(( ) ( ) ( ) ( ) ( 1 2 ( ) ( ) ( ) ( ) Two Stage Cluster Sampling and Random Effects Ed Stanek

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1 Two ag ampling and andom ffct 8- Two Stag Clu Sampling and Random Effct Ed Stank FTE POPULATO Fam Labl Expctd Rpon Rpon otation and tminology Expctd Rpon: y = and fo ach ; t = Rpon: k = y + Wk k = indx of od of mau Aumption: EW k = 0 fo all = ; t = ; k = W va k = fo all = ; t = ; k = va W W = 0 fo all = ; = ; t = ; t = and k t k t and fo all k = Paamt: k = t μ = y μ = μ β = μ μ and ε = y μ o that y = μ + β + ε t = t = = μ μ = y μ = = = t = = t= = and = Additional notation: U = U = U U L U U = U = U U L U Dfin i i i i i i L U = U = U U U and L y = y = y y y β = β = β β L β j jt j j j j L U = U = U U U y y y y y = = L L W = W = W W W SPLFCATOS W = W = W W L W k k k k Aum k = = fo all = ; t = o that k = mau of pon Agintina007-lcdoc //007 5:04 P

2 Two ag ampling and andom ffct 8- ot: f k = and w could dfin o that = y + W wh W = Wk 0 k = uitabl dfinition th modl w dicu i mo gnal PERUTATO = k and va k = EW = and W va = = = Thu with a Aum th clu and ubjct in clu a pmutd b Aum pon o pnt maumnt o on th ubjct with = fo all = and t = c Aum = fo all = Dfin % = U y j jt t = % = % = U y j % = U y j j % % % L % U y = = = = U U y = E= U U ε = W = U U W = ijk = = + W vc = vc + = + y W U U U y W U = = ij = UiU jt y = t= Wijk = UiU jt Wk = t= i = ij = i i L i To ummaiz = Xμ + ZB+ E+ W wh X= and Z= Agintina007-lcdoc //007 5:04 P

3 Two ag ampling and andom ffct 8-4 EXPECTED VALUE AD VARACE ξ ξ ξ Subcipt pnting xpctation with pct to pmutation of th clu Subcipt pnting xpctation with pct to pmutation of unit in a clu Subcipt pnting xpctation with pct to plication Expctd Valu: Vaianc: E ξξ ξ = X μ vaξξ ξ = + + J J 5 REARRAGG AD COLLAPSG THE RADO VARABLES Th andom vaiabl in copond tho that a commonly ud W aang th andom vaiabl into tho that a in th ampl and maind t i alo poibl to collap th andom vaiabl to ampl and maind PSU total W not thi fact but do not intoduc additional notation to illuat it h L To aang th andom vaiabl into a ampl and a maind lt L = L = L wh L= L L L = 0 n 0 n n m and m m n 0 0 m n n m m L = 0 n n n n gnal w pnt th ult a = L Uing popti of E ξξ ξ = X X α and va ξξ ξ = vaξξ + w valuat Thi ult in th following: Agintina007-lcdoc //007 5:04 P

4 Two ag ampling and andom ffct 8-4 X nm V V = and vaξξ X = V = nm + n Jm J nm nm V V V = V = n J 0 m m nm n J and nm nm n Jm 0 V= nm+ Jnm 0 n J 6 TARGET PARAETERS RADO VARABLES AD TEROLOG W aum that th i an int in a tagt that can b dfind a a lina combination of th xpctd valu ov pon o of th andom vaiabl W u th ampl data to imat/pdict th tagt W only conid infnc fo a ingl tagt not joint tagt W pnt th andom vaiabl by W dfin a tagt a P = g wh X E ξξ ξ = α and patition g pnting g = g g o that P = g + g X Sinc will b alizd aft lcting th ampl and obving th ult th baic infnc poblm i pdiction of g Thi i tu whn P i a fixd conant and whn P i a andom vaiabl Fo thi aon w u th tm pdicto whn dicuing infnc Fo th andom vaiabl that a aangd a abov w dfin g = b wh b = b L b a conant n paticula w limit dicuion to b= i wh i dnot an vcto with a valu of on in poition i and zo lwh Whn andom vaiabl a collapd g = i 7 DEVELOPG THE BEST LEAR UBASED PREDCTOR BLUP 976: W dfin th BLUP of P a P wh P atifi th following citia Royall P = g + a Lina in th ampl: Unbiad: E ξξ P P 0 ξ = inimum SE: va P P ξξ ξ i minimizd Agintina007-lcdoc //007 5:04 P 4

5 Two ag ampling and andom ffct 8-5 n od to dvlop th BLUP of P w fi pnt xpion fo P P and it vaianc Fi not that P P= g + a g g V V W pnt vaξ = Thn V V V V V vaξξ ξ = V V V V V V Lt u dfin V = V + and V = Thn nm R nm 0 VR 0 0 V V 0 vaξξ 0 ξ = V V Th Unbiad Conaint: W can xpand th unbiad conaint: E P P = ξξ ξ a g X α uch that = X Eξξ P P ξ ax g X α n od fo thi xpion to qual zo fo any valu of α th unbiad conaint will b alway b atifid whn ax g X = 0 Thi i intoducd a a conaint uing Lagangian multipli whn minimizing th SE Finding th inimum SE: otic that va V V V g + a = g + a g g V V V g o P P ξ R V V V g V V a vaξ R P P = a g + g + a VR g + a V V g Expanding thi xpion va = P P ava g V g V a g V g g V g R R ncluding th conaint via a Lagangian multipli w k to find th valu of a that will minimiz Agintina007-lcdoc //007 5:04 P 5

6 Two ag ampling and andom ffct 8-6 = + + f a λ ava g VR g V a ax g X λ + g Vg + gvr g Diffntiating with pct to a and λ f a λ = Va + VRg V g + Xλ and a f a λ = Xa Xg λ f a λ w t th divativ to zo imultanouly To find th valu of a that minimiz and olv fo a Th imating quation a givn by f a λ a V X a VR g + V g 0n = = f a λ X 0 λ Xg 0 λ V X a VR g + V g o quivalntly by = X 0 λ Xg To olv thi quation fo â w mak u of ult on th inv of a patitiond A B matix X = C D uch that A + A BQ CA A BQ X = wh Q CA Q Q= DCA B A a ult V V X X V X X V V X X V X = V X X 0 X V X X V X V X a= V V X X V X X V VRg + V g + V X X V X Xg W can now xp th b lina unbiad pdicto Rcall that P = g + g and w pdict P by P = g + a wh th b pdicto plac a by â Lt u dfin α = X V X X V Thn = R + + a g V g V V V X X V X X V g X X V X X V α α α =gv R V X + g X + V V X A a ult th b lina unbiad pdicto i givn by P = g R α + α + α V V X g X V V X Agintina007-lcdoc //007 5:04 P 6

7 Two ag ampling and andom ffct 8-7 W can xp th pdicto in a lightly diffnt mann by ubituting V = V V ulting in P = g α + α + α + α X V V X g X V V X Exampl Tagt: Uiμ whn i n = nm Thn α = = Xα = Pnm Jm VV Pnm = ρt nm + ρt k n nm nm m P J m m n m and V m V Pnm = k Pnmwh ρt = k = and 0 + m + + n nm a Pa = a J Uing th xpion th pdicto implifi whn i n to a m m T = i n + ρt + ρt k n + i n + k n P P Agintina007-lcdoc //007 5:04 P 7