ALLOCATION OF THE ICM SAMPLE TO THE STATES FOR CENSUS Eric Schindler, Bureau of the Census Bureau of the Census, Washington, DC 20233

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1 ALLOCATION OF THE ICM SAMPLE TO THE STATES FOR CENSUS 2000 Erc Schndler, Bureau of the Census Bureau of the Census, Washngton, DC KEYWORDS: Dual System Estmaton, Reapportonment, Jackknfe ABSTRACT: The ntroducton of Integrated Coverage Measurement (ICM) for Census 2000 requres 51 state estmates based only on data from each state. The goal s to allocate the avalable sample of 750,000 housng unts so as to acheve coeffcents of varaton for the Dual System Estmates of 0.5% n all states and standard errors of about 60,000 n the larger states. Data from the 1990 Post Enumeraton Survey are restratfed and dual system estmates wth Jackknfe varances are calculated. The need for good data qualty n both the ntal phase and the I CM phase and the effect on Congressonal reapportonment are also dscussed. I. Introducton Census 2000, as currently planned, wll ntegrate the results of a large coverage survey nto the fnal census estmates at all levels of geography. Ths paper descrbes the appled research used to determne an approprate allocaton of the Integrated Coverage Measurement (ICM) sample to the states for Census For more nformaton on Census 2000 and the desgn of the ICM program, see Hogan and Wate (1998)or Grffn and Vacca (1998). The followng basc facts are consdered by the desgn: The total ICM sample sze wll be about 750,000 housng unts. Ths sze was determned by the Census Bureau's ablty to mplement and control the ICM sample and by statstcal requrements. Rough prelmnary estmates ndcated that ths sample sze mght be enough to produce coeffcents of varaton of 0.50% n each state. Block clusters averagng about 30 housng unts wll contnue to be the prmary samplng unt. The total ICM sample wll have about 25,000 block clusters. Data from an ndependent second enumeraton of the ICM block clusters wll be compared to the Intal Phase estmate usng Dual System Estmaton. In comparson, the 1990 Post Enumeraton Survey (PES) was only about one-ffth as large. A Supreme Court rulng n March 1996 and others have expressed concern about the PES state level total populaton estmates based on data from several states. The offcal populaton of each state and the Dstrct of Columba released on December 31, 2000 wll be estmated drectly from the data from wthn the state. The prmary goal oflcm s to mprove the accuracy of the Congressonal reapportonment process. In statstcal terms, the expected value of most state. populaton estmates should be closer to the true value wth ICM than wth a tradtonal census. Wthout the ICM, the wrong states n terms of ther true populatons may be competng for the last few seats n the House of Representatves. Wth ICM, the rght states are more lkely to be n the competton. The prmary goal of ICM allocaton s to optmze the precson of the apportonment process. In statstcal terms, ICM allocaton attempts to make the state populaton estmates close to ther expected values 1. Optmzng the precson of the apportonment process for Census 2000 would requre decreasng the standard errors of those four to sx states competng for the last three or four seats n the House of Representatves as much as possble and vrtually gnorng the other states. However, precensal estmates wll not be accurate enough to dentfy these last few states. Snce census data are also used for redstrctng, for allocaton of federal and state funds, for plannng purposes, etc., reasonable precson s also requred for those states whose apportonment s certan and for synthetc estmates for substate areas and populaton subgroups. Secton II descrbes the research leadng to the 1 The 1990 reapportonment based on census counts was more precse (closer to ts expected value) but less accurate (expected value mssed the true value) than the apportonment process wll be wth ICM n Census Intal Phase estmates wll be close to ther expected values whch may be far from the true populaton. ICM state estmates wll mss ther expected values, whch wll be closer to the true values, by more than the Intal Phase estmates mss ther expected values. However, the ICM estmates wll generally mss the "true" values by less than the Intal Phase estmates mss the "true" values. The author s a mathematcal statstcan n the Decennal Statstcal Studes Dvson of the US Census Bureau. The author wshes to thank John Thompson, Henry Woltman, Rchard Grffn, and Alfredo Navarro for ther gudance and nsght over the last several years. Ths paper reports the results of research and analyss undertaken by Census Bureau staff. It has undergone a more lmted revew than offcal Census Bureau publcatons. Research results and conclusons expressed are those of the author and do not necessarly ndcate concurrence by the Census Bureau. It s released to nform nterested partes of current research and to encourage dscusson. 593

2 recommended state ICM sample szes usng data from the 1990 Post Enumeraton Survey (PES). Secton III dscusses the possble effect of changes n data qualty from 1990 to Secton IV dscusses the effect of ICM samplng errors on congressonal reappportonment. Secton V provdes a bref summary. II. Methodology Step 1 Redefne Samplng Strata: For the 1990 PES 112 samplng strata were defned based on the Census dvson, degree of urbanzaton, mnorty populaton, and tenure. Some of these samplng strata had very small sample szes. For ths work the 1990 PES samplng strata were collapsed to 39 samplng strata. In addton to a natonal stratum for Amercan Indans lvng on reservatons, each of nne census dvsons has zero, one, or two mnorty redefned strata (total 13, none n New England or the North Central dvson), and two or three non-mnorty redefned strata (total 25). Each state has PES block clusters n from two to sx of the redefned samplng strata. There are 186 samplng stratum/state substrata for non-amercan Indan Reservatons and 14 for the Amercan Indan Reservatons. Step 2: Remove Outler Block Clusters' Thrty-nne block clusters whch contrbute heavly to the error were dentfed and removed. These clusters generally have hgh samplng weghts and accounted for a large porton of the undercount or overcount n the samplng stratuna/state cell. Outlers of the magntude found n 1990 could as much as double the standard errors of the affected states. Several proposed desgn changes wll help to control the effect of outlers n Census 2000: In 1990 large block clusters (over 80 housng unts) were subsampled before PES collecton. The subsamplng resulted n hgh weghts. Increasng the number of large block clusters n the 2000 ICM wll reduce ther ntal weghts. Subsequent subsamplng wll ncrease the weghts back to a normal level. In 1990 only a very small sample of small block clusters (0-2 housng unts) was selected. Durng PES collecton some of these block clusters were found to be much larger, gvng hgh weghts to a large number of housng unts. In 2000 a two stage sample for very small block clusters wll control the weghts of those block clusters whch are found to be much larger than expected. These approaches nvolve addtonal costs and wll not succeed completely n elmnatng the effects of outler block clusters. Step 3" Adjust Weghts to Match State Totals: The PES E-Sample conssts roughly of the 1990 census reports of those persons n the PES sample block clusters. For each block cluster, weghted estmates of the E-Sample are calculated. For many states the weghted E-Sample s not a good estmate of the 1990 census count. For each state the sum of the state's weghted E-Sample estmate s rato adjusted to the 1990 census count. The weghted estmates of erroneous enumeratons (persons n the census who should not have been counted), P-Sample (persons enumerated n the second ntervew n the PES block clusters), and omssons (persons n the P-Sample who could not be matched back to the census) are multpled by the same rato. Step 4" Make State and Stratum Estmates: For each of the 39 samplng strata, dual system estmates are calculated by: ED,,,k-EED,,,k DSED,,,k = CD,,,k PD,,,k where: CDv,k s the census count n stratum k or n ths case the weghted E-sample, Ethyl, s the E-sample estmate n stratum k for the census dvson, EEt~v~, s the estmated number of erroneous enumeratons n stratum k for the census dvson, P~:, s the P-sample estmate n stratum k for the census dvson, and M~v:, s the estmated number of P-sample persons who match to the E-sample n stratum k for the census dvson. A Jacklmfe procedure droppng one block cluster at a tme from each census dvson's PES sample wthout reweghtng: s used to estmate standard errors, SEDv,k, nk.o,, ', and varances, VARDv, k,n~,,, for the E-sample person sample szes, nk.t~v, for the DSE n stratum k for dvson Dv. Snce the fnte populaton correcton factors are neglgble, for the same sample sze, r~,d,, the CV of state for stratum k s the same as the C V for the dvson. 2 The DSE and ts populaton varance can also be estmated by Taylor Seres expanson from the erroneous enumeraton and omsson rates. The results are consstent wth those of the approach used here. Ths more drect approach s preferred because t s smpler and t s consstent wth the 1990 and 2000 varance estmaton methods. Other optons consdered ncluded equal allocatons to all states and varous combnatons of the alternatves. 594

3 Therefore: SE/,k,,,~a E,k ~ = SEDv, lc, n~,z~ EDv, lc Step 5" Determne Block Cluster Sample Szes We assume n o. 2. = 10,000 E-Sample persons. Allocatng these persons proportonally to the states's E-sample populaton n the redefmed strata we have: 0 E,k n,k = ~ E~,k'. k t The standard errors for these stratum sample szes are: SE,e.,n ~ = SE~,k, nkar,, ' n k v o' n,k for each stratum and SE,no - ~~SE2e.,n ~ forthestatetotal. 0 0 The next step s to convert the rt~.,k to b, k, the number of block clusters n state stratum k, usng the observed average block cluster E-Sample person sze for stratum k wthn Census dvson. If SE,6o s the standard error for the block cluster 0 sample sze b correspondng to the 10,000 E-Sample persons, the sample sze, n block clusters, requred to,-,,-. 2 obtan a desred standard error, SE s- b = b ~z~'b~. SE~ An allocaton of 18,873 block clusters s requred to acheve the desred coeffcents of varaton (CV) of 0.5% n all states. These allocatons are shown n column 5 of Table 1.3 Step 6: Assure Mnmum Sample Sze 3 The allocatons for states n the same Census Dvson are correlated because the same populaton varance estmates are beng used. The dfferng proportons of the populaton n each samplng stratum account for the small dfferences between states. It s possble to repeat ths procedure entrely wthn each state. Ths elmnates the synthetc estmaton from the Census Dvsons to the states and the correlaton of the allocatons. Unfortunately, most stratum/state cells do not have suffcent sample to obtan relable estmates. Thrteen states have ICM samples less than 300 block clusters from step 5. These states are concentrated n several dvsons wth relatvely low estmated populaton varance. Snce the estmated populaton varances, whch are subject to hgh varance, may not be as low n 2000 as n 1990, the samples szes for these states are ncreased to 300 block clusters to be more n lne wth the remanng states. These ncreases requre about 1200 block clusters, ncreasng the total allocated so far to about The results are shown n columns 6 and 7 of Table 1. Step 7: Reduce Expected Standard Errors for States wth Populatons over 10,000,000 Reservng 350 block clusters for Amercan Indan Reservatons, about 4600 block clusters reman to be assgned. These are assgned to the largest states proportonately to the squares of ther 1990 census counts 4. Ths reduces the estmated standard errors of the largest states from 0.50% of ther populaton to These decreases are partcularly substantal n the largest states: Calforna, New York, and Texas. The sample sze for Oho was ncreased from 260 to 358, executng steps 6 and 7 smultaneously. The results are shown n columns 8 to 10 of Table 1. Columns 11 and 12 show the number of persons and occuped housng unts whch can be expected n each state. AMERICAN INDIAN RESERVATIONS In 1990 the largest reservatons, spread across fourteen states, were covered by a sample of 43 block clusters. Amercan Indans lvng on reservatons or other trbal lands have specal legal status. In 1990 varances were hgh for ths hard to count populaton of about 800,000 people wth about a 10% undercount rate. 350 block clusters, about as many as the states wth fewer than 10,000,000 resdents, 1.4% of the sample, were set asde for ths 0.3% of the populaton. III. ICM Qualty Concerns The ICM sample szes calculated above are desgned to yeld errors of 0.5% or 56,000, whchever s smaller n all states. Table 1 shows that Calforna would requre 361 block clusters to acheve a CV of 0.5%. Estmates made for the state of Calforna show a CV of about 0.45% for ts PES block clusters. On the other hand, the CVs calculated for the 1995 and 1996 tests were consderably hgher than the desgn estmates. The DSE s roughly the 4 The use of projected 2000 populatons was consdered, but the estmated allocatons for several states seemed napproprate. 595

4 Intal Phase estmate tmes the rate of Intal Phase persons who are correctly enumerated tmes the nverse of the rate of P-Sample persons who could be matched back to census reports: DSE = IP x Rc~ x 1 /R~rcn where both rates are close to 1. There s comparatvely lttle varance n IP, so (assumng equal desgn effects and even wth some correlaton) the varance s proportonal to the sum of two PQ type varances: Rc~X ( 1 -Rc~) VARDs E.~, + I'1 RMArcHX( 1 -RMArc H) where n s the ICM sample sze 5. There are several operatonal changes from 1990 n the desgn for Census 2000 whch may decrease ether the correct enumeraton rate and/or the match rate. A 3% decrease n both the correct enumeraton rate and the match rate from 97% to 94% would not change the estmate much, but t could double the estmated varance, multplyng the estmated CV by about 1.4. These changes nclude: The easy avalablty of Be Counted Forms could ncrease the number of erroneous enumeratons, decreasng the correct enumeraton rate. The use of a fve person form nstead of a seven person form could ncrease the number of persons, especally chldren and nonrelatves, mssed n the ntal phase, decreasng the match rate. The tght schedule and decreased publc cooperaton could ncrease the number of whole household mputatons n the ntal phase, decreasng the match rate. The rate was about 1% n 1990 but about 8% n the 1996 test n Chcago. Not performng a surroundng block search for addtonal matches or performng a lmted surroundng block search could decrease the match rate. There are few counterbalancng changes to mprove the data qualty 6. Therefore, t should be expected that the calculated standard errors for Census 2000 may be somewhat hgher than those predcted by the desgn. 5There s a thrd rato n the DSE formula whch adjusts for whole person mputatons n the Intal Phase. Ths term adds lttle to the varance, but t corrects for census persons who cannot be matched to by P-Sample persons because ther census data s mputed. SThe Be Counted Forms should decrease the number of nonmatches and decreased weght varaton should make the sample more effcent. n IV. Effect on Reapportonment The 435 seats n the House of Representatves are reapportoned to the states usng the Hll Algorthm whch works as follows: Assgn each state one seat. For each state calculate: R = POP / CN x(n + 1) where POP s the populaton of state beng used n the apportonment, and N s the number of seats already assgned to state. Assgn the next seat to the state wth the largest value of R. Calculate new R..s and repeat the process untl all 435 seats are assgned. Usng the 1990 PES nstead of the 1990 census counts would have gven one more seat to Calforna at the expense of Wsconsn. For the two apportonments the 435th seat was assgned as follows: 1990 census count: Washngton Massachusetts was next and would have needed more nhabtants to take the last seat nstead of Washngton PES estmate: Pennsylvana Wsconsn was next and would have requred more nhabtants to take the last seat. Montana was fourth n lne but would have requred only 3919 more nhabtants to take the last seat. The 1990 PES estmate for state can be vewed as a random draw from the normal dstrbuton about the true value. That s: PES s selected from N(T, SEpEs~: Snce we know PES~ we can reverse the stuaton and obtan 100 possble target values of the truth for each state,, by selectng T~,j=l,100 from N(PES,SEpEs~. For each target estmate T~, we can obtan 100 estmates of possble values that the ICM would produce, ICMw,, by samplng from N(Tj,SEIcM.0). Thus, t s possble to compare the apportonment from the 1990 Census to targets and the apportonments from ICM estmates to the same targets. The results are shown n table 2. Usng ether census counts or ICM, there s only a small probablty that the apportonment process wll assgn all 435 seats to the correct states. Over the smulatons of ICM estmates, the 1990 census and 2000 ICM apportonments had the same number of errors compared to the target "true" apportonments 3738 tmes. The 1990 census apportonment had fewer errors 1982 tmes. The 2000 ICM apportonments had fewer errors 4280 tmes. Over the smulatons there were nstances where a state had a dfference between ts 1990 census apportonment and ts 2000 ICM apportonment. In these nstances, the 1990 census apportonment matched the target apportonment 596

5 18270 (43.47%) tmes. The 2000 ICM apportonment matched the target apportonment (56.53%) tmes. Usng the 1990 PES estmates or the ICM estmates, the states wth the most varaton n the target apportonments; that s, the states whch may deserve one more or one less seat, are bunched around the 435th selecton for both the target and the ICM apportonments. On the other hand, even though there s only one dfference between the 1990 census and the 1990 PES apportonments, the states wth varaton n ther target apportonments are not the states clustered around the 435th selecton n the 1990 census data apportonment. Table 2: Number of Seats Shfted Compared to Target Apportonments over 100 Smulatons for 1990 Census or Smulatons for 2000 ICM Census or ICM apportonment equals target apportonment 1990 Census 2000 ICM %, about 16% of the states (8 states) wll lkely have CVs or SEs at least 20% larger than the expected values or above 0.60% or 72,000; and about 2% (1 state) wll lkely have a CV or SE 40% larger than the expected value or above 0.70% or 84,000. Any decrease n data qualty compared to 1990 may further. ncrease the CVs or SEs n The proposed ICM sample szes n 2000 wll be suffcent to assure that the correct states are n the competton for the last few seats n the House of Representatves, but they wll not be suffcent to assure that all 435 seats are apportoned perfectly. The apportonment process s very senstve to mnor populaton varatons and no affordable ICM sample sze can reduce the standard errors enough to assure perfect apportonment. However, a tradtonal census count would vrtually nsure that the apportonment would be ncorrect. It s necessary to explan ths techncal decson to nontechncal audences. Ths opton s relatvely smple. Snce t s smlar to the varance estmaton methods for 1990 and 2000, t should already be famlar to many of the nvolved partes. One seat shfted by census or ICM apportonment compared to target apportonment Two seats shfted Three seats shfted Four seats shfted Fve seats shfted Average number shned V. Summary Issues whch have not been nvestgated are: Only the varance from the ICM sample has been consdered. Varance from samplng for nonresponse follow-up and housng unts returned as vacant by the post offce, wll be small at the state level and wll have no sgnfcant effect on the estmates. A thrd source, varance due to mputaton of mssng data, could be more substantal. For the allocaton of the ICM sample wthn each state, samplng strata and estmaton poststrata wll be developed to permt adequate estmates for race, Hspanc orgn, age, sex, tenure, and geographc subpopulatons. Oversamplng small but vsble subpopulatons could ncrease the state level errors. For the allocaton proposed based on the underlyng populaton varances, t s estmated that a total ICM sample sze of 24,650 block clusters (f allocated approprately and assumng data qualty equvalent to 1990) s suffcent to (1) acheve coeffcents of varaton of 0.50% n states wth populatons less than 10,000,000, (2) allocate each state at least 300 block clusters, and (3) acheve standard errors of for states wth a populaton over 10,000,000. The expected CVs or SEs calculated above are just that: EXPECTED. The ncrease n populaton snce 1990 wll ncrease the standard errors of the largest states from to about Estmates show that the CVs of the estmated CVs or SEs are about 20%. That means that, even f the average state C V s References: Hogan, H. and Wate, J. (1998) "Statstcal Methodologes for Census Decsons Issues, and Prelmnary Results," Proceedngs of the Survey Research Methods Secton, Amercan Statstcal Assocaton, Alexandra, VA, Amercan Statstcal Assocaton, to appear. Grffn, R. and Vacca, E.A. (1998) "Estmaton n the Census 2000 Dress Rehearsal," Proceedngs of the Survey Research Methods Secton, Amercan Statstcal Assocaton, Alexandra, VA, Amercan Statstcal Assocaton, to appear. 597

6 TABLE 1: CEN DIV State (1) o (2) ICM 1CT ] 9 New ME 23 England MA 25 NH 33 RI 44 VT 50 2 NJ 34 Mddle NY 36 Atlantc PA 42 3DE 10 South DC 11 Atlantc FL 12 GA 13 MD 24 NC 37 SC 45 :VA 51 WV 54 4AL 1 East KY 21 South MS 28 Central TN 47 5AR 5 West LA 22 South OK 40 Central TX 48 6IL 17 East IN 18 North MI 26 Central OH 39 WI 55 7IA 19 West KS 20 North MN 27 Central MO 29 NE 31 ND 38 SD 46 8AZ 4 Mountan CO 8 ID 16 MT 30 NV 32 NM 35 UT 49! WY 56 _. 9AK 2 West CA 6 HI 15 OR 41 WA Census (3) ~ Szes for CV/SE Combnatons For Three Varance Alternatves Estmated 0.5% CV 300 ClusterMn SE for States> ?, ' _ % % I % % % % % % % % %! % % % % % % % F 413 ' 413 ' 0.500% F 413 ' 0.500% % ~ % % % %! % % % % % % % % % % % % % % % % % % % n. n. n % / % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 2440 I... p,, % 492! 0.500% % % % % % % % % % % % % ! % %' % % 2837~ ! % % ~ % % % %/ L %L 332U 0.500%l Persons (11) I ~ : Occ HUs (12)