n. 12/2008 A Balanced Sampling Approach for Multi-way Stratification Designs for Small Area Estimation P.D. Falorsi e P. Righi

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1 n. /008 A Balance Sampling Appoach fo Multi-way Statification Designs fo Small Aea Estimation P.D. Falosi e P. Righi

2 Le collane esistenti pesso l'istat - Rivista i Statistica Ufficiale, Contibuti ISTAT e Documenti ISTAT - costituiscono stumenti pe pomuovee e valoizzae l'attività i iceca e pe iffonee i isultati egli stui svolti, in mateia i statistica ufficiale, all'inteno ell' ISTAT, el SISTAN, o a stuiosi esteni. La Rivista i Statistica Ufficiale accoglie lavoi che hanno come oggetto la misuazione ei fenomeni economici, sociali, emogafici e ambientali, la costuzione i sistemi infomativi e i inicatoi, le questioni i natua metoologica, tecnologica o istituzionale connesse al funzionamento ei sistemi statistici e al peseguimento ei fini ella statistica ufficiale. I lavoi pubblicati in Contibuti ISTAT sono iffusi allo scopo i stimolae il ibattito intono ai isultati peliminai i iceca in coso. I Documenti ISTAT foniscono inicazioni su linee, pogessi e miglioamenti i pootto e i pocesso che caatteizzano l'attività ell'istituto. Il Comitato i eazione esamina le poposte i lavoi a pubblicae nelle te collane sopa inicate. Quelli pubblicati nei Contibuti ISTAT e nei Documenti ISTAT sono valutati peventivamente ai iigenti ell'istituto, mente i lavoi pubblicati nella Rivista i Statistica Ufficiale sono suboinati al giuizio i efeee esteni. Diettoe esponsabile ella Rivista i Statistica Ufficiale: Patizia Cacioli Comitato i Reazione elle Collane Scientifiche ell Istituto Nazionale i Statistica Cooinatoe: Giulio Bacaoli Membi: Coao C. Abbate Rossana Balestino Giovanni A. Babiei Giovanna Bellitti Riccao Cabini Giuliana Coccia Fabio Cescenzi Cala De Angelis Calo M. De Gegoio Gaetano Fazio Saveio Gazzelloni Antonio Lollobigia Susanna Mantegazza Luisa Picozzi Valeio Tea Abami Robeto Tomei Leonello Tonti Neeo Zamao Segeteia: Gabiella Centi, Calo Deli e Antonio Tobia Responsabili oganizzativi pe la Rivista i Statistica Ufficiale: Giovanni Sei e Calo Deli Responsabili oganizzativi pe i Contibuti ISTAT e i Documenti ISTAT: Giovanni Sei e Antonio Tobia ISTAT Pouzione libaia e cento stampa

3 n. /008 A Balance Sampling Appoach fo Multi-way Statification Designs fo Small Aea Estimation P.D. Falosi(*) e P. Righi(*) (*) ISTAT Sevizio Pogettazione e suppoto metoologico nei pocessi i pouzione I lavoi pubblicati iflettono esclusivamente le opinioni egli autoi e non impegnano la esponsabilità ell'istituto

4 Contibuti e Documenti Istat 008 Istituto Nazionale i Statistica Sevizio Pouzione Eitoiale Pouzione libaia e cento stampa: Cala Pecoaio Via Tuscolana, Roma

5 ABSTRACT The pesent wo illustates a sampling stategy useful fo obtaining planne sample size fo omains belonging to iffeent patitions of the population an in oe to guaantee that the sampling eos of omain estimates ae lowe than given theshols. The sampling stategy that coves the multivaiate-multiomain case is useful when the oveall sample size is boune an consequently the stana solution of using a statifie sample with the stata given by cossclassification of vaiables efining the iffeent patitions is not feasible since the numbe of stata is lage than the oveall sample size. The popose sampling stategy is base on the use of balance sampling selection technique an on a geg-type estimation. The main avantages of the solution is the computational feasibility which allows one to easily implement an oveall small aea stategy consieing jointly the esign an estimation phase an impoving the efficiency of the iect omain estimatos. An empiical simulation on eal population ata an iffeent omain estimatos shows the empiical popeties of the examine sample stategy. Key wos: Planning Sampling Size of Small Domains, Contolle Selection, Balance Sampling.. Intouction The small aea poblem is usually consiee to be teate via estimation. Howeve, if the omain inicato vaiables ae available fo each unit in the population thee ae oppotunities to be exploite at the suvey esign stage. This conition is usually met in the business suvey context whee the omain inicato vaiables ae available in the business egiste. As note by Singh et al. (994), thee is a nee to evelop an oveall stategy that eals with small aea poblems, involving both planning sample esign an estimation aspects. In this famewo, it is cucial to contol the sample size fo each omain of inteest, so that the omain is teate as a planne omain, at esign stage, fo which it is possible to pouce iect estimates with a pefixe level of pecision. In geneal, with a esign-base appoach to the infeence, the pesence of sample units in each omain allows one to compute omain estimates although not always eliably. Futhemoe, in the moelbase o moel-assiste appoach, the pesence of sample units in each estimation omain allows one to use moels with specific small aea effects, giving moe accuate estimates of the paametes of inteest at small aea level (Lehtonen, et al., 003). In fact, when the aim of the suvey is to pouce estimates fo two o moe patitions of the population, a stana solution to obtain planne sample sizes fo the omains of inteest is to use a statifie sample in which stata ae ientifie by coss-classification of vaiables efining the iffeent patitions. In the following, this esign will be enote as coss-classification esign. In many pactical situations, howeve the coss-classification esign is unfeasible since it nees the selection of at least a numbe of sampling units as lage as the pouct of the numbe of categoies of the statification vaiables.

6 In oe to explain the poblem, let us consie the population of 65 schools (Cochan, 977, pag. 4) epote in table.. Let us suppose that the paametes of inteest ae the totals of a Y vaiable - elate to school - sepaately fo (i) size of city (5 categoies: I,II,III,IV,V) an (ii) Expenitue pe pupil (4 categoies: A,B,C,D). Let us note the following: (i) the omains of inteest ae 9=5+4; (ii) the poblem efines two istinct patitions of the population, inee the size of city epesents a patition of the taget population in 5 non-ovelapping omains, an the expenitue pe pupil is an altenative patition in 4 omains. The stana coss-classification esign efines 0=5x4 stata by cossing the categoies of the omains of the two patitions; in each statum shoul be selecte at least one school (o two schools fo estimating the sampling vaiance without bias) an, consequently, accoing to this esign, the sample size shoul be of 0 (o 40 schools) at least. If the bugetay constaints limit the sample size to 0 schools the coss-classification esign becomes unfeasible. Table.. Numbe of schools by Size of City an expenitue pe pupil Size of Expenitue pe pupil Sample Totals City A B C D size I II III IV V Totals Sample size Cochan (977, pag. 4) The above bacgoun is typical of the business suvey context. Inee, the Euopean Council Regulation n 58/97 on Stuctual Business Statistics establishes that the paametes of inteest efe to estimation omains efine by thee iffeent patition subsets of the population of entepises. As we may note by table., in Italy the total numbe of estimation omains is,8; while the numbe of non-empty stata of the coss-classification esign is lage than 37,000. Table.. Numbe of omains of the Italian Stuctual Business Statistics Suvey by patition Patitions Numbe of omains Economic activity class (4-igits of the NACE ev. classification) 465 Economic activity goup (3-igits of the NACE ev. classification) by Size class () 395 Economic activity ivision (-igits of the NACE ev. classification) by Region () 96 Total numbe of estimation omains,8 () Size classes ae efine in tems of numbe of pesons employe. () Regions ae incluing autonomous povinces. In oe to ovecome some poblems of coss-classification esigns, an easy stategy is to op one o moe statifying vaiables o to goup some of the categoies. Nevetheless, some planne omains become unplanne an some of them can have small o null sample size. Many methos have been popose in the liteatue to eep une contol the sample size in all the categoies of the statifying vaiables without using coss-classification esign. These methos ae geneally efee to as multi-way statification techniques, an have been evelope une two main appoaches: (i) Latin Squaes o Latin Lattices schemes (Byant et al., 960; Jessen, 970); (ii) contolle ouning poblems via linea pogamming (Causey et al., 985; Sitte an Sinne, 994). The seminal pape of Byant et al. (960) suggests allocating the units in the sample by means of a two-way Latin Squae table anomly selecte an two estimatos of the paamete of inteest ae popose. The metho has some awbacs that limit the application in eal suvey 4

7 contexts. Fo example, it implies that the expecte sample counts in each statum isplay inepenence between the ows an columns of the two-way table; futhemoe it is not possible to implement the poceue when thee is no population in one o moe coss-classification stata. In oe to solve these poblems, Jessen (970) poposes two appoaches, both faily complicate to implement an not always leaing to a solution (Causey et al., 985). As concens the methos base on linea pogamming ae concene, Causey et al. (985) consie the contolle multi-way statification as a ouning poblem solve by means of tanspotation theoy. The metho may not have a solution in case of thee o moe statification vaiables. Following the linea pogamming appoach popose by Rao an Nigam (990, 99), Sitte an Sinne (994) suggest a metho base on linea pogamming moe flexible in iffeent situations than the metho popose by Causey et al. (985) an some futhe computational simplification of Sitte an Sinne metho have been suggeste by Lu an Sitte (00). Nevetheless, the main weaness of the linea pogamming appoach is the computational complexity. As a consequence, the awbacs of both appoaches have limite the use of multi-way statification techniques as a stana solution fo planning the suvey sampling esigns. The sampling stategy consiee in this pape oes not suffe fom the isavantages of the above mentione methos an allows one the contol of the sample sizes fo omains of inteest, which ae efine by iffeent patitions of the efeence population. Futhemoe it guaantees that the sampling eos of omain estimates ae lowe than the given theshols. The popose sampling stategy is base on the use of both a balance sampling selection technique (Deville, Tillé, 004) an a geg-type estimation (Lehtonen, et al., 003). As shown in the stuy on empiical ata, the main avantages of this solution is the computational feasibility an the efficiency, that is the sampling eos fo multiomain-multivaiate case ae easonably close to those efine by the optimal univaiate solutions. This allows one to faily implement an oveall small-aea stategy consieing jointly the esign an estimation phase an impoving the efficiency of the iect omain estimatos. It is wothwhile to note that, if a given population patition efines a too lage numbe of omains, it coul happen that the buget constaints oblige to efine a too lage pefixe sampling eos of the iect estimatos of the omains of the patition; in this situation, it coul be necessay to aopt an iniect small-aea estimato, in oe to contol the mean squae eos of patition omain estimates. Howeve, as biefly setche in section 5, the iniect estimation is stengthene by the use of an impove iect estimato. The pape is oganise as follows. Section states the poblem an intouces the essential notation; moeove it escibes the oveall sampling stategy. Section 3 shows an iteative poceue that efines the inclusion pobabilities an the coesponing planne omain sample sizes solving a non linea poblem whee the objective function is the minimization of the oveall sample size guaanteeing the sampling eos of omain estimates to be lowe than given theshols. Sections 4 an 5 illustate two extensions of the sampling stategy. In section 4 the case in which the vaiance citeion is epesente by the anticipate vaiance is stuie. An extension to the case of a simple small aea iniect estimato is pesente in section 5. The main esults of two empiical stuies conucte both on a eal population of Italian entepises an on a simulate population ae shown in section 6. Finally some bief conclusions ae uneline in section 7.. The Sampling Stategy.. Paametes of inteest In oe to efine fomally the poblem, let us enote with U a population of N elements an with b a specific patition of U (b=,, B) in which b-th patition efines M iffeent non ovelapping b 5

8 omains, U (=,, M b ), of size N being N = N M b = B an, finally let M b = Q the oveall numbe of omains. In the table. example, b= when consieing the patition of the population by size of the city an b=b= when consieing the patition by expenitue pe pupil, being M = 5 an M = 4. Continuing the example, U iniviuates the omain, of N = 6 schools, having the I-th city size; futhemoe U 4 iniviuates the omain, of N 4 = 38 schools, having the D-th expenitue pe pupil. Let y, an δ enote espectively the value of the Y ( =,,R) vaiable of inteest in the -th population unit an the omain membeship inicato, being δ = if U an δ = 0, othewise. Let us suppose that the δ values ae nown fo each unit in the population. The paametes of inteest ae the M = R Q omains totals t = y = y, U U b=, δ ( =,,R ; b=,, B; =,, M b ). (..) The expession (..) efines a multivaiate-multiomain poblem since thee ae R vaiables of inteest (multivaiate aspect) an Q > omains (multiomain aspect)... A concise esciption of the sampling stategy Let us suppose that, in oe to estimate the t paametes, a sample s of fixe size n is selecte fom population U, with inclusion pobabilities π ( U ). Let s = s U enote the sample M b of n units belonging to the U omain (with n = n ), being n = U U = λ = π, (..) with λ = if s an λ = 0 othewise. The sample is selecte by a multi-way statification technique evelope une the balance sampling famewo guaanteeing that the selecte sample espects the following balancing equations whee tˆ ht = (..) z, t z tˆ z, ht = z λ a enote the Hovitz-Thompson estimates of = U tz z, being z a vecto of auxiliay vaiables nown fo each population unit an a = / π. A suitable specification of the z vectos can assue that the ealize sample sizes, n, ae equal to fixe quantities nown in avance, as escibe in section.3. The estimates of t, enote with t ˆ, geg, ae obtaine with the moifie geg estimato (Rao, 003, page 0), given by (see section.5): U 6

9 whee: ˆ (..3) t, geg = w y, s = a δ + ( t tˆ x x, ht ) ( ax x / c ) a x s w / c enote the sampling weights, x inicates a vecto of auxiliay vaiables, c is a nown constant, being t x = x an tˆ x, ht = xa. The estimato (..3), also nown as suvey U s egession estimato (Battese, Hate an Fulle, 988), may be eive une the following woing supepopulation moel x (..4) y, = β + ε, β enotes an unnown vecto of fixe egession paametes an whee esiual. The moel expectation, E m, an moel vaiances, m ε, is the anom V, ae espectively given by Em( ε ) = 0 ; Vm( ε, ) = c σ ; Em ( ε,, ε, i ) = 0 if i. The appoximate sampling vaiance of the moifie geg estimato une balance sampling is: being V p N ( tˆ ˆ z η,, (..5), geg t, ht = tz ) = N Q U π η, ε, z = z B z, ε B z, ε fo fo U U, with B z, ε = z z z ε, δ, U π U π whee U is the subset of U complementay to U. A poof of (..5) is given in section.5. The inclusion pobabilities, π, an the omain sample sizes, n, ae etemine with a poceue which attempts to minimize the oveall sample size, n, guaanteeing that the sampling vaiances V ( ˆ ˆ p t, geg t z, ht = tz ) ae lowe than pefixe level of pecision theshols, V : V ˆ p( t, geg t, ht tz ˆ z = ) V (b=,, B; =,, M b ; =,,R) The technical etails ae escibe in section 3. 7

10 .3. The Balance sampling fo maginal statification Multi-way statification esigns can be teate in the context of the balance sampling. The efinition of a balance sample epens on the assume infeential famewo. In the moel base appoach, a sample is efine as balance on a set of auxiliay vaiables if thee is the equality between the sample an the nown population means of the auxiliay vaiables (Royall an Heson, 973; Valliant et al., 000). Following the esign base (o moel assiste appoach) consiee in this pape, a sample is balance when the Hovitz-Thompson estimates of the auxiliay vaiables totals ae equal to thei nown population totals (Deville an Tillé, 004). Fo efining the balance sampling in the esign o moel assiste appoach, let us intouce the geneal efinition of sampling esign as a pobability istibution p(. ) on the set S of all the subset s of the population U such that p( s) =, whee p(s) is the pobability of the sample s to be s S awn. Each set s may be epesente by the outcome λ = ( λ,, λ,, λ N ) of a vecto of N anom vaiables. Let π = ( π,, π,, π N ) be the vecto of inclusion pobabilities, whee π = Ep( λ) = s p( s) λ, being E ) S p( the expecte value ove epeate sampling. Let z = ( z,,z h,, z Q ) be a vecto of Q auxiliay vaiables available fo each population unit. The sampling esign p(s) with inclusion pobabilities π is sai to be balance with espect to the Q auxiliay vaiables if an only if it satisfies the balancing equations given by (..) fo all s S such that p(s)>0. Let us suppose that a vecto of inclusion pobabilities π, consistent with the maginal sampling istibutions n (b=,, B; =,, M b ), is available, that is n = π (b=,, B; =,, M b ). (.3.) U Multi-way statification esign epesents a special case of balance esign whee fo unit the auxiliay vaiable vecto is given by z b= b= B Q = ( 0,, π,,0,,0,, π,,0) = π ( δ,, δ,, BM δ B ). (.3.) The expession (.3.) efines the z as vectos of (Q-B) zeos an with B enties equal to π in the places inicating the omains which the unit belongs to. When efining the z vecto as (.3.), if conition (.3.) hols, the selection of sample satisfying the system of balancing equations (..), ( zλ ) / π = z, guaantees that the n values ae non anom quantities. U U Refeing to the -th omain, the left han-sie of the balancing equation (..) is U = = U s ( π δ λ ) / π λ = n, while the ight han-sie is 8

11 U π δ = π = n. U One elevant awbac of balance sampling has always been implementing a geneal poceue giving a multivaiate balance anom sample (see Valliant et al., 000). Deville an Tillé (004) popose the cube metho that allows one the selection of balance (o appoximately balance) samples fo a lage set of auxiliay vaiables an with espect to iffeent vectos of inclusion pobabilities. In paticula, Deville an Tillé (000) show that with specification (.3.) of the z vectos, the balancing equations (.3.3) can be exactly satisfie. A fee SAS (vesion 9) softwae coe fo the selection of balance samples fo lage ata sets may be ownloae in the website The moifie iect geg estimato Following Lehtonen et al. (003), the estimato (..3), may be expesse une the geneal fom t ˆ = ~ y + a ( y ~,, y, ) (.4.) geg, U s whee ~ y, enotes the peiction of y, une the assume supepopulation moel. The peictions{ ~ y, ; U} iffe fom one moel specification to anothe, epening on the functional fom an fom the choice of the auxiliay vaiables. The estimato (..3), is eive une the woing supepopulation moel (..4). The peictions ~ ae then obtaine by ~ y x β ˆ, =, (.4.) y, being ˆ ( a / c ) ( y, a / c ) s β = x x x. (.4.3) s Let us obseve that the linea moel (..4) allows us to efine the estimato only nowing the omain totals of the auxiliay infomation an the x values fo the sampling units. Howeve, nowing the x values fo evey U, it is possible to buil an estimatos with moe efficient peictions ~ obtaine by genealise linea moels (Lehtonen an Veijanen, 998) o non y, paametic egession techniques (Montanai an Ranalli, 003). As note by Rao (003, pag. 0) the estimato (..3) is appoximately esign unbiase as the oveall sample size inceases, even if the omain sample size n is small; futhemoe it may also be viewe as a calibation estimato (Singh an Mian, 995) with weights w minimizing a chisquae istance s c ( a δ w ) / a subject to the constaints w x = tx. s 9

12 The sum of the ˆ estimates ove all the omains of a patitions is benchmae to the usual t, geg geg estimate of the total, Mb = tˆ = tˆ, geg, geg, being ˆ t, geg = y, a [+ ( x xa ) ( ( xx a / c )) x / c ]. s U s s.5. Sampling vaiances In oe to eive the expession of the vaiance (..5), consie the esult given in expession (7) of Deville an Tillé (005), which taes into account the Hovitz-Thompson estimato t ˆ, ht = y, a of the total t = y,. This esult states that, une balance sampling, a goo s U appoximation of the sampling vaiance of the ˆ estimato is given by t, ht whee V tˆ t ˆ = t ) = V ( tˆ + ( t tˆ ) B ) = V (ˆ t tˆ B ) = p (, ht z, ht z p, ht z z, ht z, y p, ht z, ht z, y = V N p ( a ( y, z Bz, y )) ( y, z Bz, y ) s N Q U π, (.5.) B z, y = zz z y,. (.5.) U π U π Let us consie, now, the linea appoximation, * t ˆ, geg which may be obtaine accoing to Sänal et al. (99, pages ), of the geg estimato, the eivation of tˆ ˆ*, geg, geg t = U x β + a ε = x β + a ε δ. s, U s, On the basis of expessions (.5.) an (.5.), it is possible to eive the following esult V ( ˆ * p t, geg z, ht z p, geg z, ht z ˆt = t ) V ( tˆ ˆ t = t ) = = V p x β + a ( ε,, ht z U s p ( a ε, δ ˆ tz, ht = tz ) = s p ( a ε + ( ˆ, δ tz, ht tz, ht ) Bz, ε s p ( a ( ε, δ z B z, ε ) = s = V = V = V = V p s δ ˆ t z = t, ) N Q U π ) = ) = N ( a η η,. 0

13 The appoximate sampling vaiance of tˆ geg epens on the esiuals of the whole set of units, because of balance selection. Theefoe, the units not belonging to U have an influence on the sampling vaiance of the estimato. Let us examine now the univaiate uniomain case an assume that the suvey has an unique taget paamete, t. Let us suppose futhemoe that the selecte sample espects the balancing equations, ˆ t z,ht = tz, being fixe the oveall sample size n. It is tivial to emonstate that, in this sampling context, each unit coul be selecte with ( Q R ) iffeent optimal inclusion pobabilities, π& &, (b=,, B; =,, M b ; =,, R) π &. (.5.3) = π, = n η, η, i i U If the balance sample is selecte using the pobabilities V tˆ ˆt = t ) eaches its minimum value, p ( *, geg z, ht z V p π& &, * V n, expesse by, the appoximate vaiance * ( tˆ, geg ˆt z, ht = t z ) V n = η, η,. (.5.4) n U U Let us finally uneline that in Tillé an Fave (005) is given a citeion fo obtaining a peiction ˆη, of the η, values, that may be use in epeate sampling contexts. 3. Sampling algoithms fo the etemination of the sample sizes The inclusion pobabilities π an the eive omain sample sizes, n = π, ae obtaine U with a two phase poceue: (i) in the fist phase, enote, as optimization, the peliminay inclusion pobabilities, π, ae etemine solving a minimum constaine poblem; (ii) in the secon phase, enote as calibation, the inclusion pobabilities, π, ae obtaine as a slight moification of the π ; the calibation poblem is implemente fo assuing that the omain sample sizes n ae integes. As illustate in the following, the π values may be expesse as implicit functions of the unnown esiuals η,. But, in eal suvey context, the etemination of the inclusion pobabilities π may be one using the peictions η instea of η. This is a geneal poblem concening the phase of planning the sampling esigns, because the vaiances ae geneally unnown quantities that may be suitably estimate. In epeate suvey contexts the effect of using the estimates ˆ η, as a eplacement fo η, may be teste by computing the sampling vaiances afte the ata collection phase. The empiical esults may then be use fo intoucing pope ajustments in planning the next suvey esign. Howeve, as illustate in empiical analysis of section 6, the popose stategy seems to be efficient an sufficiently obust with espect to small epatues of ieal conitions. ˆ,,

14 3.. Optimization The inclusion pobabilities π can be efine as solution of the following non linea pogamming poblem with N unnowns, π, an ( N + Q R) constaints Min N N Q 0 < π U π U π η, ( =,, N ) V ( b =,, B; =,, M b. (3..) ; =,, R) A numeical solution to (3..) may be eive consieing the algoithms evelope fo the multivaiate allocation in statifie suveys. Such algoithms allow one to fin the unnown values v h > 0 (h=, ) which epesent the solution of the following non linea poblem Min ν h h une the constaints Ah ν h A, whee A h an A (=,, ) ae nown positive quantities. h Bethel (989) invoes the Khun-Tuce theoem to show that thee exists a solution to the above poblem. He escibes a simple algoithm an iscusses its convegence popeties. Chomy (987) evelops an algoithm, which is suitable fo automate speasheets, but oes not pove his algoithm always conveges. A slight moification of the Chomy s algoithm able to solve the poblem (3..) guaanteeing the inequalities 0 < π ( =,, N) ae especte is escibe heein in the following. Afte the Initialization, the algoithm fins the π values by iteating the two actions of Calculus an Chec. Initialization Let τ ( τ = 0,,,..) enote the geneic iteation. At initial iteation ( τ = 0 ), set = τ γ (=,,N). Calculus The geneic iteation ( τ =,, ) evelops the Chomy s algoithm an consists of a sequence of steps. The inex u (u=0,, ) - afte a comma on the ight of the iteation inex, τ - enotes the geneic step. τ, u At initial step ( u = 0 ), set = (fo b=,,b; =,, M b ; =,,R) an calculate N φ τ τ V 0 = η, γ. N Q U At subsequent steps (u=,, ), calculate the τ,u π values using the following equation

15 τ, u π τ τ N = ( γ ) + γ N Q B Mb R τ, u b= = = φ η, /. (3..) Calculate, futhemoe N = N Q τ, u V u η τ, U π, τ γ τ, u τ, u τ, an V = V + V0. (3..3) If the following two conitions ae especte fo all b=,,b; =,, M b ; =,,R: τ, u V V τ, u τ (, u = an φ V V ) 0, (3..4) then the action of Calculus stops an the inclusion pobabilities equation (3..). Othewise, the upate quantities τ, u+ φ τ π ae compute ae those calculate in τ, u+ φ = τ, u φ τ, u τ, u τ [ V /( V V )] (3..5) an the equations (3..) an (3..3) ae calculate at u+, ove an ove again with eplacing τ, u φ until conitions (3..4) ae especte. τ, u+ φ Chec τ π If the conition is tue fo all, then the algoithm stops an the τ τ to π = π. Othewise the γ values ae upate as π values ae set equal τ + γ if = 0 if τ τ π π >. (3..6) τ + The calculus is iteate at τ + with γ eplacing γ. The SAS maco that allows one to solve the poblem (3..) has been evelope by the authos of this pape an may be elease on eman. τ 3.. Calibation The quantities n ae efine, fist, by ouning the esults of the Q sums π (b=,,b; U =,, M ). Sometimes a futhe ata manipulation coul be necessay in oe to assue the b 3

16 conition M b M b n = nb l = = l= of the following calibation poblem Min G ( π ; π ) U π = n U π = n b=,,b; =,, M U n, fo each b b. The pobabilities π ae then obtaine as solution b -, (3..) whee, G ( π ; π ) is a istance function between π an π. Note that (3..) may be solve by the well nown IPF (Bishop et al., 975) o GIPF (Dysta (985); Dysta an Wollan, (987)) poceues. The logaithmic istance function G ( π ; π ) = π ln( π / π ) ( π + π ) avois to efine the π pobabilities lowe than 0, while GIPF pevents to obtain π values lage than. 4. The anticipate vaiance A fequently use citeion fo planning the sampling stategies is that of contolling the anticipate vaiance, which may be efine as: AV tˆ tˆ z = t ) = (, geg, ht z E m E p ( ˆ ˆ t, geg t t z, ht = tz ). (4.) The following esult may be eive une the assumptions of the moel (..4) an using the esults given in section (.5): AV ( ˆ ˆ * t, geg t z, ht z m p, geg t z, ht z = t ) E V ( tˆ ˆ = t ) = N N = Em + ( ε z Bz, ε ) ( z B z, ε ) =. N Q U π N Q U b π N N = Em ε + + N Q, ( z B z, ε U π N Q ) U π N ε, z Bz, ε = N Q U π N Q U π being σ c ( g ) + σ gjc j if U a j( ) U, = η, gj c othewise j U N a η, (4.) 4

17 j N ) = U U U U U whee: ( g,, g,, g = z ( Z Ω Z ) Z Ω matix of Em E m g, { } N z, Ω = { / π } N Z U = col z = enotes the ( N Q ) U iag =. The (4.) has been obtaine une the following two esults ( ZU ΩU ZU ) ZU ΩU V ΩU ZU ( ZU ΩU ZU ) z ( z Bz, ε ) = σ z = = σ g V g = σ gj c, j U ( ε, z Bz, ) ε = ( Z Ω Z ) Z Ω ε I ( ε δ,, ε δ,, ε ) ) = = z U U U U U Em (, N,,, N δ = g c δ, σ whee V = iag { c δ } N =, an I N = iag {} N =. The esult (4.) shows that it is possible to efine a sampling stategy which aims at contolling the a anticipate vaiances. Inee, if the quantities η, (o thei pope peictions a ˆ η, ) ae use as a eplacement fo the esiuals η,, the poblem (3..) efines a sampling esign which allows one to guaantee the following conitions that AV ( tˆ geg tˆ, z, ht = tz ) V (b=,,b; =,, M b ; =,,R). An inteesting esult is the following. In the special case of a single patition, if the inclusion pobabilities, π, an the etheoscheastic factos, c, ae quite constant in each omain, then the selection of a balance sample eceases the anticipate vaiance. This esult is emonstate in Appenix. N 5. Bief extension to the case of a simple small aea iniect estimato If a given population patition efines a too lage numbe of omains, it coul happen that the buget constaints oblige to efine a too lage pefixe sampling eos of the iect estimatos of the omains of the patition; in this situation, it coul be necessay to aopt an iniect small-aea estimato, in oe to contol the mean squae eos of patition omain estimates. Heein in the following we will show as the sampling stategy, escibe in sections an 3, may be extene to the case of a simple small aea iniect estimato. Let us consie the enough geneal case in which the vecto x of the auxiliay covaiates has an intecept, such as N = w. s Let b & enote the patition fo which it is necessay to aopt a small aea iniect estimato an let us consie the moel (7..) escibe in Rao (005, pag. 6). In the heein stuie context, fo the omains of the b & th patition, this moel may be efine as t ˆ tˆ N b&, geg = &&, geg / && = && a φ + && && + & (=,, M b &; =,, R) h v u & (5.) whee coefficients, b & a is a p vecto of aea level covaiates, φ is an unnown p vecto of egession & is a nown quantity elate to the b & th omain, & v ii 0, & ) h b & ( b & σ ν 5

18 inepenent of the sampling eo b& b& σ t & t is = V ˆ ˆ p ( && t, geg tz, ht = tz ) / Nb& ˆ, ˆ, b& u. Fo nown b & σ ν an appoximately in 0, ), being b& σ t ( b& σ t & values, the BLUP estimato of b& t blup = Nb& ( ˆ b& γ && t geg + ( && γ ) & a φ ) (5.) being b & ν && t ν b& γ = σ h /( σ + σ h ) an (5.3) b&& && b&& Mb&& M b && φ ˆ = + + b&& a && a ( && σ t b&& σ h bl blt ˆ ν && ) && a &&, geg ( && σ t b&& σ ν && h ) (5.4) = l= The MSE of the BLUP estimato is MSE & t ˆ ( b&,blup = N && ) = && γ && σ t + a M b&& ( a a t + && γ ) && && && ( && σ b&& σ ν && = h ) b&& a. (5.5) Looing at expessions (5.5) an (5.3), it is possible to note that fo a given values of the vaiance &, it is possible to contol the MSE & t ˆ ) in the sampling esign phase, by efining a ν b & σ t (,blup pope value of the vaiance b& σ. The following iteative poceue fins the π inclusion pobabilities which guaantee the minimum sample size an assue the espects of the following N /( N Q) / π η V (fo b b& ; =,, M b ; =,, R) an constaints ( ) b& blup & U, MSE ( tˆ, ) V (=,, M b ; =,, R) &. Initialization Let j ( j = 0,,,..) enote the geneic iteation. At initial iteation ( j = 0), set eff = (=,, M b & ; =,,R). By means of the algoithm escibe in section (3.), fin the j inclusion pobabilities, solution of the poblem (3..), using the η, = η (=,, Iteation M b & & ; =,,R; =,,N) as eplacement fo the The geneic iteation ( j =,, ) is aticulate as follows. Calculate j U j b& η, & values. & σ t = [ N / ( N && ( N Q))] [(/ π ) ] & η (=,, M b & ; =,,R)., && && j b& j, && j π eff 6

19 Calculate j b& γ j & an MSE ( ˆ & t,blup ) (=,, M b & ; =,,R) espectively by means of equation (5.3) an (5.5) by using the sampling vaiances j j j b& σ t & instea of b& σ t. j Calculate ( ˆ, ) / & eff = MSE && t blup σ && t ; Fin the j π inclusion pobabilities, solution of the poblem (3..), using the j j && η, = && η, eff && (=,, M b & values. & ; =,,R; =,,N) as eplacement fo the b& η, Chec If the following conition is satisfie, fo a small quantity v, j j π π v, (5.6) U then the algoithm stops an the inclusion pobabilities π ae those calculate at iteation j. Othewise, the iteation is calculate ove an ove again until conition (5.6) is especte. 6. Empiical Analysis In oe to veify the empiical popeties of the popose sampling stategies, two expeiments have been implemente, the fist one is on atificial ata, the secon expeiment is base on a simulation on eal entepise ata. In the expeiment on atificial ata the whole sampling stategy popose in section is implemente incluing the allocation phase escibe in section 3. The simulation on eal entepise ata aopts a simplifie allocation ule that may be easily implemente in eal suvey contexts but is iffeent fom that optimal escibe in section 3. Both expeiments have showe goo pefomances of the popose stategy. 6.. Atificial ata An atificial population U with,83 units has been ceate to evaluate iffeent sampling stategies. Two categoical vaiables have been ceate: U with 5 categoies (,,5) ientifying the 5 omains U of the fist patition an U with 0 categoies (,,0) ientifying the 0 omains U of the secon patition. The values u, an u, of the vaiables U an U have been assigne to each unit in the population, geneating a contingency table, assuming the categoical vaiables ae inepenent. The contingency table (table 6..) has sewe maginal istibutions, whee the maginal fequencies ecease as the levels of U o U incease. Fo each unit, the auxiliay vaiable x has been ceate awing fom the nomal istibution N( , ; u, u ). 7

20 Table 6... Contingency table of atificial population an maginal sample sizes of the multi-way statifie balance sampling Patition Totals Sample size Patition Totals ,83 Sample size The vaiables of inteest, Y, has been geneate with the following supepopulation moel y, 0. 35x + ε, = (6..) with E ε ) 0, E ε ε ) 0, fo l an m (, = m (,, l = V ε. 5x. m(, ) = Seven sampling esigns, as epote in table 6.., have been compae. The BAL esign is the one escibe in the pape. The maginal sample size have been efine with the poceue escibe in section 3, assuing that the pecent Coefficient of Vaiation (CV) of the fist an secon patition omain estimates ae lowe than 4% an % espectively: with the symbols of section 3, it is ( V t ) 00 = 4 (=,,5) an ( V t) 00 = 8 (=,,0). The oveall sample size, n, is 4 units an the maginal istibutions ae shown in table 6... The maginal sample sizes of each patition ae then aopte fo the fou one-way statifie esign, OPT, OPT, STDOM an STDOM, whee the inclusion pobabilities ae escibe in table 6... The oveall sample size n has been use to efine the Simple Ranom Sampling Without Replacement an Pobability Popotional to Size Sampling sampling esigns. Tab Sampling esigns Sampling Design Abbeviation Inclusion Pobability Multi-way statification with balance sampling BAL Define in section 3 One way statification Statifie by Patition with Optimal inc. pob. fo Patition OPT π = n ( x / x ) U Statifie by Patition with Optimal inc. pob. fo Patition OPT π = n ( x / x ) Statifie by Patition with SRSWOR* with in each statum STDOM π = n / N Statifie by Patition with SRSWOR* with in each statum STDOM π = n / N SRSWOR* SRS π = n N / U U Pobability Popotional to Size Sampling PPS π = n ( x / x ) *SRSWOR: Simple Ranom Sampling Without Replacement Knowing all the population values, the sampling vaiances V ( ˆ p t, geg ) have been compute fo each omain estimate fo the moifie geg estimato (..3) base on the moel (6..). Fo the BAL esign the vaiance has been calculate accoing to the expession (..5), while the stana textboos expession fo the othe esigns has been aopte. Table 6..3 shows the mean values an maximum values of the pecent CV by sampling esign an patition, being the CV efine as V ˆ p ( t, geg ) / t. Futhemoe, the same mean an maximum CV values have been compute consieing the oveall set of 5 maginal omains.

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