Estimating the Variance in a Simulation Study of Balanced Two Stage Predictors of Realized Random Cluster Means Ed Stanek
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1 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 clutr a W df th varac of th ut paratr a M ( µ t µ ug th gratd ut paratr Notc that w oly grat t M o t of ut ffct for th populato Th ut ffct arud to crat paratr for clutr-ut for ach clutr W grat th ut ffct th ar to forc th varac of ut wth clutr to b cotat for all clutr Th coo wth clutr varac whch w rprt by qual to th avrag wth clutr varac, N Prdctor ar cotructd for ach tral To dvlop a prdctor, w u N quatt calculatd for ach lcto of a clutr (PSU, ad avrag of th quatt ovr lctd clutr (PSU th apl (tral Th prdctor obtad by cobg th rult W dcrb th proc bad o dvlopt of th xd odl prdctor (gv by quato (4 Stak ad Sgr(003 Frt, w valuat N M ( u + ut t Th corrpod to th ralzd varac for th t th lctd clutr Th, for ach lctd clutr, w valuat th quatt Y, ad Y v + W alo for th quatt v ad v Th xt tp to u th quatt ovr th lctd clutr th tral rultg Y v ad / v, fro whch w dtr ˆ µ wy whr w, v / v quvalt to ˆ µ Y v Fally, cobg th rult back to th apl, w v obta th xd odl prdctor gv by pˆ ˆ µ k ( ˆ Y µ Varac Matrx for Sapl + C03d3doc 5//003 0:00 AM
2 Th varac atrx for a apl gv by VI ( + I + ( I J J W tat ach tr th xpro N ug a ultaou quato Th quato bad o ttg th tatd varac (( atrx gv by ( Yk Y( Y Y k qual to th thr varac copot by ply avragg th varou porto of th varac atrx J Not that ( I P YI (( Yk Y ad that P I (( Y Y Y A a rult, var ( I P YI ( + ( I P, whl J J var P I ( + + ( Y P P J Hc, E ( Yk Y ( ( +, whl E ( Y Y ( ( + + Ug th xpro, ( Yk Y ( Y Y E ( +, whl E ( ( ( + + ( Yk Y Mthod of ot tat ar gv by ˆ ( ( ˆ +, ad ( ( ˆ ( ( Y Y Yk Y Fally, w valuat ( Yk Y( Y Y k ( Yk Y Y I, for ad Sc var ( Yk Y( Y Y var ( I k Y, whch plf to C03d3doc 5//003 0:00 AM
3 ( Yk Y( Y Y var ( I var k Y ( + + ( + I J N J Now, ( I J, ad 0 J Thu, var ( Yk Y( Y Y ( + + k W ca u thod of ot tat to tat ( Yk Y ˆ ( ( ˆ + To parat tat for th varac copot, w d a xtral dpdt tat of ˆ Wh thr o rpo rror, th th tator ˆ plf to a tat of Sc, a tat of gv by olvg th quato M ( Y Y ( Yk Y ˆ ˆ, or ( ( M ( Y Y ( Yk Y ˆ + ˆ If th xpro l tha ( ( M zro, w tat by zro C03d3doc 5//003 0:00 AM 3
4 To uarz, w tat by ( Y Y ( Yk Y ˆ ˆ ax +,0, ad ( ( M vˆ ˆ ( ˆ + ( Yk Y xtral ourc, th ˆ ˆ ˆ ˆ M ( Aug w hav a tat of fro o ( Yk Y ˆ ( Suary of Varac Copot Etat Lt u df MSB ( Y Y (, ad w tat by ad MSE ( Yk Y ( Alo, lt u au w hav a dpdt tat of whch w rprt by vbb _ W tat ( by ˆ v ˆ ( MSE _ ax,0 W tat by ˆ v _ MSE W tat by v _ u ˆ ax MSB MSE +,0 M Fally, w tat by vtar _ ˆ ax ( MSB MSE,0 W uarz th tat th followg tabl ˆ C03d3doc 5//003 0:00 AM 4
5 Varabl SAS Na Dcrpto ( Y Y ( b Sapl a quar btw clutr ( Yk Y ( Sapl a quard rror wth clutr ˆ vbb_ Etatd rpo rror varac (( MSE ˆ ˆ ax,0 v_ Etatd avrag varac of ut for a clutr ˆ MSE v_ Etatd varac of lctd clutr ad rpo rror ˆ ˆ ax MSB MSE +,0 M v_u Etatd Var btw clutr ˆ ax ( MSB MSE,0 vtar_ Etatd ad varac of lctd clutr for RP odl Ug th tr, w tat th hrkag cotat uch that Varabl SAS Na Dcrpto ˆ ˆ k ˆ + ˆ k_ Etatd Mxd Modl hrkag cotat ˆ ˆ k ˆ + kˆ ˆ ˆ ˆ + + ˆ ( ˆ k_ Etatd RP Modl hrkag cotat ktar_ Etatd RP Modl hrkag cotat wth doator rpo rror C03d3doc 5//003 0:00 AM 5
6 kˆ r ˆ + ˆ + + ˆ ( ˆ ˆ ( µ ˆ ( µ krtar_ p ˆ ˆ + k Y ˆ p6 Et MM ( ( ( µ ˆ µ ( ( ( ˆ ( ˆ ( f ( Y kˆ Y Y RP Modl hrkag cotat wth urator ut varac ad doator rpo rror Pˆ fy + f ˆ + k Y ˆ p7 Et Scott&Sth Tˆ fy + f Y + k Y Y p8 Et RP Tˆ f Y + k Y Y r ( ( + + p9 Et RP + Rp Err C03d3doc 5//003 0:00 AM 6
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