An Investigation of Stratified Jackknife Estimators Using Simulated Establishment Data Under an Unequal Probability Sample Design
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1 Secti Survey Research Methds SM 9 A Ivestigati f Stratified ackkife Estimatrs Usig Simulated Establishmet Data Uder a Uequal Prbability Sample Desig Abstract Plip Steel, Victria McNerey, h Slata Csiderig tw differet samplig schemes (ille ad Paret), we preset the results f a Mte Carl simulati studyig the statistical prperties f several variace estimatrs a sythetic data set, mdeled establishmet data. We test the validity f icludig a varyig fiite ppulati crrecti i the frmulati f the stratified jackkife (as de with the Yates-Grudy-Se estimatr), ad we cmpare the effects f direct replicati f a rate t a aylr liearizati frmulati. Key Wrds: jackkife, ille, Paret, fpc, uequal prbability samplig. Itrducti he distributi f maufacturig sectr establishmets teds t be ghly skewed. Csequetly, t esure a represetative sample, a stratified, uequal prbability samplig scheme (a -ps r PI-PS desig) is preferred t a stratified, simple radm samplig scheme, ad large uits are fte icluded with certaity. reduce variace ad t limit csts, a fixed sample size is preferred as well. Althugh uequal prbability samplig prvides a measure f ctrl ver the sample, such strategies preset difficulties i variace estimati. Apprximate samplig frmula variaces require all jit iclusi prbabilities. s makes cmputati ad strage mre cmplicated, especially whe mre tha tw uits per strata are selected. hese statistics are ubiased ly uder cmplete respse ad fr estimates f ttals; liearizati techiques must be used fr -liear estimatrs. I ctrast, replicate variace estimatrs require fewer samplig parameters ad liearizati frmulae. he purpse f ur research is t determie whether a replicate variace estimatr ca be used fr key estimates prduced by the U.S. Cesus Bureau s Quarterly Survey f Plat Capacity (QSPC). he QSPC replaces the aual Plat Capacity Utilizati (PCU) Survey ad has a very similar desig. Like its predecessr, the QSPC has a stratified -ps desig, ad the key estimate is the Plat Capacity Utilizati Rate. he cmbiati f survey desig ad estimatr (a smth statistic) suggests that these data are suited t the stratified jackkife replicati variace estimati prcedure. We will cmpare that t the curret prcedure wch takes the apprximate samplig frmula estimates f the variace ad cvariace f the rate s ttals, ad iputs them it a stadard aylr series liearizati frmula t estimate the variace f the rate. s prcedure was used by the PCU ad is curretly used by the QSPC. he QSPC sample is curretly selected via a Paret samplig scheme. s replaces the illé samplig scheme used fr bth the PCU ad the iitial quarters f the QSPC. s reprt is released t ifrm iterested parties f gig research ad t ecurage discussi. Ay views expressed statistical r methdlgical issues are thse f the authrs ad t ecessarily thse f the U.S. Cesus Bureau. 5
2 Secti Survey Research Methds SM 9 Csequetly, we csider the statistical prperties f all the csidered variace estimatrs uder bth samplig schemes. s paper describes the results f a Mte Carl simulati studyig the statistical prperties f several variace estimatrs a sythetic data set. s data set was mdeled frm a subset f the idustries represeted i the PCU strical data. Secti describes the illé ad Paret samplig methdlgies ad the assciated apprximate samplig frmulae. he stratified jackkife variace estimatrs are specified i secti 3, ad secti 4 presets the data sythesis ad ur Mte Carl simulati results. We cclude i Secti 5 with sme bservatis ad recmmedatis.. PCU ad QSPC Sample Desig he PCU published idustry level estimates f plat capacity utilizati rate, defied as the rati f actual prducti t full prducti capability. he PCU survey used a illé samplig prcedure (illé, 996). I ts prcedure, uits are rejected frm the ppulati e by e util the desired sample size is reached. he apprximate samplig frmula variace f a Hrvitz-hmps estimatr uder illé is: v D y = hj hj L i y i hj ( Yˆ ) = γ i h = i = j hj j = where the data are srted i ascedig rder wit stratum by B i (Slata ad Faga, 997). here are L samplig strata, idexed by h, ad = iclusi prbability fr uit i ( = i = = $, h $ h(i+), h if < i < = $, h i = h β, h =., h he variace estimate f the plat capacity utilizati rate ( Yˆ ) is btaied with the fllwig aylr liearizati variace estimatr: () ˆ [ ˆ ˆ v ( ) ( ) ( ˆ ) ˆ cv ( ˆ ˆ D Y v Y v Y )] ˆ D + D D ˆ where ˆ is the actual plat utilizati, ˆ is the full prducti capability, Yˆ =, ˆ ad the v ad cv are calculated usig the apprximate samplig frmulae. D D he PCU sample was selected twards the ed f 4. he iitial frame csisted f maufacturig ad publicati establishmets frm the Ecmic Cesus ad was stratified by 6 digit NAICS (Nrth America Idustry Classificati System). reduce cverage bias, additial strata were added t represet establishmets that came it busiess (were br) after. Births were itrduced a aual basis after the iitial chrt. hese strata were created frm the birth frames idetified fr the Aual 53
3 Secti Survey Research Methds SM 9 Survey f Maufactures. Each birth stratum ctaied represetatives frm a variety f NAICS cdes. he illé samplig prcedures used t btai the PCU sample have bee replaced by a Paret samplig scheme fr the QSPC. Paret samplig is a particular case f rder samplig. Like illé samplig, Paret samplig is a -ps prcedure that yields a fixed sample size. Hwever, the Paret samplig prcedure is easier t implemet, ad the apprximate samplig variace frmula is quite easy t derive. Bth prcedures prduce similar samples with regards t the distributi f uits by size ad we hpe t cfirm that stratified jackkife methds fr variace estimati wrk well bth kids f samples. We use the fllwig prcedure fr selectig a Paret sample. First, fr each uit i wit stratum h, cmpute q u( ) ( u ) = where u is U(,). he sample f size h csists f h uits with the smallest q wit stratum h. A apprximate samplig frmula variace f a ttal btaied by Paret samplig (Rsé, 997) is v ( Yˆ ) ( ) = y B hy = A h i = h > > hy B C hy h = h where A ( ) B ( ) hy C hy i = = h ( ). i = y hy = h i = he apprximate samplig frmula cvariace is give as cv ( ) = h ( ) h y B hy x Bhx = BhyB h hx Yˆ, Xˆ Ahyx h i = h y > > = h where A ( ) hyx i = y x. 54
4 Secti Survey Research Methds SM 9 Nte that these frmulae apply ly t Hrvitz-hmps estimates. As with the illé samples, we apply aylr s liearizati frmula t btai a apprximate variace f the plat capacity utilizati rate: () ˆ [ ˆ ˆ v Y ) v ( ) + Y v ( ˆ ) Yˆ cv ( ˆ ˆ )] ( ˆ where v ad cv are as just defied ad ˆ, ˆ, ad Yˆ are as they appear i (). hese ttals ad rati estimates are calculated fr each dmai (NAICS). 3. Stratified ackkife Variace Estimatrs () he stadard stratified jackkife replicate estimate is created by drppig the i th uit frm the sample ad reweightig the remaiig f the h - uits wit stratum h by h /( h -). I a stratified simple radm sample, the stratified jackkife frmula is mdified t iclude the fiite ppulati crrecti factr (if ecessary) by icrpratig it it the replicate sums-f-squares crrespdig t the uits strata. With a uequal prbability sample desig, ts pre-multiplicati by a stratum-level cstat is t apprpriate. Istead, we ca multiply each replicate i's ctributi t the sums-fsquares by (- i ), where i is the iclusi prbability assciated with mitted uit, i. We call ts factr a varyig fiite ppulati crrecti. Fr each samplig scheme ad estimatr ( Yˆ ), we estimated the variace with the stadard stratified jackkife estimate (3) ad tw variatis: (3) v ( Y $ ) ( )( Yh i Y h $ $ = () ) h i= where Y$ () is the replicate estimate $Y is the full sample estimate, is the umber f uits wit stratum h, ad h ( ) is a varyig fiite ppulati crrecti fr uit i i stratum h. (4) vr ( Y $ ) ( )( Yh i Yh h $ $ _ = () ) h i= where $ _ Y h is the average f the replicates i stratum h, (5) vu ( Y $ ) = ( Y Y $ $ () ) h h i= 55
5 Secti Survey Research Methds SM 9 $ ˆ ( ) Whe Yˆ is a rati, ˆ h i ˆ ( ) _ Y h( i) =, Yˆ ˆ = h( i) ˆ, ad $ $ i= Y h = ( ) i= () (). Nte that (3) ad (4) differ ly i the secd term f the sums-f-squares -- it is the full sample estimate i (3) ad a weighted average f replicate estimates ver the stratum i (4). Equati (4) is algebraically equivalet t the apprximate samplig frmula variace estimate f a ttal frm a Paret sample, v. Equati (5) is icluded t assess the ecessity f icludig the varyig fpc-crrecti factr i the variace estimatr. Fially, we cmpute the aylr liearizati variace estimate usig the stratified jackkife variace estimates as iput. s estimatr mimics ur stadard prcedure fr -liear estimatrs. (6) ˆ [ ˆ ˆ v ( ) ( ) ( ˆ ) ˆ cv ( ˆ ˆ Y = v Y v Y )] ˆ + j where v ad cv are the variaces ad cvariace cmputed via the traditial stratified jackkife methd i (3). 4.. Geeratig A Artificial Ppulati 4. Mte Carl Simulati Study I rder t examie multiple samplig prcesses ad their effect variaces, we prduced a simulated ppulati mdeled data selected frm the Plat Capacity Utilizati survey. Fr a detailed descripti f the PCU data ad sample desig see the publicati appedices: Data fr ur simulati were btaied frm the 4-6 survey cllectis. he PCU surveys hudreds f distict idustries. Despite a detailed classificati f idustry, sme idustries display cmplex distributis f size (capacity). We cfied ur simulati t thse idustries fr wch we had sme cfidece i ur ability t mdel; that is, thse that appeared t be distributed as a pure lgrmal. We used a prgram develped by the Cesus Bureau, LgNrmSim (McNerey ad Adesya, 6), t geerate idustry level multivariate lgrmal data ppulatis, usig certaity cases frm the selected PCU data as traiig data. Our multivariate lgrmal distributis mdeled three variables: full prducti capability, actual prducti, ad payrll. he first tw variables are used i the estimate f the plat capacity utilizati rate; the trd is a measure f size used t calculate the prbability f selecti. Fr each idustry, we evaluated the fit f the simulated ppulati t the traiig data seve percetiles, kurtsis, skew, stadard deviati, mea, ad the crrelatis betwee the three variables. We als calculated the average percet differece i all percetiles frm the survey t the simulated data ad the average percet differece i the crrelatis. Frm thse withut majr defect i fit, we selected eleve subppulatis 56
6 Secti Survey Research Methds SM 9 with atteti give t variety ad the iclusi f several wch mdeled idustries kw t be f particular iterest i the fial survey publicati. After cductig a empirical aalysis f the rigial PCU data, we decided t cstruct tw additial subppulatis t study particular data phemea. I idustry ZZZZZZ, we radmly set apprximately 8% f the values t i a selected mdeled idustry. s was the ghest reprted-zer rate bserved i rigial PCU micrdata. I idustry YYYYYY, we added a utlier/ifluetial value. he utlier was a sample case, but received a weight very clse t e. I all ppulatis, certaity cases were excluded frm the traiig data used fr mdelig. Istead, after the 3 simulated subppulatis were created, we added the relevat certaity cases frm the PCU survey t ur simulated ppulati. Next, we divided ur simulated ppulati it 5 distict strata: 3 (-certaity) strata based frame idustry; e birth stratum with cases radmly recruited frm thse 3 subppulatis based the prprti f births bserved i the PCU survey; ad e stratum ctaiig all f the certaity cases. w f the dmais, 3733 ad 337, had a gh prprti f births (betwee.5 ad percet f cases). Nie dmais, icludig YYYYYY ad 33458, had births. It seems likely that ur mdelig criteria had a tedecy t exclude NAICS with active birth prcesses. Frm ts ppulati, we drew, idepedet samples usig the illé samplig methd ad, idepedet samples usig the Paret samplig methd. I each sample we calculated the variace estimates f the plat capacity utilizati rati usig the three stratified jackkife methds (v frm equati 3, v R frm equati 4, v U frm equati 5), the aylr apprximati usig the samplig frmula variace (i.e., v D, the frmer prducti methd that used v D fr illé samples (frm equati ) r v, the curret prducti methd that uses v fr Paret samples (frm equati )), ad aylr apprximati usig v ( v, wch uses equati 6 with stratified jackkife variaces f ttals--usig equati 3--as iputs). assess the perfrmace f each methd ver repeated samples, we calculated the relative bias ad relative stability fr each variace estimatr ( v s ) wit each dmai usig the frmulae belw (Sha ad u, 995, p5): v s s= Relative Bias = MSE( Yˆ) Relative Stability = s= ( v V ( Yˆ ) s MSE( Yˆ) where = ( ˆ MSE( Yˆ) Y s Y ) ad ( Yˆ) = ( Yˆ s Y ) s= / V, Yˆ s is the sample estimate, Y is the mea f all, sample estimates, ad Y is the subppulati value f the plat capacity utilizati rate. s= 57
7 Secti Survey Research Methds SM Results ables thrugh 4 preset the relative bias ad relative stability results fr the six variace estimatrs. he relative bias results preseted i tables ad ca be summarized as fllws: Uder either samplig scheme, the aylr liearizati variace apprximatis ( v D ad v r v ad v ) ted t yield gher abslute relative bias tha their fpc-crrected jackkife cuterparts (v ad v R ). I the few ppulatis where the curretly implemeted aylr liearizati methd ( v D btaied via (6)) has better prperties, the v R variaces prvide a clse secd. Nte that the majrity f these differeces i relative bias are fairly trivial; he perfrmace f the v estimatr appears t be greatly affected by existece f a utlier i the sample (see idustry ( YYYYYY ) but t des t appear t be verly affected by the prevalece f zers i the data (see idustry ZZZZZZ ); he v R replicate estimatr appears t be fairly resistat t the presece f a utlier as well as beig uaffected by a gh prevalece f zers i the data, as d all three aylr liearizati variace estimatrs ( v, v, v ); he relative bias f the ucrrected replicate variace estimatr (v U ) is uifrmly pr, with verestimati i all cases. s prvides strg evidece f the ecessity f icludig a fiite ppulati crrecti i estimates f variace fr bth f these fixed size sample desigs. D able : Relative Bias Usig illé Samplig NAICS v v R v v v U D 333.% -.7% -.38% -.% 6.9% % -3.86% -.59% -9.9% 48.5% % -.% -5.38% -5.48% 4.93% 37.7%.75% -6.9% -6.9%.74% % 3.3%.65% -.% 47.9% % 5.% 4.8% 3.7% 38.84% % 4.%.85%.% 5.6% 339.5% -.3% -3.4% -4.84% 5.54% % -.% -3.94% -3.73% 3.9% % -.% -.34% -.8% 34.64% % -.74%.4% -4.8% 56.% YYYYYY.8%.84% 4.35% -6.35% 88.5% ZZZZZZ.86%.5%.% -.87% 58.44% 58
8 Secti Survey Research Methds SM 9 able : Relative Bias Usig Paret Samplig NAICS v v R v v v U % -.8% -4.45% -5.% 57.98% % -3.4% -.% -.74% 49.75% % -.6% -3.9% -4.6% 6.7% 37.88%.56% -7.% -7.3%.57% % 4.8% 3.36%.8% 48.9% 337.4% -.5% -.63% -.9% 3.65% % -.7% -.3% -.8% 44.66% %.5% -.64% -.88% 55.47% % -3.45% -7.% -7.% 5.9% % -4.9% -6.5% -6.9% 9.% % -.3%.% -6.3% 6.73% YYYYYY 6.3% -.58% 99.77% -8.6% 84.95% ZZZZZZ -.64% -.93% -.4% -3.48% 54.57% he stability results preseted i tables 3 ad 4 ca be summarized as fllws: I mst cases, the stabilities f the variace estimates are cmparable fr the tw crrespdig aylr liearizati variace estimatrs ( vd ad v ) ad the tw crrected jackkife variace estimatrs (v ad v R ). hus, direct replicati f the plat capacity utilizati rate (a rati) des t appear t detrimetally affect the variace f the variace estimates. It appears that the additial smtg by the aylr liearizati apprximati is t ecessary with ts rati estimatr, ppulati, ad samplig scheme (illé r Paret); I the few cases where the stability is affected by the chice f variace estimatr, the v R frmulati geerally prduces the mre stable variace estimates tha the ther methds; he ucrrected jackkife replicate estimates (v U ) are the mst ustable i all dmais. able 3: Relative Stability Usig illé Samplig NAICS v v R v v v U D % 4.54% 39.3% 4.89% 78.3% % 69.9% 6.44% 64.95% 98.8% % 49.87% 45.89% 46.% 56.34% % 86.39% 66.4% 66.73% 93.43% % 4.3% 36.3% 35.36% 65.89% % 6.6% 5.76% 5.39% 4.6% % 3.86% 3.3% 3.6% 6.57% % 3.3% 8.68% 9.% 7.86% % 34.43% 3.7% 33.3% 49.% % 37.5% 35.3% 35.87% 56.5% % 6.6% 63.88% 6.% 7.% YYYYYY 3.84% 47.66% 3.% 47.48% 95.76% ZZZZZZ 5.84% 5.6% 4.9% 5.% 66.3% 59
9 Secti Survey Research Methds SM 9 able 4: Relative Stability Usig Paret Samplig NAICS v v R v v v U % 4.8% 38.96% 38.9% 75.3% % 7.% 6.9% 6.33% 99.86% % 5.75% 46.58% 46.53% 57.75% % 86.6% 66.5% 66.% 93.6% % 4.33% 37.5% 36.53% 67.47% % 4.4% 4.6% 4.4% 35.7% % 3.57% 3.5% 9.87% 57.7% % 3.64% 8.86% 8.8% 74.37% % 33.6% 3.49% 3.48% 45.39% % 35.83% 34.44% 34.4% 5.79% % 6.44% 64.8% 57.6% 76.9% YYYYYY 6.5% 46.67% 8.87% 44.9% % ZZZZZZ 5.% 5.% 4.48% 4.48% 6.67% With either samplig scheme, the v R estimates have relatively trivial bias ad are fairly stable. he estimates v R ad v differ sigificatly ly the utlier ppulati. I all cases, the replicate variace estimates have cmparable if t better statistical prperties t their aylr liearizati cuterparts. s presets a clear advatage ver the curret prducti methd: btaiig estimates with cmparable statistical prperties i a simpler fas, with assumed apprximatis (e.g., egligible gher rder derivatives fr the aylr liearizati, trivial -respse fr the iput apprximate samplig frmula variaces ad cvariaces). hese advatages apply regardless f the samplig scheme i ts simulati. 5. Cclusi s research was mtivated by a practical prblem: t determie whether the stratified jackkife variace estimatr culd be used t prduce usable variace estimates fr the U.S. Cesus Bureau s Quarterly Survey f Plat Capacity (QSPC). As a part f ts aalysis, we evaluated bth direct replicati f a rati estimate ad a variace apprximati that cmbied apprximate samplig frmula r replicate estimates with a aylr liearizati apprximati. With a ghly stratified sample, the literature supprts the usage f the stratified jackkife variace estimatr fr a smth statistic such as a rati (e.g., Sha ad u, 995). Hwever, we fud very little guidace i the literature the apprpriate fiite ppulati crrecti factr with a uequal prbability sample. We als fud little i terms f ccrete recmmedatis fr the gld stadard i the replicate sums-fsquares. he preseted aalysis prvides strg evidece fr the usage f the varyig fiite-ppulati crrecti factr i the replicate variace estimatr, demstratig that failig t iclude ts crrecti leads t csistet verestimati. [Nte that Deville (999) advcates the usage f ts adjustmet i s paper liearizati variace estimatrs.] Our research als prvides supprt fr the lcal replicate weighted average i the sums-f-squares, thugh the perfrmace is ly margially better tha the full sample estimate i all but the utlier ppulati. 53
10 Secti Survey Research Methds SM 9 he strg perfrmace f v R the ppulati with a utlier i the curret prducti variable culd t be traced t ay cmpet f the estimatr. It may ly be cicidet t whatever directi the utlier pulls the rati. Mre research is eeded t determie if i fact the v R estimatr perfrms better data with sigificat utliers. Ackwledgemets he authrs wuld like t thak Katherie. hmps ad David Kiy fr their help i the study desig ad their cmmets ts paper. Refereces Deville,.C. (999). Variace Estimati fr Cmplex Statistics ad Estimatrs: Liearizati ad Residual echiques, Survey Methdlgy, 5,, pp illé, Yves (996) A Elimiati Prcedure fr Uequal Prbability Samplig Withut Replacemet, Bimetrika, 83,, pp Slata, h G. ad Faga, ames. (997), A Mdified Apprach t Sample Selecti ad Variace Estimati with Prbability Prprtial t Size ad Fixed Sample Size, upublished reprt, MCD Wrkig Paper Number: Cesus/MCD/WP-97/. McNerey, Victria G., ad Adesya, Sams A. (6), User Guide fr Geeralized Ppulati Simulati Prgrams, upublished reprt. Rsé, Begt (997), O Samplig with Prbability Prprtial t Size, ural f Statistical Plaig ad Iferece, 6, pp Sha,. ad u, D. (995). he ackkife ad Btstrap. New Yrk: Spriger Verlag. 53
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