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On the Null-Distribution of the F-Statistic in a Randomized
Partially Balanced Incomplete Block Design ,nth Two
Associate Classes under the Neyman Model
by
Junjiro Ogavm, Sadao Ikeda and Motoyasu Ogasawara
Nihon University, Tokyo
Institute of Statistics Mim.eo Series No. lJ.l4
December 1964
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This research was SUpported by Air Force Office
of Scientific Research Grant No. 84-65.
DEPARTMENT OF STATISTICS
UNIVERSrry OF NORTH CAROLINA
Chapel Hill, N. C.
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On the Null-Distribution of the F-Statistic in a Randomized
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Partially Balanced Incomplete Block Design with Two
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Contents
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Associate Classes under the Neyman Model
By
Junjiro Ogawa, Sadao Ikeda*and Motoyasu Ogasawara
Nihon University, Tokyo
Summary and introduction
I.
~2.
Spectral decomposition of the matrix NN'
The null-distribution of the F-statistic before the randomization under
the Neyman model.
'.,3.
The calculations of the mean and variance of the quantity
e
with respect
to the permutation distribution due to randomization.
3.1
Necessary notations.
3. :2 The mean value of
3.3 The variance of
4.
e.
e·
An approximation to the permutation distribution of
e by
a certain Reta-
distribution when the number b of the blocks is sufficiently large
certain uniformity conditions are imposed on the unit errors.
'5.
The approximate null-distribution of the F after the randomization.
References.
* Sadao
Ikeda is now at Waseda University, Tokyo.
1
A.nO
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Summary and Introduction
One of the authors, J. aSawa, showed that for a complete block design [2J,
a Latin-square design [3J, and for a balanced incomplete block design [4J, the
null-distribution of the F-statistic (i.e., the variance ratio) in theiranalysis
of variance can be approximated by the familiar central F-distribution dven
under the Neyman model, i.e., an intra-block analysis model with both the unit
errors and technical errors, if they are randomized.
In the present article,
the authors consider the null-distribution of the F-statistic in a randomized
partially balanced incomplete block design with two associate classes under the
Neyman model and they reached the same conclusion as mentioned above, though
the calculations are much heavier than those for preVious cases.
P. V. Rao [6J treated the same problem under the Fisher model for a special
class of randomized partially balanced incomplete bl.ock designs with two associate classes with
~l
= 1,
~2
= O.
Thus the result presented in the present paper seems to be the most general
one among those thus far published along the line mentioned above.
The extension of the reasoning presented in this paper to a randomized
partially balanced block design with m associate classes is not difficult and
will be presented in a forth coming paper.
If one looks at those results obtained from the point of view of the usual
normal theory, this may be regarded as the so-called
"rObustness" of the usual
regression model with normal errors which ignores unit errors completely.
In
Sl
the spectral decomposition of the matrix NN', N being the incidence
matrix of the design under consideration, is given and this is useful for discussions in later sections.
The null-distribution of the F-statistic before
the randomization under the Neyman model is presented in §2 and this turns out
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to be a non-central
the quantity
F-distributio~ whose
e which
non-centrality parameter depends upon
mean and variance of the quantity
e
with respect to the permutation distribu-
tion due to randomization are calculated.
qui te complicated.
In § 3 the exact
is a quadratic form of the unit errors.
The calculations in this part are
Then in §4, if the number of blocks is sufficiently large
and a certain uniformity condition on the within-block variances of the unit
errors is satisfied, the permutation distribution of
mated by a suitable Beta-distribution.
e
is shown to be approxi-
Finally in §5, it is shown that the
null-distribution of the F-statistic after the randomization can be approximated by a familiar central F-distribution provided the two conditions mentioned
above are satisfied.
§I..
Spectral decomposition of the matrix NN' •
As for the definition of a partially balanced incomplete block design and
notations being used in the present section, references should be made to [lJ
Let the association matrices be A = 1 , A , A , and let their regular
o
2
3 l
representations be
0
r 0 = r"
p1
n
=
l
0
1
0
1
Pll
1
P2l
2
Pll
2
P2l
0
,
P2
=
0
n
2
0
1
1
P12
1
P22
2
P12
2
P22
respectively, then it is known [5] that there exists a non-singular matrix
1
1
c=
such that
3
1
I
0
0
0
zlu
0
0
0
z2u
n
C P C- l
u
( 1.1)
=
u
,
u
= 0,
J
1, 2
simultaneously, where
Z21 = (a-d)/2,
Z12
= -(a+2+d)/2,
z22
= -(a+2-d)/2
,
with
(1.3 )
2
1
a
= P21
d
12
= (P2
- P -1)
21
12
- P12- 1 ,
+ 4P
2
21
2
2
= (1
P -P -1)
12 21
+ 4P
1
"
12
One may put as follows:
C
lO
(1.4 )
C
20
= 1,
= 1,
C
c
11
21
zll
n
l
z21
=-,
=nl '
c
z12
=n12
,
2
C
z22
22
=n2
Three orthogonal idempotents of the association algebra are given by
. #
, A
0
(1.5)
\,
=-v1
G
v
,
2
zll
{ A# (1 + 1
n
l
I
I
Z 2
II
A# = (1 + 2 !
\
2
n
1
wi th respective ranks
=
°0
= 1,
01 and
z 2
z
z
12)-1 (A + 11 A +.J..g A ),
n
1
n
2
n
0
2
l
2
2
z
z
z22
+ ..§.g)-l (A +2! A +
- A ),
n
1
n
n
0
2
2
2
1
+
°2 "
Let the incidence matrix of the design by N, then it is known that
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,
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where
(1.7)
and
(1.8)
Po' PI' P2 are the characteri.tic roots or the matrix with multiplicities
°1 ,
~
respectively.
Column vectors or
I
A~
00'
are characteristic vectors corres-
ponding to the characteri8'tic root Pu or Im' •
62.
The null.distribution or F.statistic before the randomization under the
Keyman model.
We shall be dealing with a partially balanced incomplete block design with
such an
two associate classes which has v treatments with/as.ociation as explained in
the previous section, b blocks or size keach, r replications of each treatment, and the numbers of incidence of any pair of treatments Al or \~ according
as they are 1st associates or 2nd associates.
As for the notations being used
in this section, references should be made to (lJ and
[5].
Let the incidence matrices of treatments and blocks be t and , respectively,
then the Ne,man model assuming no interaction between treatments and units is
given by
x ... 11 + t
I.
I
if
+}.. A +,.. A ) ... P
+ P Ai! + P A#
01122001122 '
NN' (-r A
(1.6)
T
+
+~
+
1f
+e ,
where x' ... (~, ... , x ) 1s the observation vector, T'
n
5
= (Tl ,
... , Tv) and
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~' = (~l' ••• , ~b) are treatment-effects and block-effects subjected to the
restrictions
'1 + ••• + Tv = 0 and ~l + ••• + ~b = 0
respectively, and ~' = (~l' ••• , ~n) stands for the unit errors.
Finally,
e'= (e , ••• , e ) is the technical error vector being distributed as N(o', ~21 ).
l
n
n
Now we are interested in testing the null-hypothesis
(2.2)
H:
o
T= 0 •
Likewise one may consider the testing the partial null-hyPOtheses
./1.
Air...I
U
= 0,
u = 1, 2.
Although the arguments for these hyPOtheses are similar to the one presented
in this paper for H , we confine ourselves to the null-hyPOthesis H for the
o
0
moment and testing of H(u) will be dealt with in the forth-coming paper
o
tioned in the introduction.
Sums of squares due to treatments adjusted and errors are given by
2
S
t
2
Se
II
1
- -k B -
Jf
1
V~ -
#
V2 )x
respectively, where
1
- rk-p
(I u
1
JJ.
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J1
= x' {'t.j/1
+ V 7f)X
2'
= x'{I
men-
~
1
it B) T~{I - it B) ,
u
= 1,
2
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with
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,
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Under the null-hypothesis H , they can be expressed as follows:
o
s;
+21f
t
[I_i(clAi+C2A~)t'J(I4)1f
t+
(I4)(I-i(C l A c 2A:)t'J (I4)e
+ e' (I4)[I-W(ClAr+c2A:)t'J(I4)e,
where
(2.6)
•I
I
= 1f' (I4)
c
k
-~~
I - rk-P
I
'
c
2
k
=-:--
rk-P
2
The null-distribution of the variate
before the randomization is the non-central chi-square distribution of degrees
of freedom
v-I and with the non-centrality parameter
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Hence its probability element is
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(2.8)
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The null-distribution of the variate
before the randomization is the non-central chi-square distribution Df-?egrees
or u-eeClom
7
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n-v-b+l and
'll:i.tll
tile
~
nOl1-centralit~T parameter
,
fj, E
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:It':It •
Hence its probability element is given by
1\)-1 X'2
x2
exp( 22) d(
i)
K
(2.10)
I
I
2
exp(-2 )
Since Xl2 and X2 are stochastically independent of each other, the null2
distribution of the F-statistic
(2.11)
F =
n-v-b+l
v-I
~
7e
•I
before the randomization is, after a little aJ.aebra, given by
(E.::E.)
!::.! -1
~
'2
(V-l F) 2
(11 v-I F) '2
f(V;l)f(n-V;b+l) n-v-b+l
n-v-b+l
f
(2.12)
6
00
exp(~) l~
6)/
('?
-.er
f
I!
~
(1I n:;_b+l F)- IJ.+\)=! IJ.. \).
1
!:
eu (
I
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I
) V(V-l
.' 1-0
n-v-b+l
,
where
The null-distribution of the F after the randomization should be
)ll
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obtained as
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r(~)
!:'!-I
2
(V-l ) 2
-~l~~----:b-+~l- n_v_b+lF
)r(n-v~
r(V;
)
(
v-I
1 + n_v_b+lF )
d(
(2.14 )
_~
2
v-I F)
n-v-b+l
6 00
exp ( --:--2) !:
2CT (=a
,
where e( el-l(l_e) ~ stands for the expectation with respect to the permutation
distribution of e due to randomization.
§3.
The calculation of the mean and variance of the quantity
e with
to the permutation distribution due to randomization.
Since
with
0
=
1
=
I-l
I
)
(3.1)
1-l
I-l
2
=
k
k
2
Z2
zll
+
2)
(rk-Pl)(l~
n
l
~
kZ
1l
2
~
zll
z12
(rk-Pl)(l~ + ---)
n
n2
l
+
kZ
+
9
2l
z 2
z 2
22
z 2
Z2
(rk-p )(l~ + ~)
2
n
n
1
2
kZ
12
2
2
zll
z12
(rk-Pl)(l+-=:
n + ---)
n
2
1
2
2
z2l
z22
(rk-p 2 )(1+--n +---)
n
l
2
kZ
+
(rk-p )(l~ + ~)
2
n
n
l
2
respect
I
let
US
3.1.
~
put
Necessary notations.
Now we use the numbering of' the whole units such that the i-th unit in
the p-th block bears the number f'
= (p-1)k
+ i.
Let
Tu
= II Tpq(u~1
p,q = 1, .•• ,b
,
T (u) = I1 t (u) pq
pq
•
ij
l
i, j = 1, ••• ,k.
where
,. 1, if' the i-th unit in the p-th block and the
t(u)pq
ij
=
L
j-th unit in the q-th block receive treatments which are u-th associates,
otherwise,
and f'urther let
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where
Notations necessary f'or the calculations in this section are introduced
here f'or the case k > 4.
The cases in which k
=2
or 3 shall be examined
later.
(1)
p
Number of' the ordered pairs of' treatments being of' the u-th
associated which occur in the p-th block is denoted by
10
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~
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~(l)uu
pp
EEt(U)PP
i~j
ij'
U
= 1,
2.
one can see that
(ii)
Number of the ordered triplets of treatments in the p-th block,
p
in which the 1st and the second are u-th associates and the 1st
and the 3rd are v-th associates is denoted by
I /
U,v = 1, 2 •
One gets the following relations irmnediately:
~(2)uv = ~(2)vu
pp
PP ,
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I.
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(iii)
p
i
Number of two pairs of treatments in the p-th block, of which
two treatments of the one pair are u-th associates and those
) u
j
of the other pair are v-th associates is denoted by
~(3)uv
PP
E
~
t(u)pp.t(v)pp,
i~j~J{G
iJ
1m
Trivial relations are
11
1
u, v, = ,
n
c.
I
Other similar 12 notations with their trivial relations are listed below.
q
p
(iv)
;>..(4)uv =: E t(U)PPt(v)qq u,v
pq
i~j
ij
ij'
i
i
j
l;>..(4)21 + ;>..(4)22 = ;>..(1)22
pq
pq
11'
t
(v)
),,(4)11 + ),,(4)21
=
;>..(4)12 + ;>..(4)22
pq
pq
= ;>..(1)22
pq
pq
),,(1)11
qq ,
qq
•
;>..(5)UV =: E t(U)PPt(v)ql U v
pq i~j~l
ij
i"
q
P
2.
v . ;>..(4)11 + ;>..(4)12 = ;>..(1)11
pq
PP ,
\
pq
U
j
= 1,
,
=1
,
2 •
i
U
v
[),,(5)11 + ),,(5)12
pq
I
j
pq
i
. ;>..(5)11 + ;>..(5)21
= (k_2);>..(1)11
pq
I
v
j
!
pq
pq
pq
i~Ul
•
i
=1
. ;>..(6)11 + ;>..(6)21
= (k_2)(k_3)~(1)11
pq
pq
;>..(7)uu
pq
E!
Et
i
,
qq ,
= (k_2)(k_3);>..(1)22
qq
(u)pq
_
i i ' U - 0, 1, 2.
;>..(7)00 + ;>..(7)11 + ;>..(7)22
pq
pq
pq
12
= k.
2.
,
pp ,
= (k_2)(k_3)~(1)22
pp
;>..(6)12 + ;>..(6)22
pq
pq
q
qq
;>..(6)21 + ;>..(6)22
pq
pq
i
P
= (k_2)~(1)22
I
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•I
qq ,
;>..(6)11 + ;>..(6)12 = (k_2)(k_3);>..(1)11
m
(Vii)
pp'
;>..(6)uv =: E t(U)PPt(v)~q, U v
pq i';'j~l~
ij
m
'
q
U
pp'
= (k_2);>..(1)22
;>..(5)12 + ;>..(5)22
pq
pq
P
(k_2»),,(1)l1
;>..(5)21 + ;>..(5)22
pq
pq
~
(Vi)
=
~
•
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(viii)
p
A,(8)uu ~ ~ t(u)~~, u = 0, 1, 2.
pq i~j
~J
q
= k(k-l).
A,(8)00 + A,(8)11 + A,(8)22
pq
pq
pq
(ix)
p
A,(9)uv ~ ~ t(u)~~ t(V)p~ , u,v
pq i~j
~~
jJ
q
i /"j -
V-1l ~
A,(9)uv
pq
U
= A,(9)vu
pq
J
p
i
A,(10)uv
pq
q
A,(10)uV
pq
i
U
j
= (k_l)A,(7)uu
j
p
i
i~j
'\. (ll)uv
'"
pq
A,(11)00
pq
i
j
~j
= A,(10)vu
pq
(
1, 2.
•
X
X
= A,(8)uu
X'
;::' ~~
t(u)pq t(v)pq, u, v
i~j
11
ij
=0
= 0,
= 0, 1, 2 •
•
A,(11)01 + A,(11)02
(
pq
= (k_l)A,(7)00
pq
A,(11)uO + A,(11)ul + A,(11)u2
pq
pq
pq
pq ,
= (k_l)A,(7)uu
pq
. . A, (11) Ou + A, (11) lu + A, (11) 2u = A, (8) uu,
pq
pq
pq
pq
p
i
'\.(12)uv
'"
pq
q
u
i
~
v=O
j
"-".
I
,
A,(11)10 + A,(11)20 = A,(8)00
pq
pq
pq ,
i
{
(xii)
= 0,
, u
Ji
A,(10)uO + A,(10)ul + A,(10)u2
q
k u-t
I~
pq
;:;~ t(u)~q t(V)~q , u,v
X
(xi)
1, 2.
•
A,(O)uO + A,(9)u1 + A,(9)u2
pq
pq
pq
(x)
= 0,
~
u=O
~ ~
t(U)pq t(V)pq,
i~j~t
ii
j{ u,v
A,(12)uv
pq
A,(12)uv
pq
= (k_l)(k_2)A,(7)uu
pq
= (k_W)A,(8)vv,
pq
13
u
0
= 1, 2,
1 2
= , , .
, u=O, 1, 2.
v=O, 1, 2.
1, 2.
I
(x1ii)
P
1
~(13)uv
pq
q
u
~
1h.j.1
t(U)pq t(V)pq
J
11
1
~(13)uV
pq
",
2
!:
J
v=o
p
q
j
i
= ~(13)vu
pq
,\(14)uv
pq
~(14)uv
pq
•
= (k_2)~(8)uu
t(U)pq t(V)PCl
E!:
1/:k~1
= ~(14)vu
m
~(15)uv
pq
2
!:
v=o
u, v
= 0,
1, 2.
•
0
pq
•
pq
E!:
11tj-I:i~
1
j
1t '
1J
pq
~(14)00 =
pq
,\ (15 )uv
pq
q
,
pq
~ ~(14)uv = (k_2)~(8)uu
p
u, v = 0, 1, 2.
pq
~(13)uv
v=o
(xv)
'
1
, v
(xiv)
~
E
= ~(15)VU
t
(u)p~ (v)pq,
iJ
1m
u, v = 0, 1, 2.
•
pq
~(15)uv = (k_2)(k_3)A(8)uu •
pq
pq
We list the relationships holding among those parameters of 15 classes.
J
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I
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I
J
14
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..I
I
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I
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I
I
--I
oJ 4 )pq
22 = k(k-1)
I
pq
l
pq
..I
I
qq
pq ,
pp
qq
pq
,
= k(k-1)(k-2)(k-3)-(k-2) (k_3)A,(1)11
pp
_ (k_2)(k_3)A,(1)11 + A,(6)11
qq
A,(7)22
=
pq
pq ,
k_A,(7)00_A,(7)11
pq
pq'
A,(8)22 • k(k_1)_~(8)00_A,(8)11
pq
pq
pq ,
A,(9)22 = k(k.'1)-2(k-1)A,(7)00-2(k-1)A,(7)11
pq
pq
pq
+A.(9)00+2A,(9)01+A.(9)11
I
pq
~A,(10)22
pq
I
I
I
I
I
pp
= k(k_1)(k_2)_(k_2)~(1)11_(k_2)A,(l)11+A,(5)11
A,(5)22
'A,(6)22
_ A,(1)11 _ A, (1) 11 + A,(4) 11
pq
pq'
= k(k_1)_2A,(8)00_2A,(8)11+
pq
pq
+ A,(10)00+2A,(10)01+A.(10)11
pq
A,(11)20
pq
A,(11)22
pq
pq
pq'
= A,(8)00_A,(l1)10
pq
pq'
= k(k_1)_(k_1)A,(7)00_(k_1)A,(7)11
pq
pq
_ A,(8)00_A,(8)11+A,(11)01+A,(11)10+A,(11)11
pq
A,(12)22
pq
pq
pq
pq ,
= k(k-1) (k-2)-(k-1) (k_2)A,(7)00_(k_1)
(k_2)A,(7)11
pq
pq
_ (k_2)A,(8)00_(k_2)~(8)11
pq
pq
+ A,(12)00+A.(12)01+A.(12)10+A.(12) 11 ,
pq
15
pq
pq
pq
I
A(13)22
pq
J
= k(k-l) (k_2)_2(k_2)A(8)OO_2(k_2)A(8)11
pq
pq
+ A(13)00+2A(13)Ol+A(13)11
pq
pq
pq ,
I ~(14)22 = k(k_l(k_2)_2(k_2)~(8)OO_2(k_2)~(8)11
W
j
I
W
W
+ 2~(14)Ol~(14)11
I
W
W'
: ~(15)22 = k(k-l)(k_2)(k_3)_2(k_2)(k_3)~(e)Oo
W
~
W
_ 2(k_2)(k_3)~(8)11~(15)OO+2~(15)Ol~(15)11
pq
(A(7)OO)2
pq
pq
pq
pq ,
= A(7)OO + ~(9)OO
pq
A(7)OO~(7)11 =
pq
pq
pq ,
A(9)Ol
pq ,
(A(7)11)2 = ~(7)11 + A(9)11
pq
pq
pq ,
~(7)OO~(8)OO
pq
pq
= ~(12)OO
--I
pq'
~(7)OO~(8)11 =
pq
pq
A(l1)Ol + ~(ll)Ol + A(ll)Ol
pq
qp
pq
A(7)11~(8)OO = A(11)10 + ~(11)10 + ~(12)10
W
W
W
A(1)l1A(8)11
= A(11)11
pq
pq
qp
W'
+ A(11)11 + A(12)11
pq
qp
Pri'
(A(8)00)2,..A. (8)00~ (10)00~ (14)00~(14)00;eA (13)OO-M.(15)00
pq
pq
pq
pq
qp
pq
pq'
(A(8)11)2,..A.(8)11~(10)11~(14)11~(14)11;eA(13)11~(15)11
pq
pq
pq
pq
qp
pq
pq'
A(8)OOA(8)11,..A.(10)Ol~(14)Ol-M.(14)Ol+2A(13)Ol~(15)Ol
•
pq
pq
pq
pq
qp
pq
pq
3.2 The mean value of e.
I
I
I
I
I
I
I
I
I
I
I
I
J
Let us write
16
I
I
I
..I
I
I
I
I
I
I
lit
I
I
I
I
I
I
..I
I
e
=
@/A,
where
8
.f1:
= rrt~'rr
= rrt(~oT0
~
+
Following Ogawa [4J, we calculate
T +
1 1
e (@)
~
T )rr •
2 2
as follows:
1
=
(k~ )b
= cr
and cr
Now, since
and
where we have put cr
p
q
=",
and
1i) ) = it1 Ap ,
()2
," (rrcr
"(rr(p)
,
,0
cr(i)
rr(p)
cr(j)
(rr~11)rr~(L
) =
) =
1
'k(k-1J
f:::.
p'
tdrr~11)rr~(~) ) = 0
we get
11
,
I
J
where 6 = 1{(p)' 1{(p) •
P
Now,1t can be seen that
~
i=l
t#PP
ii
= ~ (~ t(O)pp
1=1
11
0
+
~
1
t(l)PP + ~ t(2)PP
ii
2
ii
and
E
i~j
tip~
iJ
=
E
i~j
=~
(~t(O)pp + ~ t(l)PP + ~ t(2)Pp)
ij
0
E
1 i~j
1
ij
2
ij
t(l)pp + ~ E t(2)PP
ij
2 i~j
ij
= ~ A(l)ll + ~ A(1)22
1
PP
2
PP
--I
= ~ A(l)ll + ~ [k(k_l)_A(l)ll]
1
PP
2
pp
= k(k-l)~2 + (~ _~ )A(l)ll •
1 2
PP
Hence we have
p. (6)
and consequently
As a special case, we shall examine the case of BIBD.
since the spectral decomposition of the matrix NN' is
NN'
= rk -v1
1
Gv + (r-A)(Iv - -v Gv )
18
I
I
I
I
I
I
I
In this case,
I
I
I
I
I
J
I
I
I-
..I
I
I
I
I
I
I
and therefore
Whence it follows that
Thus for a BIBD, (3.2.5) reduces
p, (0) =
which confirms the earlier result given by Ogawa [4].
3.3.
o.
The variance of
@2
= (~'U'T'U
~)2
0"
0"
=
--I
I
I
I
I
I
k
VA
(~ ~(p)' S' T1;~ S ~(p)
p=l
O"p PP O"p
~(p)' S'
S ~(q»2
O"p pq 0"q
;J/
= ( 1: ~ (P ) , S' T 1:' S ~ (p ) ) 2
P
O"p pp O"p
+ 2(1:
p
~(p)' s'
rril
S rr(p»( 1:
O"p pp O"p
rl:s
~(r)' S;
T//: S rr(S»
O"r rs o"s
+ (1: rr(r)'s' T# S ~(S»2.
rl:s
O"r rs O"s
This can be expanded as follows:
r#
= 1: rr(p)' S' T=!l S ~(p)~(p)' S' Til S rr(p)
p
O"p PP O"p
O"p pp O"p
+ 1: rr(p)' S' T# s' rr(p)~(q)' S' T): S rr(q)
PM.
O"p PP O"p
O"p qq 0"q
+ 2[ 1: ~(p)' S T F-S ~(p)(rr(p)' S' T)¥ S ~(q)+rc(q)' S' ri/,L S ~(p»)
PFq
0"P PP 0"P
0"P pq 0"q
0" q qp 0"P
..I
I
+ 1:
PM.
19
I
J
Since the terms which are linear with respect to some S
0"
/I
such as those with 7/'
in the above vanish by taking their expectations, we have
where
C= E
p~
C'=
Since C
= C',
n(p)' S' T/I S n(q)n(p)' S' Til n(q:
O"p pq 0" q
O"p pq
,
E n(P)'S'Ti'S n(q)n(q)'s' TO#S J(p).
PM.
O"p pq 0"q
0"q qp 0"1
it follows that
Now we shall calculate (~ (A),
t".
(B) and
r'.
(C) in this order.
process of the calculation of 1~(A) in detail.
We present the
I
I
I
I
I
I
I
til
I
I
I
I
I
I
J
Since
20
I
I
I
..I
I
I
I
I
I
I
--I
= (E
i
t#pp~(p)2
+
ii cr(i)
E
i~j
t1fPP~(P) ~(p)
Jin. 2 ( )4
-"
11()2 ( )2
.IL... -'I-.
3
= E t 1WP ~ p(. + E t·irPl?t1r.l:'~~ P ~ P +4 E t,r~~t'ir~p~(p) ~(p)
i
1i cr~) 1~j i1 jJ cr(1) cr(j) i~j 1i ij cr(i) cr(j)
+ 2
+ 4
E
1~j~[
E
i~j~[
+
'I)!..
()2 ()
()
t'/tl'P
t "lrl'P~ P ~ P ~ P +2
i1
t'l;lpPt
ij
E
1~jFl~
j[ cr(i) cr(j) cr([)
E t"lrPPt"li}l~
)1
-''--
(
p) 2 ( p) 2
i~j ij ij~cr(i)~cr(j)
'pp~(p)2~(p) ~(p)
i[ cr(i) cr(j) cr(l)
t0pPtf.'pp~(p) ~(p) ~(p) ~(p)
ij lm cr(i) cr(j) cr(l) cr(m)'
and
(p)4
1
~ (~ (.)) = cr ~
k
k
'I
2
t'P~ =
1=1
i~
E
rP
where
r =
P
k
E
~(p)
4
I
i=1 1
2
kj..l
0'
I
I
I
I
I
( I..l.
o
..I
I
)2
ij cr(1) cr(j)
21
t(O)PP-u, t(l)PP-u, t(2)PP)
jj'~1
jj'~2
jj
I
J
p.
_
1
(A2 _ 2r )
- - k(k-1)(k-2) P
P ,
(p)2 (p) (p»
(1\r(1)1C
(j)1C (i)
cr
cr
Z 13fPP t i;fpp
i~j~! 11 j[
=
I
I
I
I
I
I
I
Z
1~j~!
(Il t(O)PP+ll t(l)PP-+il t(2)PP)
11 1
0
11 2
11
( Il t(O)PP-+il t(l)PP+1l t(2)PPt)
J!
o
= (k-2)1l
J! 2
1
j
Z (Il t(O)PP+ll t(l) PP-+il t(2)PPt)
0
Jt 1 Jt 2 j
o j~t
=
(k-2)1I (., A(l)ll-+il A(1)22) • (k-2)1l
'-0'-1
PP 2
pp
0
( )2 ( )2
t
(1Ccr11 )1C cr1
1
j»
=
~
S:::r()fJ.S
2
k(k-1) (Ap - rp )'
Z t#'Pp2= Z (fJ. t(O)pp-fi,L t(1)PP+1l t(2)PP)2
2
1J
1~j 0
1
1~j
=
=
Z (1l2t(1)PP-+il2t(2)PP)
1
1j 2
1j
1~J
2A(1)1l-+il2A(1)22
III
(p)2 (p)
pp
2
(p»
pp
=~
fJ. 2A(1)SS
pp
s=O S
1
t (1Ccr {1)1Ccr (J)1Ccr (!) = - k{k-l)(k-f:lJ
( 2
•
)
Ap - 2rp ,
Z t#PPt 7W = Z (Il t(O)pp-fi,L t(l)PP-+il t(2)PP)
1~J~t ij it 1~j~t 0
1J 1
1j 2
1J
t(l)PP+ll t(2)PP)
( Ilot(O)pp-fi,L
it 1 it 2 it
22
A(l)SS.
PP
--I
I
I
I
I
I
J
I
I
I
..I
I
I
I
I
I
I
=~~~
~
s,t S t i~j~t
p
.
i
t(S)PPt(t)pp
ij
it
= ~ ~ ~ ~(2)st
s,t=O s t
pp.
(1t(p) 1t(p) 'J(p) 1t(p) ) _
1
2
a(i) a(j) a(t) a(m) - k(k-l)(k-2){k-3) (6p -2rp )'
J!pp
.1j~81-"ijtll'g
2
=
I: ~ ~t
I: t
s,t=O S
i'l=J~lf:m
=
~ ~ ~ ~(3)st.
s,t=O s t
PP
.1£
F F1.rw
()
S
PPt
ij
(
t)~
... m
2
Hence we obtain
~
(A)
= I: [2
~ r ~2(62-r )- (4 )
o p o p P k k-1
P
~
0
rP I:s ~SA(1) PP
ss
--I
I
I
I
I
I
I: ~2A(1)SS
4
~ ~ ~ A(2)st
P k(k-1) S S
PP - k(k-1)(k-2) s,t s t
pp
+ (62 _ 2r ) 1
p
+ k(k-l){k-2)(k-3)
In similar manner one can obtain
..I
I
23
I:t~s~tA(3);:).
s,
I
J
+
2
E ~2h(1)88 +
2
k (k-l'f
+
8·
1
2
k (k_l)2
pp
2
4
k (k_l)2
E ~ ~ h(3) at J
B,t • t
pp
and
e
(C) •
E
6 6
[~ E ~2 (h(7)BB + h(8)•• )
pk pqk
+
~
B·
~
1
E ~ ~ ~(9)Bt-M.(lO).t+2h(12)at+2h(13).t+h{l').t)
k 2(lt_l)2 s,t s t
pq
~
~
pq
pq
Theretore one obtains the variance ot
e as
Var e • 6- 2 Var (8)
•
~(A)~(B)+2e{C)~o,~
2
6
Ai 2r
E p- P [ 1
• p
62
2h {l)..
k{k-l); ~8
(3.3.6)
4
pp k{k-l}{lt-2)
t
h(2).t
s,tP.~t
pp
I
I
I
I
I
I
I
--I
I
I
I
I
I
J
24
I
I
I
..I
I
I
I
I
I
I
--I
I
I
I
I
I
..I
I
For a BIBD, (3.3.6) reduces
where
~
pq
= ~(7)OO~(8)OO
= number
pq
pq
of treatments common to the
p-th and q-th blocks,
and this confirms the Ogawa result once again [4].
§ 4.
An approximation to the permutation distribution of 9 by a certain Beta-
distribution when the number b of the blocks is sufficiently large and certain
uniformity conditions are imposed on the unit errors.
In this section, we assume that the following uniformity conditions
(4.1)
Ap
= Ao
and r
p
are imposed on the unit errors.
= r0
for p
= 1, 2, ••• ,
Under such conditions the mean of 9 and the
variance of 9, Var 9, can be presented as follows :
(4.2)
e) _
1
e ( - bk(k-l)
b
2
~ (
8=1
_
~o ~s
Var 9
25
) ~ ~(l)SS
P
pp
I
J
_
4
~ ~ ~ ~ A(2)st _
1
~ ~ ~ ~ A(3)st]
b2k2 (k_l)2 s ,t s t P
PP b2k2 (k_l)2 B,t s t P
PP
+ ~ ~ ~2 ~ (A(7)sS+A(8)ss)
b 2k2 s S p~q
pq
pq
+
~ ~ ~
~ (A(9)st+A(lO)St+2~(12)St+2~(13)st
s,t s t p~
pq
pq
pq
pq
+ A(15)st)
pq
2
b 2k2 (k_l)2
-
2 24
~ ~ ~t
b k (k-l) s,t s
In order to express Var
e
~
p~
(2A(ll)st+~(14)st).
pq
pq
in tenus of the parameters of the design under
consideration, we show that
(i)
f
I
(u)
I
(4.4)
,! (iii)
I
~ ~(l)SS
p
pp
=~
n v
s s '
~ (~(7)ss+~(8)ss) = (r2_~ )n v
pp
pp
s s'
p
~ (A(9)st+A(lO)st+2~(12)st+2~(13)st+A(15)st)
p~
pq
pq
pq
2
pq
pq
= Q~~~QA~ l:cp~p:tnrv-&stAsnsV-&so(1-&to)(k-2)AtntV
,
\
I
I
I
I
I
I
I
--I
I
I
I
I
I
J
26
I
I
I
..I
I
I
I
I
I
I
lit
I
I
I
I
I
I
..I
I
where we have put
(4.5 )
Indeed,
(4.4) (i)
I; A(l)SS
p
= I;
PP
I; t(s)pp = I; I; n G f3 nt:t • tr(N'A N)
ij P o:,f3 ap as ...p
s
P i"j
= tr(RNtA
)
S
= tr(I;0: A0:f3I;
pf3 AA)
as ...
=Ass
n v
•
(4.4) (ii)
I; n c~f3 n p, q 0:, f3 ap. as f3q
= I;
= l'N'A Nl - tr(N'A N)
s
(4.4)
(iii)
s
I; I;n G.~nA
p o:,f3 ap
...q
= r 2ns V-As ns v = (r2_As )ns v
It can be shown that
t (A(7)OO+A(8)OO)2
p#q
pq
pq
=
I; [(A(7)OO~(8)OO)
p#q
pq
pq
+ (A(9)OO+A(lO)OO+2A(12)OO+2A(13)OO+A(15)OO)J.
pq
pq
pq
pq
pq
Now, let us consider the following identity relation:
I; (I; nap ~~ nA )2 = treNt A N)2 •
p,q 0:,f3
...q
0
Since
if p#q
the left-hand side of the identity is shown to be
27
•
I
J
The right-hand side of the identity is
tr(N'A N)2 = tr(N f N)2
o
= tr(
E
0,13,1
~ hQP~A )
0 ~ ~ 1
=0E h02 n0 v
Hence one obtains the relation
2 n v
= E h0
0
- r(k-l)v •
0
It can be shown that
Z (h(7)OO+h(8)OO)(h(7)ss~(8)ss) • E [2(2h(11)OS+h(14)Os)
PM.
pq
pq
pq
pq
PM.
pq
+ (h(9)OS+h(lO)OS+2h(12)OS+eh(13)OS+h(15)OS)J.
pq
pq
pq
pq
Now,
and on the other hand
= rZq
Z n
0,1
oq
c: O'.qs-1q
1 n
= r treNt As N)
Thus one gets the relation
= rAm v
s s
pq
pq
•
I
I
I
I
I
I
I
til
I
I
I
I
I
I
J
28
I
I
I
..I
I
I
I
I
I
I
--I
I
I
I
I
I
..I
I
Since
= E(En
ap
p
) (E n G: 6 n ) + E (E nNn(7~ n Q ) ( E n ,,6 n )
)',6 i'P )'8 6q
pk Q,t3 ~l:'
IJq )',6 i'P)'8 eq
= kE~(l)88
p
pp
+ E
p~
(~(7)OO~(8)OO)(~(7)8S~(8)ss),
pq
pq
pq
pq
and
8
E (E nNnu.'NnOnQ)( E n
p,q Q,t3 ~l:' ~l:' IJq )',6 i'P
a6
)'8
n 6 ) . tr(N'A NNtA N)
q
0
S
+
One obtains the relation
~ ~ p8 n v _ (k-2)~ n v •
Qt30t3s
Similarly, the two way calculations of
yields the relation
and consequently
29
S8
2r~
n v •
8 8
I
J
Thus one obtains
(4.6)
_ 2r(k-l)
This can be rearranged as follows:
Now for cases where k
= 2,3,
we have
2f
o
E
E ~ ~ ~pa n ]
s t erst a
s,t~ ~l
I
I
I
I
I
I
I
til
I
I
I
I
I
I
J
= /),2
0
30
I
I
I
..I
and therefore the last tern in (4.7) vanishes for those cases.
If we consider the limiting process such as
b
I
I
I
I
'I
I
lit
I
I
I
I
I
I
->
CD
fixing v, nl , n2 , P~k as constants, then, since vr=bk and
and at least one ~i are of the same order as b, and
It can be seen that
~~~) ~ bk(k-l) (k-2)
~~~)
; bk(k-l)(k-2)(k-3)
and
ro /~20 <=
Thus
i'le
get
Var
(4.8)
1 •
2(v-l) (1+(; -bl ))
- b2 (k_l)2
e-
as b - >
CD
•
For a BIBD this is
which confirms Ogawa's result [4J, and for a case when ~l=O' ~2 .;0 (h.b)
reduces
Var
_ 2(v-l) (
V-l)
2
2 1 - b(k 1)
,
b (k-l)
-
e-
which can be seen from (4.7).
This confirms P. V. Rae's result [6J •
..I
I
i~l ni~i = r(k-l),r
31
I
§ 5.
The approximate null-distribution of the F after the randomization.
We take the Beta-distribution
v +v
v
-1 _ 1
e2
r( 1 2)
2
v
v
r( ;)r( ;)
as an approximation to the permutation distribution of
zation under the uniformity conditions (4.1).
e due
to the randomi-
... Var
e.
Then
= ",,(e),
Hence we get
=
v
2[f,(e)_,~2(e)_
Var
1
Var e] " (e)
e
c
Notice that
_
_
_ ~1-~2
~(e) - ~o ~2 bk{k-l)
~(l)ll...
E
P
pp
v-I
b{k-l}
and hence
1 - F', ()
e ...
n-b-v+l
b {k-l} •
I f we put
V
l
=~(v-l)
and
v
2
= ~(n-b-v+l),
then
Var
e
J
I
I
I
I
I
I
I
til
I
I
I
I
I
I
J
32
I
I
I
..I
I
I
I
I
I
I
--I
I
I
I
I
I
..I
I
Therefore, for sufficiently large value of b, the permutation distribution of
e
due to the randomization can be approximated by the Beta-distribu-
tion
n-b-v+1
- 1
(l-e)
2
~-:::--
under the uniformity condition (4.1).
Consequently the null-distribution of
the statistic F after the randomization can be approximated by the familiar
central F-distribution
v-1
n-b
V - 1 1 )- 2
v-l
)2
1
(1
+
b
(
F
n-b-v+1 F
n- -v+
d(
which is obtained by averaging (2.14) with respect to
v-1
n-b-v+1
e having
F)
'
the density
given by (5.6).
References
[lJ
Bose, R. C. (1949) Least square aspects of analysis of variance.
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