Notation for Mixed Models for Finite Populations

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Barbara Warner
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1 30- otato for d odl for Ft Populato Smpl Populato Ut ad Rpo,..., Ut Labl for,..., Epctd Rpo (ovr rplcatd maurmt for,..., Rgro varabl (Luz r for,...,,,..., p Aular varabl for ut (Wu z μ for,...,,,..., p Aular varabl dvato from ma for ut z for,..., Epctd Rpo (ovr rplcatd maurmt(otato 0 ud wth aular formato (Wu w for,..., whr w ( μ ( μ w Wght for ut. Wght for rgro paramtr (Luz r w for,..., Wghd pctd rpo for ut w μ ( for,..., Rgro approach (Luz r for,..., Rgro approach (Luz r A+ B Lac of ft rgro approach (Luz r b μ μ Slop for ut bad o coctg populato ma (Luz r Wth aurmt Error,..., r d for rplcato W or W (f r for,..., r aurmt rror for th rplcato of ut clutr-otato007.doc //007 5: P

2 30- W % or W % (f r for,..., r aurmt rror for th rplcato o th ut poto + W for,..., r aurmt for th rplcato. w Wghtd rpo for w th rplcato of ut Covto: U th ubcrpt R for vctor wh th ol radom compot maurmt rror. U th ubcrpt po for pctato ovr poto maurmt rror (Luz r U th ubcrpt ξ 3 for pctato ovr maurmt rror o ut clutr. U th ubcrpt ξ for pctato wth rpct to prmutato of ut a clutr. U th ubcrpt ξ for pctato wth rpct to prmutato of th clutr. U th ubcrpt ξ for pctato wth rpct to prmutato of th ut of a mpl populato. U th ubcrpt S for pctato wth rpct to prmutato of th ut of a mpl populato (Luz r. U a upr-crpt for vctor wh ut ad maurmt rror ar radom. t U a ovr-crpt U wh radom varabl ar padd. Vctor ( ( L Epctd rpo vctor (wth rpct to maurmt L Aular rpo for ut (Wu (( ( p ( ( L Aular rpo (Luz r (( ( L Aular rpo (Luz r (( z ( z0 z z z L p ( z z ( ( w w w w L w Vctor of wght clutr-otato007.doc //007 5: P

3 30-3 ( ( L Epctd wghtd rpo vctor (wth rpct to w w w w w ( ( R maurmt L aurd rpo vctor (for r for all,...,, ( ( L aurd wghtd rpo vctor (for r for all wr w w w w (( (,...,, W W W W L W aurmt rror vctor (for r for all,...,, α Gral vctor of paramtr. Cotat vctor wth poto ad zro lwhr Paramtr ad Cotat c c Avrag rmag coffct wh prdctg a ma ug aular + varabl ( c c Ud ESE of Rgro Prdctor (Luz r C + f Samplg fracto (Wu (but th ma b dfd dffrtl -tag amplg μ Uual dfto μ Dfto ud wth aular varabl, rgro. μ Rgro problm (Luz r μ Dfto ud wth gl rgro problm μ w w A μ Bμ Rgro trcpt clutr-otato007.doc //007 5: P 3

4 30-4 B ( μ ( μ ( μ Rgro paramtr for aocato. (Luz r ( μ or ( μ Rgro problm (Luz r ( μ Rgro problm (Luz r ( μ Rgro problm (Luz r w w ( w μ ( μ ( μ Rgro (Luz r ( ( μ μ ρ ( W var R ( W var R L aurmt rror varac aocatd wth ut % aurmt rror varac (.g. trvwr, laborator ( W var po aocatd wth poto (Ed % % aurmt rror varac aocatd wth poto (Luz (Ed r (Luz r clutr-otato007.doc //007 5: P 4

5 30-5 (Luz r w w + Radom prmutato hrag cotat for mpl radom amplg wth ut maurmt rror whr r for all,...,. % + Radom prmutato hrag cotat for mpl radom L amplg wth poto maurmt rror whr r for all,..., (Ed. + + Partall codtoal radom prmutato hrag cotat for mpl % ( ρ ( + + ρ % + ( ρ ( + ρ ( ρ ( % ρ + % % ( radom amplg wth poto maurmt rror whr r for all,..., (Ed. Cotat for Rg wth ut maurmt rror (Luz r Rgro problm (Luz r Cotat for Rg wth Poto rror (Luz r Ud b Luz r for Rgro + + D c + + c Cotat for Rgro wth maurmt rror (Luz r ( D % c + c + % Cotat for Rgro wth poto rror (Luz r clutr-otato007.doc //007 5: P 5

6 30-6 u u aurmt rror varac aocatd wth th ut ralzd poto. Vctor ad atrc (otc that w could df u u + L, whch m bttr. a a 0 L 0 0 L 0 ; O 0 0 L ; L L Ja aa ; O L a 0 L 0 0 a L 0 D a O 0 0 L ac Pa a J a a Pab, a J a b 0 R 0 P whr (( r ( c D a dagoal matr wth lmt Δ P whr w ot that L L L a a a a L a. A dagoal matr. o th dagoal. Ud to ubtract ma rgro (Luz r Δ ad w w Δw ud to collap a partto radom varabl, uch a L ( 0 L 0 ( clutr-otato007.doc //007 5: P 6

7 30-7 K K ud to partto radom varabl to ampl, rmadr, uch that K K ( 0 K 0 ( rgro (Luz r g g g cotat to dtrm targt ( c c c Cotat to dtrm targt wth aular varabl (Wu. ( ( c c c c L c Parttod cotat for ampl (Wu ( ( C c c c L c Parttod Cotat to df rgro targt (Luz r (( c + ( c + c + c c L Parttod cotat for rmadr (Wu (( c + ( c + c + c C L Parttod Cotat to df rgro targt (Luz r ( C c 0 Parttod lar combato for ampl wth aular p ( ( p varabl (Wu C c 0 Parttod lar combato for rmag wth aular varabl (Wu Parttod vro of dcator vctor for poto. ( G 0 Epro mlar to dg matr wth aular ampl radom varabl p g C 0 Parttod lar combato for ampl wth rgro (Luz ( ( ( p r G 0 Epro mlar to dg matr for rmadr radom varabl. G GK Sampl lar combato to df targt b Vvaa S R S G GK Rmadr lar combato to df targt b Vvaa R ( μ μ (Wu ( μ μ 0 (Wu z p clutr-otato007.doc //007 5: P 7

8 30-8 ( μ μ μ (Luz r X ( z( z Σ z μ z μ X ΣX Σ Varac matr (Luz r o o Σ o o vr varac for rgro ud b Luz r ( p (( L Σ ρ X X X X X X Σ Squard multpl corrlato coffct of o X ( β β β β Σ L Rgro paramtr btw rpo ad aular varabl β X X p μ, o that + ad μ β β β β L β (Doubl dfto!. Prmutato Radom Varabl-,..., Poto th prmutato. U for,..., ;,..., dcator of lcto of ut poto. u for,..., ;,..., Ralzato of th radom varabl U for a prmutato S U for,..., Radom varabl rprtg labl for ut poto U for,..., Radom varabl rprtg pctd rpo for ut w w poto U for,..., Radom varabl rprtg pctd wghtd rpo X U for,..., Rgro approach (Luz r clutr-otato007.doc //007 5: P 8

9 30-9 X X μ ( Rgro approach (Luz r tar X Rgro approach (Luz r tar + + Rpo for th ut poto (for U W + r for all,..., wh thr maurmt rror (Luz r. + u W or + + W % Partall codtoal rpo (Ed (Doubl dfto! X Rgro approach (Luz r tar + + U or Rpo for th ut poto (for r for all,..., wh thr maurmt rror (Ed. w Uw % U + W % or % (wh Rpo for th ut poto wh thr W UW W % or poto (.. trvwr rror or W UW Rplcato rror for th ut poto (for r W % (f r for all,..., aurmt rror aocatd wth Poto (.. trvwr for th ut poto. W% W% X Rgro problm (Luz r ( ( U U U U L U ( ( u u u u L u Ralzato of th vctor U. clutr-otato007.doc //007 5: P 9

10 30-0 ( L U U U U t t t W Epadd Radom Varabl U U U W for,..., ad,..., Radom varabl rprtg for ut ad poto a prmutato. for,..., ad,..., Radom varabl rprtg rpo for ut ad poto a prmutato. for,..., ad,..., Radom varabl rprtg maurmt rror for th rplcato for ut ad poto a ( ( t U U U L U or prmutato. t U (( ( ( ( t U + W U + W U + W L U + W t t t W U or + ( + W t t t t vc ( t L or vc ( + W whr r for all,..., U t t t t vc L or ( t vc U ( ( W t UW UW UW L U W or t W W U whr r for all,..., Collapd Radom Varabl t t L A B L ud to collap radom varabl, a for ampl ( (( ( L or U ( ( L or U w w w w w clutr-otato007.doc //007 5: P 0

11 (( ( L or (( ( L or W wh r for all,..., (Ed UW wh r for all,..., (Luz r % + W% Rgro problm wth poto rror, (Luz r (( ( w w w w L w (( ( W L or W W W W (( ( ( ( L w w w w w W UW wh r for all,..., W % W % W % W % L W % wh r for all,...,, poto maurmt rror (Luz X U Z U z z U r Z Uz + + Z vc( X Rgro problm wth maurmt rror (Luz r Z% vc( % X Rgro problm wth poto rror (Luz r ( E + vc E + E Rdual rgro (Luz r E μ aurmt rror (ut calld Rpo Error (Luz r + + E % % μ aurmt rror (poto (Luz r E X μ Dvato from ma (Luz r Sampl ad Rmadr,..., Sampl Sampl total of Epctd Rpo Rmadr total + Sampl total of aurd rpo whr wh r for all,..., clutr-otato007.doc //007 5: P

12 30- + Rmadr total of aurd rpo whr wh r for all,..., w Sampl total for wghtd pctd rpo w w Rmadr total for wghtd pctd rpo w + w w Sampl total wghtd maurd rpo wh r for all,..., w w + Rmadr total wghtd maurd rpo wh r for all,..., Sampl avrag Rmadr avrag + X X Sampl ma of aular varabl (Wu X X Sampl ma for o aular varabl (Luz r or Sampl avrag of aurd rpo whr wh r for all,..., + Rmadr avrag of aurd rpo whr wh r for all,..., + + Sampl ma for Rgro prdctor wth ut maurmt rror (Luz + + r Rgro problm (Luz r Vctor clutr-otato007.doc //007 5: P

13 30-3 ( ( L (( ( + + L ( ( L w w w w w ( w ( + w ( + L w w w ( ( L R ( ( ( + ( + L R ( ( L wr w w w w ( w w ( ( + ( + L wr w w ( ( L ( ( L ud b Vvaa S (( ( + + L (( ( + + L ud b Vvaa R (( ( L whr ( all,..., 0 wh r for ( ( ( + ( + L whr 0 ( wh r for ( ( L w w w w w ( w ( + w ( + L w w w (( ( L w w w w w ( w w ( ( + ( + L w w w all,..., clutr-otato007.doc //007 5: P 3

14 30-4 Z Z S R Z0 Z L Zp Z0 Z L Zp O Z0 Z Z L p Z0, + Z, + L Zp, + Z0, + Z, + L Zp, + O Z0, Z, Z L p, Z vc Z Sampl (Wu S whr S ( Z 0 Z whr Z R ( 0 Z Z vc Z Rmadr (Wu R Z K RZ Sampl radom varabl rgro (Luz r + + Z K RZ Rmadr radom varabl rgro (Luz r + + ( ( X μ ( X μ ( X μ Z % % % L % L Rgro wth poto rror (Luz r Epctd Valu, Targt, Prdctor P g Targt problm wth a mpl populato θ c c Targt problm wth aular varabl, b Wu. θ G Targt doma problm, b Vvaa θ G Altratv targt doma problm, b Vvaa c T c Targt rgro problm wth rror b Luz r P ˆ g + a Bt lar ubad prdctor of targt P ˆ wz Bt lar ubad prdctor of targt wth aular varabl (Wu ˆ ( ˆ g a Z Rgro prdctor (Luz r T a Cotat to dtrm to optmall prdct th targt (, + + ( f a λ ava g V a g V g ax g X λ Fucto to mmz for ESE λ, LaGraga multplr clutr-otato007.doc //007 5: P 4

15 30-5 {, (, } wˆ V V + G GV G G GV V C c ( f β, ESE cotat wth aular ˆ a V V X X V X X V V g + V X X V X X g ESE cotat + ˆ D + a ( C + c ESE cotat for rgro wth rror (Luz r D% a% % c( % + c Rgro (Luz r ˆ R R RS S SR R + R R RS S S RS S SR R RS S S R θ G V V V V G G X V V X V V V X V V X G -optmal var ( ˆ var ( ˆ θ θ -optmalt crtra for X α E ξ X ˆθ, a lar, ubad prdctor X β S S E R XR Epctd valu ud b Vvaa for doma problm E ξ R X var X V α V, ξ V, V S VS VSR var R VRS VR Varac ud b Vvaa for doma problm. var var ξ R V V, V, V V V V, V V V V, V, V ξ R, V V V R clutr-otato007.doc //007 5: P 5

16 30-6 V ( + J + 0 V Σ P, Rgro parttod varac (Luz r + 0 V Σ P, Rgro varac (Luz r + V, Σ Rgro varac (Luz r Othr Trm Dfd for Prdctor αˆ X V X X V αˆ X V X X V ˆ X V X X V α % μ % % Wghtd ampl ma wth maurmt rror aocatd wth poto % + + % μ % Wghtd ampl ma for partall codtoal RP modl wth maurmt + rror μ + + Rgro problm (Luz r μ X X Rgro problm (Luz r ˆ μ a ud SE for Rgro Etmator of Luz r ( X μ + + Bˆ X ( μ + + Rgro problm (Luz r clutr-otato007.doc //007 5: P 6

17 30-7 B% μ + ( μx μ Rgro problm (Luz r ( ρ ( + ρ Rgro problm, (Luz r U Rgro problm (Luz r Clutrd/Stratfd Populato Ut ad Rpo,..., Clutr Labl (ud b Ed t,..., Ut labl for clutr (ud b Ed,..., J Stratum labl (ud b Vvaa {,,, } L Ut labl for tratum (ud b Vvaa,..., Doma labl ach tratum umbr of ut doma tratum umbr of ut th populato,..., r t d for rplcato t for,..., ; t,..., Epctd Rpo for ut t clutr for,..., J ;,..., Epctd Rpo for ut tratum w Wght for th ut poto clutr (Ed w w Wght pcal ca that all poto wght ar qual clutr (Ed W t for,..., ; t,..., ;,..., r Rpo rror. for clutr th rplcato of ut t clutr-otato007.doc //007 5: P 7

18 30-8 t t + Wt for,..., r Rpo for th rplcato of ut t clutr Vctor L Epctd rpo for ut clutr (( ( t (( ( L Epctd rpo for ut tratum (Vvaa (( ( L Epctd rpo for all ut ad clutr. ε L Dvato of ut t from pctd valu of clutr (( ε ( ε ε ε t (( ( ε ε ε ε L ε Vctor of dvato of ut from clutr pctd ( ( t valu W W W W L W Rpo rror vctor (for r t for all t,..., (( ( for clutr W W W W L W Rpo rror vctor for all ut ad clutr (for r t for all,..., ; t,...,. Paramtr ad Cotat m Sampl z for clutr (Ed Sampl z for tratum (Vvaa f m K 0 0 ( m atr to form ampl rpo vctor wh m ( m all,..., for K 0 0 m ( ( m m 0 ( atr to form rmadr rpo vctor wh for all,..., clutr-otato007.doc //007 5: P 8

19 30-9 ( K K K X K X wm X Ud for ampl Partall collapd uqual z modl (Ed X X K X wm w( m Ud for rmadr Partall collapd uqual z modl (Ed g c Two-tag ubalacd problm (Ed g K g g c Coffct for partall padd ampl (Ed g K g g c c Coffct for partall padd rmadr (Ed (,, g g g 0 m m g, g 0 ( m 0 g g (, G S G R Cotat to df targt tratfd doma problm for ampl (Vvaa Cotat to df targt tratfd doma problm for rmadr (Vvaa ( ( ( g g g g g L Cotat multplr to df targt wh,..., for all clutr-otato007.doc //007 5: P 9

20 30-0 ( ( g g g g L g Cotat multplr to df targt (( ( b b b b L b Vctor of cotat to df targt b b b ( μ t a for clutr (Ed d t w μ Ud partall collapd uqual clutr z (Ed β a for tratum (Vvaa μ μ β μ μ ε t t μ o that t μ + β + εt μ μ ( μ Varac of ut clutr (Ed t t v f w f Ud partall collapd padd uqual clutr (Ed ( μ Varac of ut tratum (Vvaa Avrag ut varac ovr clutr t t Avrag rplcato varac ovr ut clutr r t t clutr-otato007.doc //007 5: P 0

21 30- Othr Varac for Othr d odl Prdctor v + varac for md modl prdctor m w / v / v Wght for varac md modl prdctor Shrag cotat for md modl prdctor v δ Varac of clutr ma for Scott ad Smth prdctor v + Varac for Scott ad Smth prdctor δ m w / v / v Wght for varac Scott ad Smth prdctor mδ mδ + Shrag cotat for Scott ad Smth prdctor Varac paramtr dfd for radom prmutato modl ρ + ρt + r tra cla corrlato of clutr tra cla corrlato of ut wth rplcato rror m m + + ( r Radom prmutato hrag cotat wth rplcato rror whr ad m m for all,..., m m + Radom prmutato hrag cotat for -tag clutr amplg wth o maurmt rror whr ad m m for all,...,. clutr-otato007.doc //007 5: P

22 30- r m + m + + r Addtoal radom prmutato hrag factor wh whr ad m m for all,..., d d v + ( Partall collapd uqual clutr cotat (Ed Avrag ug partall collapd uqual clutr cotat (Ed d Cotat for partall collapd uqual clutr (Ed d Vctor ( ( μ μ μ μ L μ Clutr ma (Ed ( β β β β L Vctor of clutr ffct (Ed ( β β β β L Vctor of trata ma (Vvaa J X Fd paramtr dg matr Z Radom ffct dg matr (( w ( w w w w L Wght for poto clutr (Ed ( m 0 m t C ( 0 m m t t t g ( p g C CC Partall Collapg matr (Ed Lar cotat for targt to appl to partall collapd radom varabl (Ed t t t t Pt C CC C C Ortho-complmt of partal collapg (Ed Prmutato Radom Varabl,..., Poto prmutato of clutr,..., Poto prmutato of ut clutr clutr-otato007.doc //007 5: P

23 30-3 υ,..., Poto prmutato of ut tratum U for,..., ;,..., dcator of lcto of clutr poto. t U for,...,, t,..., ad,..., dcator of lcto of ut t poto ( U υ for,..., clutr υ,,..., ad,..., J dcator of lcto of ut poto υ tratum (Vvaa % U Epctd rpo SSU clutr t t t ( υ U Rpo of ut poto υ of tratum υ % w U or w t t t % U Epctd wghtd rpo for SSU clutr (Ed w w U U t t t + W B U β Vctor ad atrc ( U U U U L U t t t t t U U U U L U t U U U U L U Prmut ut clutr (Ed U ( ( ( ( L U U U ( ( ( ( U U U ( ( ( ( ( U U U U υ Prmut ut tratum (Vvaa. clutr-otato007.doc //007 5: P 3

24 30-4 ( ( U U U U L U ot that th a colum of U, whl U a row of U (( ( L U U U U U U U U Parttod clutr prmutato matr (Ed ( ( U U U U L U Sampl porto of clutr prmutato matr (Ed ( ( + + U U U U L U Rmadr porto of clutr prmutato matr (Ed (( ( ( (( υ % % % % L % or % U % Prmutd ut clutr (Ed ( ( U Prmutd ut tratum (Vvaa ( ( J L Vctor of trata wth prmutd ut (Vvaa (( ( L % % % % % w w w w w ( ( L Wh ( ( for all,...,. L Wh for all,...,. or ( ( U U (( E ( E U U ε Wh for all,..., (( W ( W U U W Wh (( ( B B B B L B or B Uβ + W T g Targt for all,..., ad wh,..., r t ad r t for all,..., ; t,..., ;.. T A g Targt wth maurmt rror T g t Targt (-tag uqual (Ed w clutr-otato007.doc //007 5: P 4

25 30-5 T g t Targt (-tag partall collapd (Ed p wp Epadd Radom Varabl Vctor t (( U % ( U % U % L U % w w w w w t t t E E ξ ξ w w w w w w w w (Ed t t t t t L (Ed ( ( ( ( L t t t t t E E E E E t w w w w w wp t t C w Partall Collapd Epadd radom varabl (Ed Sampl ad Rmadr umbr of ut ampl tratum (Vvaa umbr of ut doma tratum (Vvaa m umbr of ut ampl for th PSU poto ud md modl umbr of ut th PSU poto ud md modl f m amplg fracto of SSU th PSU poto ud md modl m f whr ad m m for all,..., (Ed U dcator of cluo of clutr th ampl (Ed t t m t U Partall collapd uqual clutr (Ed Uw ˆ Wghtd ampl ma for PSU (partall collapd uqual (Ed ˆ ˆ Avrag for ampl partall collapd uqual clutr (Ed clutr-otato007.doc //007 5: P 5

26 30-6 m m whr ad m m for all,..., ad,..., r t whr r t for all,..., ; t,..., m m whr ad m m for all,..., m m whr ad m m for all,..., ad,..., r t whr r t for all,..., ; t,..., m m whr ad m m for all,..., Vctor K K K K ( ( L ( ( L Epadd Radom Varabl Vctor t t 0 ( w wp t t w 0 ( t t E 0 E ( w w wp Sampl radom varabl (Ed Rmag radom varabl (Ed Parttod ampl rdual (Ed clutr-otato007.doc //007 5: P 6

27 30-7 t t Ew 0 Ew ( Parttod rmadr rdual (Ed t V w V, varξξ t w V, V Partall collapd uqual clutr z (Ed V fd fd + f v V, J ( ( fd f d 0 J ( P (Ed fd fd P f v 0 ( J 0 J ( ( fd P fd fd ( f d P 0 ( V 0 ( + 0 f v J P ( ( f d ( f d P ( f dp fd (Ed Epctd Valu E ξξ ξ 3 X μ X S Sampl porto of dg matr for tratfd doma (Vvaa X R Rmadr porto of dg matr for tratfd doma (Vvaa var var ξξ V V, V, V V V V, V V V V, V, V ξξ ξ3, V V + r m clutr-otato007.doc //007 5: P 7

28 30-8 Etmator/Prdctor m m dcat th avrag of SSU for th PSU ralzd poto a ampl. Th otato ud wth md modl. ˆ μ w wghtd ma ud md modl. T ˆ L t Prdctor for partall collapd uqual clutr (Ed w, ˆ L g + V V X X V X X V V g + V X X V X X g Pc + c f f BLUP (Ed ˆ ˆ ˆ T c c + c w ( Partall collapd uqual clutr prdctor (Ed αˆ X V X X V ( SE r ˆ ma 0, ˆ ma 0, SB SE f ˆ + m ˆ ma 0, [ SB SE] m clutr-otato007.doc //007 5: P 8

Estimating the Variance in a Simulation Study of Balanced Two Stage Predictors of Realized Random Cluster Means Ed Stanek

Estimating the Variance in a Simulation Study of Balanced Two Stage Predictors of Realized Random Cluster Means Ed Stanek Etatg th Varac a Sulato Study of Balacd Two Stag Prdctor of Ralzd Rado Clutr Ma Ed Stak Itroducto W dcrb a pla to tat th varac copot a ulato tudy N ( µ µ W df th varac of th clutr paratr a ug th N ulatd