~~y Stat 511 Exam IIApril 7, 2004 Prof. VardemanI. Consider a 3 x 3 factorial analysis with factors A and B under an ordinary (fixed effects) linear model.The effects model under the R baseline restriction has parameter vector for mean responses1 = (,u.,a; ,a; ,/3; ,/3; ,a/3;2'a/3;J'a/3;2'a/3;J )'~ a) Write out al19 cell means in terms of the entries of 1 in the table below. ~et rows correspond to levels ofA(I to 3 top to bottom) and columns correspond to levels ofB (I to 3left to nght).,}A*- A~+ ~ p.~+ ~JJ~+fJI!- AA~+0(...+,Q~ -1- "' 1( Jll(.L.~~+ .Q'1-JY\ ).. I" 2. ,-~ II\ 2'2.. ~'" ,1 1. '--13A kt C(~ )I{ ~+ I/.: + f?>: 4- X :2.. ){if. +- ~;+ (j~/~ b) Give below a matrix C so that the testable hypothesis Ho: Cy = O is the hypothesis Ho: /3j = O 'Vi. (Asalways, /31 ='Uj-'U.) ~~ ,$ ,M.I = M.2...=-jJ1-~ t'.t'.~ ( .3)J.~ +4-+ ~~ ) = ~ ( 3A ~+ I;(: +- I)(~ -+- 3P~+ t;?(.~~2... + Kf~z-)/Jf',.t.R -j(~M.' +t><::+ ~; ) = ~ (3}A~+ ()(i -1- ~ + 3~~ 4- ~,B~3 ~ K~;3).~ + l-N:,Q~ +- J..f>/.A *" -= 0 -D .,.. ~ -Lr,(R'*' -=Ot .! .(32. 3v~2..z. 3 (...3Z. ~ ?3 + ~Z.'3. + 3~!..;>3~50 ~ y( O o o I o 3 ' D ~ 0)t=- () 00010 i O~~ c) Give below a matrix C so that the testable hypothesis Ho: Cy = O is the hypothesis that considering onlylevels 2 and 3 of A and levels 2 and 3 ofB (the lower right 4 cells in the table) "there are no interactions"(considering only these 4 cells an interaction plot of means would have the "parallelism" property).~Ms is )A2.~- ))\22.. = )A33- );(.32... i.~ ..~- A~ 'A-I: -~ -= A~ -A (+ rx:A~ -'A~~~ (""",2 + lX;;.3 C(PZ2. I--I~ r.2 13-3' ~- 32..e. R AJ ~ ~ v ~t. .-Jf.rz...z 1- ""f2.3 + ~f32.. -~~33 =- D5(.) tI( $-( \t= (() o o 0 O -I I ( -I
View Full Document
Unlocking...