N A N A ( ) We re-arrange and collapse the random variables into a set corresponding to the weighted

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1 7- trodutio Note that rom 08bx0v3.do (p6) while rom page 7, ad Title Bo Xu Y = μ T B E N N ( Y ) = ( ) UV N μ Commet: Bo, Write a brie itrodutio explaiig what this doumet will do. You a opy parts rom the paper as bakgroud. This doumet should stad aloe- right ow you have t deied ay terms, or related the otatio to aythig else. Formatted: Let Formatted: Let Formatted: Let ( ) VarUV ve Y = J N J N. Re-arragig Terms ito the ample ad Remaider We re-arrage ad ollapse the radom variables ito a set orrespodig to the weighted sample totals, Y, ad a set orrespodig to the weighted remaiig totals or eah treatmet, Y K Y suh that = Y where Y = ve( Y ). Y K We assume equal sample size or eah treatmet. Let K = δ 0 = ( N ) Y Y KY =. Let us deie ( Y... Y... Y ) Y =, theky = Y. Y Commet: sample ad remaider meas (right?) imilarly, let = N = ( ( ) ) K N δ. The Y ad where = ( Y... Y... Y) K Y = Y N = i N i= Y Y Y. C08bx04-es.do 9/5/008

2 7- δ 0 K ( N ) = K ummarizig these results, = Y ad Y =. K (( δ ) N) K Y N = Determiig the Expeted Value We use the expressio or E ( ) = ( ) Y K Y μ ad = Y Y K UV N Y expeted value ad variae o Y. Y K ie = Y ad EUV ( Y ) = μ N, Y K Y K EUV = EUV ( Y ) Y K δ 0 = ( N) = μ (( ) N) δ N = = μ. Y We express this as EUV = μ Y Determiig the artitioed Variae Next, we osider expressios or the variae: var UV Y V V, varuv =, where Y V, V V = ( ) J N Y K K varuv ( ) = Y Y K K V V, = V, V to orm the Deleted: these expressios Formatted: Lowered by 70 pt Commet: Bo, The idea o this doumet is that a perso a easily hek the results, movig rom oe step to the ext. You leave out all the steps. t ould be that the simpliiatio is orret, but you d eed to re-do the simpliiatio to see i it is orret or ot. would like you to put i the steps that lead to the results. You a assume that a perso readig it uderstads matrix algebra, but do t skip so may steps that by readig it, you a t hek the results. C08bx04-es.do 9/5/008

3 7-3 V J, = ( ) [ ] N V = [ J]. N ( ) reditig arameters orrespodig to Combiatios o Treatmet Meas Now ( ) = ( ( ) ) g g, X = () ad X = (). where Y is the overall sample mea. ˆ α = (3) Y Next we simpliy the term b V, V = V, V = ( dj) J a a ab b d bd = J J J a a ab a a ab (4). b d( a b) bd = J a a ab b ad = J a a ab ( ) Usig (), (3) ad (4), ˆ ˆ, ( ˆ = g Y g α α) X V V Y X ( Y ) b ad = g Y g Y ( ) J a a ab Y b ad ( ) ( ) = Y Y J Y Y ( ) a a ab ( ) b ad = Y ( ) Y Y J a a ab Y ( k ( k) Y) ( ) b ad b ad = ( ) Y Y Y Y Y a a ab a a ab = Y = Y k Y Y Commet: This expressio looks like it was take rom some other doumet, sie oe o the terms are deied. Do you have a doumet that develops the preditor (iludig these simpliiatios)? You should have oe. you do, you a reer to that developmet here, ad you do t eed to re-do the developmet. The questio asked was about a ompariso o the MEs. F you do t have a doumet developig the preditor, you should augmet this presetatio so that you give eough details ad deiitios that someoe a hek it. C08bx04-es.do 9/5/008 3

4 7-4 The target is = Y Y ( ) The var ˆ = var Y Y Y k Y Y ( ) ( ( ) ( ) ) = var Y ( ) Y JY ky JY = var Y JY ky JY ( ) Y = var J k J ( ) Y Y = var J ky ( ) Y Commet: do t thik you wat to ilude a subsript here, sie it ould reer to ay liear ombiatio. Commet: Your expressio is or ( ˆ ) ( ˆ ) var, ot var. uderstad that they are equal, but still, it should be kept osistet. Deie M = J k ad M = ( ) the, var ( ˆ ) var ( ) = M Y M Y (, ) = M VM M V M M V M M V M 3 = ( ) J MVM = Jk ( ) k J J = Jk ( ) k J J = ( J) k J Jk J k k = ( k) k J J k = k J J ( k) k J = ( ) ( ) ( ) ( ) - J k k k Commet: You eed to ilude more steps here. Ca you do these alulatios i your head? a t. d like to be able to ollow the alulatios ad ot have to re-do them mysel. Commet: how some o the itermediate steps that lead to this simpliiatio. Commet: How do you get rom lie to lie 3? lso, it seems that lie ad lie are the same. m ot sure i this is right or wrog, but it is ot lear. C08bx04-es.do 9/5/008 4

5 7-5, = k MV M J J N ( ) [ ] ( ) = k ( )( k) J 3 Commet: a t ollow this simpliiatio without more steps give. var ( ˆ ) var ( ) = M Y M Y ( ) ( ) ( k k) ( k) - J 3 = k ( )( k) J 3 ( ) J = - ( ) k k J Commet: how some itermediate steps so a perso a ollow it. Y ˆ Y = ad = Y ( ) Y C08bx04-es.do 9/5/008 5

6 7-6 ( ) ( Yˆ ) ( ) var = var Y Y Y ( ) var( Y Y) ( ) (, ) = = V V V ( J) = ( ) [ J] ( ) N ( ) [ ] J N ( J) = ( ) J ( ) [ ] ( ) [ J ] ( J) ( ) ( ) [ ] = J ( ) [ ] J = ( J) The dieree o the two ME is : Oe ME is ( ˆ var Y ) = ( J) ad the other ME is ( ˆ ) k ( k) var = - J. = - J k ( k) s a result, the dieree i MEs is Commet: This looks right, but the solutio does t make sese to me. What happes i you ust wat to predit the mea or the th treatmet. t seems that the variae should ilude a ompoet or the treatmet variae (sie it is a radom eet). t does t. thik the expressios or the variaes ad all the other thigs eed hekig to make sure it is right. However, maybe it is right. do t kow. ust thik you should make sure it is right by hekig thigs areully. Formatted: (sia) Chiese (RC), Lowered by pt Formatted: (sia) Chiese (RC), Lowered by 36 pt C08bx04-es.do 9/5/008 6

7 7-7 ( ˆ ) ( ˆ ) ( ) Y k k ( ) var var = J J The, sie k ( k)( k) = ad = J k J( J) ( k) = J ( k ) ( k ) ( k) = ( k ) ( k) J = ( k)( k) ( k)( k) = ( k) ( k) ( k) k =, = while k = = the ( k) ( k) = = TO HERE Formatted: (sia) Chiese (RC), Lowered by 09 pt Formatted: (sia) Chiese (RC), Lowered by 6 pt Formatted: Lowered by 47 pt Formatted: Cetered Formatted: Cetered Formatted: Lowered by 47 pt Formatted: Lowered by 36 pt C08bx04-es.do 9/5/008 7

8 7-8 ( ˆ ) ( Yˆ ) ( ) var - var = var M Y M Y = k ( k) ( ) - J J = ( k) ( k ) ( k) ( k ) = = ( k ) Deleted: C08bx04-es.do 9/5/008 8