SMALL AREA ESTIMATES FROM THE AMERICAN COMMUNITY SURVEY USING A HOUSING UNIT MODEL

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1 SMALL AREA ESTIMATES FROM THE AMERICAN COMMUNITY SURVEY USING A HOUSING UNIT MODEL Nanak Chand and Donald Malec U.S. Bureau of the Census Abstract The Amercan Communty Survey (ACS) s desgned to, ultmately, rovde census long-form nformaton on a contnuous bass. Although the am of rovdng current soco-economc data on the oulaton can be realzed, a yearly samle sze equal to the tradtonal census long-form samle sze would be rohbtvely exensve. The am of ths work s to roduce a small area estmaton method that accounts for the samle desgn and does not assume that the wthn tract varance s estmated wthout error. In future work, ths model can be easly extended to ncororate more covarates, at any of the levels, or to nclude data collected at revous tmes. Keywords: Herarchcal Model, Arcsne square root transformaton, Unt level Small Area Model Introducton In an effort to rovde estmates for census-tye aggregatons such as tracts, on a yearly bass, small area methods can be emloyed. Recent revew artcles on small area estmaton methods nclude Marker (1999) and Rao (1999). We roose a herarchcal model of ersons wthn housng unts wthn tracts for makng tract level estmates. Besdes develong estmates from ths model, we nvestgate ossble gans of ths aroach over nferences from a standard model that assumes that the estmated wthn tract samlng varances are known. The urose of modelng erson characterstcs, wthn housng unts, wthn tracts s to be able to estmate and secfy the varablty of the wthn tract samlng error and resultng effects on small area estmates. Comarsons of estmates are made assumng that the, more comlex, housng unt model s true. The amount of borrowng s also evaluated. In addton, redctons of desgn-based tractlevel summares are comared wth the actual samled data. Also, the ft of the model as a descrton of the wthn tract varablty s evaluated grahcally. Based on the above comarsons, the utlty of usng the housng unt model wll be assessed. Estmates, and ther estmated recson, are roduced usng Monte Carlo Markov Chan methods va a non-subjectve Bayesan aroach. As an llustraton of the method, we generalze the model used by Chand and Alexander (1995) for makng tract-level estmates of the ercent of ersons n overty. Ther model secfes a tract-level lnear relatonsh between the arc-sne square root of the roorton of ersons n overty and tract-level ncome characterstcs from ncome tax returns. We ncororate ths model nto one that models ersons n a housng unt va a famly (who are ether all n overty or not) and unrelated ersons (who have an ndvdual overty ndex) lvng n the same housng unt. Our model ncludes a rovson that the overty status of unrelated ndvduals may deend on the overty status of the housng unt's famly. In order to account for the samlng varablty and to make estmates at the tract level, we nclude a herarchcal multnomal model of housng unt characterstcs. The same data set, as used by Chand and Alexander, consstng of a samle contanng 163 Oregon census tracts, collected n 1996 wll be used. A samlng fracton of 15% was used for ths samle. The medan wthn-tract samle sze s 19 housng unts. About 5% of the samled tracts have 47, or fewer, housng unts n samle and about 95% have a samle sze of at least 351 Ths aer reorts the results of research and analyss undertaken by Census Bureau staff. It has undergone a Census Bureau revew more lmted n scoe than that gven to offcal Census Bureau ublcatons. Ths reort s released to nform nterested artes of ongong research and to encourage dscusson of work n rogress.

2 The Poulaton Model The Amercan Communty Survey s a systematc samle of housng unts. Because a systematc samle of housng unts s selected, t s assumed that there s no samle selecton bas at the housng unt level. There may be a selecton bas wthn housng unts. We roose a model of ersons wthn housng unt to account for a ossble selecton bas due to correlaton wthn housng unts. Snce erson characterstcs tend to cluster wthn household, a model that treats ndvduals as ndeendent observatons s narorate. A model that can account for some degree of wthn housng unt correlaton wll be used, here, to crcumvent ths roblem. An alternatve aroach s to use estmates based on a smle random samle but adjust the varance to take nto account the cluster samle. Ths latter aroach s emloyed by Chand and Alexander (1995) who use a jackknfe method to adjust varance. Although ths latter aroach rovdes an arorate adjustment of varance, an emrcal Bayes tye aroach s emloyed and the adjustments are treated as known. Any samlng error of these varance estmates s not accounted for n dervng an estmate. Snce borrowng strength s drectly related to the amount of wthn and between varance, not accountng for ths error could bas the results. By contrast, the housng unt model wll automatcally adjust borrowng based on the uncertanty of the varance estmates. Estmates from the two aroaches wll be comared. Wthn a State, a two-stage model s emloyed. A model of housng unt characterstcs s ostulated. Then, wthn a housng unt, a model of ndvdual characterstcs wthn a housng unt s rovded. In ths relmnary develoment, housng unt sze and comoston nto famly members and unrelated housng unt resdents are modeled. Subfamles are consdered as art of the famly and share famly characterstcs. In ths alcaton ersons below overty are of nterest. Here, the salent features of the model are that all members of a famly are ether n or out of overty. Unrelated ndvduals wll have ther own unque overty status however, a model s emloyed whch wll account for ossble correlaton between famly overty status and the overty status of unrelated ndvduals wthn the same housng unt. Further modelng of famly characterstcs as a functon of housng sze, demograhc characterstcs, etc. could be nvestgated n the future. As n Chand and Alexander, admnstratve records are emloyed to model tract varablty of overty rates. Heurstcally, the model for an ndvdual s overty status deends on whether he or she s a famly member, or not: P( erson sn overty n a famly) = P(famly sn overty) P(unrelatederson sn overty famly overty status) = P(erson sn overty famly overty status) P(famly overty status) In order to estmate the overty rate or count at the ndvdual level, a model for the number and comoston of housng unt resdents s needed. A multnomal model wth robabltes of the form: P ( HU contansexacly f famly membersand u unrelated ersons) = wll be used. The Wthn Tract-level Poulaton Model In order to utlze tract-level data to estmate ossble unque tract-level features, the above models wll all have tract level-secfc arameters. A herarchcal model across tracts, wthn a state, wll be secfed n order to ncrease the samle sze whle estmatng common features across tracts. Further herarches, e.g., across states, can be ncluded. However, only Oregon s used n ths analyss, so State s not secfed here. The Wthn Tract-level Housng Unt Comoston Model Formally, wthn tract, : a, K 1, a ~Multnomal ( a,,, ) T. 1 K T where T: the number of unque housng unt comostons, n samle. - The T tyes conssts of the unque ars, (f,u), of famly sze, f, and number of unrelated ersons, P fu

3 u, n a housng unt. Ths ncludes vacant housng unts k=(0,0). By conventon, occued housng unts wll have at least one famly. - It s assumed that ths set of unque housng unt tyes s dverse enough to reresent the oulaton of unque housng unt tyes. a : The number of housng unts n tract who have comoston of tye k. k k : The assocated robablty that a housng unt n tract s of comoston tye k. An alternatve but equvalent secfcaton of the housng unt model s to defne a multvarate ndcator random varable, δ = δ, K, δ ), such that h ( h1 ht 1, f hu comoston s tyek = ( fk, uk ) δ hk =, and 0, otherwse δ ~ multvarate Bernoull ( h1, K, ht ), ndeendent. h The Wthn Tract-level Poverty Status Model Wthn tract,, wthn an occued housng unt, h, of tye k=(f,u): I ~Bernoull ) bnomal u, I + (1 I ), where,,m.h h I h m.h 0 ( 0 ( h h : 1/0 ndcator of whether famly n housng unt h n tract s/s not n overty. : then number of unrelated ersons n unt h n overty : tract level robablty of famly overty status : tract level robablty of overty status of unrelated ersons n housng unts wth famles N n overty. : tract level robablty of overty status of unrelated ersons n housng unts wth famles not n overty. Note that the above model secfes two tyes of deendences wthn a housng unt; that famly member overty status s all or nothng and that the overty status of an unrelated ndvdual s deendent on the overty status of the famly resdng n the same housng unt. In summary, the lkelhood wthn tract,, s roortonal to 0, where m n 0-m 0 m n -m mn nn - mn (1- ) 0 0 (1- ) (1- ) N N n o = the number of occued housng unts n samle, n tract mo = the number of famles n overty n samle, n tract n m = the number of unrelated ersons lvng wth famles n overty n samle, n tract = the number of unrelated ersons n overty lvng wth famles n overty n samle, n tract n N =the number of unrelated ersons lvng wth famles who are not n overty, n samle, n tract m N = the number of unrelated ersons n overty lvng wth famles who are not n overty n samle, n tract. The counts of housng unt tyes a k have been defned. The Between Tract-level Poulaton Model T k = 1 ak k

4 The between tract-level model s secfed as a dstrbuton of the tract-level arameters: 0,, N and k. The Between Tract-level Housng Unt Comoston Model A herarchcal, multnomal dstrbuton wll be secfed for the dstrbuton of housng unt tyes wthn tract. A shercal transform (a generalzaton of the arcsne, square root transform) on the multnomal robabltes s used because covarates can be ncluded, relatvely easly (unlke multnomal / Drchlet models) and because t can be generalzed to accommodate robabltes wth mass at zero or one (unlke logstc transforms). Defne the shercal transformaton of the multnomal robabltes ' 1 sn θ = 1 j-1 ' ' j = sn θ 1 cos θ r r=1 T-1 ' T = Π cos θr r=1 Π, 1 < j < T Further, defne - < θ <, such that θ ' j j 0 = θ j θ 0 < θ Allowng θ j to range over the real lne enables one to model zero robabltes (and one's, too) wth j j 0 < θ ostve ont mass. It s exected that many tracts wll not have housng unts of certan tyes. Ths tye of model wll be able to reresent these cases. There wll be very lttle data n each tract to estmate the arameter of the multnomal model. A herarchcal model between tracts s utlzed to borrow data by lettng θ ~ N (, γ ), nd.,, j = 1,.., T - 1. j µ j j The secfcatons for the housng unt model are comleted wth the ndeendent rors for the γ ' j s. µ ~ N ( µ, σ µ ) j µ j j γ - j ~ Gamma (d?, d? ). The arameters, σ µ j and d? are chosen so that they have a neglgble effect on estmaton and other nference. The Between Tract-level Poverty Status Model j µ ' j s and Borrowng of nformaton data on overty status arameters across tracts wll be acheved n two ways. Frst, a regresson relatonsh across tracts based on avalable covarates wll be ostulated. Second, random tract effects wll be ncluded to catalze on any remanng smlartes of the arameters across tracts. Defne,

5 ' ' 0 = sn (x β + t ), f 0 < d0 = x β + t < ' = sn (x ) β + t + ν + z, f 0 x ' < d = β + t + ν + z < and ' N = sn (x β + t + ν N + zn ), f 0 x ' < d N = β + t + ν N + zn <. If d z ( 0, ), Defne 0, f d z 0 z =. 1, fdz The x are the known tract-level IRS covarates used by Chand and Alexander n modelng overty status: x 1 = 1, x = ln(medan ncome) x 3 = ln(er cata ncome) x 4 = ln(q L ) x 5 =ln(q U ) 1 x 6 = sn P, where Q L, Q U and P V are resectvely, the lower quartle ncome, the V uer quartle ncome and the roorton of ersons below overty level n the tract. t s a random tract effect?,? N are fxed effects denotng the nfluence of a famly's overty status on unrelated ndvduals n the housng unt z, z N are the corresondng tract-level random effects of a famly's overty status nfluence on unrelated ersons n the housng unt. The herarchcal model for overty status s comleted by defnng w = (t, z, z N )' and secfyng that w ~ N (0, 3 ). Indeendent rors for the locaton arameters, a = [ ß',?,? N ], and the scale arameters, 3-1, are secfed as a ~ N (µ a V a ), and ~ Wshart d Σ, MΣ. dσ As wth the housng unt comoston model, the arameters, V a and d Σ, are chosen so that the ror has a neglgble effect on the resultng nference. Estmaton Ultmately, tract level estmates of the erson-level overty rate, wth estmates of recson, are needed. Estmates are based on the avalablty of recse estmates of the total number of housng unts, H, n each tract (avalable as adjusted counts from the samlng frame). Gven the total number of housng unts n tract, the overty rate n tract s: H T H T POVR = fkδ hk I h + mh fkδhk + uk, where h=1 k = 1 h=1 k = 1 f k and uk are, resectvely, the famly sze and number of unrelated ersons contaned n a household wth comoston of tye, k.

6 The dstrbuton of POVR s comletely secfed from the model of secton. The osteror redctve dstrbuton wll also be roer snce only roer rors are used. The osteror mean and varance of POVR wll be determned and used as estmates of locaton and scale. Although an analytcal equaton s not avalable, these estmates are made numercally. Inference of all model arameters wll be made from ther osteror dstrbuton. Ths wll be accomlshed usng Markov Chan Monte Carlo methods successvely aled to the condtonal osteror dstrbutons of the arameters. In artcular, the adatve rejecton algorthm wll be used on the condtonal osteror of the arameters, θ, j β, ν N, ν, t, z N and z. The condtonal osteror dstrbutons of the remanng arameters are all ether Normal or Wshart dstrbutons or the Gbbs samler s used. A Tract-Level Model As mentoned n the ntroducton Chand and Alexander used a model of data aggregated at the tract level to make estmates of overty. Unlke the housng unt model, the tract-level model does not account for the uncertanty of the wthn tract varance when estmatng overty rates and assocated recsons. Ths feature that may affect the amount of borrowng and may affect the total recson of the resultng estmates. However, the extra effort of usng the housng unt model may not be necessary and estmates from both models are comared to each other to assess whether s are any ractcal dfferences between the two. For the uroses of comarson, we wll use the followng tract-level model: where ' g ~ ˆ N (x b + d, v ) d ~ N(0, τ ), ndeendent, 1 g = sn ( ˆ ), ˆ, s the weghted estmate of erson-level overty rate, n tract. v, s estmated samle varance of ˆ ˆ obtaned usng a jackknfe on housng unts. Although ths s an estmate based only on data from tract,, t s assumed to be fxed and have no varablty. Note that the tract level model mles that ˆ ' E( b, d ) sn (x b + d ) = 0 =. For comarson uroses, a Bayesan analyss wll be used for ths model, also. Keeng the tye of nference the same for both models wll rovde a more even comarson. As wth the housng-unt level model, over-dsersed rors are assgned to the remanng arameters: b ~ N (µ b V b ), and α, β. τ - ~ Gamma ( ) By defnton, the tract-level model s not secfed below the tract level. Hence, estmates overty rate based a redctve dstrbuton of unsamled housng unts cannot be obtaned. Instead, the osteror mean and varance of 0 wll be used as estmates of the locaton and scale of overty rate Note that the tract-level model, as secfed, cannot rovde tract-level estmates of the total number of ersons n overty because the oulaton sze has not been ncluded n the model. The housng unt model ncludes a model for oulaton and can also be used to estmate the total number of ersons n overty. In order to estmate overty counts from a tract-level model an addtonal model of oulaton counts wll need to be ft.

7 Model comarson The housng unt model has been formulated to model the wthn tract varance. Snce the tract-level model varances are obtaned emrcally, there s no one-to-one corresonds between the two models at the tract level. To evaluate the adequacy of the housng unt model, redctve samles are generated from the model and resultng redctve jack-knfed estmates of varances are obtaned. If the redctve dstrbuton contans the samled jack-knfed estmates wth hgh robablty, the model wll be deemed adequate. Above the tract level, the aggregate-level model s a secal case of the housng unt model. Ths can be seen as follows. Frst, defne the samle overty rate as: Pˆ = [ f + ] [ + ] h I m h h fh uh h s h s sze and number of unrelated ersons., where f h and uh are the corresondng housng unt famly By defnton, E ˆ A P 0,, ) = fh uh ( 0 for the aggregate model. For the unt-level model: E U ( Pˆ 0,,,, ) N fh u = [ + ( (1 ) ) ] [ + ] h fk N uk fk u k. h s If overty status s homogenous wthn housng unt model (.e., ν ν = 0 and Σ = 0) then 0 = =. In ths case N E U ( Pˆ 0,,,, ) = N fh uh 0. Results All lots are for samled tracts, only. In addton, each lot resents tract results sorted by the sze of the tract samle. To see clearly the results for each tract, wthout usng a lot of sace, the results are resented by the tract samle sze order. Fgure 6 show the corresondence between actual samle sze and the order resented n the other fgures. Fgure 1 show the dfferences between estmates of overty rate from the two models. Takng the housng unt model as the truth, 95% osteror robablty ntervals are calculated for each tract and comared wth the osteror mean of overty rate, usng the tract-level model. As can be seen, usng the osteror mean from the tract-level model can be far off from the osteror dstrbuton based on the housng unt model. It matters whch model s used. Fgure lots the osteror means of the overty rate from both models along wth the tract level samle mean. Ths fgure llustrates the mortance of samle sze, n that the three estmate s values become close as the tract samle sze ncreases and less borrowng takes lace. For most of the tracts the housng unt model takes values closer to the samle means than the tract level model showng that, n general, the housng model borrows less. Even though the housng unt model borrows less and ncludes more arameters, the coeffcents of varaton are comarable between the two models. The housng unt model does have larger CV s when the samle sze gets very small. Fgures 4 and 5 look more closely at the usefulness of modelng the wthn tract varablty. Based on the housng-unt model a new samle can be redcted, the arcsne square root of the samle tract-level mean and jackknfed estmate of varances can be calculated. Fgure 4 rovdes 95% robablty ntervals for the jackknfed standard devatons (.e., the square root of the jackknfed varances). As can be seen, the samle standard devatons may be very mrecse for small samle szes. Although ths s not a major roblem for ths data set, the actual ACS s exected to only take a -3% samle (nstead of the 15% target = N h s

8 taken here). Fgure 5 s an nformal check on the adequacy of the housng unt model of wthn tract varablty. Here, 95% redctve robablty ntervals for jackknfed standard devaton multled by the square root of samle sze s resented. If the model s any good, t should at least be a good redctor of the actual wthn-tract samle standard devaton (see, e.g. Gelman, et al. (1995), secton 6.3). As shown n Fgure 5, the redctons are farly good. Dscusson The housng unt model has been secfed n order to account for the wthn tract level error of tract level samlng varances, an mortant error to measure snce t a major factor n settng the borrowng strength of small area estmates. A Bayesan mlementaton has been resented here but a frequentst analyss, such as Maxmum Lkelhood Estmaton, could have been carred out usng the same model. The housng unt model can also be exanded to nclude other terms or be aled to other stuatons. It could easly be aled at the county or hgher level. Addtonal herarchcal models, such as state effects, could be added. A housng unt model, of ths tye, could also ncororate housng unt comoston rates from the decennal census, relyng on the ACS to udate changes from the census. It has been shown that the tract-level model s a secal case of the housng unt model at and above the tract level and t has been demonstrated, emrcally, the housng unt model does an adequate job of modelng wthn tract varablty. However, more model refnement could be made. Frst, deendence of overty or other outcomes on housng characterstcs such as sze, demograhc comoston, etc. could be refned. The utlty of usng transformatons other than the arcsne square root could also be evaluated. Also, related structure among tyes of housng unt characterstc may smlfy the model. References Chand, Nanak and Alexander, Charles H. (1995). Indrect Estmaton of Rates and Proortons for Small Areas Wth Contnuous Measurement. ASA Proceedngs of the Secton on Survey Research Methods, Gelman, A. and Carln, J. B. and Stern, H. S. and Rubn, D. B. (1995). Bayesan Data Analyss, Chaman & Hall. Marker, Davd A. (1999). Organzaton of Small Area Estmators Usng a Generalzed Lnear Regresson Framework, Journal of Offcal Statstcs, 15, 1-4. Rao, J. N. K.(1999), Some Recent Advances n Model-based Small Area Estmaton, Survey Methodology, 5,

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