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A Aaly of WCro, SPSS, ad Metor Proedure for Etmatg the Varae of a Weghted Mea Dr. Albert Madaky Ve Predet, The Aalytal Group, I. ad H.G.B. Alexader Profeor Emertu of Bue Admtrato Graduate Shool of Bue

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A Aaly of WCro, SPSS, ad Metor Proedure for Etmatg the Varae of a Weghted Mea Dr. Albert Madaky Ve Predet, The Aalytal Group, I. ad H.G.B. Alexader Profeor Emertu of Bue Admtrato Graduate Shool of Bue Uverty of Chago The reator of tattal proeg oftare for the marketg reearh ommuty have ofroted them th a varety of approahe dealg th gfae tetg relatg to eghted ample mea. Eah of thee approahe produe a dfferet varae of the eghted ample mea, ad thu a dfferet tet tatt. The purpoe of th ote to expla ther bae, ompare ther approahe, ad make ome reommedato. 1. Termology The formula for the eghted mea x* Ad o the varae of the eghted mea Var(* x x 1. Var( x. ( If eah of the x ha the ame varae,, the th redue to 1 1 Var(* x ( here the effetve ample ze f gve by ( 1 f. 1, f 1 . Etmato of a. WCro If eah of the x ha the ame expeted value µ ad varae, the the uual etmate of the varae, amely here 1 x ( x x 1 a ubaed etmate of. It th etmate that ued by WCro omputg the varae of the eghted mea x*,.e., the WCro etmate of the varae of the eghted mea /f. b. SPSS A alteratve etmate of the varae, ued by SPSS t omputato, baed o the eghted data, amely x,, ( x x* 1 It a be ho that th etmate a baed etmate of, that E ( ( 1 1 ( 1 1 o that a proper ubaed etmate of baed o g gve by ( g ( g ould be g, here the ubag fator SPSS doe ot perform th adjutmet, but tead ue the baed etmate. SPSS ompoud the etmato problem by etmatg the varae of the eghted mea by 1 tead of by /f. That, tead of dvdg by the effetve ample ze f t dvde by the um of the eght, the eghted ample ze, 1. Metor Frt let u etablh a gloary relatg CfMC Metor otato to our. F S Z Y So E=F /Y the effetve ample ze, hh e all f. The eghted mea M=S/F, hh e all x*. Metor alulate a adjuted um of quare A, va the follog formula: Z S / F A Y / F Th a be rertte a o that the expeted value of A x x x ( x / / 1 1 A x ( x* 1 1 / / 1 1 3 EA ( E ( / / g 1 / 1 1 ( 1 [ 1] ( f 1 1 Metor etmate of gve by V=A/(E-1, or ( A[ 1], 1 1 ad o e ee that t a ubaed etmate of. Follog a algebraally mplfed expreo for Metor the etmate the varae of x* by ( ( x x* 1 1 ( 1 1 /f.., Metor etmate of. 3. Comparo a. Varae of WCro etmate of varae of x* Se both the WCro etmate ad the Metor etmate of the varae of x* are ubaed, the ay oe mut ompare the to etmate by determg hh oe of thee etmate ha the maller varae. Se (-1 / ha a h-quare dtrbuto th -1 degree of freedom, e ko that the varae of (-1 / (-1, o that varae of the WCro etmate of, amely, 4 /(-1, ad the varae of the WCro etmate of the varae of x* 4 /f (-1. 4 b. Varae of Metor etmate of varae of x* Se both WCro ad Metor etmate the varae of x* by dvdg ther etmate of by f, oe eed oly ompare the varae of, the WCro etmate of, th the varae of, Metor etmate of. We frt etablh ome otato. Let X be the - vetor of obervato, E be the -vetor of 1', ad I be the detty matrx. The a be expreed a =axax here a=1/(-1 ad A = I - (1/EE. We a expre a =bxbx here, a above, the eghted ample ze, b 1, B=D - (1/WW, W the -vetor of eght, ad D a dagoal matrx th the eght o the dagoal. The ymmetr matre A ad B a eah be expreed a a produt of orthogoal ad dagoal matre, here the orthogoal matre are the matre of egevetor of A ad B ad the dagoal matre are matre otag the egevalue of A ad B. Let the deompoto of A ad B be expreed a A=Q A D A Q A ad B=Q B D B Q B. The =axq A D A Q A X=aY A D A Y A ad = bxq B D B Q B X=bY B D B Y B Se the ovarae matrx of X I, ad both Q A ad Q B are orthogoal matre, the ovarae matrx of Y A Q A Q A = I ad the ovarae matrx of Y B Q B Q B = I. Therefore ad are expreble a a eghted um of quare of depedet varable th ommo varae, ad here the eght are the egevalue of ad A ad bd B, repetvely. That a y A A 1 B B 1 b y ad o, e y A / ad y B / have h-quare dtrbuto th 1 degree of freedom, o that Var( y A =Var( y B = 4, e ee that the varae of the to etmate are expreble term of the um of quare of the egevalue of ad A ad bd B, amely 5 4 a A 1 Var( 4 b B 1 Var( It rema to determe thee egevalue. All but oe of the egevalue of A are equal to 1, th the -th egevalue equal to 0 (ee S.N. Roy, B.G. Greeberg, ad A.E. Sarha Evaluato of Determat, Charatert Equato ad ther Root for a Cla of Pattered Matre Joural of the Royal Stattal Soety. Sere B (Methodologal, Vol., No.. (1960, pp Thu the um of the egevalue of A -1, ad o, e a =1/(-1, e ee that Var( = 4 /(-1, a demotrated earler ug a omatral dervato. We eed ot determe the egevalue of B to alulate ther um of quare, for B =Q B D B Q B Q B D B Q B = Q B D B D B Q B, ad o the um of quare of the egevalue of B equal to the um of egevalue of B. But the um of egevalue of a ymmetr matrx equal to the trae of that matrx,.e., the um of t dagoal term. So e eed oly look at the dagoal term of B to obta th requred quatty. ad ad o Se B=D - (1/WW, B = D - (1/D W WW - (1/ WW D W + (1/ WW WW, trb W ( ( 1 1 ( ( 1 1 Var( b [ ] 3 [ ] ( ( {( ] [ ] ( ( ( ( ( . Comparo Before proeedg th a proof that Var( Var(, I ll llutrate thee omputato th a example. I eleted a eght 100 radom umber from a uform dtrbuto betee 0 ad 1. Thee eght, alog th ther quare ad ube, are gve Appedx I. The varae of, exludg the fator 4, 1/99= The varou um eeded to ompute the varae of are The varae of, aga exludg the fator 4, alulated a x ( x x ( ( x x ( ( Thu th example ue of ll produe a etmate of the varae of x* th 1.46 tme the varae ompared th the ue of. No let u ompare Var( th Var(. Oe a mplfy the expreo for Var( by aumg that the eght um to 1. Th merely reale the eght ad ll have o mpat o the omputato of Var(. The Var( redue to 3 ( ( 1 1 Var( Note that he the are all equal to 1/, the / / ( / Var( 1 / ( / 1 hh the ame a Var( that ae. Let u o determe hat are the value of the that mmze Var( ubjet to the otrat that the um of the equal to 1. To do th e form the Lagragea 4 3 log( log[ ( ] log[1 ( ] ( , L et the dervatve of L th repet to eah of the equal to 0, ad olve for the mmzg value of the ad, the Lagrage multpler. The reult of th the et of equato 7 L (1 j 6 4 (1 j j1 j1 3 j j ( j 1 j ( j j1 j1 j1 j1 j1 The oly ay for th equato to hold for eah of the he all of the are equal,.e., he =. Othere Var( ll be greater tha Var(. 4. Coluo Gve both the ba the SPSS etmate of ad t orret deomator determg the tadard error of x*, the probablte alulated baed o the t-tatt ll be orret. The probablte baed o both the WCro ad Metor tatt ll be orret, but, beaue Metor ue a etmate of the varae of x* th a larger varae tha that of the etmate ued by WCro, t more lkely that oe ll fd feer gfat dfferee ug the Metor proedure tha ug the WCro proedure. 8 APPENDIX I

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