Usng Pedcton-Oented Softwae fo Suvey Estmaton - Pat II: Ratos of Totals James R. Knaub, J. US Dept. of Enegy, Enegy Infomaton dmnstaton, EI-53.1 STRCT: Ths atcle s an extenson of Knaub (1999), Usng Pedcton-Oented Softwae fo Suvey Estmaton, whch dealt wth the estmaton of totals and subtotals and the coespondng estmates of vaance n the pesence of mssng data, whethe mssng as pat of a samplng scheme, o as a esult of nonesponse. The cuent atcle deals wth atos of totals. n example fom the electc powe ndusty would be the estmaton of evenue pe lowatthou and ts assocated vaance estmate. s n Knaub (1999), the goal s to poduce such estmates by mang use of cuently avalable softwae n whch the model can be qucly and easly modfed, and the data may be stoed n such a manne that they may be easly ecategozed fo puposes of publshng vaous aggegatons of the data wth coespondng vaance estmates. s n Knaub (1999), a blendng of suvey statstcs and econometcs can be seen. KEYWORDS: suvey samplng; pedcton; estmaton; mputaton; vaance estmaton; atos of totals SOME PPICTIONS: geat advantage wth ths new method s that t s easy to stoe and manpulate data. Statstcal agences may pesent data n dffeent fomats, n dffeent tables. It s cumbesome to aggegate data one way to estmate subtotals fo one table and anothe way to estmate subtotals fo ovelappng aeas fo anothe table. The gand totals, fo example, would dffe. In the case of publshng subtotals fo (ueau of the) Census dvson egons, consde that Census dvsons ae goups of States. Howeve, Noth mecan Electc Relablty Regons (NERC Regons) have boundaes that cut though States. Futhe, NERC boundaes ecently moved. If mputed values wee substtuted fo each mssng obsevaton, usng the lagest, elatvely homogeneous set of data avalable fo each pedcton, then they could be aggegated howeve desed, and usng the method of Knaub(1999) and ths atcle, standad eos may be estmated fo any aggegaton. Fo establshment suveys, a vey stong eason fo usng cutoff model samplng at all s that the smallest and most plentful establshments ae often unable to supply data on a fequent bass wth easonable accuacy. lot of mputaton may be necessay. Resouces ae anothe poblem. The method of ths atcle, and Knaub(1999), also apples to mputaton fo census suveys, and may be used to help publsh pelmnay subtotals/totals and/o atos of such numbes moe tmely. Fo a desgn-based sample, ths method could be used to pedct/mpute fo the mssng membes of the sample, and then the aggegate level vaances fo that pat could be added to the vaance estmates fom the desgn-based sample. (Ths technque s appaently used elsewhee. See ee, Rancout, and Saendal(1999).) NEW METHODOOGY: s shown n Knaub (1999), any statstcal softwae pacage that wll povde pedcted values, a standad eo o vaance of the pedcton eo, and the mean squae eo (MSE) fom the analyss of
vaance, wll suffce fo estmatng (sub)totals and the vaances n the pesence of mssng data, usng the method found n that atcle. The egesson weght must be suppled by means of consdeatons such as those found n Knaub (1997). Fo puposes of pedctng mssng numbes, the populaton should be categozed nto the lagest, elatvely homogeneous sets of data possble. Imputed numbes ae then each ndvdually assocated wth vaance elated nfomaton that can be egouped accodng to whateve aggegatons one may wsh to publsh. The cuent atcle goes a step fathe and assocates pas of numbes whose ato s of nteest, and then assgns covaance nfomaton to the pa fo late aggegatons. s n Knaub (1999), a gven aggegaton could contan lttle o no obseved values, yet t may be possble to estmate totals o atos of totals wth some usefulness. Thus small aea statstcs esults may be avalable. Hee we consde (T T), the vaance of the eo when estmatng a total. Ths s a multple egesson fom of n Royall and Cumbeland (1981), whch contaned some moe obust vaance estmates. Howeve, Knaub (199), page 879, Fgue 1 shows that may do vey well, and ths multple egesson fom of ths vaance estmato has pefomed well, as n Knaub (1996) and Knaub (1999). y y Now, accodng to Knaub (1999), usng ( y y ) fo the vaance of the pedcton eo (see Maddala (199)), and notng that (T T) = ( ) when thee s only one mssng value, one fnds n Knaub (1999) that n geneal, we may appoxmate as follows: ( ( ) ( ) T T) 0 < δ < 1 e e = δ N n y y +, δ w w whee, ( = 0.3 may be a fa geneal use value; futhe dscusson s found n Knaub (1999)), means to sum ove the cases wth mssng data (see Royall (1970)), e (Knaub (1996)) s the estmated vaance of the andom facto of the esdual, e 0, 1/ e = w e (see Knaub (1993, 1995)), whee the eo tem s, o w s the egesson weght, and (N - n) s the numbe of membes of the populaton that ae not n the sample. δ (Note: s (N-n) appoaches 1, appoaches 1. Howeve, wll geneally decease qucly as (N-n) becomes a lttle lage.) δ
Followng s an except fom Knaub (1999), page 8: Pctue a typcal data fle as follows, whee "EG" s a categoy fo puposes of pefomng pedctons (an "estmaton goup"), and "PG" s a categoy fo puposes of publshng subtotals (a "publcaton goup"). Each lne epesents a ecod fo a gven membe of the populaton. y value s an obseved (o "collected") value, and y s a pedcted value. et S1 = ( y ), the vaance of the pedcton eo, and y S = e w, the mean squae eo dvded by the egesson weght, fo each case,. Example of a patal fle: y o y S 1 S EG PG(a) PG(b) PG( c ) 675 0 0 1 1 5 4359 0 0 1 1 3 189 0 0 1 4 4 497 0 17 1 1 3 317 13 11 1 78 10 9 1 1 3 3 9 8 1 3 fte that, Knaub (1999) dscusses adjustments fo nonsamplng eo that would be applcable hee also, but wll not be epeated hee. fgue fom Knaub (1999), page 1, s epnted next, as t may help n the geneal vsualzaton of the mplementaton of ths method. 3
Consde, fo example, that a populaton could be subdvded nto seven categoes whch each epesent nealy homogeneous data unde one model pe categoy. Theefoe, thee ae seven EGs. Futhe suppose that the populaton s dvded nto fou pats fo whch subtotals and the vaances ae to be estmated. That would mean havng fou PGs. Ths could be epesented vsually as n the fgue below:... The dashed lnes epesent the sepaaton of the unvese nto the fou "publcaton goups" ("PGs"). The sold lnes epesent boundaes between "estmaton goups" ("EGs"). Hee, y, ( y ), and ae estmated fo each mssng obsevaton wthn a gven y e w EG goup, usng all data n that goup. Then evey pece of an EG wthn a gven PG s teated as a statum fo estmatng the total fo that PG goup. The vaance fo each statum s estmated usng the (T T) fomula above, and the total vaance estmate s found by addng the stata vaance 4
estmates. et T = y + y s be the total fo any gven statum (.e., fo a gven pece of an "EG" wthn a "PG"). s efes to summaton ove the obseved data, and efes to summaton ove pedctons fo mssng data (see Royall (1970)). The coespondng vaance fo that ( ( ) ( ) e e statum s T T) = δ N n y y +. w w The cuent poblem, howeve, s to extend ths to estmatng vaance fo the estmated atos of such (sub)totals. et the estmated ato be T / T, and the vaance sought s. In the case T / T of totals, estmates of subtotals and the vaances wthn stata ae smply added to obtan an estmate of a total and ts vaance, espectvely. In the case of an estmated ato of totals, the numeato and denomnato ae estmated sepaately, addng statum components untl the estmates of numeato and denomnato ae completed, and then the estmated ato s found. Fo the estmated vaance of ths estmated ato, an estmated vaance fo the numeato, and an estmated vaance fo the denomnato, and a covaance estmate wll each have to be constucted fom the stata estmates, and then appled to the oveall estmatons of the ato and ts vaance. To do ths, howeve, n a manne that s flexble to changes n PG categozatons, ( y ), and wll be needed fo the numeato y e w and fo the denomnato, and a ffth numbe, a covaance component, wll be needed, fo each elated pa of mssng data ponts n the populaton. Ths s vey lttle data to have to stoe and yet leave such flexblty n the publcaton pocess fo data aggegatons. T / T T / T Fo one statum, the estmaton of the ato,, desgnated, s staghtfowad. We smply use T = y + y s (Royall (1970)) fo the numeato, and epeat the applcaton fo the denomnato. Howeve, vaance estmaton s moe nvolved and wll be dscussed below. Futhe, when consdeng moe than one statum, an estmated total can be found by addng the subtotal estmates by stata, and smlaly fo the estmated vaance of the total. The estmaton of the ato of totals s also staghtfowad, but would now nvolve a moe complcated vaance fomula. Howeve, vaance estmaton would stll ely on only the fve stoed bts of nfomaton fo evey pa of mssng data ponts. 5
RINCE ESTIMTION: Statng wth the case of a sngle statum: Knaub (1994) s lagely a evew of and eles heavly upon P.S.R.S. Rao (199) fo covaance fomulae assocated wth the vaance of a ato of vaables. Hee, howeve, as n Knaub (1999), the thust s somewhat dffeent. Hee the emphass s on smplcty of opeaton, ncludng easly evsed models and the assocaton of all nfomaton at the ndvdual (pas of) pont(s) level that wll be needed to estmate atos and the vaances at any level of aggegaton. s n Knaub (1994), So, ( / ) ( T / T ) ( ) ( ) CO (, ) T T T T T T T T = + T T ( T ) T ( T ) TCO ( T, T) / 3 T T = + 4 T T T lso fom Knaub (1994) and Hansen, Huwtz and Madow (1953), pages 56 to 58, CO whch coesponds to ( ) ( ) T T CO y, y = +,... (T T) > ( ) y y n Knaub (1999). lso, by Knaub (1999) = e / + ( ) ( b0 ) + 1 ( b1) + (T T) w N n x... and 6
( ) ( b0 ) ( b1) y y = e / w + N n x ( ) + +... y Knaub (1996), n n e = e0 / d.f. = / d.f, we = 1 = 1 whee e = y y = esdual and d.f. s the numbe of degees of feedom, so CO n w e w e ( y, ) = +... = +... 0.5 0.5, j j j j e; y y y 0.5 0.5 0.5 0.5 w w w w ( d.f. ) j j j j theefoe o 1 / (T ) (T ) e; y y, (T,T ) 0.5 0.5 w w CO. e e w w 1 / (T ) (T n ) CO /. e 0.5 0.5 0.5 0.5 (T,T ) w d.f. j w j w je jw je j e e e w w y y (Note: fo, and, one can save n each case ( and ) n anothe fle.) 7
So, n addton to poducng ( y y ), and, fo each mssng numbe, also save e w e ; y y, 0.5 0.5 w w fo each pa of coespondng, mssng numbes. Once agan, pe Knaub (1994) and Hansen, Huwtz and Madow (1953), sum ove CO ( T, T ) just as s done fo (T fo the case of multple stata,. T ) Then ( T ) T ( T ) T CO ( T, T) / 3 T T = + 4 T T T T = T,, T = T ( T ) = (T T ), T = (T and T ) ( ) ( T, T ) = CO ( T T ), and emembeng that CO, = T y + y s =, T, y + y ( ) ( ) s, whee e e (T =, T ) δ N n y y + w w ( ) ( ) e e (T =, T ) δ N n y y + w w and 8
1 / (T ) (T n ) e e w w 0.5 0.5 0.5 0.5 (T,T ) w d.f. j w j w jejwje j CO /.. e ; y, y So, usng 0.5 0.5 = S3, an example of the equste data fle s as follows: w w y y Example of a patal fle: y y o S 1 o S S 1 S 3 EG PG(a) PG(b) 675 0 0 43 0 0 0 1 1 4359 0 0 30 0 0 0 1 1 189 0 0 85 0 0 0 1 4 497 0 17 35 9 8 3 1 1 3 317 13 11 6 5 1 78 10 9 17 3 3 1 1 1 3 3 9 8 14 3 1 1 3 S Example applcaton: Electcty sales and assocated evenue data fo the esdental economc end-use secto wee taen fom a census suvey of utltes n Indana and Oho fo two succeedng yeas n whch the suvey was conducted. Fo the moe ecent yea, accodng to these data (whch would, natually, nclude some nonsamplng eo), the oveall evenue pe lowatthou found fo Indana was 7.010 cents pe lowatthou, and fo Oho t was 8.704 cents pe lowatthou, fo the esdental secto. Thee wee 40 utltes n these two States whch wee used fo ths expement. They each ft unde one of thee owneshp classes (muncpally owned, pvately owned o coopeatves), and modelng by 9
owneshp code showed substantally dffeent gowth ates n evenue, dependng upon owneshp class. (Sales pobably also would show ths dffeence clealy, but multple egessos wee used fo sales, peventng dect compason of the coeffcents fo a sngle egesso, but execsng the flexblty of the methodology.) Theefoe, thee wee thee estmaton goups (EG1, EG, EG3), the owneshp classes, and two publcaton goups (PG1, PG), whch wee the two States. Data fom the most ecent census wee pepaed as follows: Evey fouth esponse was emoved as f thee wee a nonesponse ate of 5%. Fou such data sets wee pepaed, smlaly, but wth a dffeent stat (fst, second, thd o fouth ecod) fo the pocess of desgnatng whch numbes to elmnate as f they wee nonesponses. Theefoe, thee was one data set used fo egesso data, and fou othes fo the data of nteest fo fou tests. etween the fou data sets to be used fo the cuent data, evey obsevaton was elmnated once and only once. Theefoe, a patculaly favoable data set (wth espect to test esults) must be balanced by one o moe less favoable data sets. The followng tables show some esults of ths expement. What s of geatest nteest hee s the dffeence between standad eo estmates, wth and wthout a nonzeo covaance tem. Obtanng an estmate when covaance should be consdeed nonzeo was the pont of ths extenson to Knaub (1999). These data ae hghly sewed. Totals fo sales and evenue n Oho ae domnated by eght vey lage utltes. Thus, a 5% nonesponse could sometmes mean that the majoty of an estmated total s mputed. Indana Oho cents pe lowatthou fom full test data set estmated cents pe lowatthou estmated standad eo estmated standad eo f zeo covaance eo Test 1 7.010 6.984 0.06 0.0 0.080 Test 7.010 7.00-0.010 0.01 0.044 Test 3 7.010 7.011-0.001 0.05 0.060 Test 4 7.010 7.011-0.001 0.003 0.00 cents pe lowatthou fom full test data set estmated cents pe lowatthou estmated standad eo estmated standad eo f zeo covaance eo Test 1 8.704 8.695 0.009 0.00 0.004 Test 8.704 8.736-0.03 0.043 0.073 Test 3 8.704 8.706-0.00 0.004 0.01 Test 4 8.704 8.698 0.006 0.036 0.080 10
s pevously stated, the test data came fom a set of 40 obsevatons of electc utlty sales and evenue data (esdental) n Indana and Oho. State and owneshp type wee as follows: Muncpal Pvate Coopeatve Indana 4 6 73 11 Oho 7 8 84 119 69 14 157 40 Evey fouth one of the 40 obsevatons was teated as a nonesponse. The fou complementay data test sets wee pepaed by usng a dffeent stat ecod (.e., fst, second, thd o fouth) fo each. Softwae: Compute code was shown and dscussed n Knaub (1999), but the stuaton s somewhat moe complex hee. Moe manpulaton may be needed to use esduals n accodance wth the equatons shown n ths atcle, so that covaance s taen nto account. It s stll lely, howeve, to be elatvely easy to mae use of exstng softwae, as was done fo the example above. Futhe, the ease wth whch data may be stoed and used fo multple puposes by post-poducton softwae maes ths method attactve. If you softwae calculates mean squae eo, as shown n the code on page 34 n Knaub(1999), ths s no longe adequate when estmatng covaance. Each esdual needs to be dentfed. (See the covaance fomula at the bottom of page 7 n the cuent atcle.) If egesson weghts ae passed fom one compute pogam to anothe, be cetan to save enough sgnfcant dgts. It may be best to save the squae oot of these weghts. (Ths s usually not a poblem n suvey statstcs whee the poblem moe often s the pesentaton of too many dgts fo a gven statstc.) Fo moe nfomaton, please contact the autho. Genealzed δ : alues fo delta wee also dscussed n Knaub(1999). When N-n s small, the value of delta should be nceased, but n such cases, the mpact on vaance fom the tem whee delta s found s coespondngly deceased. Consdeng delta to be a constant, as n Knaub(1999), say δ = q, s ( N n) pobably adequate, but the followng could be used:. δ = q+ (1 q )(1 + q) 11
REFERENCES: Hansen, M.H., Huwtz, W.N., and Madow, W.G. (1953), Sample Suvey Methods and Theoy, olume II: Theoy, John Wley & Sons. Knaub, J.R., J. (199), Moe Model Samplng and nalyses ppled to Electc Powe Data, Poceedngs of the Secton on Suvey Reseach Methods, mecan Statstcal ssocaton, pp. 876-881. Knaub, J.R., J. (1993), "ltenatve to the Iteated Reweghted east Squaes Method: ppaent Heteoscedastcty and nea Regesson Model Samplng," Poceedngs of the Intenatonal Confeence on Establshment Suveys, mecan Statstcal ssocaton, pp. 50-55. Knaub, J.R., J. (1994), "Relatve Standad Eo fo a Rato of aables at an ggegate evel Unde Model Samplng, Poceedngs of the Secton on Suvey Reseach Methods, ol. I, mecan Statstcal ssocaton, pp. 310-31. Knaub, J.R., J. (1995), " New oo at 'Potablty' fo Suvey Model Samplng and Imputaton," Poceedngs of the Secton on Suvey Reseach Methods, ol. II, mecan Statstcal ssocaton, pp. 701-705. Knaub, J.R., J. (1996), Weghted Multple Regesson Estmaton fo Suvey Model Samplng, InteStat, May 1996, http://ntestat.stat.vt.edu. (Note shote, moe ecent veson n S Suvey Reseach Methods Secton poceedngs, 1996.) Knaub, J.R., J. (1997), Weghtng n Regesson fo Use n Suvey Methodology, InteStat, pl 1997, http://ntestat.stat.vt.edu. (Note shote, moe ecent veson n S Suvey Reseach Methods Secton poceedngs, 1997.) Knaub, J.R., J. (1999), Usng Pedcton-Oented Softwae fo Suvey Estmaton, InteStat, ugust 1999, http://ntestat.stat.vt.edu, patally coveed n "Usng Pedcton-Oented Softwae fo Model-ased and Small ea Estmaton," to appea n S Suvey Reseach Methods Secton poceedngs, 1999. ee, H., Rancout, E., and Saendal, C.-E. (1999), aance Estmaton fom Suvey Data Unde Sngle alue Imputaton, pesented at the Intenatonal Confeence on Suvey Nonesponse, Oct. 1999, to be publshed n a monogaph. Maddala, G.S. (199), Intoducton to Econometcs, nd ed., Macmllan Pub. Co. Rao, Podu S.R.S. (199), unpublshed lettes, ug. - Oct. 199, on covaances assocated wth thee Royall and Cumbeland model samplng vaance estmatos. Royall, R.M. (1970), "On Fnte Populaton Samplng Theoy Unde Cetan nea Regesson Models," ometa, 57, pp. 377-387. Royall, R.M. and Cumbeland, W.G. (1981), "n Empcal Study of the Rato Estmato and Estmatos of ts aance," Jounal of the mecan Statstcal ssocaton, 76, pp.66-88. 1