Using Markov Chain Monte Carlo for Modeling Correct Enumeration and Match Rate Variability

Size: px

Start display at page:

Download "Using Markov Chain Monte Carlo for Modeling Correct Enumeration and Match Rate Variability"

Alexia Newman
5 years ago
Views:

1 Usng Marov Chan Mont Carlo for Modlng Corrct Enumraton and Match Rat Varablty Andrw Kllr U.S. Cnsus urau Washngton, DC Ths rport s rlasd to nform ntrstd parts of ongong rsarch and to ncourag dscusson of wor n progrss. Th vws xprssd on statstcal, mthodologcal, tchncal, or opratonal ssus ar thos of th author and not ncssarly thos of th U.S. Cnsus urau. Abstract Th Cnsus urau conductd th Accuracy and Covrag Evaluaton (A.C.E.) wth th goal of producng stmats of th nt covrag rror of Cnsus 000. Th A.C.E. usd dual systm mthodology to stmat th nt covrag rror. Dual systm stmats wr cratd for populaton subgroups calld post-strata. A problm n modl-basd valuaton of covrag wth rspct to smallr gographs s that th varanc btwn blocs and wthn thos gographs nds to b spcfd n ordr to stmat a covrag corrcton factor. To mprov covrag stmats, ffort has bn mad towards advancng modls nvolvng smallr gographs. Ths papr offrs Marov chan Mont Carlo (MCMC) mthods as a computr ntnsv mthod to gnrat ths stmats. Spcfcally, random ffcts modls of th corrct numraton and match rats at th bloc lvl ar dvlopd to spcfy ths varanc wth an accompanyng statmnt of prcson. acground Th A.C.E. usd two sampls to valuat covrag for Cnsus 000, th populaton sampl (P sampl) and th numraton sampl (E sampl). Th P sampl stmatd prsons that should hav bn numratd n th cnsus at that locaton accordng to cnsus rsdnc ruls but wr not. Th P sampl consstd of popl rostrd from a sampl of housng unts n a spcfc locaton (ndpndnt of th cnsus) from a sampl of cnsus bloc clustrs (from now on rfrrd to as blocs). It was populatd basd on th rsults from a prson ntrvw, ndpndnt from th cnsus numratons n th sampl blocs. Th E sampl stmatd cnsus rronous numratons that should not hav bn ncludd anywhr n th cnsus or at th spcfc locaton. Th E sampl consstd of cnsus numratons. It was dntfd n th sam st of cnsus blocs slctd for th P sampl. E-sampl numratons who matchd to P-sampl popl wr countd as corrct numratons. Nonmatchd E-sampl numratons undrwnt a follow up ntrvw to dtrmn whthr thy wr corrct numratons for th spcfc locaton. Th A.C.E. dvdd th populaton nto 46 post-strata whr smallr groupngs wr combnd or collapsd to produc mor stabl stmats. A post-stratum was a group of popl sharng dmographc and gographc charactrstcs that wr assumd to hav th sam probablts of ncluson n th cnsus (U.S. Cnsus urau 004). A post-stratum was composd of an E-sampl post-stratum and P-sampl post-stratum par. Wthn a sngl post-stratum, th dual systm stmat (DSE) formula was dfnd as: CE / E DSE = cnsus DDRATE () M / P whr: : Post-stratum cnsus : Th cnsus count wthn post-stratum

2 DDRATE : Th rato of data dfnd cnsus rcords to all cnsus rcords wthn post-stratum. CE : Wghtd stmat of corrct numratons n post-stratum E : Wghtd stmat of numratons n post-stratum M : Wghtd stmat of matchs n post-stratum P : Wghtd stmat of P-sampl rcords n post-stratum Th modl dvlopmnt dscrbd n ths papr focuss on th thrd trm for Formula, th rato of th corrcton numraton (CE) rat and match rat. Th CE rat quantfs th rato btwn total E-sampl numratons and a smallr subst of corrct E-sampl numratons. Th match rat quantfs th rato btwn total P-sampl popl and a smallr subst of P-sampl popl who matchd to a cnsus numraton. Ths rato drvs th calculaton of DSE (), and th rsultng covrag corrcton factors, CCF =. cnsus Th goal of ths rsarch s to vntually modl CE and match rat varancs btwn blocs wthn a spcfd gography. From th modl-basd approach usd hr, covrag stmats can b mad from th modld CE rat and match rat. Th modl-basd rats, p and p, rspctvly can b usd for computaton of nw DSEs n () CE M and covrag corrctons factors. That s, for ach bloc b and post-stratum : pce, b DSE = cnsus DDRATE () b b b p M, b Th modl was appld to slctd stats by ncludng ach A.C.E. sampl bloc wth btwn 3 and 79 housng unts. Tradtonally, to approxmat manngful varancs of th CE and match rats, t has bn ncssary to mov up th gographcal hrarchy. That s, varanc stmats wr computd on largr gographs whr suffcnt data was prsnt to calculat a dsgn-basd varanc of th rspctv rats. y dvlopng th modl, th goal was to gnrat varancs wth corrspondng statmnts of prcson for smallr lvls of gography. Mthodology Th modl dvlopmnt was gard towards CE rats. In th futur, an analogous procss wll b followd to dvlop a modl for match rats. Data Dvlopmnt For th E sampl, ach cnsus rcord was assgnd a CE probablty basd on A.C.E. procssng (U.S. Cnsus urau 004). To crat th data, th CE probablty was compard to a unform num btwn 0 and, U (0,). asd on that comparson, th rcord rcvd a bnary CE valu. For ach bloc n th A.C.E. sampl wth btwn 3 and 79 housng unts, th num of corrctly numratd prson rcords and total prson rcords wr aggrgatd and a CE rat was gnratd. Modl Dvlopmnt Ths ntal modl dvlopmnt ncorporatd btwn bloc varablty usng random ffcts. Intally, th modl usd a sngl fxd ffct. Lt p (, ) rprsnt th ownr CE rat n a gvn bloc b. Smlarly, lt p (, ) rprsnt th rntr CE rat n a gvn bloc b. Snc th pb (, ) ar sampl proportons btwn 0 and, logstc In 000, th cnsus rqurd two charactrstcs for a rcord to b data dfnd. Rlatonshp, sx, rac, Hspanc orgn, and thr ag or yar of brth countd as charactrstcs. A vald nam also countd as on charactrstc. To b consdrd vald by th cnsus, a nam had to hav at last thr charactrs n th frst and last nam togthr. Ths data dfnd cnsus rcords wr lgbl for A.C.E. procssng.

3 rgrsson was usd for th modl. caus hom ownrshp has tradtonally bn a good prdctor of corrct numraton, that was ncludd n th logstc modl as a fxd ffct. Also, snc ach bloc has unqu charactrstcs, a random bloc ffct was ncludd n th modl. As a rsult, th ownr and rntr CE rats wr wrttn as pb (, ) = + µ : Intrcpt trm µ + αs + ε ( b) µ + αs + ε ( b ) α : Ownrshp ffct s : nary varabl ndcatng whthr th rcord was dsgnatd as an ownr ( s = ) or a rntr ( s = 0 ) ε ( b): loc ffct Rcall that ach prson had a bnary CE valu. Lt δ = rfr to ach corrctly numratd ownr rcord n an,, arbtrary bloc b wth n total ownr numratons. Consquntly, δ = 0 rfrs to ach rronously,, numratd ownr rcord. Analogously, lt δ = rfr to ach corrctly numratd rntr rcord n an arbtrary,, bloc b wth n total rntr numratons and δ = 0 rfr to ach rronously numratd rntr rcord.,, Assumng ownr corrct numratons follow a bnomal modl wth a CE rat p (, ), th llhood functon was wrttn as n Lp ( (, )) = p (, ) δ ( p (, )) = δ, b, w, b, w bnomal modl wth a CE rat p (, ), th llhood functon was wrttn as n + n δ L( p( b, r)) = p( b, r) ( p( b, r)) = n + δ, b, r, b, r. Smlarly, assumng rntr corrct numratons follow a. Combnng ths llhoods togthr, n n + n δ, b, w δ, b, w δ, b, r δ, b, r. (3) L( p( b, w)) L( p( b, r)) = p( b, w) ( p( b, w)) p( b, r) ( p( b, r)) = = n + Th CE rats wr modld to nvolv th paramtrs of ntrst: µ, αε, ( b). Also, substtutng th logstc xprsson for th rspctv CE rats nto th llhood functon rsultd n th followng modfcaton to (3): L( µαε,, ( b)) = ( ) ( ) ( ) ( ) n µ + αs + ε ( b) µ + αs + ε ( b) ( ) ( ) n + n µ + αs + ε b µ + αs + ε b δ,, δ,, δ,, δ,, µ + αs + ε ( b) µ + αs + ε ( b) µ + αs + ε ( b) µ + αs + ε ( b) = + + = n y applyng what s nown about s for ownrs and rntrs and applyng xponntal proprts, th modl was furthr smplfd. Thrfor, L( µαε,, ( b)) = ( ) ( ) ( ) ( ) n µ + α+ ε ( b) n + n µ + ε ( b) δ δ δ δ, b, w, b, w, b, r, b, r µ + α+ ε ( b) µ + α+ ε ( b) µ + ε ( b) µ + ε ( b) = + + = n ( b ) n δ, b, w n ( δ, b, w) + ε ( b ) n + n δ, b, r = n + n + n ( δ, b, r = n + ) = = µ α ε µ = ( ) ( ) ( ) ( ) µ + α+ ε ( b) µ + α+ ε ( b) µ + ε ( b) µ + ε( b) y dfnton of δ b,,, n n n + n n + n δ = nc, ( δ ) = n, δ = nc, ( δ ) = n,,,,,, = = = n + = n +,, whr:

4 nc : Num of corrctly numratd ownr rcords n an arbtrary bloc b n : Num of rronously numratd ownr rcords n an arbtrary bloc b nc : Num of corrctly numratd rntr rcords n an arbtrary bloc b n : Num of rronously numratd rntr rcords n an arbtrary bloc b As a rsult, th llhood functon for bloc b was fnally wrttn as: µ + α+ ε ( b) µ + ε ( b) L( µαε,, ( b)) = ( ) ( ) ( ) ( ) µ + α+ ε ( b) µ + α+ ε ( b) µ + ε ( b) µ + ε ( b) nc n nc n Th fnal llhood was: L( µαε,, ( b)). In addton, t was ntally assumd that th bloc ffcts wr b S ε ( b) normally dstrbutd. Ths ld to an augmntd llhood modl, L( µαε,, ( b)) xp( ). (4) b S πσ σ Marov Chan Mont Carlo In gnral, lt θ b a vctor of unobsrvabl populaton paramtrs and y dnot obsrvd data. Marov chan Mont Carlo (MCMC) mthods tratvly gnrat dpndnt sampls n th paramtr spac that convrg to a targt dstrbuton, p( θ y). aysan tchnqus allow nfrnc about θ condtonal on th obsrvd data y. Usng ays rul, th postror p( θ, y) p( y θ) p( θ) dstrbuton s p( θ y) = = p( y θ) p( θ). Th llhood functon from abov s anothr p( y) p( y) way of wrtng p( y θ ) snc t s proportonal to th probablty gvn unnown paramtrs. Th p( θ ) trm s calld th pror dstrbuton. Mtropols-Hastngs Algorthm Th Mtropols-Hastngs Algorthm gnrats sampls from a postror dstrbuton. Th Mtropols-Hastngs Algorthm uss an accptanc-rjcton schm to draw sampls from a canddat dstrbuton. Th algorthm wors as follows:. Start wth an ntal valu θ ( t = 0) whr th postror dnsty s gratr than 0.. For subsqunt tratons t =,,..., T, T ; sampl a canddat valu θ ( t ) from a canddat dstrbuton, C( θ( t ) θ( t )). Ths canddat dstrbuton may not b symmtrc. p( θ( t ) y) C( θ( t ) θ( t )) 3. Calculat th Mtropols-Hastngs rato, Rato =. If th canddat p( θ( t ) y) C( θ( t ) θ( t )) dstrbuton s symmtrc, thn th Mtropols-Hastngs rato abov s smplfd to a rato of th postror dnsts. 4. If Rato > U (0,), thn θ ( t ) s accptd and θ() t = θ( t ). If not, θ () t = θ ( t ). Aftr th canddats ar dtrmnd to convrg to th targt dstrbuton, nfrnc s compltd.

5 Gbbs Samplr For mult-paramtr Marov chan applcatons, th Gbbs samplr s usd to cycl through all th paramtrs. If θ s composd of multpl paramtrs, thn for ach traton, a sngl paramtr θ g s drawn from a condtonal dstrbuton gvn all othr paramtrs of θ. Th Gbbs samplr s a spcal cas of th mor gnral Mtropols- Hastngs Algorthm whr all canddats ar accptd and th targt dstrbuton s nown. Modl Spcfcs Ths analyss usd th Mtropols-Hastngs algorthm wthn th Gbbs samplr. It usd th Gbbs samplr to draw a sngl paramtr from a condtonal dstrbuton gvn all othr paramtrs. Howvr, snc th targt dstrbuton was unnown, th Mtropols-Hastngs Algorthm was usd to accpt and rjct canddats. For ach traton t, + 3 paramtrs wr procssd, whr was th num of random bloc ffcts n th modl. Th rmanng thr paramtrs corrspondd to th man ( µ ), ownrshp ffct (α ), and varanc btwn th bloc ffcts ( σ ). Th procss cycld through ach paramtr condtonal on th valus of th othr + paramtrs and th data by valuatng thr Mtropols-Hastngs ratos. Th σ paramtr was th only paramtr whr a non-constant pror dstrbuton was assumd. Its pror dstrbuton was assumd to b half- Cauchy. To proprly chc for convrgnc multpl squncs (chans) wr run. Wth rspct to followng notaton, z rfrs to a chan. loc Effcts. For th bloc ffct, th canddat valu was sampld by drawng from a normal dstrbuton, ε( bzt,, ) ~ N( ε( bzt,, ), σ ( zt, )). Th normal dstrbuton s symmtrc n ε ( bzt,, ) and ε ( bzt,, ). As a rsult, th Mtropols-Hastngs rato wth rspct to th bloc ffcts was smplfd to th rato of postror dnsts and th dcson to accpt was basd solly on th rato of th postror dnsts. Intrcpt. For th ntrcpt trm, th canddat valu was sampld by drawng from a normal dstrbuton, µ ( zt, ) = µ ( zt, ) + whr ~ N (0,). Th normal dstrbuton s symmtrc n µ ( zt, ) and µ ( zt, ). As a rsult, th Mtropols-Hastngs rato wth rspct to th ntrcpt was smplfd to th rato of postror dnsts and th dcson to accpt was basd solly on th rato of th postror dnsts. Ownrshp Effct. For th ownrshp ffct, th canddat valu was sampld by drawng from a normal dstrbuton, α( zt, ) = α( zt, ) + whr ~ N (0,). Th normal dstrbuton s symmtrc n α ( zt, ) and α ( zt, ). As a rsult, th Mtropols-Hastngs rato wth rspct to th ownrshp ffct was smplfd to th rato of postror dnsts and th dcson to accpt was basd solly on th rato of th postror dnsts. Varanc twn loc Effcts. For th varanc btwn th bloc ffcts, th canddat valu was sampld by drawng from a gamma dstrbuton, γ ~ Gamma[ ωψ, ( z, t)] ω = ( )/ ψ( zt, ) = ε ( bzt,, ) σ ( zt, ) = γ Th postror dnsty for th varanc btwn bloc ffcts ncludd a half-cauchy pror trm. Th + σ ( zt, ) condtonal postror dnsty was xprssd as:

6 ε ( bzt,, ) = = σ ( zt, ) + σ ( zt, ) p[ σ( zt, ) ε( b, zt, ),..., ε ( b zt,, ), µ ( zt, ), α( zt, ), y] σ( zt, ) xp[ ] Snc th gamma dstrbuton s not symmtrc, thn th Mtropols-Hastngs rato wth rspct to th varanc btwn bloc ffcts ncludd th canddat dstrbuton. Th canddat dstrbuton was: ε bzt σ ( zt, ) (,, ) C( σ( z, t ) σ( z, t )) σ( z, t ) xp[ ] ε ( bzt,, ) Th logstc trms had no ffct bcaus thy had no σ dpndnc. Th σ xp[ ] trms canclld σ ( zt, ) bcaus thy wr on oppost sds of th quotnt. Th smplfd Mtropols-Hastngs rato was xprssd as: p( σ( z, t ) othrs) C( σ( z, t ) σ( z, t )) + σ ( zt, ) Rato = = p( σ( z, t ) othrs) C( σ( z, t ) σ( z, t )) + σ ( zt, ) Convrgnc Analyss Ths analyss mployd th Glman and Rubn Mthod (Glman t al. 000) as ts convrgnc dagnostc. To montor convrgnc, a potntal scal rducton factor was calculatd for vry paramtr. For ths analyss, th paramtr vctor subjct to convrgnc montorng was comprsd of th random bloc ffcts of ach bloc, th ntrcpt trm, th ownrshp trm, and th varanc btwn th bloc ffcts. That s, θ ε ε ε ε µ α σ = ( ( b = ), ( b = ),..., ( b = ), ( b = ),,, ). To bgn, Z = 0 startng valus for ach paramtr wr chosn as ntal valus. For thos ntal valus, dsprsd startng ponts wr usd. Ths was don to dtrmn f problms xstd wth th modl s convrgnc and to nsur that th paramtr spac was thoroughly sarchd to uncovr possbl mods. To complt nfrnc, th potntal scal rducton factor was calculatd at ntrvals of on hundrd tratons. Whn all paramtrs had a potntal scal rducton factor clos to, th MCMC mthod was thought to hav convrgd at that traton, t = τ. Infrnc Aftr τ was dtrmnd, th paramtr valus wr usd to calculat modld ownr and rntr corrct numraton totals for vry traton btwn τ + and τ for ach bloc wthn ach chan. That s, µ ( zt, ) + α( zt, ) + ε ( bzt,, ) CEmdl ( b, z, t) = ( ) n ownrs µ ( zt, ) + α( zt, ) + ε ( bzt,, ) + (5) µ ( zt, ) + ε ( bzt,, ) CEmdl ( b, z, t) = ( ) n rntrs µ ( zt, ) + ε ( bzt,, ) + wr usd as draws from th jont postror dstrbuton. Modl Chcng nomal trals wr run to produc nw sampls. Smpl mans and standard rror stmats from th nw sampls wr compard to corrspondng statstcs from th obsrvd sampl to assss modl ft. To do ths, for ach chan/bloc/traton groupng btwn τ + and τ, n trals wr run to gt a sampl of th num of corrctly

7 numratd ownrs n that bloc. Smlarly, n trals wr run to gt a sampl of th num of corrctly numratd rntrs n that bloc. Th nputtd nomal probablts wr basd off th modld corrct numraton totals for ownrs and rntrs n (5). Th corrct numraton totals for th bnomal trals wr computd as follows: CEmdl ( b, z, t) ownrs CEbn ( b, z, t) ~ n( n, p = ) ownrs n CEmdl ( b, z, t) rntrs CEbn ( b, z, t) ~ n( n, p = ). rntrs n CEbn(, b z,) t = CEbn (, b z,) t + CEbn (, b z,) t ownrs rntrs Nxt, a man ( manceratebn) and standard rror ( sceratebn ) ovr th blocs wr calculatd for ach traton btwn τ + and τ for ach chan. Thy wr calculatd as follows: [ CEbn( b, z, t) ] manceratebn( z, t) = n + n [ ] [ CEbn( b, z, t)] CEbn( b, z, t) rpceratebn( b, z, t) = [ n + n ] [ n + n ] sceratebn( z, t) = ( rpceratebn( b, z, t) manceratebn( z, t)) Th mans and standard rrors for all tratons btwn τ + and τ for ach chan wr combnd and thn sortd from smallst to largst. That rsultd n two vctors of sz 0τ. That s, manceratevct = manceratebn(, τ + ),..., manceratebn(0, τ ) [ ] [ sceratebn(, τ ),..., sceratebn(0, τ) ] sceratevct = + sortdmanceratevct = sort( manceratevct) sortdsceratevct = sort( sceratevct) Thn, th 5% covrag valus wr cratd by tang th man and standard rror for th τ sortd traton. Smlarly, th 95% covrag valus wr cratd by tang th man and standard rror for th τ sortd traton. That rsultd n th followng covrag ntrvals: man _ cvg _ trvl = [ sortdmanceratevct( τ ), sortdmanceratevct( τ )] (6) s cvg _ trvl = [ sortdsceratevct( τ), sortdsceratevct( τ)] As an xampl, suppos a MCMC modl s run for 5000 tratons ovr 50 blocs wth 0 chans. Suppos that, from th Glman and Rubn Mthod, th modl convrgs at τ = 000. For all chans, th mans and standard rrors for tratons btwn τ + = 00 and τ = 4000 ar combnd and thn sortd from smallst to largst. That rsults n vctors of sz 0000 for th man and standard rror. Th covrag ntrvals (6) ar thn formd by:

8 man _ cvg _ trvl = [ sortdmanceratevct( ), sortdmanceratevct( )] s cvg _ trvl = [ sortdsceratevct( ), sortdsceratevct( )] Th covrag ntrvals wr compard to th valus obsrvd from th 000 A.C.E. sampl. Thy wr computd as follows: mancerateobsrvd = rpcerateobsrvd ( b) = [ nc + nc ] [ n + n ] [ + ] nc nc [ nc + nc ] [ n + n ] [ n + n ] scerateobsrvd = ( rpcerateobsrvd ( b) mancerateobsrvd) (7) Rsults Th goal of ths rsarch was to dtrmn th fasblty of applyng MCMC tchnqus to modl corrct numraton and (vntually) match rats. To assss ths, modl-basd man and standard rror covrag ntrvals of th CE rat from (6) wr compard to th man and standard rror of th obsrvd sampl from (7). On of th y aspcts n dvlopng th modl was dtrmnng f th assumpton that th random bloc ffcts had a normal dstrbuton from (4) was accurat. Th followng tabl sts compar th modl-basd man and standard rror covrag ntrvals to th man and standard rror from th obsrvd sampl. Ths scton provds rsults for ght stats n ach of th four cnsus rgons. Rcall that, wthn ach stat, only a subst of blocs was tan to modl CE rat varablty. As a rsult, modl rsults may not b llustratv of th whol stat.

9 Modl : No Random loc Effct Intally, th challng was to justfy th ncluson of a random bloc ffct n th modl to account for htrognty btwn blocs. To do ths, t was ncssary to mprcally show that omttng a random bloc ffct would not b suffcnt to modl th obsrvd sampl. Th rsults wr as follows: Tabl.A CE Rat Covrag Intrvals Doman Obsrvd Valu Covrag Intrval Lowr ound Covrag Intrval Uppr ound Ys Ys Ys Ys Ys Ys Ys Ys Tabl. SE(CE Rat) Covrag Intrvals Doman Obsrvd Valu Covrag Intrval Lowr ound Covrag Intrval Uppr ound No No No No No No No No Intrval Covrs Obsrvd Valu? Intrval Covrs Obsrvd Valu? Tabl.A ndcats that th modl-basd covrag ntrvals for th CE rat covr th obsrvd CE rat. Howvr, Tabl. shows that th modl-basd covrag ntrvals for th standard rror consstntly undrstmat th obsrvd standard rror of th CE rat. As a rsult, t was dtrmnd that modlng th random bloc ffct was ndd to account for htrognty btwn blocs.

10 Modl : Normal Random loc Effct Whn t was clar that a random bloc ffct would nd to b ncludd, th ntal thought was to modl th random bloc ffct wth a normal dstrbuton. Th rsults wr as follows: Tabl.A CE Rat Covrag Intrvals Doman Obsrvd Valu Covrag Intrval Lowr ound Covrag Intrval Uppr ound Ys Ys Ys Ys Ys Ys Ys Ys Tabl. SE(CE Rat) Covrag Intrvals ``Doman Obsrvd Valu Covrag Intrval Lowr ound Covrag Intrval Uppr ound Ys Ys Ys Ys Ys Ys Ys Ys Intrval Covrs Obsrvd Valu? Intrval Covrs Obsrvd Valu? Tabl.A ndcats that th modl-basd covrag ntrvals for th CE rat covr th obsrvd CE rat. Tabl. shows that th modl-basd covrag ntrvals for th standard rror now covr th obsrvd standard rror of th CE rat. caus of th rsults, t can b nfrrd that th assumpton that th random bloc ffcts ar normally dstrbutd was corrct. Conclusons and Futur Wor Ths wor rprsnts a bgnnng n studyng th fasblty of applyng MCMC mthods to stmat varanc of covrag stmats ovr smallr gographs. Although th ntal rsults abov ar promsng, ths modl has only bn appld to a subst of th 000 A.C.E blocs sampld wthn ach stat. It wll nd to b dtrmnd f th ncluson of small blocs wth lss than thr housholds or larg blocs wth gratr than 79 housholds wll ncsstat a chang to th modl. As mntond arlr, futur wor wll nclud modlng match rats usng th sam paradgm. Furthrmor, snc CE and match rats ar thought to b corrlatd wthn a bloc, th futur modl wll hav to ncorporat that rlatonshp. Addtonally, a modl for data dfnd rats wll b constructd usng smlar tchnqus. Rfrncs Glman, A.., Carln, J.S., Strn, H.S., and Rubn, D.. (000) aysan Data Analyss. Washngton D.C.: Chapman and Hall/CRC. U.S. Cnsus urau Tchncal Papr 004: Accuracy and Covrag Evaluaton of Cnsus 000: Dsgn and Mthodology.

Chapter 6 Student Lecture Notes 6-1

Chapter 6 Student Lecture Notes 6-1 Chaptr 6 Studnt Lctur Nots 6-1 Chaptr Goals QM353: Busnss Statstcs Chaptr 6 Goodnss-of-Ft Tsts and Contngncy Analyss Aftr compltng ths chaptr, you should b abl to: Us th ch-squar goodnss-of-ft tst to dtrmn