~~y
Stat
511
Exam
II
April 7, 2004
Prof. Vardeman
I. 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 responses
1 = (,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 of
A(I
to 3 top to bottom) and columns correspond to levels ofB (I to 3left to nght).
,
}A*-
JJ~
JY\
A~+
+ fJI!-
p.~+
AA~+0(...+,Q~
I"
2. ,-~
)..
A kt C(~
-1-"' 1(
II\ 2'2..
)I{ ~+ I/.: + f?>: 4- X
~
Jll(.L.~~
.Q'1~'" ,1 1. + '--13
:2..
){if. +- ~;+
(j~
/
b) Give below a matrix C so that the testable hypothesis Ho: Cy = O is the hypothesis Ho: /3j = O 'Vi. (As
~
always, /31='Uj-'U.)
~ ( .3)J.~ +4-+
/Jf',.t.R -j(~M.'
+t><::+
.~
50
~~
,$
~~
) =
~;
) =
+
t .! .(32.
( 3A ~+ I;(:
~
(3}A~+
3
O
o
()
00010
(...3Z.
o
I
t'.t'.
+- I)(~ -+- 3P~+
()(i -1- ~
+- J..f>/.A*"
3v~2..z.
(
~ y
~
,M.I = M.2...=-jJ1-~
l-N:,Q~
t=-
~
~
+
-= 0
o
3~~
-D
~
3
t;?(.~~2...
+ Kf~z-)
4- ~,B~3 ~ K~;3)
?3
'D
~
i
O~
.,..
+ ~Z.'3.
~
-Lr,(R'*'
+
3~!..;>3~
-=
O
0
)
c) Give below a matrix C so that the testable hypothesis Ho: Cy = O is the hypothesis that considering only
levels 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.~
.
.
~~
~-
t..e. .-Jf.rz...z
5(.)
A~
'A-I: -~
-= A~ -A (+
(""",2 + lX;;.3
C(PZ2.
I--I~ r.2
R
AJ ~
1- ""f2.3
~f32..~
+
rx:A~ -'A~
13-3'
~- 32.
v ~
-~~33
=- D
tI($-(
\
t=
(()
o
o
0
O
-I
I
(
-I
)
I