Comparison of the performance of best linear unbiased predictors (BLUP)

Comparon of h prformanc of b lnar unbad prdcor (BLUP) Pkang Yao Synh Spn 130 Wrgh Lan Ea W Chr, PA 19380 USA yao.pr@ynh.com Edward J. Sank III Dparmn of Publc Halh 401 Arnold Hou Unvry of Maachu 711 Norh Plaan Sr Amhr, MA 01003-9304 USA ank@choolph.uma.du ABSTRACT Th b lnar unbad prdcor (h Clur man, h Mxd modl, Sco & Smh prdcor and h Random Prmuaon modl) of lcd mporan publc halh varabl wr valuad n praccal ng va mulaon ud. Th varabl corrpondd o maur of d, phycal acvy, and ohr bologcal maur. Th mulaon valuad and compard h man quar rror (MSE) of ho four prdcor. I mad varanc bwn ubjc and day, and rpon rror for paramr dfnd ovr on yar prod, bad on daa from a larg-cal longudnal udy, h Saon Sudy. Thn, valuad h rlav MSE ncra bwn prdcor of h ru ubjc man n varou ng bad on horcal rul. In addon, a mulaon compard h horcal and h mulad MSE for all four prdcor. Th dffrnc n h MSE bwn prdcor wa llurad n D plo. Conac Addr: Pkang Yao Synh Spn 130 Wrgh Lan Ea W Chr, PA 19380 USA Phon: 610-719-583 Fax: 610-719-510 yao.pr@ynh.com Kyword: upr-populaon, b lnar unbad prdcor, wo-ag amplng, random prmuaon pkangyao-h_papr1 3_4_5g.doc /1/007

Comparon of h prformanc of BLUP 0-1. INTRODUCTION Many clncal ud n halh cnc fld hav maur of bologcal and bhavoral varabl on pan a h baln, and hn by h follow up v. Th maur flucua ovr m du o aonal chang. In ordr o ma h man daly aurad fa nak for a ubjc, w uually maur oal aurad fa nak on a ubjc on om ampld day, and hn avrag ho maurmn. W would xpc h mad avrag bad on ho maurmn, X, o dffr from h ru ubjc man, m, rprnng h man of h ru aurad fa nak ovr all day n a yar for h ubjc. W call h man, m, h ubjc lan valu. Thrfor, hr a dvaon bwn h mad ubjc man, X, and h ru ubjc man, m. Th papr a udy of prdcor of h ru ampld ubjc man oal aurad fa nak ovr h whol yar, n a ng whr a random ampl of ubjc lcd from a fn populaon. Th purpo of h papr o dcu h xn o whch b lnar unbad prdcor (Th Clur man, h Mxd modl, Sco & Smh prdcor and h Random Prmuaon modl) provd br prdcor. W valua propr of h prdcor for dffrn ng n h conx of a larg obrvaonal longudnal udy of aonal varaon n cholrol lvl whch w rfr o a h Saon udy. Th udy nvgad h naur and cau of aonal flucuaon n blood cholrol. Our focu wa no on aonal ffc, bu rahr on maor of pan paramr uch a h man oal aurad fa nak ovr h whol yar. W bgn wh a brf rvw of prdcor ha hav bn propod n h ng, ncludng h mxd modl, Sco and Smh Prdcor, and h random prmuaon prdcor. Thn w llura h dffrnc n nrpraon, n h hrnkag conan, and n h xpcd MSE. Nx w dcrb a vary of uaon whr prdcon of ky publc halh varabl of nr. W condr varabl on d nak, phycal acvy, and cholrol and blood

Comparon of h prformanc of BLUP 0-3 prur. For ho varabl, w dcrb h accuracy of h prdcor wh known varanc, and h rul of a mulaon udy n h praccal ng whr varanc hav o b mad.. BEST LINEAR UNBIASED PREDICTORS B Lnar Unbad Prdcor (BLUP) wr fr dvlopd by Goldbrgr n 196. H mpha wa on prdcon of a fuur obrvaon bad on pa obrvaon ung mad random ffc (Goldbrgr, 196). Snc hn, BLUP hav bn dvlopd o prdc unobrvd valu of random varabl from obrvd valu of ho random varabl n h ampl (Sco and Smh, 1969; Royall, 1976; Robnon, 1991; Sarl, Calla and McCulloch, 199; Sank and Sngr, 004). BLUP prdcor afy h followng condon. Fr, h prdcor a lnar funcon of h ampl valu. Th ampl valu ar ralzd random varabl n h ampl. Scond, h prdcor rqurd o b unbad. Thrd, h prdcor rqurd o hav mnmum varanc..1 Mxd Modl A mpl mxd modl for h rpon of h h j lcd un, j 1,..., = m for h h lcd clur (PSU, prmary amplng un), = 1,..., n gvn by Y = μ + B + E (.1) j j whr μ dno o h xpcd rpon ovr clur n a populaon, B corrpond o h dvaon bwn μ and h xpcd rpon of h h PSU, whch a random ffc, and E j rprn a random dvaon of h h PSU. Th aumpon ar ha B d N( 0, σ ) h j un rpon from h xpcd rpon of h : and Ej d N( 0, σ ) : (Sarl and McCulloch, 199). 3

Comparon of h prformanc of BLUP 0-4 In modl (.1), w aum ha varanc ar known, and h maor of h fxd ffc h wghd la quar maor, n mˆ = å wy whr = 1 w = 1/ v n *1 = 1/ v * m 1, Y = å Y, and m j = 1 j v σ m = σ +. Soluon of Hndron mxd modl quaon rul n h BLUP of B. Th xpron B ˆ gvn by Bˆ = k(y - mˆ ) (.) whr k =, + / m h varanc ( ) h varanc bwn clur,.. var( B ) =, and h varanc ( ) varanc whn clur,.. var( E ) =, h paramr m h numbr of ampld un pr ampld clur. Thn h prdcor of h lan valu of h h ralzd PSU ( pˆ ˆ ˆ m B) = + a lnar combnaon of m and h prdcor of B gvn by j pˆ = mˆ+ k( Y - mˆ) (.3) (Sarl and McCulloch, 199 and Sank and Sngr, 004).. Sco and Smh Prdcor Sco and Smh (1969) propod a wo-ag amplng modl o prdc lnar combnaon of lmn of a fn populaon from a upr-populaon modl. Th fn populaon cond of N clur, ach wh M un. A h fr ag, n clur wr lcd from N clur. A h cond ag, m dnc un wr lcd from ach of h n lcd clur. Thy aumd ha: (1) Th M lmn, Y j n h h clur, wr ndpndn obrvaon from a drbuon wh man m and varanc σ. 4

Comparon of h prformanc of BLUP 0-5 () Th clur man μ 1,, varanc δ. m N wr uncorrlad from a drbuon wh man m and Th aumpon lad o a mulvara drbuon for h lmn wh EY ( j ) = μ and ( j kl ) δ σ cov Y, Y = + whn = k; j = l = δ whn = ; = 0 ohrw. k j l (.4) whr d varanc bwn clur, and varanc whn a clur. Sco and Smh aumd h populaon wa a ralzaon of h upr-populaon, and a prdcor bcam a lnar funcon of h fn populaon un. Bad on mnmzng h xpcd MSE of a lnar prdcor, hy dvlopd h prdcor m M m [ * ( *)] M M μ μ * pˆ ˆ ˆ = Y + + k Y (.5) whn PSU n h ampl, whr n * * = 1 ˆ μ = wy, w * = 1/ v n = 1 * 1/ v *, σ = +, and m * v δ k * = mδ mδ + σ. Th prdcor for PSU n h ampl. If h PSU no n h ampl, h man prdcd by pˆ = ˆ*. μ Th fr rm n h prdcor (.5) h ampl man for h h PSU n h ampl. Th cond rm h prdcor of h rmanng condary amplng un (SSU) for h PSU. Th wgh facor wr h rao of obrvd SSU and h unobrvd SSU. If PSU wr no n h ampl, h prdcor mplfd o b h wghd ampl man (Sank and Sngr, 004)..3. Th Fn Populaon Rpon Error Modl and Paramrzaon A hrd prdcor wa dvlopd bad on a wo ag random prmuaon of h populaon (Sank and Sngr, 004). Aum ha a fn populaon compod of N clur, 5

Comparon of h prformanc of BLUP 0-6 ndxd by = 1,..., N, and ach clur conan a lng of M un, ndxd by = 1,..., M. Aum h whr h k rpon for un n clur gvn by Yk = y + Wk. (.6) y dona a fxd conan rprnng h xpcd rpon for h un, and Wk rprn rpon rror (wh zro xpcd valu). Modl (.6) rfrrd o a a rpon rror modl. If σ rprn h rpon rror varanc for un n clur, hn h avrag rpon rror varanc gvn by σ r σ =. Th man and h varanc of h xpcd N M = 1 = 1 NM 1 M rpon for un n clur ar dfnd a μ = y and M M 1 1 M σ = M ( y μ ) for 1,..., M = 1 1 N σ N = 1 varanc dfnd a σ =. μ = 1 = N, rpcvly. Th avrag whn clur Smlarly, h populaon man and h varanc bwn clur can b dfnd a 1 N μ N = 1 N 1 1 N N N = 1 = and σ = ( μ μ) valu of clur,, rpcvly. Th dvaon of h lan μ, from h populaon man rprnd a β ( μ μ) = ; h dvaon of h xpcd rpon for un (n clur ) from h lan valu of clur donad a ε ( y μ ) =. Thn, h rpon for un n clur gvn by y = μ + β + ε. (.7) Aumng y ( y 1 y... y N ) = whr ( y y y ) y = 1... M, modl (.7) can b ummarzd for all un a 6

Comparon of h prformanc of BLUP 0-7 y = Xμ + Zβ+ ε (.8) whr X= 1N 1 M, Z= IN 1 M, b' = ( b1 b... bn). Non of h rm n modl (.8) ar random varabl. 1 a an a 1column vcor of on, and ε dfnd mlarly o y..4 Two Sag Random Prmuaon Modl L u ndx a clur n a prmuaon by = 1,,..., N. A random varabl U on whn PSU clur, zro ohrw. W dfn h un n a clur a condary amplng ( ) un (SSU). Smlarly, a random varabl U ak valu of on whn SSU j n clur un, and zro ohrw. Whn all prmuaon l ar qually lkly, h random vcor Y= ( Y,..., Y )' a random prmuaon of h populaon, and ach lmn of 1 N j Y = (Y 1 Y Y 3... Y M)'. Th random varabl rprnng h Y j, n h prmuaon a follow: h j SSU whn h h PSU, whr N M ( ) j j = 1 = 1 Y U U Y = åå (.9) U ak a valu of on whn PSU clur, and a valu of zro ohrw, and ak on a valu of on whn SSU j n clur un, and zro ohrw. W aum a oal of m lmn n ach of n clur ar lcd by a wo-ag amplng chm from a populaon. Th populaon oal compod of hr componn: 1) h oal for obrvd lmn; ) h oal for unobrvd lmn n ampld clur; and 3) h oal for unampld clur. Sank and Sngr (004) dvlopd an unbad prdcor of a ralzd PSU man, whch wa a lnar combnaon of h random varabl n h ampl. Th prdcor mnmzd h xpcd valu of h man quar rror (MSE). If hr wa no rpon rror, h man of PSU can b prdcd by U j 7

Comparon of h prformanc of BLUP 0-8 ( )( ( )) Tˆ = fy + 1 f Y + k Y Y (.10) whr f m = (ampld fracon), Y h man of h lcd clur, Y h ovrall man M dfnd a Y 1 n Y n = 1 * mσ = å, k = * mσ + σ and = -. M *.5 Comparon Bwn Prdcor Tabl 1 prn h prdcor llurad prvouly. Th prdcor appar o b algbracally mlar, bu hr conn ar dffrn. Each prdcor a wghd lnar combnaon of wo rm. Th fr rm prdc h lan valu of h SSU n h ampl. Th cond rm prdc h lan valu for h rmanng SSU no n h ampl. Unlk ohr prdcor, h mxd modl prdcor a prdcor of h unobrvd SSU for a PSU, and plac all h wgh on h cond rm. Tabl 1. Prdcor of h lan valu of PSU whn n n wo-ag clur amplng (Sank, 003) Modl Prdcor Pˆ = fy + 1 f Y Clur Man ( ) Mxd Modl pˆ ( ˆ ( ˆ = μ + k Y μ )) * * * Sco & Smh Pˆ ( )( ˆ ( ˆ = fy + 1 f μ + k Y μ )) Random Prm. Tˆ = fy + ( 1 f )( Y + k ( Y Y) ) * * RP + Rp. Err. Tˆ = f ( Y + kr ( Y Y) ) + ( 1 f )( Y + k ( Y Y) ) 8

Comparon of h prformanc of BLUP 0-9 In h xpron, Y 1 m = Yj m, 1 n Y Y j = 1 n = 1 =, ˆ μ = wy, n = 1 w = 1/ v n * = 1 1/ v *, v σ σ * * = +, ˆ = n μ wy ; m = 1 w * = 1/ v n * = 1 * 1/ v * *, σ = +, m * v δ k = mσ mσ + σ, k * = mδ mδ + σ, k * mσ = mσ + σ *, k * r = mσ + σ * * + + r ( m ) σ σ σ, and k * * mσ = * mσ + σ + σ ( r ), = -. No ha n h Mxd Modl, or n Sco and m * Smh Prdcor whou rpon rror and aumng qual un varanc for ach clur, h xpron for σ qual o ( σ σ r ) + (Sank, 003). Sco and Smh prdcor and h random prmuaon modl ar narly dncal. Howvr, h dffrnc bwn h prdcor du o dffrnc n varanc componn and hrnkag conan (Sank and Sngr, 004). Bcau h random prmuaon modl prmud PSU, u a ngl SSU componn of h varanc rprnng h avrag of h SSU whn clur varanc. Thrfor, h varanc whn a clur pcfc componn, (a uppod o h clur, n h Sco and Smh prdcor). Smlarly, h varanc bwn σ PSU n h random prmuaon modl dfnd aσ = σ, a oppod o M and Smh prdcor. W no ha h aumpon of Sco and Smh prdcor for h varanc componn do no corrpond o h varanc componn ha drvd from prmuaon clur and un n a fn populaon (Sank and Sngr, 004). d n Sco Wh addonal aumpon ha h varanc whn a clur dncal for all clur, and h rpon rror varanc h am for all un and qual o r, and σ = σ + σ, r 9

Comparon of h prformanc of BLUP 0-10 hn h hrnkag conan k * mσ = k = mσ + σ + σ r. Each prdcor n Tabl 1 can b rprnd a T ˆ Y c( Y Y) ( ) = +. Tabl how valu of c for Prdcor Tˆ = Y + c Y Y of h Lan Valu of PSU whn n n Two-ag Clur Samplng wh Homognou Un and Rpon Error Varanc. A hown n Tabl, h dffrnc bwn h prdcor ar du o h conn of hrnkag conan. Tabl. Valu of c for prdcor T ˆ Y c( Y Y) = + of h lan valu of PSU Modl Mxd Modl cmm = k Sco & Smh css = f + ( 1 f ) k Random Prmuaon. crp = f + ( 1 f ) k Random Prmuaon wh c = f ρ + ( 1 f ρ ) k * RPR Rpon Error.6 Smulaon A mulaon udy program wa dvlopd by Sank (003a) o valua h prdcor n wo-ag clur amplng conx n dffrn ng. Th mulaon udy cond of hr ub-modul, craon of a fn populaon, lcon of wo ag ampl from h fn populaon, and comparon of h mulad man quar rror (SMSE) and h horcal man quar rror (TMSE) of h prdcor. Fr, h populaon wa dfnd by a of valu bad on prcnl of a hypohcal drbuon. W ud prcnl of uch drbuon o cra a populaon of un and clur. Th bac drbuon from whch h fn populaon wr gnrad wr normal. Dffrn drbuon could b lcd for un and clur. Howvr, w ud h am drbuon (normal) o gnra h un ffc for all clur. 10

Comparon of h prformanc of BLUP 0-11 For ach mulaon, h populaon cond of N clur wh M un pr clur. Each ndvdual clur paramr wa rprnd by h lan valu for clur, μ and hr man, by μ. Th varanc bwn clur, σ, wa fxd, and hn N nal valu vnly pacd wr gnrad bad on h prcnl of h pcfd drbuon. Th valu wr h nal valu of h clur paramr. Bcau h numbr of clur n h populaon wa fn, h avrag of h clur paramr would no ncarly qual o b h populaon man, μ. Th man of clur paramr wa rdfnd by cnrng hm a μ and r-calng hr valu o N ( μ μ) ha h varanc machd σ =. N 1 = 1 Nx, w gnrad un ffc for h M un for ach clur. Th ffc wr o b forcd o avrag o zro. In addon, h un ffc wr gnrad ung prcnl of a pcfd drbuon. Th drbuon wa normal. Th varanc of h un ffc may b hr o b conan for all clur or vary proporonally o max 0.1, μ. Th paramr for μ h clur wr formd by addng h un ffc o h clur man and wr rprnd by y. M (y - μ) Th varanc of h un paramr n clur wa gvn by σ =. Th common M-1 whn clur varanc, rprnd by σ, wa qual o h avrag whn clur varanc, =1 N σ σ =. Un ffc wr r-cald o ha hy had zro man for ach clur, and hr N = 1 avrag varanc wa qual o b σ. Th paramr n h mulaon program wr h numbr of clur n h populaon (N), h numbr of un n a clur (M), h numbr of clur n h ampl (n), h numbr of un n a ampld clur (m), h populaon man ( μ ), h varanc of clur man ( σ ), h 11

Comparon of h prformanc of BLUP 0-1 varanc of un n a clur ( σ ), h rpon rror varanc ( r ), h clur drbuon, and h un drbuon. W adopd and modfd h mulaon for h rarch. Th avrag dffrnc bwn h prdcd PSU man and h ru PSU man, and h MSE wr mad bad on known and unknown varanc. If varanc componn wr known, hr wr gh dffrn aumpon abou mxd modl, ladng o gh mxd modl prdcor, and gh analogou prdcor bad on Sco & Smh modl. Alo w oband h prdcor corrpondng o h clur man, and o h Random Prmuaon modl. Th ruld n oal o 18 prdcor bad on known varanc. For ach prdcor, w mad h MSE by valua h avrag quard dvaon from h ralzd clur man ovr many ampl (whr ach ampl corrpond o a ral). If varanc componn wr unknown, Sank (003a) propod wo dffrn mhod of mang varanc componn, whch lad o wo mxd modl prdcor, wo Sco and Smh prdcor, and wo random prmuaon prdcor. 3. MATERIALS AND METHODS 3.1 Saon Sudy Daa for h projc wr from h Saon Sudy and provdd by Dr. Ockn. Th varabl ncludd cholrol, d, lgh, acvy and ohr varabl wh hr ngl or mulpl maurmn on ach ubjc. Daa wr collcd from volunr (N=5000) rcrud from Fallon Halh Mannanc Organzaon (HMO) mmbr wh ag bwn 0 and 70 yar old. Pan wr nrolld bwn 1994 and 1998. Maurmn wr conducd on concuv hr-monh nrval ovr a wlv-monh prod on ach ubjc (Sank and Sngr, 004). Th daa cond of hr ub-daa : quarrly daa, 4-hour rcall daa, and baln daa. Th Quarrly daa conand lpd daa, 7DDR (7 day dary rcord bad on 1

Comparon of h prformanc of BLUP 0-13 ubjc rcall, 1 ach pr quarr), and hormonal daa. Maurmn wr mad on a mpl random ampl of 1 day ( m = 1) durng a quarr (3 monh prod, M = 90 ). Th 4-hour rcall daa conand phycal acvy, d, and lgh xpour. Thy wr maurd on randomly lcd wkday and on randomly lcd wknd day pr quarr( m= 3, M = 90). Nurn varabl valuad wr oal aurad fa nak, oal fa, oal carbohydra, and cholrol. Phycal acvy varabl wr maurd o valua h 1-yar avrag of rpord phycal acvy nrgy xpndur (MET-hour/day), by acvy doman (.., houhold, occupaonal and lur m) and nny. Sandard mabolc quvaln (MET) valu wr ud o calcula ma of phycal acvy nrgy xpndur. A wghd um of daly phycal acvy nrgy xpndur (MET-hour/day) wa compud ung h m rpord (hour/day) n acvy of ach nny and h followng MET wgh: lgh acvy, 1.5 MET; modra acvy, 4.0 MET; vgorou acvy, 6.0 MET; and vry vgorou acvy, 8.0 MET. On MET-hour/day wa approxmaly quvaln o 1 kcal/kg body ma/hour or o h rng mabolc ra of a pron wghng 60-70 kg (Mahw, al., 001). Phycal acvy varabl o b valuad wr lgh nny acvy, modra nny acvy, and vgorou nny acvy. Up o 3 day of 4-hour acvy wr collcd pr quarr. Th baln daa ncludd dmographc facor, uch a ag and gndr. Boh ubjc and da of maurmn ovr a quarr wr no randomly lcd. Subjc parcpang n h udy wr aumd o b comparabl o a mpl random lcon from a fn populaon. Thrfor, wo-ag B Lnar Unbad Prdcor n h udy wr aumd o b applcabl for h daa. 3. Emang Varanc Componn Varanc componn uch a varanc bwn clur (ubjc), rdual varanc (a combnaon of varanc bwn un (day) and rpon rror) for on yar (.. M=365) wr 13

Comparon of h prformanc of BLUP 0-14 mad ung a mxd modl wh rrcd maxmum lklhood mhod ung h Saon udy daa. Th varanc bwn day ( d ) wa calculad by ubracng rpon rror ( ) from h rdual varanc. Th nra-cla corrlaon of rpad maur on a un ( r ), and h nra-cla corrlaon of un n a clur ( r ) wr mad bad on h varanc æ d ö componn.. r =, r =. Rpon rror of varabl ar bad hr ç è + + ø d d on lraur rvw or hrough mulaon wh coffcn varaon (Tabl 4). For xampl, Hgd and Ncolo (1987) mad rpon rror for oal rum cholrol wa 5 (mg/dl). If rpon rror of h varabl wr no avalabl n h lraur, hy wr mad by mulang wo rpon on ach ubjc. Fr, rpon rror of a maur on ach ubjc wa calculad bad on aumd coffcn of varaon. Thn, h avrag rpon rror wa mad by poolng ndvdual rpon rror on ach ubjc. Two mulad rpon wr bad on mprcal maur of ach ubjc a h quarr 1 of h quarrly daa (cbdq6) n h Saon Sudy and ung h coffcn of varaon aumd (Tabl 4). W mulad wo rpon of wgh for ach ubjc wh random normal funcon gnraor gvn by Y = m + m *c.v.*rannor(d) (3.1) dk d d whr Y dk rprn wgh of ubjc a h day d and a h h k maur, m d dno h ru wgh of ubjc on a day d, whch aumd o b h man wgh of ubjc a quarr 1 of quarrly daa n Saon Sudy,.. 78.44 (kg), h rm rannor(d) a random numbr funcon bad on a normal drbuon. m *c.v. d h pobl rpon rror allowd o b dffrn for dffrn ubjc and day. Th rpon rror of maur n wgh wa mad ung a mxd modl ( Appndx A). 14

Comparon of h prformanc of BLUP 0-15 A mpl mxd modl, whch wa ud o ma varanc componn for a varabl, for h rpon of h h j lcd day, j = 1,...,m, for h h lcd ubjc, = 1,...,n, gvn by Y = m + B + D + E (3.) jk j jk whr μ dno o h xpcd rpon ovr ubjc n a populaon, dvaon bwn μ and h xpcd rpon of h h lcd ubjc. dvaon bwn xpcd rpon of h h lcd ubjc a h h xpcd rpon of ubjc, and h h lcd ubjc and a h ar ha B d N( 0, σ ) :. B corrpond o h D j ndca h h j lcd day, Y j, and E jk rprn h random rpon rror of maur for h j lcd day. B, D and E ar random. Th aumpon Varabl chon o ma varanc componn ung h mxd modl (3.) wr body compoon and lpd varabl (body ma ndx, yolc blood prur, daolc blood prur, LDL, HDL, oal cholrol, rglycrd, wgh), nurn varabl (oal aurad fa nak, oal fa nak, aurad fa a prcn of oal calor, oal fa a prcn of oal calor) and phycal acvy (lgh nny acvy, modra nny acvy and vgorou nny acvy). Th quarrly daa conand lpd daa, collcd by on maur pr ubjc pr quarr, and nurn daa, collcd by 7 day dary rcall wh 1 ach pr quarr, and nurn and phycal acvy daa, collcd by 4 hour rcall lphon nrvw gvn up o 3 day pr quarr. On of h aumpon o ma varanc componn for hr of varabl wa ha a ubjc wa lcd a random. All ubjc wh fv maur (on pr quarr) wr rand. Th varanc componn of lpd, nurn varabl (varanc bwn ubjc, and h j jk 15

Comparon of h prformanc of BLUP 0-16 rdual) wh a prod of 365 day wr mad wh h mxd modl hown n (3.) ( Appndx B). Smlarly, 365-day-prod varanc componn of nurn and phycal acvy varabl collcd by 4-hour rcall wr mad agan wh h mxd modl llurad n (3.) (S Appndx B). Bcau a rdual rror wa compod of varanc bwn day and a rpon rror, a 365-day-prod varanc bwn day wa h dvaon bwn h rdual rror and rpon rror. All analy wr prformd ung SAS 8.0. 3.3 Th Smulaon Sudy Th mulaon udy program addrd n Scon.6 wa modfd o compar h prformanc of B Lnar Unbad Prdcor. Modfcaon ncludd calculang h rlav MSE ncra bwn prdcor of h ru ubjc man n varou ng wh or whou known varanc componn. Th rlav MSE dfnd a m1 m x100%, whr m1 m hr h horcal or h mulad MSE of h prdcor, h prdcor ar h Clur man, h Mxd modl, Sco & Smh prdcor; m h horcal MSE of h Random Prmuaon modl. In addon, modfcaon nabld u o compar h horcal and h mulad MSE for all four prdcor wh or whou known varanc componn. Snc hr wr gh pobl way h varanc componn for h Mxd modl and Sco & Smh prdcor could b dfnd whn varanc componn wr known, lad o gh Mxd modl and gh Sco & Smh prdcor. Th ghh mxd modl and h ghh Sco & Smh prdcor wr ud o compar h prformanc wh ohr prdcor. Whn varanc componn wr unknown for ho wo prdcor, hr wr wo mhod of mang varanc componn ladng o wo Mxd modl and wo Sco & Smh prdcor. Th 16

Comparon of h prformanc of BLUP 0-17 cond Mxd modl and h cond Sco & Smh prdcor wr ud o compar h prformanc wh h Clur man modl and h Random Prmuaon prdcor. 4.1 Ema Varanc Componn 4. RESULTS Tabl 4 how h paramr for coffcn of varaon (c.v.) and h man valu of h varabl of nr. Th proc of mang h coffcn of varaon and h rpon rror of varabl dcrbd nx. L u condr an xampl of mang nra-ndvdual varably of phycal acvy lvl n mnu on on day for a ubjc. Suppo ha wo maur of phycal acvy on h am day wr collcd for ach ubjc. Each ubjc wa akd wc h numbr of mnu h/h xrcd durng h day. Suppo, for xampl, ha wo rpon of h fr ubjc ar 4 and 36 mnu. Alo, uppo ha h acual numbr of mnu h fr ubjc xrcd 30 mnu. Smlarly, for h cond ubjc, h wo rpon ar 16 and 4 mnu. Th acual m h cond ubjc xrcd 0 mnu. Thn, h rang (or nrval) of wo rpon for ach ubjc accoun for 40% of h acual valu. If uch daa wr avalabl for n ubjc, hy could b ud o ma h varanc componn of phycal acvy lvl. Bcau uch daa ar no avalabl, w aum ha a 95% confdnc nrval for rpon ha a wdh of 40% of h acual mnu. Thn, h h k rpon for ubjc n day d, Y dk gvn by Y = m + E (4.1) dk d dk whr m d rprn h ru amoun of phycal acvy of ubjc on a day d, whch aumd o b h man amoun of phycal acvy of ubjc a quarr 1 of quarrly daa (cbdq6) n h Saon Sudy. E dk dno h rpon rror. On andard dvaon of rpon rror of phycal acvy lvl of ubjc a day d, ( ) 0.1 m d a a rul of 17

Comparon of h prformanc of BLUP 0-18 95% C.I. wdh aumpon bcau = 0.** md/4 =0.1* md. Thrfor, a whn-ubjc coffcn of varaon (c.v.) of phycal acvy lvl qual o b 0.1 bcau cv.. = / m. d Smlarly, w can ma h rpon rror by mulaon wh aumd coffcn of varanc of lpd varabl. L u ma h coffcn of varanc of wgh and body ma ndx, followd by h proc of mulaon. Th h k rpon of wgh for ubjc, Y k, gvn by Y = m + E (4.) k k whr m rprn h ru wgh maur of ubjc. E k dno h rpon rror of wgh. Two rpon of wgh maur ar aumd o b 0.5 kg abov or 0.5 kg blow h ru wgh. Thrfor, h andard dvaon of rpon rror of wgh maur of ubjc a day d 0.5*/ 4 = 0.15. Th m aumd o b qual o h man wgh of ubjc a quarr 1 of quarrly daa (cbdq6) n h Saon Sudy,.. 78.44 kg. Thn a whn-ubjc coffcn of varaon (c.v.) of rpon rror of wgh qual o b 0.0016 bcau cv.. = / m = 0.15/ 78.44. d Nx, w ma h coffcn of varanc of body ma ndx. Suppo h ru ubjc hgh ( m ) aumd o b 1.7 mr. Thn h ubjc gvn by h k rpon of body ma ndx for Yk m Ek = + (4.3) 1.7 1.7 1.7 W aum hgh can b maurd o h nar cnmr (.. 4*andard dvaon = cm;.. wo hgh maur can b 1.69 mr and 1.71 mr). Th maxmum rpon rror for body æ E 0.5 ma ndx var k ö æ ö = = 0.0875 ç 1.69 çè.8561. Thrfor, h andard dvaon of rpon è ø ø rror of body ma ndx maur of ubjc 0.0875. Th body ma ndx of ubjc aumd o b h man body ma ndx of ubjc a quarr 1 of quarrly daa (cbdq6) n 18

Comparon of h prformanc of BLUP 0-19 h Saon Sudy,.. 7.34 kg/m. Thn a whn-ubjc coffcn of varaon (c.v.) of rpon rror of body ma ndx qual o b 0.003 bcau cv.. = / m = 0.0875/ 7.34. d Tabl 5 how varanc componn of lpd, nurn and phycal varabl and nracla corrlaon of clur and un a on yar. A ndcad n Tabl 5, h varanc of lpd varabl uch a oal cholrol bwn ubjc for h m prod of 365 day 1463.33 mg/dl. Th rdual of varanc componn (.. day o day and rpon rror) of oal cholrol 86.3 mg/dl. Bcau h rpon rror of rum cholrol 5 mg/dl, h amoun of oal cholrol bwn day for h m prod of 365 day 61.3 mg/dl. Tabl 5 alo rpor ha h nra-cla corrlaon of un n a clur (ubjc), and of rpad maur on a un (day) for oal cholrol a h m prod of 365 day ar 0.960 and 0.14. In addon, Tabl 5 ndca ha a 365-day-prod varanc of nurn varabl collcd ung 4 hour rcall uch a oal aurad fa nak (SFA) and rdual rror ar 113.4 gm and 169.57 gm. Bcau rpon rror of oal SFA 151.01 gm, a 365-day-prod varanc of oal SFA bwn day 18.56 gm. Furhrmor, Tabl 5 how ha h nra-cla corrlaon of un n a clur (ubjc), and h nra-cla corrlaon of rpad maur on a un (day) for oal SFA a h m prod of 365 day ar 0.859 and 0.109. Tabl 6 how clur and un nra-cla corrlaon of om halh varabl a 365- day prod n abular forma. Th prformanc of h prdcor (h Mxd Modl, Sco and Smh Prdcor, h Clur Man Modl, and h Random Prmuaon Modl) wr valuad by comparng h mulad man quar rror (SMSE) and h horcal man quar rror (TMSE) of h modl a wo common ng of clur and un nra-cla corrlaon ( r r r r = 0.67, = 0.83; =0.67, =0. ). 4. Th Smulaon Rul 4..1 Known Varanc And Equal Whn Clur Varanc 19

Comparon of h prformanc of BLUP 0-0 Fgur 1 and llura h prcn ncra n h dffrnc bwn h mulad man quar rror (SMSE) and h horcal man quar rror (TSME) of h prdcor (h Mxd Modl, Sco and Smh Prdcor, h Clur Man Modl, and h Random Prmuaon Prdcor) undr an aumpon of known varanc, and wh qual whn-clur varanc n wo common ng of clur and un nra-cla corrlaon hown n Tabl 6 r = 0.67, r = 0.83; r =0.67, r =0. ) and a wo mulaon run (1000, 10000). Wh 1,000 ( ral, h rlav dffrnc bwn -.5% and 3.5%. Whn h un amplng fracon (f) ncra o 0.7, h rlav dffrnc rach h pak (Fgur 1, op). Wh 10,000 ral n boh Fgur 1 and, h rlav dffrnc hav bn rducd bwn -1.0% and 1.0%. In addon, hr ar no obvou pak n 10,000 ral a occur wh 1,000 ral. Furhrmor, wh 10,000 ral, hr ar no prdcor howng h conn rul wh h mall rlav ncrmn n SMSE ovr TMSE hrough all h un-amplng fracon. Fgur 3 how h prcn ncra n h horcal man quar rror (TMSE) of prdcor (h Mxd Modl, Sco and Smh Prdcor, and h Clur Man Modl) ovr TMSE of h Random Prmuaon Modl undr an aumpon wh known varanc, and wh qual whn-clur varanc n wo common ng of clur and un nra-cla corrlaon coffcn ( r = 0.67, r = 0.83; r=0.67, r =0. ) and a on mulaon (numbr of run=10,000). Svral parn mrg from Fgur 3 (op), whr boh clur and un nra-cla corrlaon coffcn ar largr ( r = 0.67, r =0.83). Fr, h Random Prmuaon Prdcor ha h mnmum TMSE, followd by Sco and Smh prdcor. Scond, a h un amplng fracon bcom larg, h magnud n prcn ncrmn n TMSE of h Mxd Modl ovr TMSE of h Random Prmuaon Prdcor ncra, bu h magnud n prcn ncra n h TMSE of h Clur Man Modl ovr h TMSE of h Random Prmuaon Prdcor dcra. Whn un-amplng fracon g ovr 0.63, h Mxd Modl prform wor han h Clur Man Modl. Howvr, vral dffrn parn mrg from Fgur 3 (boom), whr 0

Comparon of h prformanc of BLUP 0-1 un nra-cla corrlaon mallr ( r = 0.67, r = 0.). Fr, h Random Prmuaon Prdcor ha h mnmum TMSE, bu followd by h Mxd Modl nad of Sco and Smh Prdcor. Scond, a h un-amplng fracon bcom larg, h magnud of ncrmn n TMSE of Sco and Smh prdcor ovr TMSE of h Random Prmuaon Prdcor ncra gradually, bu h magnud of ncrmn n h TMSE of h Clur Man Modl ovr h TMSE of h Random Prmuaon Prdcor dcra dramacally. Whn h un amplng fracon rach 0.9, howvr, h Clur Man Modl ll ha hghr prcn ncrmn n TMSE ovr TMSE of Random Prmuaon Prdcor han Sco and Smh Prdcor. Fgur 4 llura h prcn ncrmn n h mulad man quar rror (SMSE) of h prdcor ovr h TMSE of h Random Prmuaon Prdcor undr an aumpon wh known varanc, and wh qual whn-clur varanc n wo common ng of clur and un nra-cla corrlaon ( r = 0.67, r = 0.83; r=0.67, r =0. ) and a on mulaon (numbr of run=10,000). Thr ar mlar parn bwn Fgur 4 (op) and Fgur 3. For xampl, whn h un-amplng fracon g ovr 0.6, h Clur Man Modl prform br han h Mxd Modl. Howvr, TMSE of h Random Prmuaon Modl no alway mallr han h SMSE of Sco and Smh prdcor a hown n Fgur 3, h Clur Man Modl prform much wor han ohr prdcor n Fgur 4 (boom). Th rao of ncrmn n SMSE of h Clur Man Modl ovr TMSE of h Random Prmuaon Prdcor ovr 95% whn un-amplng fracon 0.0. A h un amplng fracon ncra, h rao g mallr. Bu 0% whn h un amplng fracon (f) ncra o 0.9. 4.. Unknown Varanc And Equal Whn Clur Varanc Fgur 5 llura h prcn ncra n h dffrnc bwn h mulad man quar rror (SMSE) and h horcal man quar rror (TSME) of h prdcor (h Mxd Modl, Sco and Smh Prdcor, h Clur Man Modl, and h Random Prmuaon 1

Comparon of h prformanc of BLUP 0- Modl) undr an aumpon wh unknown varanc, and wh qual whn clur varanc n wo common ng of clur and un nra-cla corrlaon r = 0.67, r = 0.83; r =0.67, r =0. ) rpcvly and a on mulaon (numbr of ( run=10,000). Svral parn mrg from Fgur 5 (op). Fr, h prcn ncrmn n SMSE ovr TMSE of four prdcor (xcludng h Clur Man Modl) dcra a h un amplng fracon ncra. Scond, h Mxd Modl ha h hgh ncrmn n SMSE o TMSE n all un-amplng fracon and ovr all ohr prdcor. I rach ovr 30% whn h un-amplng fracon 0., and dca o 8% whn h un-amplng fracon g o 0.9. Among h Clur Man Modl, Sco and Smh Prdcor, and h Random Prmuaon Prdcor, h Clur Man Modl ha h mall ncrmn n SMSE ovr TMSE. Th rao of prcn ncrmn n SMSE o TMSE of Sco and Smh prdcor ovr h rao of h Random Prmuaon Modl abou whn h un-amplng fracon 0.. Th rao dcra o 1 a h un-amplng fracon g 0.9. Smlar parn mrg form Fgur 5 (boom) whn ρ = 0.67, and ρ = 0.. Howvr, hr ar om dffrnc. On of hm ha h dffrnc n prcn ncrmn n SMSE rlav o TMSE bwn Sco and Smh prdcor and h Random Prmuaon Prdcor dpnd on h un-amplng fracon (f). Whn f l han 0.35, Sco and Smh prdcor ha hghr ncrmn n SMSE ovr TMSE han h Random Prmuaon Prdcor. Howvr, whn f grar han 0.35, h Random Prmuaon Prdcor ha hghr prcn ncrmn n SMSE o TMSE han Sco and Smh Prdcor. Fgur 6 llura h prcn ncra n h mulad man quar rror (SMSE) of h prdcor ovr h TMSE of h Random Prmuaon Modl undr an aumpon wh unknown varanc, and wh qual whn-clur varanc n wo common ng of clur and un nra-cla corrlaon ( r = 0.67, r = 0.83; r=0.67, r =0. ) and a on mulaon (numbr of run = 10,000). Svral parn appar n Fgur 6 (op). Fr, h TMSE of Random

Comparon of h prformanc of BLUP 0-3 Prmuaon Prdcor l han SMSE of ohr prdcor almo all h m. Scond, Sco and Smh prdcor ha h mallr prcn ncra n SMSE o TMSE of h Random Prmuaon Prdcor han h Mxd Modl and h Clur Man Modl mo of m. Thrd, h ncrmn n SMSE of h prdcor ovr TMSE of h Random Prmuaon Prdcor dcln a h un amplng fracon (f) ncra. Howvr, whn un nra-cla corrlaon ( ρ ) 0.83 (Fgur 6, op), h Mxd Modl ha h hgh prcn ncra n SMSE o TMSE of h Random Prmuaon Prdcor, followd by h Clur Man Modl. Whn un nracla corrlaon ( ρ ) 0. (Fgur 6, boom), h Clur Man Modl ha h hgh prcn ncra n SMSE o TMSE of h Random Prmuaon Prdcor, followd by h Mxd Modl. Fnally, whn un nra-cla corrlaon ( ρ ) 0. n Fgur 6 (boom), h prcn ncra n SMSE of all four prdcor ovr TMSE of h Random Prmuaon modl abou 175% a h un-amplng fracon of 0.1; howvr, whn un nra-cla corrlaon ( ρ ) 0.83 n Fgur 6 (op), h prcn ncra n SMSE of all four prdcor ovr TMSE of h Random Prmuaon modl abou 47%. Tha may mply ha h hghr rpon rror n maur of h varabl, h bggr h prcn ncra n SMSE o TMSE of h Random Prmuaon modl whn h un-amplng fracon vry lowr uch a 10%. 3

Comparon of h prformanc of BLUP 0-4 Tabl 4. Th paramr for h coffcn of varaon of om halh cnc varabl Varabl Toal SFA Prcn of SFA Toal Fa Inak Prcn Fa Inak Lgh nny acvy Modra nny acvy Vgorou nny acvy Collcon nrval C.V. 4hr 0.10 Man 5.19 gm 7ddr 0.0 9.0 gm 4hr 0.10 7ddr 0.10 11.0 % of calor 1.50 of % calor 4hr 0.10 70.01 gm 7ddr 0.0 85.6 gm 4hr 0.10 7ddr 0.10 31.1 % calor 36.91 % calor 4hr 0.10 0.1 MET hr 4hr 0.10 0.86 MET hr 4hr 0.10 0.87 MET hr Body wgh Quarrly 0.0016 78.44 kg Body ma ndx Quarrly 0.003 TG Quarrly 0.6 HDL Quarrly 0.074 7.34 kg/m 140.90 mg/dl 47.8 mg/dl Sourc and aumpon A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 80% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 80% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. A 95% confdnc nrval for rpon ha a wdh of 40% of h ru rpon. abolu dffrnc bwn wo rpon 0.5 kg, abolu dffrnc bwn wo rpon 0.5 kg, h accuracy of hgh maur 0.01 mr. Smh al, 1993. Smh al, 1993. LDL Quarrly 0.09 144.0 mg/dl Smh al, 1993. SBP Quarrly 0.03 DBP Quarrly 0.03 119.83 mm Hg Andr al., 1987. Cavlaar al., 004. Rpoll al., 001. 77.15 mm Hg Andr al., 1987. Cavlaar al., 004. Rpoll al., 001. TC Quarrly 0.0688 18.58 mg/dl Hgd and Ncolo (1987) (Connud on nx pag) 4

Comparon of h prformanc of BLUP 0-5 Tabl 4. Connud *4hr: Bad on a 4 hour rcall lphon nrvw gvn for up o 3 day n a quarr. *7ddr: 7-day dary rcord bad on ubjc rcall, 1 ach pr quarr. *bad on Saon udy daa unl ohrw nod. Sourc: h04py_abl4.a Tabl 5. Varanc componn of om halh varabl and nra-cla corrlaon (ICC) of clur and un Varabl Collcon Tm Varanc componn ICC Inrval Prod Subjc day rp. rror Clur Un % FAT(%calor) 7ddr 365 39.79 10.99 15.60 0.784 0.413 % SFA(%calor) 7ddr 365 6.40.9 1.79 0.737 0.561 Toal FAT(gm) 7ddr 365 1397.83 1037.74 401.61 0.574 0.71 Toal SFA(gm) 7ddr 365 156.7 73.38 47.40 0.681 0.608 % FAT(%calor) 4hr 365 9. 4.15 56. 0.876 0.069 Lg. Acv.(MET hr) 4hr 365 0.0 0.00 0.57 1.000 0.000 Mod. Acv.(MET hr) 4hr 365 0.69.01 4.13 0.56 0.38 In. Acv.(MET hr) 4hr 365 1.83 7.16 0.01 0.03 0.999 % SFA(%calor) 4hr 365 6.63 1.34 11.93 0.83 0.101 Toal FAT(gm) 4hr 365 695.95 138.14 946.99 0.834 0.17 Toal SFA(gm) 4hr 365 113.4 18.56 151.01 0.859 0.109 BMI (kg/m) quarr 365 30.06 0.47 0.01 0.984 0.984 DBP (mm Hg) quarr 365 64.77 36.14 6.19 0.64 0.854 HDL (mg/dl) quarr 365 17.97 13.30 13.55 0.906 0.495 LDL (mg/dl) quarr 365 1135.58 44.58 18.30 0.96 0.196 SBP (mm Hg) quarr 365 31.14 70.78 13.40 0.766 0.841 TC (mg/dl) quarr 365 1463.33 61.3 5.00 0.960 0.14 TG (mg/dl) quarr 365 14606.16 3354.3 873.3 0.813 0.793 Wgh (kg) quarr 365 98.35 3.81 0.0 0.987 0.995 *4hr: Bad on a 4 hour rcall lphon nrvw gvn for up o 3 day n a quarr *7ddr: 7-day dary rcord bad on ubjc rcall, 1 ach pr quarr. Sourc: py04p0j.a, py04p019h.a, py04p3h.a, py04p07l4.a, py04p08.a 5

Comparon of h prformanc of BLUP 0-6 Tabl 6. Clur and un nra-cla corrlaon of om halh varabl wh a m prod of 365 day un nra-cla corrlaon ( r ) [0-0.10) [0.10-0.35) [0.35-0.65) [0.65-0.87) [0.87-1) 1 ( r ) clur nra-cla corrlaon [0-0.10) [0.10-0.35) [0.35-0.65) [0.65-0.87) %FAT ** Toal Fa** Modra Ac. Toal SFA** %SFA** [0.87-1) LDL, TC %SFA*, %FAT*, Toal SFA* HDL Toal FAT* DBP SBP TG Inn Ac. BMI, WT 1 Lgh Ac. *:7ddr, 7 day dary rcord bad on ubjc rcall, 1 ach pr quarr **:4hr, bad on a 4 hour rcall lphon nrvw gvn for up o 3 day n a quarr 6

Comparon of h prformanc of BLUP 0-7 Fgur 1. Incrmn n SMSE rlav o TMSE of Clur Man Modl, Mxd Modl, Sco and Smh, and Random Prmuaon a wo mulaon run (op=1000, boom=10000) wh ρ = 0.67, ρ = 0.83 7

Comparon of h prformanc of BLUP 0-8 Fgur. Incrmn n SMSE rlav o TMSE of Clur Man Modl, Mxd Modl, Sco and Smh, and Random Prmuaon a wo mulaon run (op=1000, boom=10000) wh ρ = 0.67, ρ = 0.0 8

Comparon of h prformanc of BLUP 0-9 Fgur 3. Incrmn n TMSE of prdcor ovr TMSE of Random Prmuaon a on mulaon (numbr of run=10000) a wo ca (op: ρ = 0.67, ρ = 0.83, boom: ρ = 0.67, ρ = 0.0) bad on known varanc 9

Comparon of h prformanc of BLUP 0-30 Fgur 4. Incrmn n SMSE of prdcor ovr TMSE of Random Prmuaon a on mulaon (numbr of run =10000) a wo ca (op: ρ = 0.67, ρ = 0.83, boom: ρ = 0.67, ρ = 0.0) bad on known varanc 30

Comparon of h prformanc of BLUP 0-31 Fgur 5. Incrmn n SMSE rlav o TMSE of Clur Man Modl, Mxd Modl, Sco and Smh, and Random Prmuaon a on mulaon (numbr of run=10000) a wo ca (op: ρ = 0.67, ρ = 0.83, boom: ρ = 0.67, ρ = 0.0) whn varanc ar unknown 31

Comparon of h prformanc of BLUP 0-3 Fgur 6. Incrmn n SMSE of prdcor ovr TMSE of Random Prmuaon a on mulaon (numbr of run=10000) a wo ca (op: ρ = 0.67, ρ = 0.83, boom: ρ = 0.67, ρ = 0.0) whn varanc ar unknown 3

Comparon of h prformanc of BLUP 0-33 5. CONCLUSIONS Whn varanc wr known, and h whn-clur varanc wr qual, hr wa no much of a dffrnc n h prcn ncra n SMSE rlav o TMSE for four prdcor a boh ng of clur and un nra-cla corrlaon ( r = 0.67, r = 0.83; r=0.67, r=0. ). Howvr, whn h varanc wr unknown, and whn-clur varanc wr qual, hr wr dffrnc n SMSE rlav o TMSE for four prdcor n boh ca of clur and un nracla corrlaon, pcally a mallr un amplng fracon ( r ). Whn varanc wr known, whn-clur varanc wr qual, and boh clur and un nra-cla corrlaon wr bggr han 0.5, h Random Prmuaon Prdcor had h mnmum TMSE compard o TMSE of ohr prdcor, followd by Sco and Smh Prdcor. Whn h un nra-cla corrlaon wa l han 0.5, for xampl, 0., and ohr ng wr h am, h Random Prmuaon Prdcor ll had h mnmum TMSE compard o TMSE of ohr prdcor, bu wa followd by h Mxd Modl. Th wa conn wh h rul gvn by Sank and Sngr (004), n whch h Random Prmuaon Prdcor had h mall TMSE. Whn varanc wr known, and h whn-clur varanc wr qual, and boh clur and un nra-cla corrlaon wr grar han 0.5, SMSE of Sco and Smh Prdcor wa h clo o h TMSE of h Random Prmuaon Prdcor (Fgur 3A). Howvr, whn un nra-cla corrlaon wa 0., h SMSE of h Mxd Modl wa h clo o h TMSE of h Random Prmuaon Prdcor (Fgur 3B). Marno, Sngr and Sank (004) rpord mlar rul whn varanc wr known, and h whn clur varanc wa qual, n whch h Random Prmuaon Prdcor had h mall SMSE, followd by h Mxd Modl whn ρ and ρ wr mall (up o 0.) or by Sco and Smh Prdcor whn ρ and ρ wr modraly larg (bwn 0.5 and 0.8). Whn varanc wr unknown, and whn-clur varanc wr qual, h Random Prmuaon Prdcor wa no alway h b among hr prdcor (xcludng h Clur 33

Comparon of h prformanc of BLUP 0-34 Man) bad on h rlav MSE h mulad MSE-h horcal MSE x 100% (pag 3). h horcal MSE Mnmum MSE wa oband for h prdcor undr h Random Prmuaon whn h Clur amplng fracon (F) and h un-amplng fracon (f) wr mall. Th wa conn wh h rul gvn by Marno, Sngr and Sank (004). Th Clur Man Modl had h mall prcn ncra n SMSE rlav o TMSE among four prdcor a boh ng of clur and un nra-cla corrlaon. Furhrmor, Th Clur Man had a conan rlav MSE ovr h un-amplng fracon (f). Th raon wa ha h Clur Man Modl dd no dpnd on h un-amplng fracon (f) a hown n Tabl 1. In addon, whn rpon rror ncra,.. ρ, wa mall, h rlav ncra n h prdcor SMSE ovr h horcal MSE of h Random Prmuaon wa nflad pcally whn h un-amplng fracon wa mall. Tha may ndca ha prformanc of prdcor bcam poor a h crcumanc. 6. DISCUSSION Thr wr vral lmaon o h udy. Fr, daa n Saon Sudy wr oband from volunr, and boh ubjc and da of maurmn ovr a quarr wr no randomly lcd. W aumd ha ubjc parcpang n h udy o b comparabl o a mpl random lcon from a fn populaon. In addon, nc h rpon rror of mo of h varabl n h udy wr no avalabl n h lraur, hy wr mad by wo mulad rpon on ach ubjc. Th mulad rpon of h varabl on ach ubjc dpndd on boh h man valu of varabl a quarr 1 of h quarrly daa n h Saon udy and h coffcn of varaon of h varabl. W had mad up coffcn of varaon corrpondng o mpl plaubl aumpon. Morovr, w only aumd ha daa wr normally drbud. In addon, w aumd ha numbr of un n ach clur wa qual, and an qual numbr of un wr lcd from ach lcd clur. And w had no mulad 34

Comparon of h prformanc of BLUP 0-35 rul whn h un-amplng fracon (f) wa undr 0.1. Dp ho lmaon, h udy had vral faur worhy of our rarch ffor. Th udy may b ud o a publc halh rarchr n valuang BLUP wh varabl of publc halh mporanc n dffrn praccal ng. Thrfor, h rarch had provdd praccal gudanc a o how o b prdc om common arbu uch a aurad fa nak. Fnally, h prdcor could b valuad by h horcal and h mulad MSE. 35

Comparon of h prformanc of BLUP 0-36 APPENDIX A SAS CODE SPECIFICATION 1 Daa mu b ranformd a wo row/maur pr ubjc wh column valu of ubjc (d), wgh. Proc mxd daa=quarr mhod=rml; Cla d; Modl wgh=/oluon; Random d; Run; Th rdual rror n h varanc componn h rpon rror of wgh. APPENDIX B SAS CODE SPECIFICATION Sacal analy for modl (3.1) ung SAS Daa mu b ranformd a fv row/maur pr ubjc wh column varabl dnfyng h ubjc (d), oal cholrol (c). Proc mxd daa=quarr mhod=rml; Cla d; Modl c=/oluon; Random d; Run; 36

Comparon of h prformanc of BLUP 0-37 REFERENCES Andr J.L., J.C. P, R. Gugun, and J.P. Dchamp. 1987. Varably of arral prur and har ra maurd a wo prod of 15 mnu o 15 day nrval. Arch Mal Cour Va. Jun:80(6):1005-1010. Cavlaar M., J.H. Tuln. J.H. Van Bmml, P.G. Muldr. And A.H. Van Dn Mrackr. 004. Rpoducbly of nra-arral ambulaory blood prur: ffc of phycal acvy and pour. Journal of Hyprn. Jun:(6)1105-111. Goldbrgr, A. S. 196. B lnar Unbad Prdcon n h Gnralzd Lnar Rgron Modl. Journal of h Amrcan Sacal Aocaon. 57: 369-375. Hgd, D.M. and R.J. Ncolo. 1987. Indvdual varaon n rum cholrol lvl. Proc Nal Acad Sc., USA. 84(17) 659-661. Marno, S. S., J. Sngr, E. J. Sank. 004. Prformanc of balancd wo-ag mprcal prdcor of ralzd clur lan valu from fn populaon: A mulaon udy. unpublhd rpor. Mahw, C.E., J.R. Hbr, P.S. Frdon, E. J. Sank III, P.A. Mrram, C.B. Ebblng, and Ira S. Ockn. 001. Sourc of varanc n daly phycal acvy lvl n h aonal varaon of blood cholrol udy. Amrcan Journal of Epdmology. Volumn 153. Numbr 10. 987-995. Rpoll, Or M, Roboo E. Marn., Morno A. Daz., Bana B. Arangurn., Smon M. Murca., Mdna A. Mdna., and Dl Pozo FJ Fonca. 001. Agrmn n h maurmn of blood prur among dffrn halh profonal. Ar mrcury phygmomanomr rlabl? An Prmara. March 15;7(4)34-43. Robnon, G. K. 1991. Tha BLUP a good hng: h maon of random ffc. Sacal Scnc 6(1): 15-51. Royall, R. M. 1976. Th lnar la quar prdcon approach o wo-ag amplng. Journal of h Amrcan Sacal Aocaon 71:657-664. Samango, F. 003. Commn o Prdcng Random ffc from fn populaon Clurd ampl wh rpon rror auhord by Sank and Sngr. Sarndal, Carl-Erk, Bng Swnon and Jan. Wrman. 1991. Modl ad urvy Samplng. Sprngr-Vrlag. p14-150. SAS Inu Inc. 1999. SAS/STAT Ur Gud. Vron 8.0. Gary, NC. Sarl, S.R., G. Calla and C.E. McCulloch.199. Varanc componn. Wly Sr n Probably and mahmacal ac. John Wly & Son, Inc. Sco, A. and T.M.F. Smh. 1969. Emaon n mul-ag urvy. Journal of h 37

Comparon of h prformanc of BLUP 0-38 Amrcan Sacal Aocaon 64(37): 830-840. Smh, S.J., G.R. Coopr, G.L. Myr, and E.J. Sampon. 1993. Bologcal varably n concnraon of rum lpd: ourc of varaon among rul from publhd ud and compo prdcd valu. Cln Chm. Jun; 39(6)101-10. Sank, E. J. III., A. Wll. And I. Ockn. 1999. Why no rounly u b lnar unbad prdcor (BLUPS) a ma of cholrol, prcn fa from ka and phycal acvy? Sac n Mdcn 18: 943-959. Sank, E. J. III, 003. Evaluang h MSE of Prdcor n Balancd Two Sag Prdcor of Ralzd Random Clur Man Wh Rpon Error. (hp://www.uma.du/clur/d/bblo-papr.hml). Sank, E. J. III., Julo Sngr. 004. Prdcng random ffc from fn populaon clurd ampl wh rpon rror. Journal of h Amrcan Sacal Aocaon. Dcmbr 004, Vol. 99, No. 468. Thory and Mhod. Sank, E. 003a. Noaon ud o Conruc Prdcor and Ema of Prdcor n h Smulaon Sudy for Prformanc of Balancd Two Sag Prdcor of Ralzd Random Clur Man. (hp://www.uma.du/clur/d/bblo-papr.hml). Sank, E. 003b. Emang h Varanc n a Smulaon Sudy of Balancd Two Sag Prdcor of Ralzd Random Clur Man. (hp://www.uma.du/clur/d/bblopapr.hml). Sank, E. 003c. Prdcng random ffc n group randomzd ral. (hp://www.uma.du/clur/d/bblo-papr.hml). 38