Institute for Policy Research Northwestern University Working Paper Series WP-15-05

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1 Insttute for Polcy Research Northwestern Unversty Workng Paper Seres WP Effects of Census Accuracy on Apportonment of Congress and Allocatons of Federal Funds Zachary H. Seeskn Graduate Research Assstant, Insttute for Polcy Research Doctoral Canddate, Department of Statstcs Northwestern Unversty Bruce D. Spencer Faculty Fellow, Insttute for Polcy Research Professor of Statstcs Northwestern Unversty Verson: May 18, 2015 Address correspondence to Bruce D. Spencer, Department of Statstcs, Northwestern Unversty, 2006 Sherdan Rd., Evanston IL Emal: Ths research was supported by NSF grant SES , NCRN-SN: Census Bureau Data Programs as Statstcal Decson Problems. DRAFT Please do not quote or dstrbute wthout permsson Sherdan Rd. w Evanston, IL w Tel: Fax: w pr@northwestern.edu

2 Abstract How much accuracy s needed n the 2020 census depends on the cost of attanng accuracy and on the consequences of mperfect accuracy. The cost target for the 2020 census of the Unted States has been specfed, and the Census Bureau s developng projectons of the accuracy attanable for that cost. It s desrable to have nformaton about the consequences of the accuracy that mght be attanable for that cost or for alternatve cost levels. To assess the consequences of mperfect census accuracy, Seeskn and Spencer consder alternatve profles of accuracy for states and assess ther mplcatons for apportonment of the U.S. House of Representatves and for allocaton of federal funds. An error n allocaton s defned as the dfference between the allocaton computed under mperfect data and the allocaton computed wth perfect data. Estmates of expected sums of absolute values of errors are presented for House apportonment and for federal funds allocatons.

3 1. Introducton The U.S. Consttuton requres that the populaton be enumerated decennally, for purposes of allocatng Representatves among the states. Representatves shall be apportoned among the several States accordng to ther respectve numbers, countng the whole number of persons n each State, excludng Indans not taxed. The actual Enumeraton shall be made wthn three Years after the frst Meetng of the Congress of the Unted States, and wthn every subsequent Term of ten Years, n such Manner as they shall by Law drect. The Consttuton of the Unted States, Artcle I, Secton 2, as amended by the 14 th amendment Although the Consttuton requres a census, t does not say how accurate the census should be. Accuracy and cost are closely related. The Census Bureau can ncrease accuracy by spendng more money, at least up to a pont. As the great demographer Nathan Keyftz (1979, 46) noted, Askng why the census cannot [accurately] count 100 percent of the populaton n a free socety s lke askng why books contan typographcal errors, why manufactured products often have defects, or why the polce cannot catch all crmnals. Accuracy can be ncreased through nvestment of more resources n the census, but the accuracy wll never be perfect. The current strategy for choosng census accuracy s to specfy a cost target and optmze the accuracy that can be attaned for that cost. The cost target, as we understand t, s consstent wth recommendaton of the Natonal Research Councl (2011, Recommendaton 3) that the cost per housng unt for 2020 be kept at the same (nflaton-adjusted) level as for The 2010 census was estmated to be qute accurate for the total U.S. populaton, so that the natonal net undercount was estmated to be nearly zero (Census Bureau 2012). However, most uses of the census depend on populaton szes for geographc areas or demographc subgroups, and the census was estmated to have a 0.8 percent natonal overcount of non-hspanc whtes, a 2.1 percent natonal undercount of blacks, a 1.5 percent undercount of Hspancs (Census Bureau 2012). There were net overcounts for some states and net undercounts for others. The queston of what accuracy s attanable for the specfed s cost s complex and s beng studed by the Census Bureau. In ths paper, we address the related queston of the consequences of a gven 1

4 profle of census accuracy. We refer to profles of accuracy rather than levels, because census accuracy, lke census statstcs, s mult-dmensonal. For example, census populaton numbers are produced for state and local governments and much smaller areas, and for demographc subgroups both natonally and by geography. To understand the consequences of mperfect accuracy, one needs to know how census data get used. The most vsble uses of the census results nclude ntergovernmental allocaton of funds by formulas usng populaton statstcs, apportonment of the U.S. House of Representatves, and redrawng of Congressonal dstrct boundares. When the census populaton numbers contan errors, the fund allocatons, Congressonal apportonment, and dstrct szes are dfferent than what they would be f the census numbers had no error. We wll refer to the dfferences as, respectvely, errors n allocaton (msallocatons) of funds, errors n apportonment (malapportonment), and errors n dstrct szes. Not all uses of the census are or can be known, and t s mportant to acknowledge that some of the most mportant uses of the census may be the least vsble, ncludng research n socal, economc, behavoral, medcal, and polcy areas and applcatons of that research. The role of census data n polcy development and decson-makng by the Congress and the Whte House, by state and local governments, and by busnesses and other organzatons has not receved suffcent study, but we conjecture that t s mportant. For example, surveys are wdely used sources of nformaton, and almost all natonal populaton surveys whether government or prvate sector, whether by nternet, mal, phone, or n-person drectly or ndrectly use decennal census numbers for adjustng ther results. In ths paper we measure the dstortons n allocatons of representaton and fundng among states that are projected to occur at alternatve profles of accuracy. The fundng formulas and the apportonment algorthm are treated as fxed, and the allocatons that would occur wth error-free statstcs are treated as true values for the allocatons. When actual statstcs are used to compute the allocatons, the resultng allocatons are sad to be emprcal or estmated. The dfference between the estmated allocaton and the true allocaton s called the error n allocaton or, more smply, the msallocaton. The term error s standard usage n statstcs and does not mply that someone made a mstake. The value of mprovng accuracy to reduce msallocatons s a poltcal queston that relates to the queston of how much t s worth spendng on the census, but that we do not address here. We represent accuracy n terms of the multvarate dstrbuton of errors n census estmates for states and the Dstrct of Columba (DC). The focus on the states and DC level s consstent wth the uses of census data for apportonment of House seats among states and for allocaton of federal funds to states 2

5 and DC. The error n an estmate or statstc s the dfference between the statstc and ts true value. The mean squared error (MSE) s the expected value of the squared error, and t s equal to the square of the bas plus the square of the standard devaton. The root mean squared error (RMSE) s the square root of the MSE, and the relatve root mean squared error (relatve RMSE) s the RMSE expressed as a percentage of the true value beng estmated. If the estmate s unbased, the RMSE s equal to the standard devaton or standard error of the estmate, and the relatve RMSE s equal to the coeffcent of varaton (c.v.). Intally, we wll consder several alternatve accuracy profles for the census, as shown n Table 1. Alternatve profles are dscussed n Secton 4. Accuracy Profle base case Descrpton errors for all states (and DC) are ndependently normally dstrbuted, wth zero bas, and equal relatve RMSE correlated case same as base case, except estmates for all areas have constant correlaton of 0.5 accurate small states case dfferental bas case errors for all states (and DC) are ndependently normally dstrbuted, wth zero bas; the smallest 25 states and DC have zero relatve RMSE; the largest 25 states have constant relatve RMSE, such that the weghted average of RMSEs for all 50 states and DC, weghted by the 2010 state census populatons, equals the correspondng relatve RMSE for the base case errors for all states (and DC) are ndependently normally dstrbuted, wth equal relatve RMSE and wth absolute value of the relatve bas equal to the c.v.; the bas for the smallest 25 states and DC has one sgn, and the bas for the largest 25 states has the opposte sgn Table 1. Alternatve Accuracy Profles for the Census The Consttuton requres that seats n the U.S. House of Representatves be allocated to the states accordng to ther respectve Numbers, and the law snce 1941 has requred that the so-called Method of Equal Proportons be used to calculate the apportonment (Balnsk and Young 1982); Secton 3 provdes further explanaton. Defne the true allocaton of seats true a j as number of seats gven to state j when the Method of Equal Proportons s appled to the true populaton numbers (.e., f the census were perfectly accurate). The emprcal or estmated allocatons est aj are the numbers of seats gven when the apportonment method s appled to census populaton numbers. The dfference a est j a true j s the error n allocaton, or msallocaton, to state j. The number of msallocated seats s defned as the sum of absolute errors (.e., sum of absolute values of errors n apportonments), est true aj aj. (1) j 3

6 We used computer smulatons to analyze the dstrbuton of the sum of absolute errors. Fgure 1 shows the strong lnear relatonshp between the relatve RMSE of state populaton estmates and the expected number of seats gong to the wrong states n House apportonment Expected Number of Malapportoned Seats 0% 1% 2% 3% 4% Average Relatve RMSE of State Populaton Base Case Correlated Case Accurate Small States Case Dfferental Bas Case Fgure 1. Expected sum of malapportoned seats n the U.S. House of Representatves under alternatve profles of census accuracy. It should be noted that Fgure 1 shows the expected number of seats msallocated, and that for any partcular census wth a gven accuracy profle the number of seats msallocated could be much greater than the expected number. For example, n the base case wth relatve RMSE equal to 1%, the expected number of malapportoned seats s estmated to be 3.4, but there s a 1 n 6 chance that 6 or more seats are malapportoned; wth relatve RMSE equal to 4%, there s a 3 n 4 chance that 12 or more seats are malapportoned. Blumerman and Vdal (2009) dentfed 140 federal grant and drect assstance programs that apparently dstrbuted funds at least partly on the bass of populaton and ncome data. These programs dstrbuted 4

7 approxmately $446.4 bllon n FY For analyss, we selected the 8 largest programs (n terms of FY 2007 oblgated amount) wth certanty, whch accounted for approxmately 4/5 of the total FY 2007 oblgatons of the 140 programs, and we selected a dsproportonate stratfed sample of 10 of the remanng 132 programs, so that larger programs had a hgher chance of selecton. We used sampleweghtng methods to get unbased estmates of totals for all 140 programs. For each sampled funds allocaton program we estmated the expected sum of absolute errors n state allocatons that would arse from alternatve profles of census accuracy. We found that for 4 of the 18 selected programs, the allocatons would not be affected by error n the most recent census, ndcatng that, although census numbers have been used to determne past allocatons, 2020 census error wll not affect subsequent allocatons. For a few programs, the role of census numbers was lmted to affectng samplng weghts for estmates of nflaton and unemployment. We multpled the estmated sum of absolute errors by 10 to approxmate the effect for the ten-year perod untl the 2030 census, usng the smple but perhaps unrealstc assumpton that the legslated allocaton programs now n exstence wll contnue n ther current form and current fundng levels for , and no new allocaton programs dependent on census results wll be ntroduced. The weghted estmates of the expected absolute errors n federal fund allocatons to states for ten years are shown n Fgure 2. 5

8 Expected Msallocaton of Federal Funds to States over 10 Years Bllons of Dollars % 1% 2% 3% 4% Average Relatve RMSE of State Populaton Base Case Correlated Case Accurate Small States Case Dfferental Bas Case Fgure 2. Expected sum of absolute errors n federal fund allocatons to states over 10 years, as related to census accuracy. 2. Methods 2.1. Overvew Smlar but not dentcal methods are used to estmate the effects of census accuracy on apportonment and fund allocaton. For the former, the true apportonment occurs when the 2020 census numbers for states have zero error. Let P and ˆP denote the vectors of true and estmated populaton szes of states for the census year, The future populaton szes P of states are unknown, and our belefs about them can be summarzed by a pror dstrbuton, developed on the bass of populaton forecasts. Ths s descrbed n Appendx 4. From specfcatons of the accuracy profles (Table 1), and condtonal on the true populaton szes P, we can derve the probablty dstrbuton census estmates, P ˆ. House apportonment s determned purely by the populaton szes, and the dstrbuton of the errors n apportonment,.e., apportonment based on ˆP mnus apportonment based on P, can be derved. 6

9 Unlke apportonment, whch depends only on state populaton szes n 2020, formula-based allocatons of funds depend on a wde varety of populaton statstcs and other statstcs. It would be very complex to jontly forecast the values of all such statstcs ahead to 2020, and the results would lkely be uncertan. Therefore, we took the smpler approach of obtanng the latest values we could of the statstcs used to calculate allocatons for the 18 programs we studed, and treatng those as f they were error-free. To allow for error n the census, we used the accuracy profles (Table 1) to frst develop a dstrbuton of census error, and we developed a dstrbuton of error n the populaton statstcs used to compute the allocatons Apportonment Snce 1941, House apportonment has been determned from census populatons usng the Method of Equal Proportons (also known as Hll s Method and Huntngton s Method). If fractons of seats could be allocated, then state j could smply receve ts quota,, q j defned as the number of seats h n the House of Representatves (currently h 435 ) tmes the fracton of all 50 states census populaton held by state j. Lettng p j denote the populaton of state j and p denote the populaton of all 50 states, we have q ( p / p) h. The allocatons a j of seats to state j j j must be whole numbers, however no fractonal allocatons are allowed. The Method of Equal Proportons chooses postve ntegers 2 a ( p / a p / h ) when the quotas j j j j q j a j that mnmze are gven (Balnsk and Young 1982, 1980, 1975; Spencer 1985). The apportonments are computed by the Census Bureau and provded to the Presdent, who transmts them to Congress. Computaton of apportonment s descrbed by Balnsk and Young (1977) and by the Census Bureau at The Method of Equal Proportons s computed n practce by frst awardng the frst ffty seats one to each state. Seats 51 to 435 are awarded teratvely, each one to the state wth the largest value of p / n ( n 1), where awarded. p s the census populaton of state and n s the number of seats already The senstvty of the apportonment to census accuracy depends n part on the values of the underlyng true populatons of the states. The requrement that the numbers of seats held by states must be ntegers mples that, for some confguratons of states populatons, a change of just a sngle person can cause the numbers to shft (Keyftz 1979). For such confguratons, even the smallest errors n census numbers wll shft the allocatons of seats. To analyze how senstve apportonments are to changes n 7

10 census qualty, we consdered a jont dstrbuton for the true state populatons and the census numbers for states. To formulate the jont dstrbuton, t s suffcent to consder the dstrbuton for the true populaton szes and the condtonal dstrbuton for census error gven the true populaton. The error dstrbuton was specfed usng an accuracy profle from Table 1, wth alternatve levels of average relatve RMSE. The mean of the dstrbuton of true populaton szes was set equal to populaton projectons prepared by the Weldon Cooper Center (2013a, 2013b). The true populaton szes were taken to be ndependent, and the coeffcent of varaton for each state's populaton was chosen to be consstent wth the past level of error n state populaton forecasts wth smlar tme horzon. See Appendx 4 for detals. We estmated the dstrbuton of sums of absolute errors n states allocatons of seats by drawng populaton numbers (true and census numbers) from the jont dstrbuton, as descrbed above. Then the true apportonments true a j and estmated apportonments est a j were calculated for each state j, and the sum of absolute msallocatons (1) was calculated. Ths process was repeated 5,000 tmes ndependently, and the average sum of absolute msallocatons was used to estmate the expected number of malapportoned seats Allocaton of federal funds As we have mentoned, Blumerman and Vdal (2009) dentfed 140 federal grant and drect assstance programs that dstrbuted approxmately $446.4 bllon n funds n FY 2007 at least partly on the bass of populaton and ncome data from the U.S. Census Bureau. The largest of these s the Medcal Assstance Program, also known as Medcad. Grants to states are equal to state medcal expendtures tmes the Federal Medcal Assstance Percentage (FMAP). The FMAP depends on per capta ncome, whch s calculated as the rato of census populaton to Bureau of Economc Analyss (BEA) personal ncome. The formula can be wrtten as 2 I / I FMAP mn max ,0.50,0.83, P / P (2) where I s the BEA personal ncome, P s the census populaton of state, I I, and P P j. j j j For analyss of total msallocated funds across all 140 programs, we selected the 8 largest programs (n terms of FY 2007 oblgated amount) wth certanty, whch accounted for about 80.1% of the total FY 8

11 2007 oblgatons. From the remanng 132 programs, we selected a dsproportonate stratfed sample of 10 so that larger programs had a hgher chance of selecton. The programs we selected are shown n Table 2. We used sample-weghtng methods to get unbased estmates of totals for all 140 programs. For each program, results are weghted by the rato of N h, the number of total programs n stratum to n h, the number of programs sampled n the stratum. We formed the weghted sum of estmates for sampled programs to estmate the total for all 140 programs. Thus, n total, the 18 programs represent $445.6 bllon n funds dstrbuted n 2007 on the bass of populaton and ncome data from the U.S. Census Bureau. Note that ths number s slghtly dfferent from the $446.4 n funds actually dstrbuted n FY 2007 due to the random samplng of programs. For each program studed, we estmated the expected annual amount of absolute errors n state allocatons that would arse from alternatve levels of census data qualty. Samplng errors and approxmate 95% confdence ntervals were also estmated usng theory for stratfed samples. Table 2 shows the sampled programs. h, 9

12 Strat h Nh nh CFDA No. Program FY 2007 Oblgaton ($Bllons) Weghted FY 2007 Oblgaton ($Bllons) Medcal Assstance Program (Medcad) $203.5 $ Unemployment Insurance $35.9 $ Hghway Plannng and Constructon $34.2 $ Suppl. Nutrton Assstance Program (SNAP) $30.3 $ Temporary Assst. for Needy Famles (TANF) $16.5 $ Federal Pell Grant Program $13.7 $ Ttle I Grants to Local Educ. Agences (LEAs) $12.8 $ Specal Educaton Grants to States $10.8 $ Head Start $6.9 $ State Chldren s Insurance Program (CHIP) $5.9 $ Specal Supplemental Nutrton Program for $5.5 $16.6 Women, Infants, and Chldren (WIC) Chld Care Mandatory and Matchng Funds $2.9 $ Chld Care and Development Block Grant $2.1 $ Socal Servces Block Grant $1.7 $ Englsh Language Acquston Grants $0.6 $ Specal Ed. Grants for Infants and Famles $0.4 $ Nonpont Source Implementaton Grants $0.2 $ Ttle V Delnquency Preventon Program $0.1 $3.0 Total $445.6 Table 2. Sampled programs allocatng federal funds. Table 3 presents the varety of the knds of statstcs used to allocate funds across the 18 sampled programs. Annual md-year populaton estmates from the Populaton Estmate Program are used n 9 of the 18 programs. Two programs use model-based estmates for small-area populatons that nclude Census Bureau populaton data n the models. Ttle I Grants to Local Educaton Agences uses Small Area Income and Populaton Estmates for school dstrct school-age chldren n poverty. The Supplemental Nutrton and Assstance Program for Women, Infants and Chldren uses a model-based estmate of the number of chldren age 1 to 4 below 185% of the poverty lne. Two programs use Amercan Communty Survey (ACS) estmates. Specal Educaton Grants to State uses nformaton on state Free Approprate Publc Educaton age chldren n poverty from ACS Publc Use Mcrodata. Englsh Language Acquston Grants uses ACS data on Lmted Englsh Profcency chldren and foregn-born chldren. Current Populaton Survey (CPS) unemployment rates help determne whether states are elgble for addtonal Unemployment Insurance assstance. The CPS uses decennal census nformaton for ts samplng frame. Three programs, Supplemental Nutrton and Assstance Program, Pell Grants and Head Start, all make 10

13 awards based on poverty thresholds. The poverty thresholds are estmated usng the Consumer Prce Index for all Urban Workers (CPI-U), whch s estmated n part wth a samplng frame that uses the decennal census. Program (Dept.) Mdyear Pop. Est. Modelbased Pop. Est. ACS Pop. Est. CPS Unempl. Rate CPI- Urban Non- Census Stats Used Latest Census Not Used Medcad CHIP Chld Care Mandatory & Matchng Chld Care and Development Socal Servces Block Grant Specal Ed. Infants & Famles Ttle V Delnquency Preventon Ttle I Grants to LEAs Specal Ed. States WIC Englsh Language Acquston Unemployment Insurance SNAP Pell Grants Head Start Hghways TANF Nonpont Source Implementaton Table 3. Statstcs used n formulas for allocatng federal funds. Many of the programs that we study use multple census-based statstcs. Further, fve programs also use non-census statstcs n formula-based allocaton. For example, Medcad awards use both census populaton numbers and BEA personal ncome. Surprsngly, we found that for 3 of the 18 selected programs, the allocatons would not be affected by error n the most recent census: Hghway Plannng and Constructon, Temporary Assstance for Needy Famles, and Nonpont Source Implementaton Grants. These three programs have used census data for past allocatons, but future allocatons are fxed to prevous shares. 11

14 Several analytc smplfcatons were necessary for analyzng the effect of census error on the allocatons. Except as noted, the smplfcatons were chosen to have the effect of overstatng the effect of census error on error n allocaton. () Unlke apportonment, whch depends only on census populaton, the fund allocaton programs nvolve other statstcs n addton to census populaton. To fully model the dverse sources of error s too vast an undertakng for ths project. Spencer (1980a, ) demonstrates the knd of nvestgatons that would be needed. For example, BEA personal ncome s used n multple allocaton formulas, but ts accuracy s unknown. We use an approxmaton that condtons on the observed values of the non-census statstcs. If we represent the allocaton to a state by f( x, y), where y denotes the census estmates and x denotes other statstcs, then the true true expected absolute msallocaton may be expressed as E f( x, y) f( x, y ), where true x and true y denote the true values of x and y. true We approxmate ths by E f( x, y) f( x, y ), condtonng on the observed values of x. Work n progress suggests that the approxmaton overstates the effect of census error n some general scenaros and that the potental understatement s smaller than the potental overstatement. () For cases where populaton enters the allocaton formula as a md-year populaton estmate, whch adjusts the census estmate for brths, deaths and net mgraton snce the census, we approxmated the relatve error n the postcensal estmate by the relatve error n the underlyng base census number. Ths approxmaton overstates the effect of census error on the postcensal estmate, snce the errors n estmates of change due to brths, deaths, and net mgraton are only somewhat dependent on the census base (Spencer 1980b). Specfcally, the relatve effect of census error on the census base overstates the relatve effect of census error on the sum of the census base and other components only somewhat affected by census error. () Model-based and ACS populaton estmates are used to calculate the proporton of the populaton n a group or area. The proporton s multpled by a census or postcensal estmate of total populaton to estmate the number n the group or area. Here too, we approxmated the relatve error n the model-based or ACS estmate of populaton of the subgroup by the relatve error n the underlyng base census number. Snce the errors n model-based and ACS estmates of fractons are largely ndependent of the census base, the effect of census error on the census base approxmates the effect of census error on the product of the census base and the modelbased or ACS estmate of the populaton proporton. 12

15 (v) To model the effect of census error on CPS unemployment rates, we used dfferental net undercount estmates by age, sex and race n 2010 and appled these to unemployment estmates for these three groups to study the relatonshp between census error and unemployment rate error. We made the smplfyng assumpton that the effect of undercount by age, race and sex on unemployment rate estmates s proportonal to the effect of state census errors on unemployment rate estmates. For CPI-U, we proceeded smlarly usng dfferental prce ndces for renters and owners together wth nformaton on renter and owner net census undercount. More detals about the procedures are descrbed n Appendx 5. (v) Ttle I Grants to LEAs provde grants to sub-state areas, namely school dstrcts. We take the smplfyng approach of studyng errors n allocaton at the state-level alone. Our models apply the state relatve errors to each LEA populaton estmate wthn the state, whch we conjecture slghtly to some extent understates the effect of census error on the LEA-level Ttle I allocatons (v) For programs that depend upon multple census-based statstcs, we assume the same relatve errors apply to all statstcs, whch overstates the effect of census error. We estmated the expected sum of absolute errors n allocatons for the year for whch the most recent data was avalable. In order to obtan estmates correspondng to FY 2007, we rato-adjusted the estmates of sum of absolute errors by the rato of the FY 2007 program oblgatons to the allocatons for the year for whch allocatons were analyzed. Typcally, ths was a downward adjustment. We conducted 5,000 ndependent smulatons of census numbers and found absolute errors for each federal program analyzed. As prevously descrbed, we used samplng theory to estmate the total expected msallocated funds for FY 2007 for all 140 programs. 3. Results 3.1. Apportonment Table 4 presents estmates of the expected number of malapportoned House seats, or House seats gong to the wrong state, under the alternatve accuracy profles. Fgure 1 s based on Table 4. Table 5 shows that the probablty dstrbuton of number of malapportoned seats for the base case accuracy profle wth relatve RMSE equal to 1%. The number of malapportoned seats can substantally exceed the expected number. 13

16 Estmated Expected Number of Malapportoned Seats Average Relatve RMSE of State Populaton Numbers Accuracy Profle 0.0% 0.5% 1.0% 2.0% 3.0% 4.0% Base Case Correlated Case Accurate Small States Case Dfferental Bas Case Table 4. Estmated expected number of malapportoned seats n the U.S. House, wth varous census accuracy profles. (Estmated standard errors for all numbers do not exceed 0.05.) Relatve RMSE of census numbers Probablty that number of msallocated seats equals or exceeds k k = 2 k = 4 k = 6 k = 8 k = 10 k = %.714 (.006).166 (.005).014 (.002).001 (.000).001 (.000).001 (.000) 1.0%.934 (.004).561 (.007).168 (.005).024 (.002).002 (.001) -- (--) 2.0%.998 (.001).956 (.003).760 (.006).427 (.007).147 (.005).034 (.003) 3.0% (--).998 (.001).975 (.002).867 (.005).640 (.007).339 (.007) 4.0% (--) (--).998 (.001).983 (.002).914 (.004).751 (.006) Table 5. Probablty dstrbuton of number of House seats msallocated, base case census accuracy profle. -- sgnfes number < 0.05%. Number n parentheses s estmated standard error of probablty. 14

17 3.2. Formula-based allocatons of federal funds Estmates of the expected sum of absolute errors n total FY 2007 federal funds allocatons are shown n Table 6. We used the latest avalable values of x and y n estmatng the expected sum of absolute true errors n allocatons. Then, we rato-adjusted the estmates of E f( x, y) f( x, y ) by the rato of the FY 2007 program oblgatons to the allocatons for the year whose allocatons based on x and y were analyzed. Typcally, ths was a downward adjustment. Estmated Expected Msallocated Funds n One Year ($ Bllons) Average Relatve RMSE of State Populaton Numbers Accuracy Profle 0.5% 1.0% 2.0% 3.0% 4.0% Base Case (standard error) (0.04) (0.07) (0.13) (0.19) (0.25) Correlated Case (standard error) (0.03) (0.05) (0.09) (0.14) (0.18) Accurate Small States Case (standard error) (0.04) (0.07) (0.14) (0.20) (0.27) Dfferental Bas Case (standard error) (0.04) (0.06) (0.11) (0.16) (0.22) Table 6. Estmated expected absolute msallocatons of federal funds n one year, wth varous census accuracy profles. 4. Senstvty Analyss The base case accuracy profle specfes that census numbers are ndependent, normally dstrbuted, unbased and have equal coeffcents of varaton across all states. Work n progress wll estmate a response surface and wll provde estmates of expected amounts of msallocated funds and malapportonment as a functon of parameters descrbng the dstrbutons of census numbers. The 15

18 response functon wll account for means and varances of state census numbers, the correlaton between state estmates, and whether or not dstrbutons have heavy tals ( t dstrbuton as compared to normal dstrbuton). Ths wll allow the assessment of the senstvty of dstortons n allocatons to a varety of dstrbutonal aspects of census error. Here, we examne varous assumptons n the base case one at a tme and study how results change for alternatve dstrbutons of census error. For each analyss, we study the mpact on one example of formula-based allocaton, Medcad for FY 2012, and on apportonment. Attenton s restrcted to small relatve RMSE levels Correlated errors To study the senstvty of fndngs to the assumpton of ndependently dstrbuted census numbers across states, we consdered scenaros where state relatve errors were multvarate normally dstrbuted wth a constant correlaton coeffcent. We examned constant correlatons of -0.02, 0.00 (correspondng to ndependence), 0.25, 0.50, 0.75 and Note that the smallest possble constant correlaton for an nn non-negatve defnte correlaton matrx s 1 /( n 1). When n s 51 (for 50 states and D.C.), then the mnmum possble constant correlaton s All scenaros are constructed so that the average relatve RMSE s 1.0%. Fgure 3 presents results for Medcad, and Fgure 4.2 presents results for apportonment. Both fgures demonstrate that sums of absolute errors n allocaton and n apportonment decrease sharply wth ncreased correlaton of state relatve errors. For Medcad, $1.6 bllon n funds s msallocated on average wth ndependent errors, but when relatve errors have a constant correlaton of 0.9, only $0.5 bllon s msallocated on average. Smlarly, 3.4 seats are expected to be malapportoned wth ndependent errors, but only 1.2 seats are expected to be malapportoned when relatve errors have a constant correlaton of

19 Sum of Absolute Errors ($ Bllons) Expected Msallocaton of Medcad Funds to States FY 2012 (1% Relatve RMSE Scenaro) Correlaton between State Relatve Errors Fgure 3. Expected sum of absolute errors n Medcad allocatons FY 2012, as related to the correlaton between state relatve errors. 17

20 Malapportoned House Seats Expected Number of Malapportoned House Seats (1% Relatve RMSE Scenaro) Correlaton between State Relatve Errors Fgure 4. Expected sum of malapportoned seats n the U.S. House of Representatves, as related to the correlaton between state relatve errors. We beleve the reason for ths relatonshp s that when relatve errors are hghly correlated, there s less varaton n the census estmates of state populaton shares. The populaton shares are the quanttes truly of nterest for dstrbutng Medcad funds or for apportonng the House of Representatves. Thus, we fnd that msallocaton and malapportonment decrease wth the correlaton n relatve errors for state census numbers. It s also possble that the fndngs understate the effects of census error f state errors are negatvely correlated. However, the understatement would lkely be small due to bounds on negatve correlatons requred for a non-negatve defnte correlaton matrx. 18

21 4.2. Heavy-taled dstrbutons Our man analyses only consder normally dstrbuted state census numbers. For the 1.0% relatve RMSE scenaro, f a state has populaton p, then the state errors are normally dstrbuted wth mean 0 and standard devaton 0.01 p. We consdered how results would change for dstrbutons wth heaver tals. In partcular, we examned the t dstrbuton wth 4 degrees of freedom. In order to compare dstrbutons wth the same varances, the state errors are dstrbuted t4 (.01 p/ 2), as the t 4 dstrbuton has a standard devaton of 2. Results are presented n Table 7. For Medcad, expected msallocated funds change from $1.56 bllon n the scenaro wth normally dstrbuted census numbers to $1.40 bllon n the scenaro usng the scaled t 4 dstrbuton. Smlarly, expected malapportonment changes from 3.41 seats n the scenaro wth normally dstrbuted census numbers to 3.09 seats n the scenaro usng the scaled t 4 dstrbuton. From ths, we conclude that usng normally dstrbuted census numbers when the dstrbutons have heavy tals leads to overstatng the effects of census error on uses. Dstrbuton Medcad ($ Bll.) Apportonment Normal Scaled t Percentage Dfference -10.7% -9.6% Table 7. Estmated expected sum of absolute errors n allocaton and apportonment wth dfferent dstrbutons of census error 4.3. Bas n state estmates We nvestgated evdence for bas n census populaton numbers by examnng estmates of net undercount rates for the 2000 and 2010 censuses. 1 If bas exsts, then we expect to see that the 2000 net undercount estmates for each state predct ther 2010 net undercount estmates well. Fgure 5 presents the net undercount estmates n a scatterplot wth the dameter of each crcle proportonal to the state's populaton. The seven largest states are labeled. The relatonshp s stronger for large states 1 Data from and accessed February 12,

22 than for small states. Whle the correlaton between the 2000 and 2010 net undercount rates s 0.19, the correlaton weghted by the populaton s Ths suggests more evdence for bas n census populaton numbers for large states than for small states. Fgure 5. Comparson of estmated 2000 and 2010 census net undercount rates by state. Dameter of each crcle proportonal to state populaton. To study the senstvty of our fndngs to bas n state census numbers, we consdered models where the largest 25 states could have bases, but the smallest 25 states dd not. For all scenaros, mean squared error s the same and equvalent to the mean squared error for the 1.0% c.v. scenaro. As mean squared error can be decomposed nto bas squared and varance, we vared the weght on bas squared across the scenaros. We examned weghts on bas squared for the 25 largest states of 0%, 10%, 25%, 50%, 75%, 90% and 100%. The 0% weght on bas squared scenaro corresponds to the 1.0% c.v. scenaro used n our man analyss. We chose the sgns of the state bases to have the same sgns as the bases estmated for the 2010 census. Results are presented for Medcad n Fgure 6 and for apportonment n Fgure 7. We do not fnd much senstvty of ether Medcad msallocaton or malapportonment to bas n large states when the weght 20

23 on bas squared vares between 0% and 50%. In ths range, we expect between $1.5 and $1.6 bllon msallocated and between 3.4 and 3.5 seats malapportoned. However, error n uses ncreases wth larger amounts of bas relatve to varance. When the errors are only due to bas and not varance, we expect $1.8 bllon n funds to be msallocated and 3.9 House seats to be malapportoned. These analyses show that modelng census error wth only varance and not bas could lead to understatng the effects of census error, but the understatement wll only be large f bas s large relatve to varance. Expected Sum of Absolute Errors ($ Bllons) Expected Msallocaton of Medcad Funds to States FY 2012 (1% Relatve RMSE Scenaro) % 20% 40% 60% 80% 100% Bas 2 as Fracton of Mean Squared Error Fgure 6. Expected sum of absolute errors n Medcad allocatons FY 2012, as related to the fracton of mean square error for 25 largest states due to bas squared. 21

24 Expected Number of Malapportoned House Seats (1% Relatve RMSE Scenaro) 4.5 Expected Malapportoned House Seats % 20% 40% 60% 80% 100% Bas 2 as Fracton of Mean Squared Error Fgure 7. Expected sum of malapportoned seats n the U.S. House of Representatves, as related to the fracton of mean square error for 25 largest states due to bas squared Unequal coeffcents of varaton among states The man analyses use equal coeffcents of varaton across all states. We nvestgated the senstvty of fndngs to dfferng coeffcents of varaton across states. We consdered fve sets of scenaros. In all scenaros, ten states were assgned a c.v. of 5.0% and forty states were assgned a 0.0% c.v. Thus, n all scenaros, the average c.v. s 1.0%. We compare scenaros where the ten largest states have the hgh 22

25 c.v., the 11th through 20th largest states have the hgh c.v., the 21st through 30th largest states have the hgh c.v., the 31st through 40th largest states have the hgh c.v. and where the ten smallest states have the hgh c.v. Results are presented n Fgures 4.6 and 4.7. Fndngs should be compared to results from the constant 1.0% c.v. scenaro, where we expect $1.6 bllon n funds to be msallocated for Medcad n FY 2012 and 3.4 House seats to be malapportoned. When the ten largest states have the hgh c.v., sums of absolute errors are very large. We expect $4.9 bllon n funds to be dstrbuted to the wrong state and 10.5 seats to be apportoned to the wrong state. On the other hand, when the ten smallest states have the hgh c.v., we expect only $0.3 bllon to be msallocated for Medcad and 1.1 House seats to be malapportoned. Ths nvestgaton shows that results are hghly senstve to whether large or small states have hgh c.v.'s. 23

26 6 Expected Msallocaton of Medcad Funds to States FY 2012 (Average 1% Relatve RMSE for All scenaros) Expected Sum of Absolute Errors ($ Bllons) Largest 11th-20th Largest st-30th Largest st-40th Largest Smallest States wth Hgh C.V. Fgure 8. Expected sum of absolute errors n Medcad allocatons FY 2012, for scenaros where 40 states have no error and 10 states desgnated have 5% relatve RMSE. 24

27 12 Expected Number of Malapportoned House Seats (Average 1% Relatve RMSE for All Scenaros) 10.5 Expected Malapportoned House Seats) Largest 11th-20th Largest 21st-30th Largest 31st-40th Largest 10 Smallest States wth Hgh C.V. Fgure 9. Expected sum of malapportoned seats n the U.S. House of Representatves, for scenaros where 40 states have no error and 10 states desgnated have 5% relatve RMSE. 5. Conclusons Fgures 1 and 2 show how that census naccuracy can have a large effect on apportonment of the U.S. House of Representatves and on the dstrbuton of more 4 trllon dollars n federal funds over the decade. Our analyss suggests that, f the average percentage error n census numbers for states s 4%, 25

28 the expected number of House seats gong to the wrong state (relatve to havng perfect data) s n the range of 10 to 14, and between $60 bllon and $80 bllon dollars n federal grants n ad wll go to or from the wrong states. It s mportant to understand the lmtatons of the scope of the current analyss. In ths paper we study only drect and specfc nstrumental uses census statstcs for allocatng funds and House seats. We have not studed the effect of census qualty on conceptual uses of census data for scentfc research or for polcy-makng, uses whch are vastly more dffcult to dentfy and descrbe (Beyer 1977). Part of the reason conceptual uses resst study s that they are hdden n chans of analyss. For example, polcy X s adopted or theory Y s accepted on the bass of cted research that depended n part on supportng research that depended on past census data, but the role of the census s not apparent. For another example, former OMB Drector OMB Drector Peter Orszag (2009, 40) noted that the educatonal polcy goal of ncreasng the number of postsecondary educaton was developed to reduce socal nequalty, based on emprcal research of Goldn and Katz (2008, 2007) that reled n key ways on decennal census data from and on Iowa State Census data from Not only s t dffcult to dentfy such uses of census data after they have occurred, but t s even more dffcult to antcpate them ahead of tme. As noted by J. G. March (1994, 246), Havng knowledge when t s needed often requres an nvestment n knowledge that s not known to be needed at the tme t s acqured. The returns from knowledge may occur n a part of the system qute dfferent from the part where the costs are pad. Another knd of use statstcs s for wndow dressng, or usng research results to legtmate and sustan predetermned postons. (Beyer 1977, 17) Symbolc uses of data can be senstve to data qualty, as explaned by Boruch (1984) among others. Suppose that a decson maker smply wants to use data as wndow dressng to defend a decson already made. If the data are hgh qualty, then they wll more accurately descrbe the true state. If the decson maker needs false nformaton to justfy the decson, whch by tself rases questons about the valdty of the decson, then that wll be more dffcult wth hgh qualty data. Unquely, the consttutonally mandated census s also mportant as a natonal ceremony (Kruskal 1984, 49-50). The decennal census s a natonal ceremony and a symbol of the relatonshp between ctzen and government. Whatever one s vew of the census, whatever one s phlosophcal 26

29 poston about the Federal Government, t may be argued that the census s one of our relatvely few natonal, secular ceremones. It provdes a sense of socal coheson, and a knd of nonrelgous communon: we enter the census apparatus as ndvdual denttes wth a handful of characterstcs; then later we receve from the census a group snapshot of ourselves at the ceremony date. Lke many famly pctures, the snapshot s a lttle blurry n spots, but recognzable and fascnatng to compare across the decades... Kruskal s comment provdes background for understandng the observaton made 4 decades ago by Representatve Wllam Lehman (Congress 1973, p.2812):... confdence of the people n the census data s transferred to the confdence of the people n ther poltcal process. In addton, the census s used to adjust or calbrate the results of vrtually all natonal sample surveys of the U.S. populaton n the publc and prvate sectors. In concluson, the effects of naccuracy n the 2020 census on apportonment and allocaton of federal funds are apprecable, but there are also mportant effects of census accuracy. 27

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