salvat, 2007). SAR models allow for spatal correlaton n the error structure. An alternatve approach for ncorporatng the spatal nformaton n the regresson model s va Geographcally Weghted Regresson (GWR) (Brunsdon et al., 1996; Fotherngham et al., 1997). GWR extends the tradtonal regresson model by allowng local rather than global parameters to be estmated. In ths paper we nvestgate the use of GWR n small area estmaton based on the M-quantle modellng approach (Chambers and Tzavds, 2006). In dong so we frst propose an M-quantle GWR model that s a local model for the M-quantles of the condtonal dstrbuton of the outcome varable gven the covarates. Ths model s then used to defne a predctor of the small area characterstc of nterest that accounts for spatal assocaton n the data. An mportant spn-off from ths approach s more effcent (1) Dpartmento d Statstca e Matematca Applcata all Economa, Unverstà d Psa, Va Rdolf 10, Psa 56124, Italy e-mal: salvat@ec.unp.t prates@ec.unp.t (2) Centre for Census and Survey Research, Unversty of Manchester, Manchester M13 9PL, UK e-mal: Nkos.Tzavds@manchester.ac.uk (3) Centre for Statstcal and Survey Methodology, School of Mathematcs and Appled Statstcs, Unversty of Wollongong, Wollongong, NSW 2522, Australa e- mal: ray@uow.edu.au 1

3 synthetc estmaton for out of sample areas. We demonstrate the usefulness of ths framework through both model-based as well as desgn-based smulaton, wth the latter based on a realstc survey data set. The paper concludes wth an applcaton to envronmental data for predctng average levels of the Acd Neutralzng Capacty at 8-dgt Hydrologc Unt Code level n the Northeast states of the U.S.A. Keywords: Borrowng strength over space; Envronmental data; Estmaton for out of sample areas; Robust regresson; Spatal dependency. 1. INTRODUCTION Unt level random effects models are wdely used n small area estmaton. See Rao (2003) Typcally, such models assume ndependence of random area effects and ndvdual effects. Ths assumpton of unt level ndependence s also mplct when M-quantle models (Chambers and Tzavds, 2006) are used n small area estmaton. 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 modelled by extendng random effects models to allow for spatally correlated area effects, e.g. va a Smultaneous Autoregressve Regresson (SAR) random effects model (Anseln, 1992; Cresse, 1993), and Sngh et al. (2005) and Prates and Salvat (2007) have nvestgated the use of the Spatal Emprcal Best Lnear Unbased Predctor (SEBLUP) for small area estmaton n ths stuaton. 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) (Brunsdon et al., 1996; 2

4 Fotherngham et al., 1997; 2002; Yu and 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 outcome varable. In ths paper we explore the use of GWR n small area estmaton based on the M-quantle modellng approach. In dong so we frst propose an M-quantle GWR model,.e. a local model for the M-quantles of the condtonal dstrbuton of the outcome varable gven the covarates. Ths model s then used to defne a predctor of the small area characterstc of nterest (here we focus on the small area mean) that accounts for spatal assocaton n the data. An mportant spn-off from ths approach s more effcent synthetc estmators for out of sample areas. The structure of the paper s as follows: In secton 2 we brefly revew unt level mxed models wth random area effects and M-quantle models for small area estmaton. In secton 3 we descrbe GWR and extend ths to defne the M-quantle GWR model. In secton 4 we show how the M-quantle GWR model can be utlsed for small area estmaton. In secton 5 we dscuss mean squared error estmaton for small area predctors defned under the M-quantle GWR model. In secton 6 we present a seres of model-based and desgn-based smulaton studes for assessng the performance of the dfferent small area predctors consdered n ths paper. In secton 7 we use data from the U.S. Envronmental Protecton Agency's Envronmental Montorng and Assessment Program (EMAP) to predct average levels of the Acd Neutralzng Capacty at 8-dgt Hydrologc Unt Code (HUC) level n the Northeast states of the U.S.A. Fnally, n secton 8 we summarze our man fndngs. 3

5 2. AN OVERVIEW OF UNIT LEVEL MODELS FOR SMALL AREA ESTIMATION In what follows we assume that the target populaton can be dvded nto d small areas, each contanng a known number N of unts, wth the value x of a vector x of p auxlary varables known for each unt n small area and wth the value y for the varable of nterest y known for each unt n the sample. The overall sample sze s n, wth the sample sze n area equal to n (ths can be zero). The am s to use these data to predct varous area specfc quanttes, ncludng (but not only) the area mean m of y. The most popular method used for ths purpose employs lnear mxed models. In the general case such a model has the form T T y! x! " z" " #, = 1,, n, = 1,, d, (1) where # s an ndvdual random effect, " s a vector of area level random effects and z s a vector of auxlary contextual varables whose values are known for all unts n the populaton. The role of the " n (1) s to characterse dfferences n the condtonal dstrbuton of y gven x between the small areas. The emprcal best lnear unbased predctor (EBLUP) of where s denotes the m (Henderson, 1975; Rao, 2003) s then ' ( MX % 1 T ˆ # ˆ T m ˆ! N y " x! " z " ) * + & s & r, n sampled unts n area, - - (2) r denotes the remanng N % n unts n the area and ˆ!, ˆ " are defned by substtutng an optmal estmate of the covarance matrx of the random effects n (1) nto the best lnear unbased estmator of! and the best lnear unbased predctor (BLUP) of " respectvely. 4

6 An alternatve approach to small area estmaton s based on the use of M-quantle models (Brecklng and Chambers, 1988). A lnear M-quantle regresson model s one where the q th M-quantle Q ( x ; ) of the condtonal dstrbuton of y gven x satsfes q T Qq ( x; )! x! ( q). (3) Here denotes the nfluence functon assocated wth the M-quantle. For specfed q and contnuous, an estmate ˆ! ( q) of! ( q) s obtaned va an teratve weghted least squares algorthm. The M-quantle coeffcent q of populaton unt was ntroduced by Chambers and Tzavds (2006) and s the value q such that Qq ( x; )! y. These authors observed that f varablty between small areas s a sgnfcant part of the overall varablty of the populaton data, then we expect unts from a partcular small area to have smlar M-quantle coeffcents. When (3) holds, wth! ( q) a suffcently smooth functon of q, they suggested a predctor of m of the form. / MQ % 1 T mˆ # ˆ ( ˆ! N 0 y " x! % ) 1 02 & s & r (4) where % ˆ 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, where these unt level coeffcents are estmated by solvng Qˆ q ( x ; )! y for q. Here Q ˆq denotes the estmated value of (3) at q. When there s no sample n the area % ˆ! 0.5. Tzavds and Chambers (2007) refer to (4) as the nave M-quantle predctor and note that ths can be based. To rectfy ths problem these authors propose a bas adusted M-quantle predctor of m of the form 5

7 6. / 1 ˆ ˆ ˆ ˆ N % n / MQ CD % T m () ( ) 4 ˆ! 7 tdf t! N 0 - y " - x! % " - y % y 51, (5) & s & r n %6 02 & s 13 T where yˆ ˆ ( ˆ! x! % ). Note that the superscrpt CD n (5) refers to the fact that t s derved from the expected value functonal of the area verson of the dstrbuton functon estmator proposed by Chambers and Dunstan (1986). Tzavds and Chambers (2007) note that, under smple random samplng, predctor (5) s also derved from the expected value functonal of the area verson of the Rao-Kovar- Mantel (1990) dstrbuton functon estmator, whch s a desgn-consstent and model-consstent estmator of the fnte populaton dstrbuton functon. 3. M-QUANTILE GEOGRAPHICALLY WEIGHTED REGRESSION In ths secton we defne a spatal extenson to M-quantle regresson based on GWR. Snce M-quantle models do not depend on how areas are specfed, we also drop the subscrpt from our notaton. Gven n observatons at a set of L locatons # u ; l 1,..., LL ; n values # y, x ; 1,..., n follows l l l l! 8, wth n l data! observed at locaton u l, a GWR model s defned as y! x! ( u )" #. (6) l l l l The value of the regresson functon! ( u) at an arbtrary locaton u s estmated usng weghted least squares n % 1 L l L nl T l l l l l l l! 1! 1 l! 1! 1 9 : 9 : ˆ(! u )! ; w ( u, u ) xx < ; w ( u, u ) xy< = > = > , where w( ul, u) s a spatal weghtng functon whose value depends on the dstance from sample locaton u l to u n the sense that sample observatons wth locatons 6

8 close to u have more weght than those further away. One popular approach to defnng such a weghtng functon puts 2 9 2? exp. 1 % ( du, / ) l u b / f du l, u8 b w( ul, u)! ; 2 3? = 0 otherwse, (7) where d u, udenotes the Eucldean dstance between u l and u and b s the bandwdth, l whch can be optmally defned usng a least squares crteron (Fotherngham et al., 2002). In what follows we wll use (7) to defne the weghtng functon. However, t should be noted that alternatve weghtng functons, for example the b-square functon, can also be used. The GWR model (6) s a model for the condtonal expectaton of y gven x at locaton u. Ths s easly generalsed to a model for the M-quantle of order q of the condtonal dstrbuton of y gven x at u by allowng (3) to depend on u. That s, we wrte T Qq ( x;, u)! x! ( uq ; ) (8) where now! ( uq ; ) vares wth u as well as wth q. That s, (8) allows the entre condtonal dstrbuton (not ust the mean) of y gven x to vary from locaton to locaton. The parameter! ( uq ; ) n (8) can be estmated by solvng n l L T - w( ul, u) - q # yl % xl! ( u; q) xl! 0. (9) l! 1! 1 1 where () 2 ( % )# ( 0) (1 ) ( 0) q t! s t qi " % q I t 8. Here s s a sutable robust estmate of the scale of the sample y values, e.g. the MAD estmate T s! medan y % x ˆ ( u ; q ) / and we wll typcally assume a Huber Proposal l l! 2 nfluence functon, () t! ti (% c8 t 8 c) " csgn( t) I( c). Provded c s bounded away from zero, an teratvely re-weghted least squares algorthm that combnes the 7

9 teratvely re-weghted least squares algorthm used to ft spatally statonary M- quantle model (3) and the weghted least squares algorthm used to ft a GWR model can then be used to solve (9), leadng to estmates of the form T A % # 1 ˆ ( ; ) ( ; ) T A uq! X W u q X X W ( u ; q ) y. (10)! s s s s s s Here y s s the vector of n sample y values and X s s the correspondng matrx of A order nb p of sample x values. The matrx W ( u; q) s s a dagonal matrx of order n wth entry correspondng to a partcular sample observaton 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 ˆ! ( q). One may argue that (8) s over-parametrsed as t allows for both local ntercepts and local slopes. An alternatve spatal extenson of the M-quantle regresson model (3) that has a smaller number of parameters s one that combnes local ntercepts wth global slopes and s defned as T Q ( x;, u)! x! ( q) "& ( uq ; ). (11) q Here & ( uq ; ) s a real valued spatal process wth zero mean functon over the space defned by locatons of nterest. The model (11) s ftted n two steps. At the frst step we gnore the spatal structure n the data and estmate! ( q) drectly va the teratve re-weghted least squares algorthm used to ft the standard lnear M-quantle regresson model (3). Denote ths estmate by ˆ! ( q). At the second step we use geographc weghtng to estmate & ( uq ; ) va T # ˆ L n l ˆ % 1 ( uq ; )! n w( ul, u) q yl % xl ( q) l! 1! (12) &! 8

10 Choosng between (8) and (11) wll depend on the partcular stuaton and whether t s reasonable to beleve that the slope coeffcent n the M-quantle regresson model vares sgnfcantly between locatons. However, t s clear that snce (11) s a specal case of (8), the soluton to (9) wll have less bas and more varance than the soluton to (12). Hereafter we refer to (8) and (11) as the MQGWR and MQGWR-LI (Local Intercepts) models respectvely. 4. USING M-QUANTILE GWR MODELS IN SMALL AREA ESTIMATION In ths secton we descrbe how the spatal extensons of the M-quantle model can be used for small area estmaton. In addton to the assumptons made at the start of secton 2, we now assume that we have only one populaton value per locaton. That s, we can drop the ndex l. We also assume that the geographcal coordnates of every unt n the populaton are known, whch s the case for example wth geo-referenced data. The am s to use these data to predct the area mean quantle GWR models (8) and (11). m of y usng the M- Followng Chambers and Tzavds (2006), we frst estmate the M-quantle GWR coeffcents # q s s; & of the sampled populaton unts wthout reference to the small areas of nterest. A grd-based nterpolaton procedure for dong ths under (3) s descrbed n Chambers and Tzavds (2006) and can be drectly used wth (11). We adapt ths approach to the GWR M-quantle model (8) by frst defnng a fne grd of q values over the nterval (0,1) and then usng the sample data to ft (8) for each dstnct value of q on ths grd and at each sample locaton. The M-quantle GWR coeffcent for unt wth values y and x at locaton u s fnally calculated by nterpolatng over ths grd to fnd the value q such that Qq ( x;, u )! y. In ether case, provded there are sample observatons n area, an area specfc M-quantle GWR 9

11 coeffcent, % ˆ can be defned as the average value of the sample M-quantle GWR coeffcents n area. Followng Tzavds and Chambers (2007), the bas-adusted M- quantle GWR predctor of the mean m n small area s ˆ # ˆ. / 1 ˆ ˆ N % n / MQGWR CD % m! N 0 y " Q ˆ ( x ;, u ) " y % Q ( x;, u ) % % 1 02 & s & r n & s (13) where Qˆ ( x;, u ) s defned ether va the MQGWR model (8) or va the MQGWR- % ˆ LI model (11). There are stuatons where 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) the synthetc mean predctor for out of sample area s m ˆ =N - x!. Under the M-quantle model (5) the synthetc mean MX / SYNTH % 1 T ˆ '& U MQ/ SYNTH % 1 T predctor for out of sample area s ˆ ˆ! 4()* 5 m! N - x. We note that wth & U synthetc estmaton all varaton n the area-specfc predctons comes from the areaspecfc 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. In ths case a synthetc-type mean predctor for out of sample area s defned by 4 5 m! N Qˆ - x u. MQGWR / SYNTH % 1 ˆ 0.5 ;, & U We expect that when a truly spatally non statonary process s present, ˆ MQGWR/ SYNTH m wll mprove the effcency of the other synthetc mean predctors. Emprcal results that address the ssue of out of sample area estmaton are set out n secton 6. 10

12 5. MEAN SQUARED ERROR ESTIMATION A robust estmator of the mean squared error of (3) was proposed n Tzavds and Chambers (2007). Here we extend ths argument to defne an estmator of a frst order approxmaton to the mean squared error of (13) under the MQGWR model (8). Our argument s easly extended to the MQGWR-LI model (11). A more detaled dscusson of ths approach to mean squared error estmaton s set out n Chambers et al. (2007). To start we note from (10) that (13) can be expressed as a weghted sum of the sample y-values mˆ! N w y, (14) MQGWR/ CD % 1 T s s where N N T % n T ws! 1s " H x % H x n n - -. (15) & r Here 1 s s the n-vector wth th component equal to one whenever the correspondng sample unt s n area and s zero otherwse and % T A # ˆ 1 & s T A H! X W ( u ;% ) X X W ( u ;% ˆ ). s s s s s Gven the lnear representaton (14), 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 and Cumberland, 1978). Put w! ( w ). Ths estmator s of the form s # ˆ 2 % ˆ vmˆ -- y Q x u (16) % where + # MQGWR/ CD ( )! + k % (,, ) k kn : 0 & sk k ( w 1) ( n 1) ( N n ) Ik ( ) wk I( k )! % " % %! " C. 6. SIMULATION STUDIES In ths secton we present results from smulaton studes that were used to examne the performance of the small area estmators that were dscussed n the precedng 11

13 sectons. In secton 6.1 we employ model-based smulatons n whch small area populaton and sample data were smulated based on dfferent parametrc assumptons about the dstrbuton of errors and the spatal structure of the data. In secton 6.2 we present a desgn-based smulaton that s based on real survey 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. 6.1 MODEL-BASED SIMULATIONS Two methods were used to smulate populaton data. In both, N = populaton values of x and y n J = 30 small areas were frst smulated. For each area we then ndependently selected a smple random sample (wthout replacement) of sze n! 20, leadng to an overall sample sze of n = 600. Ths process was repeated 200 tmes. The sample values of y and the populaton values of x obtaned n each smulaton were used to estmate the small area means. The frst method of smulaton generated populaton values of y and x n small area accordng to the two-level model y! 1" 2x " " " # where x ~ U [0,1], wth random effects generated under two scenaros: (a) " ~ N(0, 0.04) and # ~ N(0,0.16) and (b) " % and 2 ~ X (1) 1 # %. The second method of smulaton generated 2 ~ X (3) 3 populaton values wth random effects smulated under the same scenaros (a) and (b) but n addton allowed the ntercept, and slope! of the lnear model for y to vary accordng to longtude and lattude. In partcular, these locaton coordnates were ndependently generated as U [0,50] wth and,! 0.2B longtude " 0.2Blattude 12

14 !! % 5 " 0.1B longtude " 0.1B lattude. Four dfferent types of small area lnear models were ftted to these smulated data. These were () a random ntercepts verson of (1), () the lnear M-quantle regresson specfcaton (3), () the MQGWR model (8), and (v) the MQGWR-LI model (11). The random ntercepts model used n () was ftted usng the lme functon (Venables and Rpley, 2002, secton 10.3) n R (R Development Core Team, 2004). The M- quantle lnear regresson model () was ftted usng a modfed verson of the rlm functon (Venables and Rpley, 2002, secton 8.3) n R (R Development Core Team, 2004). The MQGWR models n () and (v) were ftted usng a straghtforward modfcaton of the functons used to ft (). Estmated model coeffcents obtaned from these fts were then used to compute the EBLUP (2), the bas-adusted M- quantle predctor (5), denoted MQ below, and the MQGWR and the MQGWR-LI versons of correspondng bas-adusted M-quantle predctor (13). Bases and root mean squared errors over these smulatons, averaged over the 30 areas, are set out n Table 1. For Gaussan random effects and a spatally statonary regresson surface, we can see that the EBLUP s the best predctor, as one would expect. The MQ, MQGWR and MQGWR-LI predctors all have smlar bas and RMSE n ths case. In contrast, when the underlyng regresson functon s nonstatonary we see that the MQGWR and MQGWR-LI predctors are consderably more effcent than the MQCD predctor and the EBLUP. Under Ch-squared random effects ths relatve performance s unchanged, although here the absolute dfferences n performance between the varous predctors s much smaller. Fnally, n Table 2 we show key percentles of the across area dstrbutons of the area level true and estmated mean squared errors (the latter based on (16) and averaged over the smulatons) of the MQGWR and MQGWR-LI predctors, as well as the 13

15 correspondng area level coverage rates for nomnal 95 per cent predcton ntervals. In general the proposed mean squared error estmator (16) provdes a good approxmaton to the true mean squared error. These results also show that when M- quantle GWR fts are used n (16), then ths estmator provdes some underestmaton of the true mean squared error of the correspondng predctor that also results n some undercoverage of predcton ntervals. Ths s consstent wth both the MQGWR and the MQGWR-LI models overfttng the actual populaton regresson functon. However, ths bas s not excessve, beng more pronounced n the case of the MQGWR model. 6.2 A DESIGN-BASED SIMULATION The actual 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. 2001). 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.A. For ths study, a sample of 334 lakes was selected from the populaton of 21,026 lakes n these states usng a random systematc desgn. The lakes makng up 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 and assocated survey weghts for the sampled locatons, the EMAP data set also 14

16 contaned the elevaton and geographcal coordnates of the centrod of each lake n the target area. The am of the desgn-based smulaton was to compare the performance of dfferent predctors of mean ANC n each HUC. 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 to the greater of fve and the orgnal EMAP sample sze n the HUC. No samples were taken from HUCs that had not been sampled by EMAP, leadng 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 th non-sample locaton by selectng one of the ANC values of the EMAP sample locatons wth probablty proportonal to # 2 u, u w( u, u)! exp % 0.5( d / b). Here d u, u s the Eucldean dstance from the th nonsample locaton u 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. 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 for the same four predctors that were the focus of the model-based smulatons. These results are set out n Table 3 and show that the M-quantle GWR predctors are much more effcent 15

17 that the EBLUP and M-quantle based predctors that gnore the spatal structure n the data. In partcular, we see that for the non-sampled HUCs the use of the synthetctype predctors that borrow strength over space, defned n secton 4, substantally mprove predcton. Fgure1 shows how dfferent mean squared estmators tracked the true mean squared error of the dfferent predctors n ths smulaton. Here we see that mean squared estmator descrbed n Tzavds and Chambers (2007), and ts GWR form (16), perform well n terms of trackng the true mean squared error of the M- quantle predctors. Some downward bas of (16) when used wth the MQGWR model s reported, however. Ths s much less of a problem when (16) s combned wth the MQGWR-LI model. Fnally, we see that the Prasad-Rao estmator of the mean squared error of the EBLUP performs poorly as far as trackng area-specfc mean squared error s concerned. Ths phenomenon has also been reported n other desgnbased studes (e.g. Chambers et al., 2007). 7. APPLICATION: ASSESSING THE ECOLOGICAL CONDITION OF LAKES IN THE NORTHEASTERN U.S.A. In ths secton we show how the methodology descrbed n ths paper can be practcally employed for estmatng the average acd neutralzng capacty (ANC) for each of the dgt HUCs that make up the EMAP dataset descrbed n secton 6.2. Fgure 2(a) shows the regon of nterest and the locatons of the sampled lakes. ANC s a measure of the ablty of a soluton to resst changes n ph and s on a scale measured n meq/l (mcro equvalents per lter). A small ANC value for a lake ndcates that t s at rsk of acdfcaton. Fgure 2(b) shows the dstrbuton of ANC n the EMAP data. Ths s skewed and may contan nfluental data ponts. Furthermore, the Brunsdon et al. (1999) ANOVA test for spatal statonarty ndcates 16

18 that the EMAP data are consstent wth a process charactersed by spatally varyng relatonshps. Predcted values of average ANC for each HUC were calculated usng the M- quantle GWR predctor (13) under the MQGWR model (8) and the MQGWR-LI model (11), wth x equal to the elevaton of each lake and wth locaton defned by the geographcal coordnates of the centrod of each lake (n the UTM coordnate system). The spatal weght matrx used n fttng these M-quantle GWR models was constructed usng (7), wth bandwdth selected usng cross-valdaton. Fgure 3 shows contour maps of the estmated HUC-specfc ntercepts and slopes from the ftted MQGWR model (8),.e. when ths model s ftted usng the HUCspecfc M-quantle coeffcents % ˆ. These maps support the assumpton of nonstatonarty n the data. Fnally, n Fgure 4 we show maps of estmated values of average ANC for each HUC usng the MQGWR model, the MQGWR-LI model, the spatally statonary M-quantle model (3) and the lnear mxed model (1). The two M- quantle GWR models provde smlar estmates of average ANC for each HUC and are consstent wth the patterns produced by other analyses of the EMAP data usng non-parametrc models (Opsomer et al., 2005). There are also substantally dfferent from the estmates produced by the spatally statonary models (1) and (3), whch show lower levels of average ANC (and hence greater rsk of water acdfcaton) for these HUCs. 8. SUMMARY In ths paper we propose a geographcally weghted regresson extenson to M- quantle regresson that allows for spatally varyng coeffcents n the model for the M-quantles. These M-quantle GWR models have the potental to lead to sgnfcantly better small area estmates n mportant applcaton areas where geo- 17

19 referenced data are avalable, such as fnancal and economc statstcs, envronmental and publc health modellng. Lke the M-quantle regresson model of Chambers and Tzavds (2006), the M-quantle GWR model descrbed n ths paper allows modellng of between area varablty wthout the need to explctly specfy the area-specfc random components of the model. In partcular, ths model does not explctly depend on any partcular small area geography, and so can be easly adapted to dfferent geographes as the need arses. One problem that arses wth specfyng an M-quantle GWR model s decdng whch parameters of the model vary spatally (.e. are local parameters) and whch do not (.e. are global parameters). In ths paper we have explored two M-quantle GWR models that exemplfy ths ssue the MQGWR model where both ntercept and slope parameters n the model vary spatally and the MQGWR-LI model where only the ntercept parameter vares spatally. Further research s necessary n order to develop approprate dagnostcs for decdng between them. AKNOWLEDGEMENTS The work n ths paper has been supported by proect PRIN Metodologe d stma e problem non-camponar nelle ndagn n campo agrcoloambentale awarded by the Italan Government to the Unverstes of Cassno, Florence, Peruga, Psa and Treste. The authors are grateful to the Space-Tme Aquatc Resources Modellng and Analyss Program (STARMAP) for provdng access to the data used n ths paper. The vews expressed here are solely those of the authors. 18

20 REFERENCES Anseln, L. (1992), Spatal econometrcs: Method and models, Kluwer Academc Publshers, Boston. Brecklng, J. and Chambers, R. (1988) M-quantles, Bometrka, 75, 4, Brunsdon, C., Fotherngham, A.S. and Charlton, M. (1996) Geographcally weghted regresson: a method for explorng spatal nonstatonarty, Geographcal Analyss, 28, Brunsdon, C., Fotherngham, A.S. and Charlton, M. (1999) Some notes on parametrc sgnfcance tests for geographcally weghted regresson, Journal of Regonal Scence, 39, Chambers, R. and Dunstan, R. (1986). Estmatng dstrbuton functon from survey data. Bometrka, 73, Chambers, R. and Tzavds, N. (2006). M-quantle Models for Small Area Estmaton, Bometrka, 93, Chambers, R., Chandra, H. and Tzavds, N. (2007). On robust mean squared error estmaton for lnear predctors for domans. [Paper submtted for publcaton. A copy s avalable upon request]. Cresse, N. (1993), Statstcs for spatal data, John Wley & Sons, New York. Fotherngham, A.S., Brunsdon, C. and Charlton, M. (1997) Two technques for explorng non-statonarty n geographcal data, Geographcal Systems, 4, Fotherngham, A.S., Brunsdon, C. and Charlton, M. (2002) Geographcally Weghted Regresson, John Wley & Sons, West Sussex. Henderson C. (1975): Best lnear unbased estmaton and predcton under a selecton model, Bometrcs, 31, Larsen, D. P., Kncad, T. M., Jacobs, S. E. and Urquhart, N. S. (2001) Desgns for evaluatng local and regonal scale trends, Boscence, 51, Opsomer, J. D., Claeskens, G., Ranall, M.G., Kauermann, G. and Bredt, F.J. (2005). Nonparametrc small area estmaton usng penalzed splne regresson. In I. S. U. Department of Statstcs (Ed.), Preprnt Seres. 19

21 Prates, M. and Salvat N. (2007). Small Area Estmaton: the EBLUP estmator based on spatally correlated random area effects. Forthcomng n Statstcal Methods & Applcatons. R Development Core Team (2004). R: A language and envronment for statstcal computng. R Foundaton for Statstcal Computng, Venna, Austra. URL: Rao, J.N.K., Kovar, J.G. and 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, John Wley & Sons, New York. Royall, R.M. and Cumberland, W.G. (1978). Varance estmaton n fnte populaton samplng, Journal of the Amercan Statstcal Assocaton, 73, Sngh, B.B., Shukla, G.K. and Kundu, D. (2006): Spato-Temporal Models n Small Area Estmaton, Survey Methodology, 31, 2, Tzavds, N. and Chambers, R. (2007). Robust predcton of small area means and dstrbutons, CCSR Workng Paper , Unversty of Manchester [avalable from Venables, W.N. and Rpley, B.D. (2002). Modern Appled Statstcs wth S, Sprnger, NewYork. Yu, D.L. and Wu, C. (2004) Understandng populaton segregaton from Landsat ETM+magery: a geographcally weghted regresson approach, GISence and Remote Sensng, 41,

22 Table 1 Medan values of Bas and RMSE over areas and smulatons. Predctor Statonary process Non-statonary process Bas RMSE Bas RMSE Gaussan random effects EBLUP MQ MQGWR MQGWR-LI Ch-squared random effects EBLUP MQ MQGWR MQGWR-LI Table 2 Across areas dstrbuton of true (.e. Monte Carlo) root mean squared errors, area averages of estmated root mean squared errors and area coverage rates (CR%) for nomnal 95% predcton ntervals. Predctor MQGWR MQGWR-LI MQGWR MQGWR-LI MQGWR MQGWR-LI MQGWR Indcator Percentle of across areas dstrbuton medan Mean Statonary process,gaussan errors True RMSE Est. RMSE CR(%) true RMSE Est. RMSE CR(%) Non-statonary process, Gaussan errors true RMSE Est. RMSE CR(%) true RMSE Est. RMSE CR(%) Statonary process, Ch-squared errors true RMSE Est. RMSE CR(%) true RMSE Est. RMSE CR(%) Non-statonary process, Ch-squared errors true RMSE Est. RMSE CR(%)

23 MQGWR-LI true RMSE Est. RMSE CR(%) Table 3 Desgn-based smulaton results usng the EMAP data. Results show medans of Relatve Bas (RB) and Relatve Root Mean Squared Error (RMSE) over areas and smulatons. Predctor RB (%) RRMSE (%) 86 sampled HUCs EBLUP MQ MQGWR MQGWR-LI non-sampled HUCs EBLUP MQ MQGWR MQGWR-LI

24 Fgure 1 HUC-specfc values of actual desgn-based RMSE (sold lne) and average estmated RMSE (dashed lne). Top left s the EBLUP predctor (2) wth RMSE estmator suggested by Prasad and Rao (1990). Top rght s the M-quantle predctor (5) wth RMSE estmator suggested by Tzavds and Chambers (2007). Bottom left s MQGWR verson of (13) wth RMSE estmated usng (16) and bottom rght s the MQGWR-LI verson of (13) wth RMSE also estmated usng (16). 23

25 Fgure 2 (a) Locatons of the sampled lakes n Northeastern U.S.A. (b) Hstogram of ANC values n the EMAP data. (a) (b) Fgure 3 Maps showng the spatal varaton n the HUC specfc ntercept and slope estmates that are generated when the MQGWR model s ftted to the EMAP data. Intercepts Slopes 24

26 Fgure 4 Maps of estmated average ANC for all 113 HUCs. The frst map shows estmates computed usng (13) and the MQGWR model (8), the second map shows estmates computed usng (13) and the MQGWR-LI model (11), the thrd map shows estmates computed usng (5) and the statonary M-quantle model (3) and fnally the fourth map shows estmates computed usng (2) and the lnear mxed model (1). 25

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