ALLOCATING SAMPLE TO STRATA PROPORTIONAL TO AGGREGATE MEASURE OF SIZE WITH BOTH UPPER AND LOWER BOUNDS ON THE NUMBER OF UNITS IN EACH STRATUM

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1 ALLOCATING SAPLE TO STRATA PROPORTIONAL TO AGGREGATE EASURE OF SIZE WIT BOT UPPER AND LOWER BOUNDS ON TE NUBER OF UNITS IN EAC STRATU Lawrece R. Erst ad Cristoper J. Guciardo Bureau of Labor Statistics, 2 assacusetts Ave., N.E., Room 360, Wasigto, DC KEY WORDS: Stratified probability proportioal to size samplig, Sample allocatio, Costraits, Natioal Compesatio Survey. Itroductio Cosider te followig commo sample desig. A sample of uits is to be selected from a frame cosistig of uits tat is partitioed ito strata, wit uits i strata. Te uits witi eac stratum are to be selected wit probability proportioal to size, witout replacemet. Let T i,,...,, i,...,, deote te measure of size (OS) for uit i i stratum ; i let T T i deote te aggregate OS for stratum ; ad let T T. A commo metod of allocatig te sample amog te strata is proportioal to te aggregate OS. Tat is, if deotes te umber of sample uits allocated to stratum, te T (.) T Tere are two problems associated wit (.). First, it does ot geerally yield a iteger-valued allocatio, tat is, some form of roudig is required of te allocatios i (.). We will ot focus o tis problem. Te oter problem is tat we must ave for all (.2) owever, te allocatio give by (.) does ot ecessarily satisfy (.2). Te stadard approac to adlig tis problem (Cocra 977, Sec. 5.8) is to reallocate for all for wic > (.3) ad te reallocate te remaiig sample to te remaiig strata proportioal to T (.4) owever, te ew allocatio to te remaiig strata still may ot satisfy (.2) for all te strata, i wic case tis process of fixig te sample size at for all strata for wic > ad reallocatig te remaiig sample to te remaiig strata proportioal to T is repeated util (.2) is satisfied for all strata. To illustrate cosider Table. (I all of te tables, 72, ad 0.) For te iitial allocatio give i te fourt colum, (.2) is violated for stratum sice 40.9 ad 9. Terefore, for te secod allocatio we let 9 ad reallocate te remaiig 63 uits to te oter 9 strata proportioal to T. (Tose strata wose sample size is fixed at are idicated i bold.) Sice (.2) is violated for stratum 2 for te secod allocatio, we let 2 0 for te tird allocatio. For te fourt allocatio, te sample sizes for strata 3, 4, ad 5 are additioally fixed at teir maximum values. Te fourt allocatio is te fial urouded allocatio sice (.2) is te satisfied for all strata. I te ext colum we obtai a iteger-valued allocatio by roudig up a sufficiet umber of te urouded values wit te largest fractioal remaiders to preserve te sample total of 72 ad roudig dow te remaiig values. Tis is oly oe of a umber of roudig metods discussed i Balisi ad Youg (982). Te fial allocatio before roudig obtaied troug tis recursive process is as close as possible to beig proportioal to te aggregate OS give te costraits (.2) i te followig sese. Tere is a commo ratio r / T for all strata for wic <, wile / T r for all for wic (.5) I tis sese te fial allocatio is optimal. To illustrate, (.5) olds for te fial urouded allocatio i Table wit r Te fial values of / T are give i te last colum of te table wit te values i bold for tose for wic. Similarly, suppose a lower boud, m, is placed o te sample size for eac stratum ad it is still desired to allocate proportioal to T as closely as possible subject ow to te costraits m for all (.6) Te, if te iitial allocatio (.) does ot satisfy (.6) but does satisfy (.2), a aalogous recursive algoritm

2 ca be used i wic we repeatedly reallocate m for all for wic < m (.7) ad te use (.4). If (.2) olds for te iitial allocatio, it will also old for every subsequet allocatio i te recursio, sice te allocatio is cotiually beig lowered for all for wic m. ece, tere is o eed to reallocate to satisfy te upper bouds. Cosequetly, te recursive algoritm used to satisfy (.6) will yield a allocatio as close as possible to beig proportioal to te aggregate OS give te costraits (.6) i te sese tat tere will be a commo ratio r / T for all strata for wic > m ad / T r for all for wic m (.8) Tis situatio is illustrated by Table 2. (Tose strata wit sample size fixed at m are italicized i te tables as is te fial value of / T for eac suc stratum.) ere tree iteratios are eeded ad r for te fial allocatio. Next, wat if te iitial allocatio violates (.2) for some strata ad (.6) for oter strata? It migt appear tat, aalogously to te previous situatios, we would use a recursive process were at eac iteratio after te first we would reallocate to te former set of strata usig (.3) ad te later set of strata usig (.7), ad te use (.4). owever, tat algoritm does ot yield a fial allocatio tat geerally meets te desired criteria tat tere is a commo ratio r / T for all for wic m < < (.9) ad tat (.2), (.5), (.6), ad (.8) all old. To illustrate, cosider Table 3. ere for eac iteratio we reallocated usig (.3) ad (.7). It required four iteratios to satisfy (.2) ad (.6). owever, altoug (.9) olds for te fial allocatio i tis table wit r ad (.5) also olds, (.8) is violated for 3,5,7,8. I Table 4 we preset a differet approac to te same example tat does satisfy all of te coditios (.2), (.5), (.6), (.8), ad (.9). ere i te secod iteratio we reallocated usig (.3), tat is let 9, ad te used (.4) witout applyig (.7) first. I iteratio 3 we repeated tis process. owever, i iteratio 4 we reallocated usig (.7) but ot (.3). I iteratio 5 we used (.3) oly ad fially i iteratio 6, (.7) oly. Sice te allocatio give by iteratio 6 satisfies (.2) ad (.6) we stop. Te for tis fial allocatio (.9) is satisfied wit r 0. 00, ad (.5) ad (.8) also old. Note i Table 3, wic did ot wor, we applied bot (.3) ad (.7) for eac iteratio after te first before usig (.4), wile i Table 4 we applied oly oe of tese two sets of costraits. owever, applyig oly oe of (.3), (.7) for eac iteratio is oly oe of te eys to te solutio. I geeral, we must be careful wic oe of (.3), (.7) we apply. To illustrate, cosider te iterative allocatio i Table 6 for te same example cosidered i Tables 3 ad 4. ere for iteratios 2 ad 3 we used oly (.7) ad for iteratios 4 ad 5 oly (.3). Te first tree iteratios are idetical to tose i Table 2 ad ece are omitted. I tis table (.8) is violated for te fial allocatio for strata 3 ad 5-9. Eve more iterestig would be a sligt modificatio of Table 6 for wic 0 is reduced to 7 wit o oter cages. If iteratios -5 remai te same, tere would ow be a iteratio 6 for wic 0 is reduced from 8 to 7 ad ece te fial allocatio would ot satisfy (.0) I te ext sectio we demostrate ow a specific iterative algoritm produces a fial sample ad a fial value r tat satisfies (.2), (.5), (.6), (.8) (.9), ad (.0). I order for (.2), (.6), ad (.0) to be satisfied simultaeously it is clearly ecessary tat. m (.) Tis is also sufficiet. Te geeral idea of te algoritm is tat at eac iteratio eiter (.3) or (.7) is used but ot bot. Furtermore, if summed over tose violatig (.2) is greater ta or equal to m summed over tose violatig (.6), te (.3) is used; oterwise (.7) is used. ore details are provided i te ext sectio. Te algoritm described was recetly applied to te sample allocatio for te itegrated Natioal Compesatio Survey program coducted by te Bureau of Labor Statistics. Tis applicatio is described i detail i Erst et al. (2002). 2. Te ai Algoritm We first itroduce some additioal otatio. For te most part te otatio will follow te otatio of te previous sectio, wit modificatios to idicate te umber of te iteratio. Let,,...,, deote te umber of sample uits allocated to stratum for iteratio. Let S, s deote te set of strata for wic te sample size as bee fixed to be, m, respectively for iteratio, ad let

3 R,..., } ( S s ) (2.) { tat is te set of te remaiig strata. Note tat, i particular, is te iitial, directly proportioal to aggregate OS allocatio ad S, s are prior to fixig te sample size of ay strata; tat is, S, s R {,..., }. For eac, te strata i R are to ave a commo ratio, deoted r, for / T, ad cosequetly we must ave + m s r (2.2) T m r T S R if S if s if R (2.3) It ow remais to sow te followig. We first explai ow S, s are obtaied recursively for 2 i terms of S, s ad ( ),,...,. Tis is ey to te algoritm sice (2.2) ad (2.3) are defied i terms of S, s. Te we establis tat tere exists a smallest iteger K for wic bot S K S K, K s K s (2.4) ad ece K ( K ) for all. Te we first prove tat te set of ad r defied by K ( K ),,...,, ad r r K rk (2.5) satisfy (.2) ad (.6); ext tat tis set of satisfies (.0); ad fially tat te ad r satisfy (.5), (.8) ad (.9). To recursively defie S, s for 2, let S s max{ ( ) R D, 0}, (2.6) d, 0} (2.7) S S s s max{ m ( ) R { : if D { : if d ( ) ( ) D > < d < m }if D }if d d > D (2.8) (2.9) Te calculatios of (2.6), (2.7) for te example of Table 4 are give i Table 5. To illustrate its use, sice D d we ave by (2.8), (2.9) tat S 2 { }, s 2, from wic, by (2.2), (2.3), te secod iteratio i Table 4 is obtaied. Tis is equivalet to applyig (.3), (.4) to te iitial allocatio. To establis tat tere exists a iteger K for wic (2.4) olds, observe tat S S, s s for eac 2, ad cosequetly R R by (2.). It follows from tis last relatio ad te fact tat R {,..., }, tat eiter R R for some,..., + or else R + 2 R +. Cosequetly, tere is a smallest iteger K + 2 suc tat R K R K ad (2.4) olds for tis K. It follows from (2.2)-(2.4), (2.6)-(2.9) tat te set of,,...,, defied by (2.5) satisfies (.2), (.6). To sow tat tis set of satisfies (.0), observe tat uless R K, (.0) is satisfied by (2.2), (2.3) wit K, ad (2.4), (2.5). owever, we will sow tat R by provig tat ad K ( K 2) for some R K 2 ( K 2) m for some R K 2 (2.0) (2.) sice (2.0), (2.) combied wit (2.), (2.6)-(2.9) establises tat R K. Tis is because if tere is some satisfyig bot (2.0), (2.), te R K for tis ; wile if tere is a pair of strata, oe satisfyig (2.0) ad te oter (2.), te oe of tese strata must be i R K by (2.), (2.6)-(2.9). We will establis (2.0) by provig tat for 2,..., K 2 if ( ) te R R Te sice by (.) it follows tat R R (2.2) R R (2.3) we combie (2.2), (2.3) to obtai by iductio tat,,..., K 2 (2.4) R R ad ece tat (2.0) olds sice R K 2. Te proof tat (2.) olds, wic is omitted, is aalogous. To establis (2.2) we cosider two cases, first S S ad te s s. I te former case it ca

4 be sow tat R R ( ) ( ) R( ) + D ( R( ) R ) R (2.5) ( g ) rg T r T (2.23) wile if S S g S g S g Rg te g ( g ) rg T r T (2.24) ad i te latter case tat R R R ( ) + D d d R (2.6) ad ece (2.2) olds i bot cases. Observe tat te first relatio of te cai (2.5) follows from (2.)-(2.3), (2.6)-(2.9); te secod from (2.), (2.6)-(2.9); ad te last relatio from te ypotesis of (2.2). Te first relatio of (2.6) follows from (2.)-(2.3), (2.6)-(2.9); te secod from (2.), (2.6)-(2.9); ad te last relatio from (2.9). Fially, we will sow tat,,...,, ad r defied by (2.5) satisfies (.5), (.8), (.9) by provig tat for all 2,..., K, Sice if rjt for all S j, j,..., te r T for all S (2.7) if j r jt for all s j, j,...,, te r T for all s (2.8) j, s rt for all S, r T for all s S, it is vacuously true tat. Cosequetly, oce (2.7), (2.8) are establised, it follows by iductio tat Note tat te first relatio i te cai (2.23) follows from (2.3) ad S g S, ad te secod relatio by te ypotesis of (2.7). Te first two relatios of (2.24) follow from (2.3) ad S S g, ad te tird relatio from (2.8). Te fourt relatio of (2.24) follows from (2.3) ad R g. To establis (2.22) we eed oly sow tat ad Rg Rg R ( g Rg Rg ( g max{ m D d ) g g + + ( g ) Rg ( g ) Rg ( m Rg R ( g ) ( g, 0} d sice it follows from (2.3), (2.25), (2.26), tat ) g r T g ) rg T R R R R (2.25) (2.26) ( (2.27) To obtai (2.25) ote tat te first relatio i (2.25) olds by combiig K rk T all S K K rk T all s K for (2.9) for (2.20) g (2.28) Fially, (2.3), (2.5), (2.9), (2.20) establis (.5), (.8), (.9). Tus we eed oly establis (2.7), (2.8). We will oly prove (2.7) sice te proof of (2.8) is similar. To sow (2.7) we let g deote te largest iteger satisfyig g ad S g S (2.2) If tere is o g satisfyig (2.2) te S S ad (2.7) is vacuously true. We will oterwise prove tat wic follows from (2.2), (2.3), wit te fact tat g for all Rg, wic follows from (2.3), (2.8), (2.9). Te secod relatio follows from (2.)-(2.3), (2.6)-(2.9), (2.2). Te fial relatio follows sice D d by (2.8), (2.2). g g To obtai (2.26), ote tat te first relatio follows from (2.3) ad te fact tat R R s by (2.2); te secod relatio from R g R Rg ; ad te fial relatio from (2.7). g r (2.22) r g wic establises (2.7) sice if S g te

5 3. Refereces Cocra, W. G. (977). Samplig Teciques, 3rd ed. New Yor: Jo Wiley. Balisi,. L. ad Youg,. P. (982). Fair Represetatio. New ave: Yale Uiversity Press. Erst, L. R., Guciardo, C. J., Poiowsi, C.., ad Teoica, J. (2002). Sample Allocatio ad Selectio for te Natioal Compesatio Survey Proceedigs of te America Statistical Associatio, Sectio o Survey Researc etods, [CD-RO], Alexadria, VA: America Statistical Associatio. Ay opiios expressed i tis paper are tose of te autors ad do ot costitute policy of te Bureau of Labor Statistics. Table. Example of Allocatio wit Costraits o aximum Sample Sizes Stratum T Iteratio Iteger T alloc Total Table 2. Example of Allocatio wit Costraits o iimum Sample Sizes Stratum T m Iteratio Iteger 2 3 alloc. T Total Table 3. Nooptimal Allocatio for Example wit Bot Sets of Costraits Stratum T m Iteratio Iteger alloc. / T Total

6 Stratum T Table 4. Optimal Allocatio for Example of Table 3 m Iteratio Iteger alloc. / T Total Table 5. Cotributio of Eac Stratum to Value of D, d for Example of Table 4 Stratum D d D 2 d 2 D 3 d 3 D 4 d 4 D 5 d Total Table 6. Aoter Nooptimal Allocatio Stratum T m Iteratio T Total