Balancing 2020 Census Cost and Accuracy: Consequences for Congressional Apportionment and Fund Allocations

Transcription

1 Workng Paper Seres WP Balancng 2020 Census Cost and Accuracy: Consequences for Congressonal Apportonment and Fund Allocatons Zachary Seeskn Statstcan NORC at the Unversty of Chcago Bruce Spencer Professor of Statstcs IPR Fellow Northwestern Unversty Verson: May 11, 2018 DRAFT

2 ABSTRACT The researchers queston how accurate the 2020 census needs to be, gven that accuracy s expensve but naccuracy dstorts dstrbutons of congressonal seats and federal funds. Although the 2010 census had small measured errors for states, 0.6% on average (as measured by root-mean-square error, RMS), the researchers project that Texas loses and Mnnesota gans a seat f the 2020 census has the same errors. Projectons further show that f 2020 census error for state populatons ncreases to 0.7% RMS, an addtonal seat s lost by Florda and ganed by Oho, and f error ncreases to 1.7% RMS, Texas loses a second seat, to the beneft of Rhode Island. The researchers fnd expected dstortons n fund allocatons ncrease about $9 $13 bllon for each 0.5% ncrease n average error. Correspondence to: bspencer@northwestern.edu. * Ths work was supported by the US Natonal Scence Foundaton (grant SES to Northwestern Unversty). We are grateful to Conne Ctro, Mke Cohen, Jame Druckman, Chuck Mansk, Dane Schanzenbach, and John Thompson for comments. They should not be held responsble for the vews expressed. Any errors are the responsblty of the authors alone. Data and software code used n the analyses are at

3 Balancng 2020 Census Cost and Accuracy 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. (Art. I, Sec. 2, as amended) The Consttuton requres a census but does not say how accurate the census should be. Accuracy and cost are closely related. Perfect accuracy s unattanable at any cost. As demographer Nathan Keyftz 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. (1, 46) Accuracy can be ncreased through nvestment of more resources n the census. Understandng the cost-accuracy tradeoff s crtcally mportant for choosng and evaluatng a census desgn. Assocated wth any desgn s a cost-accuracy curve ( cost curve ) that specfes the cost of attanng a gven profle of accuracy. The cost curve s determned by census technology and socal behavor, ncludng the cooperaton of the publc wth provdng nformaton requested. Fgure 1 shows an llustratve example of the cost curve. Emprcal determnaton of the curve s challengng, and ndeed s a reason for testng and development actvtes at the Census Bureau. Our study analyzes the effects of alternatve levels of 2020 census accuracy on apportonment of the House of Representatves and on allocaton of bllons of dollars of federal funds. We argue that payng attenton to census cost alone, wthout concern for accuracy, leads to large and perhaps counter-ntutve shfts n allocatons and apportonment. 2

4 Accuracy Cost Balancng 2020 Census Cost and Accuracy A Accuracy B Cost Fgure 1. The cost-accuracy curve shows the cost of attanng accuracy and the accuracy attanable at gven cost. (A) Accuracy typcally s attaned at ncreasng margnal cost and (B) addtonal spendng yelds decreasng returns n accuracy. For at least the last fve censuses, hgh accuracy was sought and spendng was adjusted to try to attan t. Ths s evnced by the successful requests by the Census Bureau for addtonal funds n the years just pror to those censuses. 3

5 Balancng 2020 Census Cost and Accuracy By contrast, for the 2020 census, Congress adopted a cost target nstead of an accuracy target, and the Census Bureau s held responsble to acheve acceptable accuracy at that cost. The target was set so that the 2020 cost per housng unt remans at the same (nflaton-adjusted) level as attaned n 2010, or about $12.5 bllon n 2020 dollars (2, Recommendaton 3). Ths s almost 30% below the projected cost of repeatng the 2010 census methods, and s attanable only wth successful nnovatons, notably use of nternet as the man venue for census reportng, use of modern geospatal magng to update malng addresses, use of moble devces by census takers to collect data from households not completng a census form, and use of admnstratve data to remove vacant housng unts and compensate for lack of data from non-respondents. Such nnovatons are stll under development and requre testng under realstc condtons (3, 4). The underfundng of requested census testng and development n the years leadng up to the 2020 census demonstrates lack of concern for accuracy relatve to cost (4, 5). Indeed, although the accuracy attanable for that cost s uncertan at ths pont, the concerns outsde the Census Bureau have focused almost exclusvely on cost (6-10). The present domnatng focus on cost leaves open the possblty that the accuracy attaned by the census may be unsatsfactory for socety s needs (just as a domnatng focus on accuracy would run the rsk of excessve spendng to obtan nconsequental mprovements n accuracy). Statstcal decson theory s a framework that jontly consders both costs and benefts of census accuracy and quantfes the tradeoff. Ths prevents excessve emphass on ether cost or accuracy. The benefts of the census arse from how ts products are used. Reductons n census cost necesstate reductons n census accuracy, and reductons n accuracy lead to dstortons n census uses. In certan stuatons, the beneft of a good can be reflected by ts value n the market. However, the market does not properly value data, as data are a publc good and wll not be 4

6 Balancng 2020 Census Cost and Accuracy adequately provsoned by the free market (11). 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 from what they would be f the census numbers had no error. Hstorcally, census counts understated true sze of populaton, and census error was quantfed by net undercount rate, whch equals the dfference, true mnus census, dvded by true. Although the estmated net undercount rate for 1990 was 1.61%, the censuses n 2000 and 2010 were estmated to exceed true populaton sze natonally, wth net undercount rates estmated at 0.49% and 0.01%, respectvely (12). For census uses that nvolve dvdng a fxed total, ncludng apportonment of the House of Representatves ( House ) and programs that use statstcal formulas to allocate fxed amounts of fund total among states, what matters are the states dfferental undercount rates, defned as the net undercount rate for the state mnus the rate for all states combned. Dfferental net undercount rates are defned analogously for demographc groups, wth estmates shown n Table 1. The dfferental rates are farly consstent across the three censuses, wth non-hspanc Whtes overcounted relatve to the naton as a whole, and Hspancs and non-hspanc Blacks undercounted. Inaccuracy n the census can dstort the reapportonment of the House, where states can gan or lose a seat after only small changes n populaton (1). The dstrbuton of House seats depends on the states shares of populaton and s calculated by the equal proportons method (13-16). Projectons of House reapportonment followng the 2020 census can be calculated from projectons of 2020 state populaton shares (17). To llustrate effects of census naccuracy on apportonment, we modfy the projectons of 2020 state populaton by allowng for census errors. 5

7 Balancng 2020 Census Cost and Accuracy Table 1. Estmated dfferental net undercount rates for demographc groups n last 3 censuses. Source: (12) Estmated Dfferental Net Undercount (%) Group 1990 Census 2000 Census 2010 Census Non-Hspanc Whte Non-Hspanc Asan Hspanc Non-Hspanc Black Non-Hspanc natve Hawaan or other Pacfc Islander Amercan Indan on reservaton Amercan Indan off reservaton n.a Table 2 shows llustratve projectons of wnners and losers under three alternatve levels of census error. The frst column shows the effect on apportonment f errors n 2020 census state populaton shares equal errors measured for the 2010 census (18) Texas loses a House seat to Mnnesota. The last two columns show shfts n House seats f the patterns of error n the 2020 census resemble those measured for states n the 2010 census, but the overall error n populaton shares s exaggerated n 2020 due to underfundng. If the szes of errors n 2020 are 20% larger than for 2010 (RMS sze 0.71 versus 0.59), Florda also loses a seat and Oho gans one; f the 6

8 Balancng 2020 Census Cost and Accuracy RMS szes of the errors n 2020 s 1.67, Texas s projected to lose a second seat, to the beneft of Rhode Island. In relyng on 2010 census error estmates, these projectons may be conservatve due to changng demographcs. For example, Hspancs comprse a larger proporton of Florda s populaton now than n 2010, and Hspancs tend to be undercounted relatve to non-hspanc Whtes. Table 2. Projected gans and losses of House seats at dfferent levels of 2020 census error. RMS 1 relatve error n state 2020 populaton shares State Florda lose 1 lose 1 Mnnesota gan 1 gan 1 gan 1 Oho gan 1 gan 1 Rhode Island gan 1 Texas lose 1 lose 1 lose 2 Every other state Seats shfted ndcates no change. 1 RMS relatve error s root-mean-square relatve error. 2 The measured errors for states the 2010 census had RMS sze As ndcated n Fgure 2, the expected number of changes n House seats due to error n the 2020 census tends to ncrease by about when the root-mean-square (RMS) sze of state errors ncreases by 1%. The RMS sze of state errors s the square root of the mean of the states squared undercount rates; columns 1, 2, and 3 n Table 2 correspond to 2020 census error RMS 7

9 Balancng 2020 Census Cost and Accuracy szes of 0.59%, 0.71%, and 1.67% respectvely. We consdered a varety of parametrc error models, ncludng state undercount rates multvarate normally dstrbuted wth zero mean, equal varance, and constant correlaton, as well as other models (20). The rght-hand axs of Fgure 2 shows the expected number of shfts n House seats for the models wth correlaton 0 and 0.5 as well as the error dstrbutons used n Table 1, whch were patterned on the measured errors for the 2010 census. When the errors are random, the actual number of malapportoned seats can be less than or apprecably greater than the expected number; e.g., n the model wth uncorrelated errors, the actual number of malapportoned seats has about a 1 n 7 chance of beng at least 20 wth RMSE at 4%, at least 16 at 3%, and at least 10 at 2% (20). 8

10 Balancng 2020 Census Cost and Accuracy Fgure 2. Expected funds msallocatons and malapportoned House seats. (FY2015 dollars) Census data affect the dstrbuton of many bllons of dollars of funds more than $675 bllon n allocatons from 132 programs n FY 2015 accordng to a recent Census Bureau study (21). In fact, the cost-beneft analyses that have been carred out to date have focused on uses of census data for allocaton of funds (22-27). Wth so many programs, t s not feasble to study the effects of census error on each program, and we selected a dsproportonate stratfed sample of 18 programs that accounted for 80% of the total oblgatons n FY 2007 (28). Sample weghtng estmates were used to obtan unbased estmates reflectng all allocaton programs lsted n both (21) and (28), and samplng varances were relatvely small (c.v. < 4%). The expected amount of 9

11 Balancng 2020 Census Cost and Accuracy msallocated funds due to census error (f the same programs are n place at the same fundng level for the decade followng the 2020 census) s estmated at $80 bllon for the decade f the RMS sze of the census errors s as large as 4%. As seen n Fgure 2 (left hand axs), the expected amount msallocated ncreases lnearly wth the RMS sze. Actual msallocatons can be hgher or lower than expected amounts. Apportonment and allocatons of funds, along wth redstrctng followng each census, are hghly vsble uses of census data, but they are not the only mportant uses. It s noteworthy 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. 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. Publc health mpacts of census error are dscussed n (29). In concluson, naccuracy n the 2020 census can cause qute large and counterntutve dstortons n dstrbutons of federal funds to states and local governments. If the average rootmean-square error of state populatons s 2%, the expected shfts n fund allocatons s on the order of $40 $50 bllon over ten years and the expected shfts n House apportonment s around 6 seats; f the average RMS error s as large as 4%, the expected shfts double n sze. The actual shfts could be smaller or even greater than the expected values. We hope the average error s much smaller than 2% or 4%, as appears to be the case for prevous censuses (30), but the realty wll strongly depend on the level of census fundng. 10

12 Balancng 2020 Census Cost and Accuracy References 1. N. Keyftz, Informaton and allocaton: two uses of the 1980 census. Am. Stat. 33, (1979). 2. Natonal Research Councl, Change and the 2020 Census: Not Whether But How (Natonal Academy of Scences, Washngton, DC, 2011). 3. U.S. Census Bureau, 2020 Census Operatonal Plan: A New Desgn for the 21st Century, Verson 2.0 (Department of Commerce, Washngton, DC, 2016). 4. B. Tarran, Intervew wth John Thompson. Sgnfcance 14(4), 6-7 (2017). 5. T. Bahrampour. Census watchers warn of a crss f fundng for 2020 count s not ncreased. The Washngton Post (Aprl 18, 2017; 6. U.S. Government Accountablty Offce, (GAO), Hgh-rsk seres: key actons to make progress addressng hgh-rsk ssues (Report GAO R, GAO, 2017; gao.gov/assets/680/ pdf). 7. U.S. Government Accountablty Offce, (GAO), 2020 census: Census Bureau needs to mprove ts lfe-cycle cost estmatng process. (Report GAO , GAO, 2016; gao.gov/assets/680/ pdf). 8. U.S. Government Accountablty Offce, (GAO), 2020 census: addtonal actons would help the bureau realze potental admnstratve records cost savngs. (Report GAO-16-48, GAO, 2015; gao.gov/assets/680/ pdf). 9. U.S. Government Accountablty Offce, (GAO), 2020 census: addtonal steps are needed to buld on early plannng. (Report GAO , GAO, 2016; gao.gov/assets/600/ pdf). 10. U.S. Government Accountablty Offce, (GAO), 2020 census: sustanng current reform 11

13 Balancng 2020 Census Cost and Accuracy efforts wll be key to a more cost-effectve enumeraton. (Report GAO T, GAO, 2012; gao.gov/assets/600/ pdf). 11. C. R. Sms, Can we measure the benefts of data programs? Proc. Am. Stat. Assoc., Soc. Stat. Sec., (1984). 12. H. Hogan, P. J. Cantwell, J. Devne, V. T. Mule Jr, V. Velkoff, Qualty and the 2010 census. Popul. Res. Polcy. Rev. 32, (2013). 13. M. L. Balnsk, H. P. Young, Far Representaton: Meetng the Ideal of One Man, One Vote. (Yale Unversty Press, New Haven, CT, 1982). 14. Census Bureau, (2013). 15. L. Glford, B. Causey, The effect of undercount adjustment on the census. (Census Bureau, 1981). 16. B. D. Spencer, Statstcal aspects of equtable apportonment. J. Am. Stat. Assoc. 80, (1985). 17. K. W. Brace, Some change n apportonment allocatons wth new 2017 census estmates; but greater change lkely by 2020 (Electon Data Servces, Manassas, VA, Dec. 26, 2017; electondataservces.com/wp-content/uploads/2017/12/nr_appor17c3wtablesmapsc2.pdf). 18. T. Mule, 2010 Census coverage measurement estmaton report: summary of estmates of coverage for persons n the Unted States (DSSD 2010 census coverage measurement memorandum seres #2010-G-01, Census Bureau, 2012; census.gov/coverage_measurement/pdfs/g01.pdf). 19. Z. Seeskn, Topcs on Offcal Statstcs and Statstcal Polcy, thess, Northwestern Unversty (2016; 12

14 Balancng 2020 Census Cost and Accuracy 20. Supportng Informaton (appended to ths document). 21. M. Hotchkss, J. Phelan, Uses of Census Bureau data n federal funds dstrbuton, verson 1.0. (Census Bureau, Washngton, DC, 2017; P. Redfern, The dfferent roles of populaton censuses and ntervew surveys, partcularly n the UK context. Int. Stat. Rev. 42(2), (1974). 23. B. D. Spencer, Beneft-Cost Analyss of Data Used to Allocate Funds: General-Revenue Sharng. (Sprnger, New York, 1980). 24. J. Aldrge, 2011 census busness case (General Regster Offce for Scotland, 2006; whatdotheyknow.com/request/8345/response/20302/attach/3/busness%20case.pdf). 25. Merts of Statutory Instruments Commttee, Draft census (England and Wales) order 2009 etc. (Parlament of the Unted Kngdom, 2009; publcatons.parlament.uk/pa/ld200809/ldselect/ldmert/176/17606.htm). 26. C. Bakker, Valung the census (Statstcs New Zealand, Wellngton, 2014; archve.stats.govt.nz/methods/research-papers/topss/valung-census.aspx). 27. B. D. Spencer, J. May, S. Kenyon, Z. Seeskn, Cost-beneft analyss for a qunquennal census: the 2016 populaton census of South Afrca. J. Off. Stat. 33, (2017). 28. L. M. Blumerman, P. M. Vdal, P. M., Uses of populaton and ncome statstcs n federal funds dstrbuton wth a focus on Census Bureau data (Governments Dvson Report Seres, Research Report #2009 1, Census Bureau, Washngton, DC 2009; R. T. Wlson, S. H. Hasanal, M. Shekh, S. Cramer, G. Wenberg, A. Frth, S. H. Wess, C. 13

15 Balancng 2020 Census Cost and Accuracy L. Soskolne, Challenges to the census: nternatonal trends and a need to consder publc health benefts. Publc Health 151, (2017). 30. Natonal Research Councl, Envsonng the 2020 census. (Natonal Academy of Scences, Washngton, DC, 2010). 31. Data and software code are avalable for download from Census Bureau, Annual estmates of the resdent populaton for the Unted States, regons, states, and Puerto Rco: Aprl 1, 2010 to July 1, 2017 (NST-EST ) (Populaton Dvson, Census Bureau, December 2017; Census Bureau, Methodology for the Unted States populaton estmates: vntage 2017 naton, states, countes, and Puerto Rco Aprl 1, 2010 to July 1, 2017 (2017; B. D. Spencer, Models for error n postcensal populaton estmates. Pp n Natonal Research Councl, Estmatng populaton and ncome of small areas. (Natonal Academy of Scences, Washngton, DC, 1980). 35. Census Bureau, Congressonal apportonment (2010 Census Bref, 2011; Census Bureau, Table 2-rev, comparson of populaton estmates and census counts for the Unted States, regons, states, and Puerto Rco: Aprl 1, 2010 (March 2012; 2-rev.xls). 14

16 Balancng 2020 Census Cost and Accuracy 37. Weldon Cooper Center for Publc Servce, Natonal and State Populaton Projectons (Demographc Research Group, Unversty of Vrgna, Charlottesvlle, 2013). 38. Socal Securty Admnstraton, Table, CPI for Urban Wage Earners and Clercal Workers (2018; Catalog of Federal Domestc Assstance (CFDA) (Offce of Management and Budget, Washngton, DC, 2015; Unted States Code Offce of the Law Revson Councl of the U.S. House of Representatves, Washngton, DC; Natonal Research Councl, Statstcal ssues n allocatng funds by formula. (Natonal Academy of Scences, Washngton, DC, 2003). 15

17 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Supportng Informaton S1. Introducton Ths materal provdes addtonal detals about estmates of the dstrbutons of dstortons n allocatons of representaton and fundng among states that arse at alternatve profles of accuracy n the 2020 census. The apportonment algorthm as well as fundng formulas and total fundng amounts as of FY 2007 are treated as fxed. Allocatons (of funds or representaton) that would occur wth error-free statstcs are treated as true values for the allocatons, n contrast to emprcal or estmated allocatons based on naccurate statstcs. The dfference, estmated mnus true allocaton, s the error n allocaton or, more smply, the msallocaton; the absolute value of the dfference s called the absolute msallocaton. Dscusson and results for measures of dscrepancy other than sum (across states) of absolute values, ncludng sum of square errors, mean absolute percentage error, maxmum absolute error, and maxmum absolute percentage error are n (19). The term error s standard usage n statstcs and does not mply that someone made a mstake. Relatve error s defned as the error dvded by the quantty beng estmated. The calculatons of errors n apportonment and n fund allocaton nvolve jont specfcaton of the true populaton and the census populaton numbers for states, or equvalently the true populaton numbers and the census errors. (For fund allocatons, we nclude Washngton D.C. as a state.) Dfferent specfcatons were used for errors n 2020 apportonment n Table 2 and n Fgure 2, and for errors n fund allocaton n Fgure 2. (Note: Tables and fgures are dentfed as Table 1, Fgure 1, etc. when they appear n the man text and as Table S1, Fgure S2, etc. when they appear n ths Supplementary Informaton.) The followng materal dscusses the methods 16

18 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy and data for the results n the man text, and provdes supplementary results. For addtonal data and software code, see (31). The organzaton of the Supplementary Informaton s as follows. Methods, data, and results are dscussed n Sectons S2 S4 for apportonment and n Secton S5 S6 for fund allocatons.. Secton S2 dscusses the data and models used to project ndvdual states errors n apportonment, as shown n Table 2. The true 2020 populaton numbers were projected by short-term lnear extrapolaton of postcensal estmates from 2017, and 2020 census errors were modeled by scalng the measured errors n the 2010 census (18).. Secton S3 dscusses an alternate specfcaton for true 2020 populaton numbers and census errors, whch was used for errors n apportonment reported n Fgure 2. The vector of 2020 true state populaton szes was consdered to be random, wth mean vector equal to state populaton projectons based on the 2010 census and constant relatve varances based on emprcal dfferences between 2010 census numbers and projectons for Aprl 1, 2010 (19). A varety of alternatve parametrc models were developed for 2020 census errors condtonal on the true 2020 populaton.. v. Secton S4 provdes supplementary results. For errors n fund allocaton as dsplayed n Fgure 2, we used a dfferent approach, whch s dscussed n Secton S5. 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. Rather than jontly forecast the values of all such statstcs ahead to 2020, whch would nvolve complexty and uncertanty of forecasts, we obtaned the latest values avalable of the statstcs used to calculate allocatons for the 18 17

19 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy programs studed, and we treated those as error-free. Thus, the true state populaton numbers used n our analyss of allocaton of funds are based drectly or ndrectly on the 2010 census, but not on projectons or forecasts of the 2020 populaton szes. S2. Projected Gans and Losses of House Seats for Indvdual States Shown n Table 2 Frst, we created a projecton of the state populaton szes for apportonment after the 2020 census. Second, we adjusted the projectons accountng for three alternatve levels of 2020 census error. Thrd, we compared the apportonments based on the populatons regardng the projectons as true wth the apportonments based on populatons ncorporatng alternatve error specfcatons for 2020 census error. S2.1. Projecton of true 2020 apportonment populaton szes The projecton of 2020 apportonment populatons s developed n two steps. The frst step took the Census Bureau s postcensal estmates x for 7/1/2016 and y for 7/1/2017 and lnearly extrapolated (projected) forward 33 months (2.75 years) to 4/1/2020 as z = y ( y x ). The Census Bureau develops postcensal estmates by accountng for change snce the prevous census due to brths, deaths, and net movement n and out of the state. The Census Bureau s estmates are avalable n (32) and the underlyng methodology s descrbed n (33). Although undercount n the pror census does affect postcensal estmates (34), for the purposes of ths analyss we are not modfyng the projectons to account for undercount, as such modfcaton would be both complex and uncertan. The second step nvolved modfyng the projecton, z, for dfferences between the census populaton and the apportonment populaton. The modfcaton for state nvolves multplcaton of the projected populaton z by the rato r of the 2010 apportonment 18

20 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy populaton (35) to the 2010 census populaton (36). The projected 2020 true apportonment populaton sze for state s v = r z. Denote the sum of v across the 50 states by v + S2.2. State-level dfferental undercount n 2010 apportonment populatons Three steps were followed to use the estmated net undercount rates for the 50 states n the 2010 census to calculate dfferental net undercount rates for the states. Frst, we calculate undercount-adjusted populaton szes. Second, we use those to calculate the undercount rate for all 50 states combned. Fnally, we calculate the dfferental net undercount rate.. For state, denote the undercount rate n the 2010 census by, u the 2010 census apportonment populaton sze by 2010 v and the true 2010 apportonment populaton sze by t We assume state apportonment populatons have the same undercount rates as the state census populatons. Ths mples u = ( t v ) / t, or t = v ( u ) / 1.. Denote the sum of 2010 v and 2010 t across the 50 states by 2010 v + and 2010, t + respectvely, u t v / t. We may rewrte ths as and defne = ( ) u t u / t =. The dfferental undercount rate for state s defned as d = u u. The dfferental + undercount s a lnear approxmaton to the relatve error of the state share of the apportonment populaton. Substtutng the estmated undercount rate u ˆ and u ˆ+ (18) for u and u +, we estmate the 2010 dfferental undercount rate for state by d ˆ ˆ ˆ = u u +. 19

21 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy S2.3. Modelng state-level undercount n the 2020 census from measured 2010 undercount The projected 2020 apportonment populaton sze 2020 v of state s adjusted for llustratve profles of net undercount n the 2020 census. To do ths, we ntroduce a multpler to apply to the dfferental undercount as n the 2010 census. Ths leads to projected apportonment enumeratons a for The formula for ths s ( ˆ ˆ = + ) a v 1 u d. One can nterpret 1 as less accuracy (larger state dfferental undercounts) than 2010, =1 as the same accuracy as 2010, and 0 1 as more accuracy. If = 0 then there s no error n the state share of apportonment populaton, a / a = v / v. It may be noted that uˆ+ was so close to zero, at %, that smlar results are found f s appled to undercount rather than dfferental undercount. v. Notce that the RMS szes of the dfferental undercount also scale by. Choces of equal to 1, 1.2, and correspond to RMS szes of 0.59, 0.71, and 1.67 as shown n Table 2. Fnally, the apportonments are then calculated usng the Equal Proportons apportonment method wth the 2020 a values as the populaton szes of the states. For 1 < < 1.2 there were 2 House seats msallocated, for 1.2 < < there were 4 seats msallocated, and for = there were 6 seats msallocated; see Table 2. 20

22 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy S3. Jont Dstrbuton of 2020 Populaton True Values and Estmates as Appled to Apportonment n Fgure 2 S3.1. Probablty dstrbuton for 2020 populaton true values For analyss of apportonment as reported n Fgure 2, true populaton szes of states were taken to be multvarate normal wth means equal to projectons for We used projectons made by the Unversty of Vrgna s Weldon Cooper Center for Publc Servce based on the 2010 census results (37) because the Census Bureau stopped producng state projectons. We chose a dagonal covarance matrx wth varances consstent wth emprcal errors n past tenyear projectons for 2010, as dscussed below. Apportonments are ntegers, and t s theoretcally possble that a change n populaton of 1 person can cause a state to gan or lose a seat (1). Specfyng a varance for the true values prevents our estmates from beng senstve to true populaton szes beng necessarly near or far from values that would change apportonments. Smulatons showed the varance around the means to have lttle f any effect on the estmates of malapportonment arsng from census naccuracy. No adjustment was made for dfferences between state populaton and state apportonment populaton. The numercal values are shown n Table S1. 21

23 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Table S1. Specfcaton of moments of state populatons n Coeff. of Varaton Coeff. of Varaton (%) State Mean (%) State Mean.Alabama 5,066, Montana 1,055, Alaska 811, Nebraska 1,908, Arzona 7,604, Nevada 3,328, Arkansas 3,120, New Hampshre 1,446, Calforna 41,715, New Jersey 9,252, Colorado 5,733, New Mexco 2,307, Connectcut 3,723, New York 19,952, Delaware 997, N. Carolna 10,736, Florda 21,784, North Dakota 678, Georga 11,078, Oho 11,763, Hawa 1,489, Oklahoma 3,986, Idaho 1,772, Oregon 4,223, Illnos 13,277, Pennsylvana 12,961, Indana 6,804, Rhode Island 1,085, Iowa 3,085, S. Carolna 5,118, Kansas 3,011, South Dakota 853, Kentucky 4,558, Tennessee 6,919, Lousana 4,635, Texas 28,738, Mane 1,394, Utah 3,193, Maryland 6,282, Vermont 662, Massachusetts 6,806, Vrgna 8,871, Mchgan 10,074, Washngton 7,576, Mnnesota 5,704, W. Vrgna 1,817, Msssspp 3,111, Wsconsn 6,004, Mssour 6,336, Wyomng 594, The varances of the 2020 populaton szes were specfed to be consstent wth the observed levels of error n state populaton projectons prepared a decade earler by the Census Bureau. Specfcally, n 2005 the Census Bureau used 2000 census results to project state populatons for July 1, The error n those projectons was estmated by the dfference between the projecton, Y, and the Census Bureau s populaton estmates for July 1, 2010, X, whch are equal to the 2010 census enumeraton adjusted for brths, deaths, and net mgraton over the 3 22

24 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy month nterval from Aprl 1 to July 1. The relatve error was computed as the error Y X dvded by X, or equvalently Y / X 1. The relatve errors were observed to be approxmately normally dstrbuted about zero, and the relatve errors tended to be closer to zero for the larger states than the smaller states. To model the squared relatve error as a functon of the true populaton sze, a lowess ft of ( Y / X 1) 2 aganst X was conducted n Stata 11 usng a bandwdth of 0.8 and preservng the mean. The lowess ftted values were used as estmates of both the relatve varances of the populaton projectons for 2020 and the relatve varances of the future 2020 state populaton szes. The assumpton of ndependence for the dstrbuton of true populaton szes of states was motvated by the followng consderatons. State populaton projectons typcally are controlled to sum to natonal forecasts, whch account for brths, deaths, and net mmgraton snce the last census. The latter lkely nduce a source of postve covarance among state populaton projectons (f the projectons are treated as random varables). However, the domnant source of error n forecasts of 10 years or shorter wll be uncertanty about nterstate mgraton. Snce the nterstate mgraton flows must sum to zero, the covarances cannot all be postve, but wll have a more complex pattern. For smplcty, the 2020 populaton szes are taken to be ndependent, knowng that only the populaton shares matter for apportonment, and that the shares mplctly nclude some negatve covarances because the sum of shares s always 1. S3.2. Condtonal dstrbuton of 2020 census errors gven true populaton szes S Uncorrelated errors model and correlated errors model Varous parametrc error models were examned to explore the senstvty of fndngs to alternatve error dstrbutons. Two such models were used to construct Fgure 2. Both models 23

25 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy assume relatve errors had a multvarate normal dstrbuton wth zero mean, common standard devaton, and constant correlaton. In the uncorrelated errors model the correlaton was zero, and n the correlated errors model, the common correlaton was 0.5. S Dfferental bas model and accurate small states model Two addtonal error models are the dfferental bas model and the accurate small states model. The dfferental bas model s lke the uncorrelated errors model except that bases are present, wth one sgn for the 25 most populous states ( large states ), and opposte sgn for all others ncludng Washngton, D.C. ( small states ), and equal magntudes of relatve bases for all states; relatve standard devatons of errors for all states were equal to each other and to the absolute value of the relatve bases. The accurate small states model s lke the uncorrelated errors model except that errors for small states were dentcally zero (zero means and standard devatons). For each of these models, specfcaton of the average root-mean-square-error (RMSE) was suffcent to completely specfy the model. S More general error models We also consdered more general models. In these models, each state s relatve error was assumed to be dstrbuted as a lnear functon of a Student s t random varable wth the same degrees of freedom. The error dstrbutons were characterzed by sx parameters: the common correlaton of the errors for each par of states, L the common standard devaton of the relatve error for large states, S the common standard devaton of the relatve error for small states, L the common mean of the relatve error for large states, and S the common mean of the relatve error for small states, and degrees of freedom,. The square of the RMSE 24

26 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy 2 of any state s relatve error equals + 2, and so the average RMSE can be derved drectly. Smlar to the prevously dscussed error models, specfcaton of the average root-mean-squareerror (RMSE) was suffcent to completely specfy the model. S3.3. Smulatng from the jont dstrbuton To conduct smulatons, we frst selected a vector of populaton szes from the dstrbuton descrbed n Secton S3.1 and then selected a vector of relatve errors from the dstrbuton descrbed n Secton S3.2. Ths jont selecton specfes a par consstng of the true populaton vector and the vector of errors. For each par, House apportonment by the Equal Proportons method was computed twce, once for the true populatons and once for the populaton numbers ncorporatng the errors, and the dfferences n apportonment for each state were recorded. The process was repeated, ndependently, 5,000 tmes. S4. Number of Malapportoned Seats n the House of Representatves under Alternatve of 2020 Census Error Models S4.1. Expected number of malapportoned seats Fgure S1 and Table S2 show the expected number of malapportoned House seats under the alternatve jont dstrbutons of populaton and census error presented n Sectons S3.1 S The numbers are derved from the smulatons descrbed n Secton S3.3. Standard errors for all estmates of malapportonment n Fgure S1 and Table S2 are less than 0.05 House seats (19, 37). 25

27 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Fgure S1. Estmated expected number of malapportoned seats under alternatve 2020 census error dstrbutons. 26

28 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Table S2. Estmated expected number of malapportoned seats n the U.S. House, wth alternatve error models. Expected Number of Malapportoned House Seats Average Relatve RMSE of State Populaton Numbers Error model 0.5% 1.0% 2.0% 3.0% 4.0% Uncorrelated Errors Correlated Errors ( = 0.5) Accurate Small States Dfferental Bas Estmated standard errors for all numbers do not exceed Estmates of the expected number of malapportoned House seats under the more general census error models of Secton S3.2.3 can be readly computed usng lnear regresson models that we ftted. The coeffcents of the equatons are shown n the frst row of Table S3. To obtan the coeffcents, we ftted the regresson models to smulaton-based estmates of sums of expected absolute devatons for 973 dfferent possble combnatons of the sx parameters defned n Secton S3.2.3: rangng between 4 and 60, between 0.0 and 0.8, L and S between 0.2% and 5.0%, and L and S between -3.0% and +3.0%. For each combnaton of parameters, the sum of expected absolute devatons was estmated by the average of the sum of absolute errors over 2,500 smulatons. To avod extrapolaton outsde the range of the parameter values used to ft the regresson, the regresson models should only be used to approxmate expected absolute loss wthn the above ranges of parameters. If one wshes to study normally dstrbuted census statstcs, usng = 60 s recommended. For the regresson ft, 2 R =

29 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy The nomnal p-values (assumng normalty) for all regresson coeffcents were below Further detals are n (19). Table S3. Coeffcents for lnear regresson predctons of expected numbers of malapportoned House seats and sums of msallocated funds ($ bll.). const. 2 L S 2 L L 2 S S L L S Seats Funds Note: = ( 60) /10; = 100 1; = 100. Regressor values should be used only n the followng ranges:.0.8;.2 5; 3 3; 4 60, wth 60 used for normal dstrbuton. The followng results are mpled by the regresson model. (a) A census error dstrbuton wth greater kurtoss than normal ( 4 60) leads to smaller absolute errors for constant varance. Wth each ncrease of 10 degrees of freedom, malapportoned House seats ncrease on average by (b) The predcted sums of absolute errors are senstve to the constant correlaton between state census number relatve errors, decreasng by about by 1.60 House seats as ncreases from 0 to 0.8. (c) The sum of expected absolute errors I apportonment s senstve to the coeffcent of varaton of the state populaton numbers, ncreasng by about 2.7 wth each 1.0% ncrease n the c.v. for large states and by 0.3 wth each 1.0% ncrease n the c.v. for small states. 28

30 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy (d) The effect of the coeffcent of varaton for state census numbers on expected sums of absolute errors decreases as the constant correlaton between the state census relatve errors ncreases. For each 0.1 ncrease n the correlaton, the effect of a 1.0% change n the coeffcent of varaton for large states decreases by 0.19 for House seats. Although negatve correlatons are possble, whch would ncrease the effect of coeffcent of varaton, the negatve correlatons cannot be too large n magntude because the correlaton matrx s non-negatve defnte. For example, the mnmum possble constant correlaton for the census numbers of the 50 states and D.C. s (e) The sum of expected absolute errors n apportonment s senstve to the relatve bases of state census numbers, although less than to the coeffcent of varaton. As L and S vary between -3.0% and +3.0%, expected House malapportonment vares by about 1.5 House seats up or down. The relatonshp s convex, reflectng ncreased malapportonment wth the magntude of census bas. S4.2. Probablty dstrbutons of number of malapportoned seats The number of malapportoned seats s random and can be much greater than the expected number. Table S4 Table S7 dsplay the estmated probablty dstrbutons for the number of malapportoned seats under the alternatve error models and alternatve levels of relatve RMSE of census numbers for states. 29

31 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Table S4. Estmated probablty dstrbuton of number of House seats msallocated, uncorrelated errors accuracy profle. 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).013 (.002).000 (.000) -- (--) -- (--) 1.0%.934 (.004).561 (.007).168 (.005).025 (.002).002 (.001) -- (--) 2.0%.998 (.001).956 (.003).761 (.006).428 (.007).147 (.005).035 (.003) 3.0% (--).998 (.001).975 (.002).867 (.005).640 (.007).340 (.007) 4.0% (--) (--).998 (.001).983 (.002).914 (.004).752 (.006) Relatve RMSE Probablty that number of msallocated seats equals or exceeds k of census k = 14 k = 16 k = 18 k = 20 k = 22 k = 24 numbers 0.5% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 1.0% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 2.0%.004 (.001).001 (.000) -- (--) -- (--) -- (--) -- (--) 3.0%.132 (.005).037 (.003).009 (.001).001 (.001) -- (--) -- (--) 4.0%.520 (.007).296 (.006).133 (.005).048 (.003).012 (.002).002 (.001) -- sgnfes number < 0.02%. Number n parentheses s estmated standard error of probablty. Table S5. Probablty dstrbuton of number of House seats msallocated, correlated errors accuracy profle. 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 = %.578 (.007).077 (.004).004 (.001) -- (--) -- (--) -- (--) 1.0%.846 (.005).330 (.007).052 (.003).003 (.001).000 (.000) -- (--) 2.0%.984 (.002).812 (.006).427 (.007).124 (.005).019 (.002).001 (.001) 3.0%.999 (.000).968 (.002).816 (.005).500 (.007).206 (.006).055 (.003) 4.0% (.000).996 (.001).958 (.003).807 (.006).539 (.007).257 (.006) Relatve RMSE Probablty that number of msallocated seats equals or exceeds k of census k = 14 k = 16 k = 18 k = 20 k = 22 k = 24 numbers 0.5% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 1.0% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 2.0% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 3.0%.008 (.001).001 (.000) -- (--) -- (--) -- (--) -- (--) 4.0%.084 (.004).020 (.002).003 (.001).001 (.000) -- (--) -- (--) -- sgnfes number < 0.02%. Number n parentheses s estmated standard error of probablty. 30

32 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy Table S6. Estmated probablty dstrbuton of number of House seats msallocated, accurate small states case accuracy profle. 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 = %.744 (.006).179 (.005).015 (.002).000 (.000) -- (--) -- (--) 1.0%.955 (.003).611 (.007).196 (.006).031 (.002).003 (.001).000 (.000) 2.0%.999 (.000).970 (.002).813 (.006).490 (.007).190 (.006).045 (.003) 3.0% (--).998 (.001).982 (.002).900 (.004).699 (.006).424 (.007) 4.0% (--) (--).998 (.001).988 (.002).934 (.004).804 (.006) Relatve RMSE Probablty that number of msallocated seats equals or exceeds k of census k = 14 k = 16 k = 18 k = 20 k = 22 k = 24 numbers 0.5% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 1.0% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 2.0%.007 (.001).001 (.000) -- (--) -- (--) -- (--) -- (--) 3.0%.196 (.006).062 (.003).014 (.002).003 (.001).002 (.001).000 (.000) 4.0%.607 (.007).377 (.007).185 (.005).074 (.004).029 (.002).008 (.001) -- sgnfes number < 0.02%. Number n parentheses s estmated standard error of probablty. Table S7. Estmated probablty dstrbuton of number of House seats msallocated, dfferental bas accuracy profle. 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 = %.662 (.005).123 (.003).008 (.001).001 (.000) -- (--) -- (--) 1.0%.902 (.003).462 (.005).106 (.003).011 (.001).001 (.000) -- (--) 2.0%.995 (.001).911 (.003).610 (.005).256 (.004).066 (.002).011 (.001) 3.0% (.000).990 (.001).919 (.003).707 (.005).409 (.005).169 (.004) 4.0% (--).999 (.000).988 (.001).936 (.002).788 (.004).544 (.005) Relatve RMSE Probablty that number of msallocated seats equals or exceeds k of census k = 14 k = 16 k = 18 k = 20 k = 22 k = 24 numbers 0.5% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 1.0% -- (--) -- (--) -- (--) -- (--) -- (--) -- (--) 2.0%.001 (.000).000 (.000) -- (--) -- (--) -- (--) -- (--) 3.0%.050 (.002).009 (.001).001 (.000).000 (.000) -- (--) -- (--) 4.0%.294 (.005).124 (.003).039 (.002).009 (.001).001 (.000).000 (.000) -- sgnfes number < 0.02%. Number n parentheses s estmated standard error of probablty. 31

33 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy S5. Expected Sums of Errors n Fund Allocatons Due to Census Error A recent study (21) found census data affect the dstrbuton of hundreds of bllons of dollars n allocatons from more than 100 dfferent programs (132 programs allocated more than $675 bllon n FY 2015). Ths updated an earler study s fndng (28) that 140 federal grant and drect assstance programs dstrbuted approxmately $450 bllon n FY 2007 at least partly on the bass of populaton and ncome data. Analyzng the effect of census error on fund allocatons s more complcated than for apportonment. There are many allocaton programs and they typcally are complex, nvolvng statstcs other than just state populaton numbers and usng census numbers n dfferent ways. To model the accuracy of the other statstcs n even a sngle program can tself be a major undertakng even for a retrospectve analyss (23). The varous statstcs n the allocaton formulas change over tme, and f we were to use 2020 populaton numbers for smulatng fund allocatons, we should also use future values of the other statstcs n computng the allocatons. We had no confdence that we could forecast the future values of the other statstcs accurately even f we had the resources to carry out the forecastng, and so we used the latest avalable numercal values of all of the statstcs the government used to compute the allocatons as f they were true numbers. To analyze the effect of census error, we used the models descrbed n Secton S3.2. Ths approach of condtonng on observed statstcs as f they were true and addng error to the census populaton numbers may, dependng on the extent of bases n the other statstcs, lead to overstatement of the effect of census error (19). A stratfed smple random sample of 18 formula-based fund allocaton programs was selected from the 140 lsted n (28) as usng Census Bureau populaton or ncome data to determne the allocatons. We selected wth certanty the 8 largest programs, whch accounted for 4/5 of the total FY 2007 oblgatons, and we selected a dsproportonate stratfed sample of 32

34 Supplementary Informaton for Balancng 2020 Census Cost and Accuracy 10 of the remanng 132 programs. The sample desgn and selected programs are shown n Table S8. The samplng ncluson probablty for a program n stratum h s equal to nh / N h, wth nh the sample sze and N h the populaton sze n the stratum. Samplng weghts were set equal to N / n, the recprocal of the ncluson probablty. h h For each selected program, we analyzed the effect of census error on allocatons, as descrbed n Secton (19, 31-37). For any gven parametrc model of census error, the sum of msallocatons for the selected program was smulated, just as descrbed for apportonment n Secton S3.3 except that the true value for the populaton was held fxed. The average across smulatons was calculated for each program. The average was then multpled by the rato of the FY2015 oblgaton from (21) to the total amount allocated for the year for whch the data were avalable and analyzed. The rato-adjusted amount provdes an estmate of the sum of FY 2015 msallocatons due to census error for the selected program. Fnally, results were multpled by ten to reflect estmates of the effect of the decennal census on the sum of msallocatons over a decade n 2015 dollars. The weghted sum of the latter (rato-adjusted amounts) was calculated, usng samplng weghts equal to N / n, wth h h h denotng the stratum to whch the program belonged. The weghted sum estmates the sum of the expected values of msallocatons for all 140 allocaton programs n (28) f ther allocated amounts were equal to the FY2015 oblgatons n (21). However, the populaton sampled (a) excludes 7 programs that came nto beng between FY2007 and FY2015, and whose FY2015 oblgatons totaled $93.9 bllon, and (b) ncludes 15 FY2007 programs that dd not exst n FY2015, totalng $2.3 bllon n 2015 dollars (19). (FY2007 dollars were converted to FY2015 dollars accordng to the Consumer Prce Index for Urban Wage Earners and Clercal Workers (38), yeldng an adjustment factor of / = ) 33