A SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION


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1 REVSTAT Statistical Journal Volume 9, Number 2, June 2011, A SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION Authors: Pero S. Coelho ISEGI Universiae Nova e Lisboa, Portugal Faculty of Economics, Ljubljana University, Slovenia Luís N. Pereira Escola Superior e Gestão, Hotelaria e Turismo, Centro e Investigação sobre o Espaço e as Organizações, Universiae o Algarve, Portugal Receive: May 2010 Revise: April 2011 Accepte: May 2011 Abstract: This paper approaches the problem of small area estimation in the framework of spatially correlate ata. We propose a class of estimators allowing the integration of sample information of a spatial nature. Those estimators are base on linear moels with spatially correlate small area effects where the neighbourhoo structure is a function of the istance between small areas. Within a Monte Carlo simulation stuy we analyze the merits of the propose estimators in comparison to several traitional estimators. We conclue that the propose estimators can compete in precision with competitive estimators, while allowing significant reuctions in bias. Their merits are particularly conspicuous when analyzing their conitional properties. KeyWors: combine estimator; empirical best linear unbiase preiction; small area estimation; spatial moels; unit level moels. AMS Subject Classification: 62D05, 62F40, 62J05.
2 156 P.S. Coelho an L.N. Pereira
3 A Spatial Unit Level Moel for Small Area Estimation INTRODUCTION Sample survey ata are extensively use to provie reliable irect estimates of parameters of interest for the whole population an for omains of ifferent kins an sizes. When the omains were not originally planne, they usually are poorly represente in the sample or even not represent at all. These omains are calle small areas an they usually correspon to small geographical areas, such as a municipality or a census ivision, or a small subpopulation like a particular economic activity or a subgroup of people obtaine by crossclassification of emographic characteristics. Traitionally, sample sizes are chosen to provie reliable estimates for large omains an the lack of sample ata from the target small area seriously affects the precision of estimates obtaine from areaspecific irect estimators. This fact has given rise to the evelopment of various types of estimators that combine both the survey ata for the target small areas an auxiliary information from sources outsie the survey, often relate to recent censuses an current aministrative ata, in orer to increase precision. Uner this context, the use of inirect estimators has been extensively applie. Such inirect estimators are base on either implicit or explicit moels that provie a link to relate small areas through auxiliary ata. Although traitional inirect estimators base on implicit moels, which inclue synthetic an composite estimators, are easy to apply, they usually present unesirable properties. For that reason, other moel base methos of small area estimation have been suggeste in the literature. These methos can make specific allowance for local variation through complex error structures in the moels that link the small areas, can be valiate from the sample ata an can hanle complex cases such as crosssectional, time series an spatial ata. Such methos are often base on explicit Linear Mixe Moels. The Best Linear Unbiase Preiction (BLUP) approach, using Henerson s metho ([13]), is the most popular technique for estimating small area parameters of interest (usually the mean or the total). Uner this approach an from the moel point of view the small area parameters of interest are functions of fixe (β) an ranom (u) effects. Consequently, the preiction of small area parameters of interest is base on the estimation/preiction of these moel effects. In practice this type of moels always involves unknown variance components in the variancecovariance structure of ranom effects. When these unknown components are substitute by consistent estimates the resulting estimator is usually name as Empirical Best Linear Unbiase Preictor (EBLUP). In the context of unit level spatial ata, little work has been one on moelbase methos of small area estimation. [25], [26] an [5] propose a spatial unit level ranom effects moel with spatial epenence incorporate in the error structure through a simultaneous autoregressive (SAR) error process.
4 158 P.S. Coelho an L.N. Pereira Other finings are ue to [4], [36], [30], [31], [32] an [40]. All these approaches consier a contiguity matrix to escribe the neighbourhoo structure between small areas. Nevertheless, there has been a lack of work regaring the explicit moeling of spatial correlation as a function of the istance between observations or small areas. The main aim of this paper is to propose an approach to the problem of small area estimation in circumstances in which the sample ata are of a spatial nature (or in other contexts in which it is possible to establish some kin of proximity between the omains of stuy), using an estimator that explicitly consier spatial correlation as a function of istance between small areas of stuy. This estimator, applicable to unit level ata, exploits both auxiliary information relating to other known variables on the population an structures of spatial correlation between the sample ata through the specification of an aequate noniagonal structure for the variancecovariance matrices of ranom effects. It is base on a general class of moels that inclues some of the existing moels as special cases an can be unerstoo as an EBLUP of the small area totals. Consequently, it oes not require the specification of a specific prior istribution for moel ranom effects. We also aim to evaluate this estimator in comparison with traitional synthetic an composite estimators that o not explicitly consier spatial variability. The paper is organize uner five sections. Section 1 introuces the context of the small area estimation an the goals of the paper. Section 2 reviews some traitional inirect estimators. Section 3 proposes an EBLUP estimator for small area totals base on spatial unit level ata. The estimator is assiste by a class of moels that fits into the general linear mixe theory. Section 4 escribes the esign of the Monte Carlo simulation stuy an presents empirical results. This stuy analyzes the performance of the propose estimator over the irect an inirect estimators using a real ata set from an agricultural survey conucte by the Portuguese Statistical Office. Discussion of the main finings of this stuy, along with some of its limitations an possible future evelopments are the subject of Section INDIRECT ESTIMATORS One possible approach for borrow information in the context of small area estimation base on implicit moels is to use irect moifie estimators. These estimators maintain certain esignbase properties such as approximately unbiase. This is the case of the regression estimator ([37]) (2.1) ˆτ,reg = ˆτ + (τ x ˆτ x ) ˆβ, = 1,..., D, where ˆτ is an estimator of the th omain total of the interest variable, usually the Horvitz Thompson or a poststratifie estimator an ˆτ x have the same
5 A Spatial Unit Level Moel for Small Area Estimation 159 meaning in relation to the vector of auxiliary variables x i = (x i1,..., x ip ), ˆβ = [ U i s υi 2 πi 1 x i x ] 1 i U i s υi 2 πi 1 x i y i are estimators of the regression coefficients obtaine using ata from the whole sample, υi 2 are regression weights an π i the inclusion probabilities resulting from the sampling esign. Estimator (2.1) is approximately unbiase, since E(ˆτ,reg ) = τ + τ x E(ˆβ) E(ˆτ xβ) τ (supposing E [ (ˆτ x τ x ) (ˆβ b) ] τ x b, where b = E(ˆβ) from the esignbase perspective 1 ). Although using information from outsie the omain for estimating the regression coefficients, usually these estimators still show low precision. An alternative is the synthetic estimation (whose properties epen on the assumptions of a postulate moel). From the esignbase point of view these estimators can be biase an inconsistent. A synthetic regression estimator can be presente as: (2.2) ˆτ,sreg = τ x ˆβ, = 1,..., D, where ˆβ is obtaine as before. A more extreme attitue uner a pure moelbase approach woul ignore the inclusion probabilities in estimating the regression parameters. The esignbase bias of estimator (2.2) is B(ˆτ,sreg ) τ x (b b ), assuming the regression weights are such that υi 2 p j=1 a j x ij, i U, where a j, j = 1,..., p, are arbitrary constants. This conition is always assure in the most typical situation of a nonweighte regression where the parameters υi 2 are assume constant. Further, typically estimator (2.2) has smaller variance than the irect moifie regression estimator but it is biase from the esignbase point of view. When ealing with small areas the reuction in variance associate with the synthetic estimator can be such that it will assure a mean square error (MSE) lower than the one obtaine through the use of the irect moifie estimator. There will always be a risk of a high bias an consequently the invaliity of any confience intervals obtaine uner repeate sampling. A significant avantage of synthetic estimation lies in the fact that it is always possible to obtain omain estimates, even in situations where the sample is very small or even zero. In orer to prevent the quality of the estimator being totally epenent on the postulate moel, some combine or composite estimators have been propose. A combine estimator typically presents the form of the weighte average of a esignbase estimator (approximately unbiase but with high variance) an a synthetic estimator (biase but with low variance): (2.3) ˆτ,com = λ ˆτ,es + (1 λ ) ˆτ,syn, = 1,..., D, with 0 λ 1. These estimators can be classifie in two main types (accoring to the way the weights λ are chosen): samplesize epenent weights an ata epenent weights. It is also possible to assume that the weights are chosen in 1 This conition supposes there is a sufficiently week correlation between ˆτ x an ˆβ what is usually easily achieve as ˆτ x an ˆβ are estimate at ifferent aggregation levels.
6 160 P.S. Coelho an L.N. Pereira a eterministic way using, for example, some previous knowlege or an informe guess. This woul result in what [39] call weights fixe in avance. A goo example of a combine regression estimator where the weights epen on sample size is the ampene regression estimator ([38]): (2.4) ˆτ,reg = λ ˆτ,reg + (1 λ ) ˆτ,sreg, = 1,..., D, where λ = 1 if ˆN N an λ = 0 otherwise, where N is the th omain population size an h is a positive constant. The authors suggeste to use h = 2. The basic iea for choosing h is to assure that the bias contribution from the synthetic component of the estimator is kept within acceptable limits. Another possible approach for borrow information in the context of small area estimation is to use a ataepenent combine estimator, through the moeling of the bias of the synthetic part of the estimator, thus proucing inirect estimates for the weights. Many of the moels that have been propose inclue ranom area effects an can be seen as particular cases of linear mixe moels. One of the best known moels applicable at unit level is the neste error regression moel ([8], [3]). All these approaches implicitly consier some kin of sectional correlation an the omain estimators are obtaine through EBLUP, empirical Bayes or hierarchical Bayes approaches. The wellknown neste error regression moel ([3]) has the form y i = x i β + u + ǫ i, = 1,..., D, i = 1,..., N, where u an ǫ are assume to be ii with zero means. It is also assume that u an ǫ i are mutually inepenent, V m (u ) = σ 2 u an V m (ǫ i ) = σ 2 υ 2 i where υ i are known constants. Here a common covariance between any two observations in the same small area is assume, Cov m (y i, y j ) = σ 2 u (i j). In this kin of moels it is assume that there is no sample selection bias, resulting that they are assume to hol both for the population an for the sample. This may be a very limiting assumption since small omain estimation is frequently neee in the context of informative sampling esigns. An alternative to the EBLUP in the context of informative sampling esigns is the pseuoeblup estimator ([29]). This estimator, base on the neste error regression moel, epens on survey weights an it is esignconsistent. 3. A COMBINED ESTIMATOR FOR SPATIAL DATA 3.1. A class of moels Let y i be the value of the interest variable for unit i (i = 1,..., n ) in small area ( = 1,..., D) an let x i = (x i1,..., x ip ) a vector of p unit level explana
7 A Spatial Unit Level Moel for Small Area Estimation 161 tory variables referring to the same unit. Consier the following class of moels: (3.1) y i = x i β+ H h=1 x (1)i q h,iu (1) h +x (2)i u(2) +ǫ i, =1,...,D, i =1,...,n, where β is a vector of p fixe effects; x (j)i is a vector of p j explanatory variables (typically a subvector of x i ) for the ith unit in small area ; q h,i are esign variables use to take into account the sampling esign an inicate that unit i belongs to a stratum or a sampling unit h (h = 1,..., H); u (1) h = col 1 j p 1 (u hj ) is a vector of p 1 ranom (or fixe) esign effects associate with stratum (or sampling unit) h; u (2) = col 1 j p2 (u j ) is a vector of p 2 ranom effects associate with omain ; l represents the geographical location associate to the centroi of omain ; f(l l e ) is a function of the vector l l e an ǫ i is the resiual term associate with unit i. We assume that E m u ( (1)) ( (2)) h = 0, Em u = 0, Em (ǫ i ) = 0, { ( (1) ) Σ E (1) {, h = g σ m u h u (1) g = 0, otherwise, E 2 m(ǫ i ǫ ej ) = i, = e, i = j 0, otherwise, E ( (2) ) m u u (2) e = { } Σ (2) f(l l e ), with Σ (1) = σ 2 (j, k =1,..., p U (1) 1 ), σ 2 = E(u jk U (1) hj u hk ), Σ (2) = { } jk σ 2 (j, k =1,..., p U (2) 2 ), an σ 2 = E(u jk U (2) j u k ). The moel is applie to ata jk from a sample of total size n = D =1 n, where n is the number of sampling units in area. It is also assume that ranom effects associate with ifferent aggregation levels are not correlate, E ( u (j) u (k)) = 0 for j k, an that the errors are noncorrelate with ranom effects, E ( u (j) ǫ ) = 0, j. In the propose moel, omain effects show a structure of spatial variability. The covariance between the ranom effects associate with omains an e epens on the vector efine by their geographical coorinates l l e. Some functions that can be applie to this context are presente in [28]. When the spatial covariance only epens on the istance l l e, then the function f( l l e ) is sai to be isotropic ([6]) an is typically such that lim l l e 0 f = 1. As the omains are not points in space, but areas, these coorinates are efine by their centrois. The assumption is that the lowest level of aggregation for which the georeferencing is available is the omain. Situations where the level of aggregation for which the referencing is available oes not coincie with the omains of stuy can generate special cases of moel (3.1). In particular, when georeferencing is possible at unit level, spatial variation can be moele through variancescovariances of the errors vector ǫ. Domain ranom effects represent the characteristics specific to the omain of stuy that affect the values of the interest variable an are not represente by the fixe effects at a higher level of aggregation. They can be thought of as moeling the bias of the synthetic part of the moel. Moreover, these omain ranom effects will now have the aitional role of bringing information from other omains, to explain the values of the interest variable in each omain.
8 162 P.S. Coelho an L.N. Pereira Design effects are use to take into account the sampling esign. The goal is to allow the moel to be applie to contexts with informative sampling esigns, overcoming the limitations ([27], [20]) of other ataepenent combine estimators that implicitly assume that the sampling esign is ignorable. The methoology propose can therefore be seen as moel assiste. The sample s is the result of a twostep proceure. First it is suppose that the finite population can be approximately escribe by a superpopulation moel. In the secon step it is assume that a sample is rawn from the finite population through a specific sampling esign. It is assume that the sample can be approximately escribe by moel (3.1), which has taken into account the existence of these two steps Estimation of moel parameters The moel 3.1 can be presente as a special case of the general linear mixe moel, grouping the unitspecific moels over the population: (3.2) y = Xβ + Zu + ǫ, where y is a vector of the target variable, X is a esign matrix of explanatory variables with rows given by x i, Z = [ ] Z (1) Z (2) is a esign matrix, u = ( col 1 j 2 u (j) ) is a vector of ranom effects an ǫ is a vector of errors. The covariance matrix of u is given by G = V m (u) = blockiag 1 j 2 G [ (j) ], where G (1) = { blockiag 1 h H Σ (1) } an G (2) =F Σ (2) with F = { f(l l e ) },,e = 1,..., D. Further, R = V m (ǫ) = iag 1 i n {σi 2 }, E m( u (1) ) ( =0, E m u (2) ) =0 an E m (ǫ) =0. 1 D Both covariance matrices G an R involve unknown variance components, represente by θ. Also the flexibility of the propose class of moels recommens proceeing in each application to the selection of a specific moel, i.e. to the choice of the explanatory variables an appropriate variancecovariance structures to u an ǫ. This step in moel selection an iagnosis is crucially important to obtaining a moel that can aequately escribe the behavior of the target population an can be performe as a systematic proceure like those propose by [7] or [43]. Once a specific moel has been selecte, the variance components θ nee to be estimate in orer to assess the variability of estimators or to preict the fixe an ranom effects. Several methos are available for estimating variance components, such as the analysis of variance (ANOVA) metho ([12]), the minimum norm quaratic unbiase estimation (MINQUE) metho ([33], [34], [35]) an the likelihoo methos. Some references about the maximum likelihoo estimation (MLE) metho ue to Fisher may be foun in [9], [1], [23], [21], [15] an [18]. On the other han, references about the resiual maximum likelihoo estimation
9 A Spatial Unit Level Moel for Small Area Estimation 163 (RMLE) metho propose by [41] an its extensions can be foun in [24], [10], [11], [2], [42], [14], [16], [17], among others. For etails about estimation general linear mixe moels see [19]. Now consier the ecomposition of all matrices into sample an nonsample components, where the subscript s is associate with the n sample units an r is associate with the (N n) nonsample units. The omission of the subscript inicates that the respective matrices allue to the whole population U s r. Assuming moel 3.2 hols an variance components are known, the best linear unbiase estimator of β an the best linear unbiase preictor of θ are given by (3.3) (3.4) β = ( X sv 1 s ũ = GZ sv 1 s ) 1 X s X s Vs 1 y s, ( ys X s β), where V s = E [ (y s X s β)(y s X s β) ] = Z s GZ s + R ss. The vectors u (j) may be preicte using ũ (j) = G (j) Z (j)s V 1 s (y s X s β), while the preictors of the errors ǫ, can be obtaine as ǫ = R s Vs 1 (y s X s β), where R s = [R ss R rs]. [ ] Rss R When the covariance matrix R = sr is blockiagonal, i.e. when R rs R rr there is no correlation between errors associate with the observations insie an outsie the sample, then R rs = 0 an ǫ r = 0. This is the case for moel (3.1). Nevertheless, it shoul be note that some situations can be evise, particularly when the spatial correlation can be establishe at unit level, where there is a correlation between moel errors that can be use in the preiction of ǫ r Estimation of omain totals The objective of the inference can be seen as to preict the total of an interest variable, τ, that uner the moel correspons to the summation of the realizations of the variable of interest over all the elements in the small area : (3.5) τ = i U y i = τ x, β + H h=1 τ x(1),h u(1) h + τ x(2), u(2) a + τ ǫ,, where τ ǫ, = i U ǫ i. It shoul be note that, from the moelbase point of view, (3.5) is a preictable function proucing inference in the narrow inference space [22]. An estimator for the small area total, τ, can be obtaine as (3.6) τ = 1 N ỹ = ỹ i = τ β x + v τsv 1 ( s ys X s β), i U where ỹ is the EBLUP of the vector y an v τs = τ z GZ s + 1 N R,s is the line vector of the moelbase covariances between the small area total τ an
10 164 P.S. Coelho an L.N. Pereira the observable vector y s, R,s = E(ǫ a ǫ s ), an 1 N is a unit vector of size N. It shoul be note that the estimator τ is the EBLUP of τ, given the observable ranom vector y s (cf. Appenix 1). When R,rs is a null matrix, then the EBLUP of the total is τ is given by a simplifie expression: (3.7) τ = Ẽ(τ,r u) = τ y,,s + τ x,,r β + τ z,,r GZ svs 1 ( ys X s β), where τ y,,s is the observe sample total in small area (cf. Appenix 2). It shoul also be note that many of the regression estimators that have been propose for small area estimation may be viewe as EBLUP of omain totals for particular cases of the class of moels (3.1). For instance, the form of the neste error regression moel an the ranom coefficient moel presente in Section 2 accor with class (3.1), with u (2) scalar, G (1) =0, G (2) = σu 2 I D an R = σ 2 I n. Also, the moel unerlying the synthetic regression estimator (2.2) is equivalent to consiering (3.1) with u (1) h scalar an taken as a fixe effect, G (2) =0 an R = σ 2 I n. Moreover, the irect moifie regression estimator (2.1), can be obtaine consiering u (1) h an u (2) scalars an taken as a fixe effects an R = σ 2 I n Domains not represente in the sample Situations may arise where some omains are not represente in the sample. If no sample falls into small area, then the respective ranom effects u (2) may still be preicte if there is covariance between u (2) an at least one of the small area ranom effects represente in the sample u (2) e (e = 1,..., D; e ). We have then (3.8) ũ (2) = G (2) (, Z (2)s V 1 s ys X s β), where G (2), = col ne 0 1 e D [ G (2),e ] [ = E u (2) u (2) ] (2) an G,e = Σ(2) f(l l e ). In an extreme situation where the small area effects u (2) are not correlate with any other small area effect for a omain represente in the sample, i.e. when G (2),e = 0, e : n e 0, then ũ (2) =0. The estimator τ is then reuce to a form similar to a following synthetic estimator: (3.9) τ = τ y,,s + τ x,,r β + H h=1 τ x(1),h,rũ(1) h. It may be note that this estimator may be written in the same generic form (3.10) ˆτ = τ x, ˆβ + f ( y s X s ˆβ),
11 A Spatial Unit Level Moel for Small Area Estimation 165 where f is use to weight the regression resiuals. This form puts in evience that estimator (3.9) can be seen as a combine estimator where the weights in f allow a correction of the synthetic part of the estimator τ x, β through the preiction errors in the omain that is the target of inference, but also in other omains spatially correlate. When no correlation between omains is specifie, the correction factor epens only on the preiction errors in the target small area an the estimator is reuce to a similar form to the ataepenent combine estimators presente in Section 2. These characteristics seem to be particularly interesting when estimating in small omains, where the available sample size is small, since it borrows information from outsie the omain of stuy in orer to assist the estimation. Moreover, taking avantage of the potential spatial correlation of ata it is possible to avoi the reuction of the propose estimators to pure synthetic estimators even when the sample size in the omain is null. 4. MONTE CARLO SIMULATION STUDY 4.1. Generation of the pseuopopulation For the simulation a pseuopopulation is use. This population is obtaine from a real ata set containing the responses to the 1993 wave of the Agricultural Structure Survey. It is an agricultural survey conucte by the Portuguese Statistical Office in the perio between agricultural censuses. The responses for the variable total prouction of cereals were extracte an circumscribe to the NUTSII of Alentejo. The total sample size in this region is 7,060 an the population size 47,049. The esign for the Agricultural Structure Survey is base on stratifie sampling. The sample is first stratifie using the região agrária as the level for geographic stratification. A região agrária is an aministrative ivision use for agricultural purposes. In each região agrária a new stratification is establishe base on Use Agricultural Surface (UAS) classes. In the same região agrária some other strata are efine base on the value of other variables consiere weakly correlate with UAS. In Alentejo there are 19 strata. For simulation purposes a pseuopopulation is generate by replicating the agricultural establishments in the sample proportionally to the inverse of their inclusion probabilities. The sampling frame resulting from this replication inclues the value of prouction of cereals for each establishment in 1993, an the same value reporte for 1989 (year of the agricultural census). The prouction in 1989 is use as an auxiliary variable in the moels use in the simulation. Also, geographical coorinates associate with the centrois of freguesias were recore.
12 166 P.S. Coelho an L.N. Pereira This was the lowest level of aggregation for which geographical referencing was available. This means that geographical ifferentiation between establishments inclue in the same freguesia is not available. A freguesia is an aministrative ivision that segments the Alentejo into 284 subregions Description of the simulations Using the sampling frame corresponing to the pseuopopulation of agricultural establishments we have run a Monte Carlo simulation. The goal is to evaluate the esignbase properties of a set of alternative estimators. Note that the approach followe in this paper is to evaluate the properties an relative merits of the propose estimators through simulation. In fact, ue to the complexity of these estimators their esignbase properties (e.g. bias, variance) are impossible to obtain through analytical methos. Also, their moelbase properties woul be of limite interest from the point of view of a benchmark with alternative irect estimators use in this simulation whose properties only make sense to evaluate from the esignbase perspective. The target parameter is the total of the variable prouction of cereals at freguesia level. The number of simulations performe is 560. In each simulation a sample is rawn from the pseuopopulation U, using a stratifie esign similar to the one use in the Agricultural Structures Survey. The only ifference in relation to that survey esign is that the sample size by stratum was reuce to 30% of the original size (2,118 establishments). The goal is to simulate a framework similar to that survey, but with a smaller sample size, enabling the evaluation of the estimators behavior in critical situations where the omain sample size is very small (sometimes only a few units or even none). This sampling esign leas to a relative precision of 7.5% (for a 95% confiencelevel) in the estimation of the total of the interest variable at the population level, using the Horvitz Thompson estimator. The expecte sample sizes for the 284 omains of interest vary from 0.3 to 45.8 units Estimators The estimators analyze in the simulation are presente in this section. They are mainly implementations of the irect, synthetic an combine regression estimators presente in Sections 2 an 3. It shoul be note that all the regression estimators inclue the same auxiliary variables (associate with the fixe effects), allowing a fair comparison of their relative merits. In what follows, the notation a is use to represent the small area of region a, where the regions correspon to the level of aggregation of NUTSIII an the small area of interest to freguesia. Table 1 summarizes the estimators use in the simulation.
13 A Spatial Unit Level Moel for Small Area Estimation 167 Table 1: Estimators use in the simulation stuy. Estimator Description ˆτ a1 Horvitz Thompson estimator ˆτ a2 Direct moifie regression estimator (2.1) ˆτ a3 Dampene regression estimator (2.4) ˆτ a4 ˆτ a5 τ a6 Pure synthetic regression estimator (2.2) with fixe effects estimate ignoring the sampling esign Synthetic regression estimator where the sampling esign is explicitly consiere through the inclusion of a vector of esign variables E hi inicating the belonging of each establishment i to the strata h =1,..., H Dataepenent combine regression estimator base on the neste error unit level regression moel τ a7 τ a8 Dataepenent combine regression estimator, similar to τ a6 but incluing fixe strata effects β h0, h =1,..., H Dataepenent combine regression estimator, base on a moel inclue in the propose class of moels (3.1), with ranom small area effects presenting a spatial covariance structure following an isotropic exponential moel. The isotropic exponential moel use to represent spatial variability in ˆτ a8 was suggeste in the moel iagnosis phase. We have teste several structures (exponential, spherical, linear, loglinear an Gaussian), through the evaluation the significance of covariance parameters (using Wal tests) an information criteria (such as AIC an BIC). Among the structures that showe statistical significance we retaine the one that minimize the several information criteria. Although we have chosen the exponential moel, some of the other structures resulte in very similar ajustments. Also note that for the ataepenent regression estimators the variance components are estimate through REML metho. The only exception regars estimator τ a8, where the parameter c e was estimate a priori through the ajustment of an exponential semivariogram to an empirical semivariogram. Note that the estimators inclue in the simulation vary in nature: ˆτ a1 an ˆτ a2 are esignbase estimators, ˆτ a4 an ˆτ a5 are synthetic estimators, while the others can be classifie as combine estimators as escribe in previous sections. The inclue estimators also ifferentiate in the way the sampling esign is (or not) taking into account: in ˆτ a1, ˆτ a2 an ˆτ a3 the sampling esign information is taking into account using sampling weights, ˆτ a5, ˆτ a7 an ˆτ a8 inclue fixe strata effects, while in ˆτ a4 an ˆτ a6 the sampling esign information is ignore.
14 168 P.S. Coelho an L.N. Pereira 4.4. Precision an bias measures The estimators uner consieration are evaluate using a set of precision an bias measures. In what follows K represents the number of simulations, an ˆτ k the th small area estimate of the total obtaine from the simulation k (k = 1,..., K) Unconitional analysis Taking into account the high number of small areas in the population (284) an in orer to facilitate the presentation of the simulation results, the small areas are ivie into six groups. Each group g contains D g small areas. Table2 presents the efinition of each group an the number of small areas involve. Table 2: Small area groups in the simulation stuy. Group Expecte sample size Number of small areas [0; 2] 20 2 [2; 3.5] 43 3 [3.5; 5] 49 4 [5; 10] 87 5 [10; + ] 65 Groups 1 to 5 were efine accoring to the expecte sample size of the small areas. Group 0 inclues small areas for which the total of the interest variable is zero (freguesias where there is no cereal prouction) regarless their size. The goal is to separate these small areas from the other groups to prevent them from changing the conclusions regaring the relative merits of the estimators. The Monte Carlo relative error for the estimators expecte value is on average 8.0% in group 1 an varies between 3.4% an 4.1% in groups 2 to 5. For the unconitional analysis the following measures were consiere for each group g: D g Average absolute bias: AAB g = Dg 1 AB, where AB = K 1 K ˆτ j τ ; Average MSE: AMSE g = D 1 g D g =1 j=1 MSE, where MSE = K 1 K (ˆτ j τ ) 2 ; =1 j=1
15 A Spatial Unit Level Moel for Small Area Estimation 169 Average variance: AV g = D 1 g D g V, where V = K 1 K (ˆτ j ˆτ ) 2 ; =1 Average absolutebiasratio: AABR g =D 1 g D g j=1 ABR, where ABR =AB / V ; Average coverage rate for a esignbase D g 100(1 α) confience interval: ACR g = Dg 1 TC, =1 where TC = 100 R /K an R represents the number of simulations for which the confience interval ˆτ j ± t α/2 V contains the true parameter τ. = Conitional analysis A conitional analysis was also conucte using a set of precision an bias measures for each small area. The superscript (n ) inicates that the respective measure is conitione to the realize sample size, n, in small area. They are: Conitional relative bias: CRB (n ) K n = Kn 1 (ˆτ j τ ) / τ ; j=1 Conitional relative stanar error: CRSE (n ) = Conitional variation coefficient: CVC (n ) = where V (n ) K n Kn 1 j=1 V (n ) / τ, (ˆτ j τ ) 2/ τ ; = K 1 (ˆτ j ˆτ ) 2 is the conitional variance ; n K n Conitional bias ratio: CBR (n ) = B (n / ) V n, where B (n K ) = Kn 1 n (ˆτ j τ ) is the conitional bias; Coverage rate of the conitional esignbase confience interval: CR (n ) = 100 R (n ) /K n, j=1 j=1 where R (n ) represents the number of simulations for which the confience interval ˆτ j ± t α/2 contains the true parameter τ. V (n )
16 170 P.S. Coelho an L.N. Pereira 4.5. Results Unconitional analysis Table 3 summarizes the unconitional results of the simulation stuy. The values for absolute bias, variance an MSE are presente relatively to the respective value associate with ˆτ a2. Table 3: Unconitional results. Group ˆτ a1 ˆτ a2 ˆτ a3 ˆτ a4 ˆτ a5 τ a6 τ a7 τ a8 Absolute bias Variance MSE Bias ratio Coverage Rate
17 A Spatial Unit Level Moel for Small Area Estimation 171 As expecte the two esignbase estimators ˆτ a1 an ˆτ a2 are approximately unbiase, even for very small areas. For these estimators the average coverage rate is very near the nominal confience level (95%). Nevertheless, especially in the smallest omains, they show variance an MSE substantially higher than those observe for the synthetic an combine estimators. It is also to be note that ˆτ a2 brings significant precision gains when compare with the Horvitz Thompson estimator. On the other han, the two synthetic estimators show very ifferent behavior. ˆτ a4, which can be viewe as a pure synthetic estimator shows isastrous behavior both in terms of bias an precision. These results are clear evience of the effects of ignoring an informative sampling esign. On the other han, the synthetic estimator with fixe strata effects, ˆτ a5, shows significant precision gains when compare to the irect regression estimator. These precision gains show a tenency to ecrease as the expecte sample size in the small areas increases. Its major rawback is relate to the high bias, which originates average bias ratios that are always above 2.6, compromising the construction of esignbase confience intervals. In fact the average coverage rate for this estimator is always below 0.46, an in many cases near The combine regression estimator with samplesize epenent weights, ˆτ a3, shows a systematic precision gain when compare to the irect regression estimator (with the exception of group 0, the ratio between the average MSE of the two estimators is between 0.79 an 0.87). Nevertheless, these gains are always moerate an substantially lower than the ones observe for the synthetic regression estimator. This estimator also shows a very goo behavior in what regars bias. In fact, although having an absolute bias higher than those observe for the irect estimators, the average bias ratio is always lower than 0.2, which originates an average coverage rate very near the nominal confience level. With regar to the combine estimators with ataepenent weights it can once again be observe that isregar of the sampling esign, as in estimator τ a6, prouces unesirable properties both in terms of bias an precision. In fact, τ a6 systematically shows higher MSE than the irect regression estimator (with the exception of group 1). It also exhibits ramatic biases (of the same magnitue as the synthetic estimator ˆτ a4 ) an bias ratios that on average are situate between 1.76 an The combine estimators τ a7 an τ a8, which explicitly consier strata effects, show very ifferent behavior. These estimators show average MSE that for the smallest small areas (groups 0 to 3) is near those observe for the best synthetic estimator, while allowing significant reuction in bias an bias ratio. In particular, τ a7 base on a neste error moel with fixe strata effects reveals important precision gains when compare to the irect regression estimator, ˆτ a2, or the samplesize epenent regression estimator, ˆτ a3, in groups 0 to 3.
18 172 P.S. Coelho an L.N. Pereira It is to be note that for groups 4 an 5 (corresponing to expecte sample sizes higher than 5 units) τ a7 shows some precision loss regaring the irect regression estimator, which is particularly important in group 5. It is also significant that, in the groups 0 to 3, τ a7 exhibits a MSE that is similar or even smaller than that associate with the synthetic estimator ˆτ a5. In these groups the increase in variance in τ a7 is more than compensate by the reuction in bias. In what regars bias measures, the estimator τ a7 shows a behavior that is situate between those recore for the irect an the synthetic estimators. The average absolute bias ratios vary from 0.40 to 2.58 (increasing with the reuction of the expecte sample size) an are strikingly lower than the ones associate with the synthetic estimators (varying between 15% an 50% of those obtaine for the best synthetic estimator). This results in average coverage rates for a esignbase confience interval that are substantially higher than those observe for the synthetic estimators. The estimator τ a8 consiers a spatial covariance base on an isotropic exponential structure at the small area level. For all small area groups (with the exception of group 0) it shows a small loss of precision when compare to τ a7, but still allows important precision gains regaring the irect estimators an the samplesize epenent regression estimator, ˆτ a3, in groups 0 to 3. It is worth noting that this ecline in precision is mainly inuce by an increase in variance, since τ a8 shows average absolute bias that is very near or even smaller than for τ a7 (mainly in the smaller areas). The bias ratios for τ a8 are, for groups 0 to 4, substantially smaller than those observe for τ a7, varying now from 0.42 to The reuction in the bias ratio tens to iminish with the increase in the expecte sample size, resulting that in group 5 the bias ratio of the two estimators is similar. In fact it is in smaller areas that τ a7 approximates more closely a synthetic estimator, allowing the aitional sample information use in τ a8 (from other spatially correlate small areas) to contribute to bias reuction. Between the combine estimators that allow precision gains in small areas groups 0 to 3, τ a8 is the one that shows the best behavior in terms of average bias ratio, which varies from 16% to 37% of those obtaine for the best synthetic estimator Conitional analysis Figure 1 an Table 4 summarize the simulation s conitional results for one small area in the stuy. Consiering the large number of small areas in the stuy (284), these ata are only intene to illustrate typical results associate with one of the smallest areas. The results refer to a small area with expecte sample size of 4.2 units, thus belonging to group 3.
19 A Spatial Unit Level Moel for Small Area Estimation 173 Figure 1: Conitional results. Table 4: Unconitional coverage rates. Sample size ˆτ a1 ˆτ a2 ˆτ a3 ˆτ a4 ˆτ a5 τ a6 τ a7 τ a The two esignbase estimators ˆτ a1 an ˆτ a2 show ba conitional properties, namely in what regars bias. In fact, both estimators show important conitional biases an bias ratios when the effective sample size eparts from the expecte sample size. This phenomenon is particularly notable in the Horvitz Thompson estimator. This bias tens to be negative for effective sample sizes that are smaller than expecte an positive for expecte sample sizes that are larger than expecte. Also, when effective sample size is smaller than the expecte sample size, the conitional variation coefficients ten to show a rising pattern with the increase in the effective sample size. The combine result of this bias an variance behavior is a conitional relative stanar error that for both estimators tens to increase with the effective sample size (for sample sizes above the expecte). When the effective sample size is significantly smaller or greater than the expecte sample size, these estimators (an mainly ˆτ a1 ) show a significant egraation in precision. These patterns are particularly notable in the small areas with a very small sample size.
20 174 P.S. Coelho an L.N. Pereira On the other han, although the synthetic estimators show very high bias an bias ratios (resulting in very low conitional coverage rates for a esignbase confience interval), they are seen to be approximately constant an therefore inepenent of the effective sample size for each small area. The conitional variation coefficient is clearly constant showing inepenence from the sample size in the small area. The combine result of these patterns is a relative stanar error which is also approximately invariant with the effective sample size. From this conitional point of view, the synthetic regression estimator can still be consiere one of the most precise estimators. For effective sample sizes that are smaller than the expecte sample size the combine regression estimator with samplesize epenent weights, τ a3, shows important conitional biases an bias ratios that for significant epartures are similar to those observe for the synthetic estimators. For sample sizes that are greater than expecte τ a3 tens to show a behavior similar to the irect regression estimator ˆτ a2. Therefore, the resulting conitional coverage rates for a esignbase confience interval also show a behavior similar to a synthetic estimator for sample sizes that are smaller than expecte an similar to the irect regression estimator when they are higher than expecte. The relative stanar error also tens to show the ba property observe for the irect estimator, characterize by an increase with the effective sample size, mainly for the smallest areas. The combine estimators τ a7 an τ a8 show interesting conitional properties as they show a mixe behavior between the irect regression estimator ˆτ a2 an the synthetic regression estimator ˆτ a5. This behavior is characterize by a significant resistance of precision an bias to epartures between the effective sample size an the expecte sample size. It can be observe that τ a7 shows a conitional bias an bias ratio that are approximately constant, although with a slight tenency to increase with the reuction of the effective sample size. As to bias, this estimator shows a clear avantage when compare to the synthetic estimators an even when compare to the irect estimators an the samplesize epenent combine estimator, particularly when the sample size eparts from the expecte sample size. This results in conitional coverage rates which in extreme situations are closer to the confience level than those associate with some esignbase estimators. The conitional variance only shows a very slight tenency to rise with an increase in the effective sample size, resulting in a conitional relative stanar error that is approximately constant. From the precision point of view it can be seen that this estimator is still competitive with the best synthetic estimator an maintains important precision gains when compare to the irect estimators an the samplesize epenent combine estimator (especially when the effective an expecte sample sizes are ifferent).
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