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Unformatted text preview: STAT 8260 Final Exam — Thursday, May 1, 2008
SHOW ALL WORK I. Consider a randomize complete block design in which two blocks are used with
four treatments per block. The four treatments are combinations of two factors:
factor A (with levels A1 and A2), and factor B (with levels 131 and B2). Let yijk
be the response in the ith block, at the jth level of A combined with the kth level
of B. The design and data can be represented schematically as follows: Block I Block 2 Bl BZ ym yin The classical model for such a design would be gigk s: n+n+aj +ﬁk+(aﬁ)jk +e¢jk.
However, we will consider a model in which we assume that factors A and B do
not interact. That is, consider the model ytjk: W #+Tz' "toe +15% +ij» i: 1333‘ = 1,2;16 m 1,2;
with the usual assumption of independent errors where E(eijk) = 0, var(ezjk) = a2
for all t, j, It. Here 73v, 013 and 516 are effects for blocks, levels of factor A, and levels of factor B, respectively. a. (7 pts) Write down the model matrix X and parameter vector 6 for this
situation. What is rankCX) here? llOlOlO ﬂ“
llOlOO‘ '57
X:/100110 _TZ
~1100l0liﬁ”a€1
lOllOlO 96L
(01100! ,
)010l10 f3;
[@lOlO/ b. (6 pts) Write down an appmpriate set of side conditions for reparameter—
izing this model as a full rank mode}. 3712:“0
04+063 :0 we 0 c. (6 pts) Write down the reparameterized model matrix and parameter vec—
tor based on your side conditions from part (b). 111' M
:1!*' 1g;
[1“{l N 94
[NH ﬁg ’
1*il‘ I
I’ll"! f
li‘ll lvl’i‘l d. (9 pts) Find the BLUE of 041 w 052, which is an estimable quantity in this
model. Your estimator should be written as a simple, non—vector/ma’srix function of the 142 ks. r ‘
um»: N074“: %/ 5? 445 Df5%%ana/ 29:: ~23; 3'— “%[31M312.> ‘ M'\)
"2% are/>22, =— ﬁ/E"‘3'2'3*3 ‘3)“ “'5” Nut 33 at (n. e. (9 pts) Obtain a simple formula in terms of the 9'5ij for the Type I sum
of squares for block effects in this model (you may assume that terms are
entered into the mode} in the order in which they appear in the model
equation above . )(W‘Q '3 jﬂrﬂaeet) :23: I'J' an fag/)‘mﬁ/ %r :3)? JAM: ) Z: {Jan
_) ‘ZT: ( J j > )
2: “:2 (O f. (6 pts) Consider now the classical model yijk m psin +05 +[3k + (01,8) jk +
eggk for this design. For this model, how would you represent the Type II
sum of squares for block effects in terms of the SSH.) notation? §S(Z’“//M a ﬁr 5%“ B 2. Suppose we collect data on fifty 10 to 13—year—old children, half of whom are male,
and the rest female. Let yij be the height of the jth child of the ith gender, and let wij be the corresponding age of that child (in months). Consider the following
model for these data: yijmai+(#+ﬁi)méj+eij i=1=2;j=1a,25a (’3‘) where en, e12, . . . “22,25 are independent, each with mean 0 and variance 02 a. (7 pts) Write this model down in y = Xﬂ+e form, specifying all matrices, vectors and assumptions that deﬁne the model for these data. (Use 3’s in your matrices and vectors to deal with the large number of rows resulting
from the iarge sample size here.) Th. 3.1...
g :(ylliyll) yl3J"') y’ﬂﬁ’f) 3151,1227 .. ")gZJZS’ O—l—h ‘éT:(o([)od1,J/lAJ/3))/32,3T 0 X1: XII ‘3 ,U 3
X 3 X17, X12, e _ f J 2.0,:5'0
.— . : 1 '. — ’ r ’0. (7 pts) Is the hypothesis H0 : a1 —— 0:2 2 ,u m 0 testable? Why or Why not? /A. If «€197Z @JZI/Ikﬁgé 50 %f 1/0%/j ,1 40%;és7éé/éﬂ
'7: See 2.: @0/ €J/m/é ﬂay/g )ZZ/ ﬂ: gr A 5‘ is: '
gamma 53:53:45; J Dev MUIIL ’% AWL AgtAY‘L/x? : 5’” Aue, A3:IJA~1:’OJ/lr’:o A") 0+0 0. (6 pts) Based on model (*), snppose we want to test'whether a simple
linear regression of y on :1: holds for these data with common latercept and
slope for the two genders. Express such a hypothesis as a. testable general
linear hypothesis on the parameters of model ﬁe, a In” /5 4/303; “4x3 ﬂ+ﬂ;3'/blv%a (or t: a} Mm Uf’/%/l 6L} : 9 f f
OVAL/c ﬂ is as m QM.) appI000)
«ﬂu“000‘”? d. (6 pts) Show that 051 is an estimable parameter in model (It really is;
I promise.) [an 5A0w ﬁf 5; 146:,» J%/ a?! f! a: new
Comémq/{art mfg {M74 04/ / 63/ 60M; 7;
in oa/ﬂcJ/ag «a, Cam J‘l M) A 4’14 m 0‘ a}? [(47.03 “d1+/"“‘/3I\Xn 5“) €®A= “I+/"+/g')){n.
Nam. 44% ﬁgwwmxuyE: (,g,+{/M/a,)>(,,) T: i v 2i" 1 a?! 962*?“
XII X12, )(HXH
X _..2.._.,
5’0 Xu :1 “Lu 4};( 4— 3x  (Ogi+01A+/(,)XJZ® (
My“ p” { /A /1, I. X];
z: 03‘ 3. (12 pts) Consider the analysis of variance mode} for the balanced oneway Iayeut: yéj = p+ai “kegj, where z' = 1,. . . ,a, j m 1,...,n, and E(e¢j) = 0, var(ezj) = 02
for all i, 3'. Show that 23le Gian; is estimable if and only if I: 0
5:“ % ﬂ 7'" “L‘ 0
Ci. a’ll' 15 ‘9; _ rm NINQ' [g L“: 03) >‘ r :t
(“'51 “a. J " :3
a; fccoﬂ‘ If esgmézé . 3W5?— CO/L/Mﬁf‘af S1)? 3'52, ‘3
r L # %5majz() ' or 0 i a
61,
NC”, SJ/PDSE 16cm!)
“1 q a
On 61* ,. EC +08)
My... :2. C’. 4’ a ., t/A t
5 EM ‘ 5'0 I gma‘7lha
a, Unﬁt”, COM 4. (9 pts) Consider the model yij m p, + on + 133' ~+~ eij, z' = 1,2, 3' m 1,2, Where the
eij are independent with sphericai variance—covariance structure. Suppose that I
Wish to use a; + 052 = $1 — )62 z 0 as a set of side conditions for this model. Is this a valid and complete set of side conditions here? Justify your answer. fﬁ ﬂO’f 7$ 51—96.; W; {I Mew 59; Maw“ POL dwelt—AM; ﬂat/5‘72 A: rang/mom; 0n .ﬂ04'85%M4J/¢ 41174045), % 6&4 1/ 0/9— A éomﬂ/a7é I {1&‘0 W (0110 0 M4
.u. ‘00! z 0 4, 0(1
#6?) 2(#(:o:lo>) HIE o OI A 04*
Iota: f}
(a;
 X ,. “Ax “Mu—r)
Neeckl)faajt[,r)~ (TL) 7””
~‘ X ; ﬂag/anr an: (
aw) (It) fank(_r\
[lot o W
(I 00! ; L/ 7: g
k X)“: {Mk 0! l 0
row I(010$
011001
Goat“
’9 m/aw/Im’éﬁg/ﬂ: é“/7g 5. (10 pts) Consider the linear model y 2 Xﬁ ~§~ e where e ~ N (0, 0'21”) and X is
n><p with rank(X) = k < 10. Suppose that 31 m G1XTy and 52 2 GZXTy are two
different least squares estimators of ['3 corresponding to G1 and Gg, resPectively,
Where G1 and G3 are different, non—symmetric generalized inverses of XTX.
Suppose we are interesteci in two distinct estimable functions of 3: X58 and Agﬁ where Am ¢ Age. Show that covmffal, A532 = aQXfGQAQ. M >133. ) >713 r /‘ at; )
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This note was uploaded on 11/13/2011 for the course STAT 8260 taught by Professor Hall during the Summer '10 term at UGA.
 Summer '10
 HALL

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