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ON THE NON-NULL DISTRIBUTION OF THE F-STATISTIC FOR TESTING
A PARTIALNULL-tcrPOTHESIS IN A RANDOMIZED PBIB DESIGN WITH
m A,sSOCLI\TE CLASSES UNDER THE J:)lEYl1A!:T J.lODEL
by
Sadao Ikeda
University of
and
N~rth
Institute::Jf Statistics M.i.meo Series No. 1.:-66
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Junjiro Ogavra
Nihon University
Carolina
March 1966
Contents
Summary
1.
Introduction
2.
The non-null distribution
~f
the F-statistic before the
randomization
,.
Asymptotic behavior of the pernmtation distributions of
(S,
'i1,~)
and
(g,,,)
due to the randomization
4. A probability distribution which is
valent
(3)
aS~l~totically equi-
to the non-null distribution of the F-stat-
istic after the randomization
Acknowledgements
Reference,s
This research 'Has supported by the Mathematics Division of
the Air Force Office of Scientific Research Contract No.
AF-AFOSR-76o-65
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLTIITA
Chapel Hill, N. C.
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1:... Introduction
In this section vTe S11all give a brief sketch of the problem which is
treated in the, present article.
!he notations and
as those of the preceding papers [1], [2], unless
ter~nologies
other1~.se
are the same
stated.
As for
the noticn:ls of asymptotic equivalence, references should be made to [3], [2~].
As in [1], ire are concerned with a PBIB design vnth
.
i~1ich
ill
associate classes,
has v treatments 1dth the association, b blocks of size keach, r
replications of each treatment, and the number of incidence of any pairs of
treatments,
Au,
if they are u-th associates.
Let us take a special numbering of the whole n(= vr = bk) plots such
that the i-th plot of the p-th block receives the number f = (p-l)k + i,
and this numbering is fixed throughout the present paper.
Let
~
and '" be the incidence matrices of treatments and blocks respec-
tively, and put B = """, and N = ~'''', N being the incidence matrix of the
design.
and ~'= (~l' ••• , 1\) be the treatmentl
effect vector and the block-effect vector satisfying the restrictions
Further let
or' = ('t , ••• , 'tv)
v
I
or
ex
=0
I
and
p
=0
p=l
ex=l
respectively, and let
~
~'= (~l'
••• , ~n)
be the plot-effect vector subject
to the restraints
k
I
p
= 1,
i=l
1
••• , b
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S2:nmary
This article treats the problem of finding a probability distribution
\:11:1. ell
is asyrrq;>totically cquivalent in the sense of type (8) under some limit-
tiES conelitions to the n:m-null distribution of tIle F-statistic for testing
a 'partial' null-hypothesis ( and hence, the 'total' null-hypothesis in a
spodal case) ina randor:uzccl. partially balanced. incomplete blocl~ design lr.ttll
~1(2
1) associate classes under the Neyman model with both unit and technical
errors.
~le
result is
satisfacto~J
to some extent.
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T11e Neyman model asslrrJdng the existence of plot-effects which have no
interaction with treatments is given by
(1.1)
= '11
x
c1l1"
+ 'ltl3 +
+ e,
1(
••• , xn ) is the observation vector, 'lis the general mean,
l' = (1, "" 1), and e' = (e l , ••• , en) stands for the technical errors,
vnlere x'
=
+
(~,
which is assumed to be distributed according to
We consider the
(1.3)
(1.2)
u
l/11ere
h
= 1,
•• "
with unknovm
11,
is a positi ve integer not greater than m.
H'
o'
1"
= In.
TU.s is· called the 'partial'
tlns reduces to the 'total' nUll-hypothesis
= 0.
To test the nu11-11ypothesis He(h)' ire take the F-statistic
(1.4.)
F
n-b-v+l
=;;.;....,;.;.....,;~
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n
null-l~othesis
Ie
.-
I)
2
null-11;YI>othesis [lJ, and \'1'11en 11
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r:;2
r:; •
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N(O,
and
= x'
(1. 5)
82
e
= x '(I
1 B - -1:
"i-There
V#
U
= (I - 1.l\. B)<l)(cUA#)<l)
, (I - 1. B)
u
k'
2
u
= 1,
••• , m
c
u
k
= --:--rl\: - p
u
Here, A# = ...l:.... G , A#l' ••• ,
o
v v
respective racl\:s
= 1,
1;1
,
A~m
0 o (=1),01.~ ,
I
I
••• , m.
are mutually orthogonal ia.ll1potents, "!,'lith
0m
, and p (=rk),P
, "', P are the characterol m
••• ,
0,
, ••• , 0,
o a..
-l
m
isti.c roots of N N' with respective multiplicities
it
for "Thicll
Imown that
~_s
TO.
ou
and
= v,
u=o
u=o
N N'
=
I
p
u
Au# .
u=o
In the special case i'1'11en 11
= lU, F
in (1. 4) reduces to the usual F-statis-
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tic
(loG)
F
"
= n-b-v+l
v-l
"ri tIl
evI
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m
s~ = x'(
I
v!,:>
x
u
u=l
(1.7)
111
s2 = x' (I - .1 B e
k
\'
L
V#u)x,
u=l
"!,~1ich
is used to test the total null-hypothesis (1.3).
NovG if the existence of the plot-effects is not assumed, it is
kno~m
that the non-null distribuJcion of tl1e F-statistic given by (1. 4) under the
Neyr;~1
model (1.1) "!,'lith
of freedom
(a,
n-b-v+l)
~
=0
is the non-central F-distribution of degrees
with non-centrality parameter
3
"if/;',
"There
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8.
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11
T=(
(1.0)
,-J
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.I
A~
l' ) 1 (1'
I ...
~ N N' )
(
u=l
Hence the probability element of the
if
is given by
00
T
exp(- 2
2CT
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)
I
t=o
ex
_ -2 +
ex
t -1
ex , F7)
(J- + n-b-v+l
(n-b-v+l F)
N(
ex
F-)'
"'" n-b-v+l' ' )
a = v-l..a •
'\filel'e I'Te have put
In the case w11en
11::: n) this reduces to
r(n-b + t)
00
(1.10)
_(n-b-O: + t)
2
T
exp(- - )
I
2
2;' t=o
v-1
+ •
v-l
F)2
n-b-v+l
J(/
(
-
1
_(_n-b _'.• )
(1 +
J(/
v-I
F) 2
n-b-v+l
"(
0,
v-1
n-b-v+l
F)
)
I'T11e1'e
(1.u)
T
= 1'1(1'
I -lHH')-r.
k
This gives the p1'obabilit;)r element of t11e nan-null distribution of the F-statistic
given in (1. 6) under the Ney:man model (1.1) vTitl1
H::nlever) once
'ITe
1C
= O.
assv:oe 'Glle existence of the plot-effects) t11ese are no
longer true) and) as :i.8 shOI-n1 in
tl~
follovTing section) the non-null distributi.::m
of t~le
if or F is a non-central F-distribution, '<Those n::>u-centrality parameter
depends on bothT
and~.
It is interesting to investiGate 'Dlether the influ-
ence of ,r::m tIle non-null c_i_s"cribution of the F or F can be eliminated under
certain 1irnting conditio11s
l)~r
yielding a randomization procedure in tl'e a110-
ca:c,iJl1 of treatments to tIle pl::Jts witIlin each block, as i t
case (2J.
'I'TaS
so in tIle null
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8.
5
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II
In this section, l1e cic:;::.ve the non-null a.istribution of the
al1.c~
(1.5),
and of the F given by (1.6) witll (1.7), before '\;lle randomization.
11
('12
0
t
(11) = x'(
I
vI)
u
x
u=l
11
=
I
(,,1 + q)-r
-I-
W -I-
1f
-I- e)'(
11
= (llll+:rc+e)'(
I
v#u )(\il -r
-I-
n + e)
u=l
II
= (lll-r
-I-
:rc)'(
I
v~
)(q)-r -I- n) -I-2(1ll-r
u=l
ane:. -::'11e ran1: of
~.
u=l
V#
is
a,
u=l
u=l
the non-null distribut:i.onoi' the variate
u
(2.1)
bei'·::ll'e tIle randomization is the non-central chi-square o.istribution of degrees
of f:teeclon
Ci
""itll the non-centrality parameter
51/ri,
Hhose probability eler:1ent
being given by
(2.2)
e:z:p(-
-2
°
2
1
2cr
eAl? (- -
r( ~. +
,.
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u=l
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b~r (1. i-j.)
S:i.nce
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if given
6
1.1 )
Xl
2
-2
Xl
) d (- )
2'
I
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-,
h
\' V# )(~h
L
u
u=l
+ 1t).
Hence, in the case vn1en 11 = m, the non-null distribution of the variate
(2
. • ,.~
1 )
is
'G~le
2 _ s2/
2
t (j
Xl -
nan-central chi-square distribution of degrees of freedom v-l with non-
centrality parameter
0/
2
(j
,
Where
&1 =
(~'r
+ 1t)I(
I
v~ )(~'r
+ 1t).
u=l
Similarly we obtain
S2 =
e .
Xl
I
I
I
m
(I - - 1 B
k
-I
u=l
ill
= (~'r
+ 1t)'(I -
~
B-
I
vt)(~'r
+ 1t)
u=l
m
1
+ 2(~'r + 1t)'(I - ~ B
-I
V~) e
+ e I (1 -
~
m
B -
u=l
1
m
#:
- E V is an idempotent of rank m-b-V+l.
u=l u
non-mull distribution of the variate
vn1ere the matrix I -
~
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_.
ill
(2.6)
I
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v~)e,
u=l
Hence
tl~
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-,
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,,,
(2.7)
bej:::ll"C the ran(bmizati:::>n :i_s tIle non-central chi-square dist.ribution of degrees
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with non-centrality parameter 02/rr
of :;:'1'coo.:::>l:", n-b-v+l
°2
be:LnG
(-2) v -
00
°2
eA'}J( - - )
(2.8)
2i
I
2rr
vI
11=0
2 n-b-v+l
+ v - 1
X
2
(-2)
2
(n-b-v+l
r
2
+ v)
2
e:::p (-
~)
2
d
~
(2)'
02
= «h"
+ 1t) f (1
-~;
m
I V~) «b'r
B -
+ 1t).
u=l
~,as
This is seen to be independent of
".
. t es
·
.S lnce
ene varla
-2
Xl
X:22
anCl"
is
sho~n
later.
are mutually indepenc.ent in the stochastic
sense before the randomization, the non-null distribution of the
F statistic
gj.vel1 by (J.)~) with (1. 5) bef8re the rand·::nuizat:bn is a n;:ln-central F-distr::tbut7.<J).1
n-b-v+l)
'ir1108e probabilH~1 elc;:lent being given by
6
(;;;)1
(2.l0)
IJ.
°2
v
(;))
v:
IJ.:
1"(
n-b-CX
2
+ IJ. + v )
r(~ + lJ.)r(n-b-v+l + v)
2
2
)
n-b-Ci
- ( -+IJ.+v
0:
2
0:
+ n- 1)-V-f-L.. F)
0:( n ..·o-;r-I~_L
,
I "
(I
1:1
(2.11)
I
probability element
,-r11e1"e
I
.e
~Ll:)se
,6
1
= 01
-
~\
:::
(~~
.,- 1l)
I
V~) (¢J~
u=h+l
s
+ :It),
F).
the
ab~ve
I
I
can be rewritten as
"I
L
1J+v+r=J
(2. J2)
r(n-~-<i
+IJ + v)
•_.-;;;;...
(
r(..E... + lJ)r(n-b-v+l + v)
2
ex
n-b-v+l
]f)
~IJ-l
(] +
-
ex]i)
_(n-~-allJ+V)
,(
n-b-v+l
c.. n-b-v+l _'
2
(2.13 )
In the case
11
= 1n,
putting
the n:m-null distribution:):f -:.;11e F-statistic given by (1. G) 1'li tl1 (1. 7) before
t~le
rando::11ization is a non-central F-distribution of degrees of freedom
(v - 1,
~
L
(2.15)
1
(;;)
v r(n;b + I)
11)
I:
1=0
•
Since the quantities
of treatments,
ments to
or
(g, 11)
~,
r(V-l + lJ)r(n-b-v+l + v)
2
(n-~=;+l F)
(S,
~,~)
V-l'IJ_ 1
2
or
(1 +
(~, 11)
n-~=;+l
2
_(n;b l/ )
F)
d(n_b~~~J. F)
contain the incidence matrix
if we a&opt a randomization procedure in allocating
1>: plots 'VTithin each block, then
~, and hence tIle quantities
k
tr~at-
(g,
Ti,
becol~e the random variables under the permut~tion distribution due
9
.-I
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n - b- v + 1), whose probability element bej.ng given by
~
ex
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F")
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~)
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••I
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to -clle rancLomization
adoptc(~•
TIle null-distribution of the F-statistic, F
F, after tbe rando:m,izat:L::m
can beob:bcdnec1 b~r takinG c:;:pcctation of tl1e probabilit:,/ clcnent (2.12) or
(2. ~5) ITltll- respect to the pcr:-,lUte/cion distributi::m ·of
d:~1.C
to the
anc1 so
11C
l~andor:Ji.zation.
(g,
'Ti,~)
or
(s, 1))
For this, exact calculations 8.,1'e d.ifficult to CLa
shall seek for anoUler probability distribution 1;11ic11 isasym;ptotico..Hy
eQ.u:L voJ.cnt to tIle non-null
c~_:'_ stribution
of the F after tllc randomization in
tIle Gcnsc:)f type (S) Unclcl~ certain limiting conditi::ms, iThich is done in tIle
fa :L~..:)\iing two sections.
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or
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3.
AS~lWt?tic behavior of the permutation distribution~ of (~, ~, ~) and (S,~)
due to the randomization
The purpose of this section is to derive the aSYdwtotic distributions of
(S, Ti,~)
the nargina1s of
(~, ~)
or of
under a certain limiting process,
ano. give some results which are useful in the next section.
Since
01
= (~T) I(
h
I
V~)«('Jn)
+
2(~T) I(
u=l
h
I
11
v~)re
+
re
u=l
l
(
I
v#)re
u
u=l
'
ancL
(~T) 1(
h
I
V~)(~1.")
=
1."1~1 (I - ~ B)~(
u=l
h
I
C
u
A~)~I (I
~ B)~T
-
u-1
1
~... N N1)( L
\'
(r I -
c
A#)(r I -
uu
1.l~
N N1) T
u=l
h
m
=T
I
(
L
(1' -
f
"..
P )A#)( \' C
U
u
u=l
L
U
A#)( r I U
u=l
A#)(1' I u
u=l
h
=
(L A~
.1:... )\T
l~
NI) T
h
1")1(1' I -
~)\T)\T')(L A~
u=l
u=l
K
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T
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I,
h
= 1." 1( L
\'
f. N )\T
-.
,.
h
=T
I
I
T),
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11
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••I
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I
= -r' (r
~_"
I -
11
iT N')( \
L
C
u
u=l
11
= -r' ( \
A#)W'rr.
L
u
u=l
11
=(
I
A~
1:)'W'rr.,
u=l
'we get
(3.1)
~.
h
=
T + 2(
h
I
A: -r) 'W'1t +1t
u=l
I
1
(
v~
)1t,
u=l
and sim:L1ar 1y
L A~
ill
(3.2)
61 = T + 2 (
-r)' ~ '1t + 1t
m
I
I (
u=h+l
V~) 1t)
u=h+1
'i,here "lTe have put
h
(3.3)
T
=
CI A~
-r)'(:c I -
~ N N')(
u=1
h
I A~
1:)
u=l
and
ill
T=(\
L,
u=h+l
m
A#1:)'(:cI-~NN')(
\
u
L
A#u1:),
.l\
u=h+1
f::>r Ifi1ic'J.l it 11010.8 t11at
.1
I
A#) , ' 1t
u
12
(3.5)
T+
I
I
T = T,
T and T being the same as th0se given by (1.8) and (1.11).
F1'am (2.1) and (2.2), :Lt follaevrs that
ill
(3.6)
51 = 51 + 81 = T -:- 2 T'
~'
L v~)rr.
rr + rr'(
u=l
The quantity 52 is easi1jr calculated as
~ B)T+ 2T'~frr + rr'rr - 51'
= T' (1' I -
~.
52 = 6. - rr'(
I
v~)rr,
u=l
~n1ere
6. = rr'rr.
This is independent of the values of T, vn1ich means that the
~ = s;/ (j2 given by (2.7) before the randomization
distribution of the variate
is independent of the hypotheses.
From (2.13), (2.14), (3.1), (3.2), and (3.7), ive have
S = 6. +
T + 2T'~
-1
rr,
L v~)rr)/(6.
ill
11 = (T +
2T'~'rr
+ rr' (
+ T +
2T'~frr)
u=l
(3.8)
11
11
= (T
+ 2(
LA~
T)
'~frr
h
+ rr' (
u=l
-11
=:
(T + 2(
I
v~)rr)/(6.
+ T
u=l
! A~T)'~' ~ ~'( ! V~)~)/(t'.
+
u=h+l
u=h+1
13
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,.
and U1e1'efa1'e
m
-.
+T
I
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I.
1.1
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NOH, the randomizati::)11 ad.optec. is represented. by tile
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u(j
,
=
o
is a
x }c
J.-~
per:,·'1.utation matrix corresponding tJ t;,he randomization
For
(3.10 )
U(jB
Let
=B
-c}.l:'cS
:matrix it is clear that
U(j :::: B.
be an arbitl'I1l7 but f'ixed incid.ence f:latrix
<I>
any :)·c.lwr incidence matrix ::J; tile treatments is given by
I
~i
treatments.
<r? (j
= U'( j~'
T:i1en
Since
n H'
is inva:cie..n t uncier this transf'::JJ.'nation, i. e.,
(N N') (j
the
\~cla.nt:tties
buti::m of'
~
HOi-i,
(j
= <fl'(j t
1jr'<!>::::
(j
<fl't
t'~:::: N N',
T, T and T a.re ::'°ega.rcLed as constants unc',er the permutation distri-
•
let us put
11
(3.n)
X
= 1"
~'1l
(j ,
Xl
=
(L
:m
A#'r)'~
'11
u
(j'
v
-"-2
and
t'=
1C'
(1
=
(L
A#
U
u=11+1
u=l
(3. J2)
.'
n x n
o
m
I
:::~llowing
... f;~~ B)~ ( L
\
(j
u=l
and consio.er the follm'ling liniting process:
14
-r)'~(j ,~
(3.13)
b.... oo
,
I
I
'.I
i
keeping v, l:, n , ••. , n , Pjl\: fixed.
1
m
FurJcJ.ler:i'lore, the fo110"ring c:mditions are assumed to be satisfied:
b
and
1
\' (
L
b
-)2
bop - bo
.... 0
p=l
unc.. . c:i: tIle limiting (3. 13), ~.fj.lel'e bo 's are the same as in [1], [2].
P
IJe has been Imovm [2J that the permutatian distribut:tan. of the variate
~v~'zes ~n the sense of ~~~ (s) to the central chi"8~l~~~e~istributionof
degrees of freedom v~l, as b-r~~ovides that the coqditions ~3.14) are ~atisfiec1..
Let us consider the as~rnptJtic distributions of Xl' ~ and X given by
(3.1J.),
for "i·rll:!. cll X = Xl + X •
2
Since
E stands for the expectation with respect to the
pe~~tation
distribution
unc1el' consid.eration, it is evident that
(3.1S)
It its als·:) clear that tIle cava::ciance of Xl and X
t~le':::atrices
of
;;
( ./
~ u=l
h jj=
u
L.J
D..JJC'
".
~
m A#
u=h+l u·
2
vanislles from the
orthagona1it~r
Hence
1 ,.. )
•
-<-
I
T:Lle variance Val' (Xl) 'Irith respect to the permutation distribution is ca1ct'.lo:l:.ecl as f0110v18:
15
,.
I
and
i~1ere
I
I
I
I
I
r
I
I
I
I
I
..
I
I
I
.e
I
I
I
I
I
I
-.
I
I
I
I
I
I
lI
I
11
Var (X))
= E(-r'
I A~t ~~
1('
11
I
1(1~cr
u=l
A~ -r
)
u=l
h
Lu -r)'~'E[U
=( \' A#
1(.1('
U
cr
cr
,]~(
u=l
h
\ ' A#-r)
Lv..
u=l
~.
;::-1
....
~
(I
1 G)
1-\. - -;:....\.
.1\.
o
(). J.G)
o
1
k-l
p( 1",
~,
~.
T -(- p(-r,
t\),
~, .. "
h
t1) ::
, .. ,
.k:1
(I
A!1")
I
~'
~-z
-=:(I,. - ~
] - ,.n.
.)
1.,-_ 1
i..",,-
o
o
vTe l1ave
)var
(x) =
(vax
(X2)
lL
=
k:i
AT
+ p( T,
is T + p( T,
16
"1' ... , ~)
'\'
•••
).
•
u=l
Li~:e;'Tise,
.J.\,
,~)
~-6.
-1~-1
I
I
-.
1
(T.
- -it G1: )
1~
o
o
t\-6.
1
(~,
- -k GLJ
""
--;:::L ].
anC.
-6
k"='l"
A.
L
... ,
(3.22)
(T_
x~ -
2. G )
k
E
o
m
~(
'\
L
A#-r).
u
u=:~+J..
o
t\-6
- k-1 (1_~
1'>. k
G,J
J ..
From the practical po:Ln"Gof vim." there is no harn in assuming that there
e:::i.st constants TO' To and To such tllat
-
TO
1-
= Um b
T,
b -+00
f,ol'"
1hlJ.CJ1,
=o
T
= l ':LIn b1
T=
an d T0
= l'
b1 T '
].111
b.-+oo
b-+oo
= TO
of course, it lwlc.s that To
+ To and these values are all posi tj.ve.
Since
o
0
t\ -6.
k:I(Ik . nip - n
b
=
k:l ~
(6. -6)
p
-n
lp
n
~l{)
lP
2p
p=
-n Ip n vp
17
n2p
- n Ip
11
n
-
11
2p
.........
11
2p
11vp
vp
vp
,
I
I
I
I
I
I
.I
I
I
I
I
I
~
I
I
I
••
I
I
I
I
I
I
it :Ls casi{ tJ see tlmt, unCeJ." t11e cJnditions (3.l~.),
~ i(1", t\, ... , ~)
lim
b~co
i
.
•
lim~ p(1",
0,
b~QO
t\' ... ~)
p( 't",
J
:::
t\, ... ,t1) : :
0
and
O.
He;.lcc lie :Jbtain
Val' (Xl)
1 .
~.lm
T
!:::.
Val' (X )
:)
'"
=
b
b~co
J
1
.1:..--.
1,.
2
lim
b
b~oo
N:m, 'lIe sl1all sI1J"i'! the ["sJrtlpt:)tic normality
!:::.:)
T
= .t-l
Jf
:)
Vax (X)
c'J1d lim
b~co
\'
11 A#
L
11
these l'o,ndom variables.
>
1"= T = [, -•
1..1=1
'
]
!-
1"V
TJ.le:l Xl is the sum of b inclependent random variables:
b
k
v
= \' ( \' \' :rt-(~),/:P~ )
\'T11ere
~ (~)
aJ.
1
L
L L
p=l
i=l a=l
a~a1
0"(1)
J
are tJ.1e elements ·:)f the incidence matrix of' treatments J
S
k
v
(L I
1"
a
,.53.).
~""'"'
2
rc (p) ) , <
0"(1)
-
1=1 a=L
v
< 1:
L~
!:::.pJ
a=l
18
~.
~To
=~
b
F :)1" tlli S J put
-.
I
I
I
I
I
~
=:
Since
.
I
I
-.
and, under the conditions (3 .11~),
b
\
1
(b -+10),
TL
p=l
fr::n (3.24) it follows tJ~~t
b
k'
[p~ E<li~
\
L
1'ex~) "~il)llr
v
var(\ ,\
b
[
v
k
p=l
L L
-:r ~(~) 1C(p~
a en
1
)
l2
0-(2).,
i=l a:l
under the conditions (3.V~).
This ShOWB that Liapounaff's condition is satisfied.
Hence from (3.16) and (3.21-1-) it follows that, ~ .t)1e condition~~:I:.ltl,
th.~J2.9r,ml1tation distributioll.2!
x/..'/l:l.o To b/ (k-l) ~~~es in
la.,:tv.J and henc.e,
ill.,th~ sense of type (3), to the standard mormal N~J.1.L as b-+ 10.
under 'bhe _c.Q!lQ.j. tions (3 .l~'),,--ih.e ]?ermutation distribution of
Sirni1~E1:~>
x2 /.Jl:l. o T b/ (k-l)
all..o....,gJ:_x/..Jl:l. To b/ (1-'-1) .£.9..11Y~F£.e~~_in the sense of typ~_~~l to t11e standard normal
o
N(O, 1).
!rhus, from (3.8) we can easily obtain the follow:i.ng
Lermaa 1
(a)
Under the conditions (3.14),
the permutation distribution of S/(l:l.+T) converGes in probability to
unity as b -+00,
(b)
the permutation distribution of ~/(T/(l:l.+T» converges in probability to
unity as b -+00 ,
(c)
the permutati::m c",istributi::m of ~/ (T/ (l:l.+T»
c,onv'erges in probability to
unity as b-+ 00 ,
(d)
the permutation distribution of ~/(T/(l:l.+T» converges in probability to
unity as b -+ 110 , anc1.
19
I
I
I
I
I
I
.I
I
I
I
I
I
~
I
I
I
••
I
__...--;.TIL-- )) or of ( LS ~ T )
fi/(6+T)
converGes in pro~_m:):n:Lty to (1,1,1) or to {:1_,:1.} ::-cspectively as~)
I
I
I
I
I
-.
I
I
I
I
I
I
l.I
I
20
0.+00
•
~A probability distribution ~~h is asymptotically e~uivalent
I
I
(s) to the
nan-null distribution
tIle F-statistic after the ranc.o:;.Jization
-<
. of ---'---'-'-"-';"-'
In this section, ''7e 6e1°ive a probability distributi::m 1<lhicl1 is asymptaticalJ.~r
to the n::m-null distribution of tIle F-statistic,
eClui valent
uncler 'c11e conditions
F or
-.
F,
(3.14).
AS';Ias L1entioned in the J_ast of Section 2, the n::m-null d.istribution of
F after
the randornzation is .:)iJ"cained. by integrating out tI1e quantities £,
ii
and ~ ii''!. (2.12) '"ith respect t.:) their permutation distribution due to randomization,for \<rllicll theorem 3.2 or Corollary 3.2 of [4] and Leal~la 1 of the prece,ling
section 'can be applied t:"' obtain a probability distribution 1fllich is
equ:ixalent
(s)
to the non-n1111 distribution of
bution 1'711ose probability e1enent being
l
00
(6. + T)
exp(-
)
\
L
l=O
2(l2
l~
F,
asymptoticall~{
i.e., a probabiiity distri-
I
I
I
I
I
I
I
Il+v+r=l
(h.l)
=
I
I
n-b-a
)
-( 2
+Il +v
+ n-b-v+l
a
F)
I
+ v)
d
is asY,;l]?totically equivalent :i.n the sense of type
but:L:>n of' F
after the ranclo:,:tLzation, as
b-+ 00,
a
(n-b-v+l
(3)
-)
F
'
t.:) the non-null distri-
prcvldec~.
tIlat the conditions
(3. lJ:.) are satisfied.
It is easily C11ecll:ed that U~.l) gives a probability ele::,lent of some
probab:L1it~l
o.istribution.
Q;u.ite analogously, a probability distribution, uhicl1 is asymptotically
equivalent in the sense of t~~e
(s)
to the non-null distribution of
21
F after
I
I
I
I
J.
I
I
I
.e
I
I
I
I
I
I
-.
I
I
I
I
I
I
~
I
the randc:mrl.zation, is gi ven
b~/
(6.
co
6. + T \ '
e2q) (-
/;
-I-
T)
2 2
/;f
2) LJ
2~ /;=0
\'
L~:
~+v=/;
/;!
v!
(4.2)
v-l
~ _ 1
_(n;b + /;)
r(n;b + /;)
(
v-l
)2-1v-l F)
( ,-'. + n-b-v+l
- - - - - - - - - - - n-b-v+l F
r(v-l + ~)r(n:1?-v+l + v)
2
2
d(
V-l-F)
n-b-v+l ' ,
under tl1e concHtions (3.14).
These are the results
irill
IJC
He '\rantec~
to derive, for
,;'"hic~.1
s::me discussions
given beloi1.
(i)
TIle results s11O\;'" tlle/c, as far as ,;'"e use these r)}~obability distributions
as the asyc:q;Jtotic distributi::ms ·af
iii"
and of F, t11e influence Df the existence
of pbt-effects on the non-null distributions of these F-statj.stics can not be
elem.inated
b~r
the randomization pr.ocedure adopted, even if
t~le
strang limiting
condit:lons (3.14) are assurneC. tD be satisfied.
(li)
In the case ,-rhen 6.
= 0,
(2~.2) gives the same pr.abability element as
that eiven by (1.10), i.e., as that of the non-null distribution of the F-statistic
under t1le usual Neyman model (11ithout plot-effects).
LikewJ.se, if we. put 6.
=:
0
in (2~.l), tile resulting probabiHty element
co
eX];)
(
-
2T2 )
co
L
-T 11
-T v
(~) (1 - ~) •
I
/;=0
••
1
'-, )
( Lfr . _)
-
r(n-b-P I /;)
_=2
.1L +
.,,--__ (
a ) (n-b-v+l
v)
r (2 +11 r
2
+
ex
n-b-v+l
22
F)
2
J
..
( '.,- -l.
_
--
ex
n-b.-v+1
.
-F)
(!2.:E.- a /;)
. 2
'( . b.~ +1 -F) '
n- -v
(.1
gives the probability
of the exact nan-null distribution of F given by
ele~ent
w'itIl ( 1. 5) under the usua 1 Ne;:,rraan made 1.
... )
( ~ll
There may be
atT1el~
equivalent in the sense of tY?c
One of them ,'Till be deri vecl
b~r
I
I
(1.4).
I
I
probability distributions i'lhich are asymptotically
(3)
to the nan-null distribution of
F or
of
F.
investigating the conditional distribution af
or of 71 given S = 6. + T and. applying theorem 3. 1 of [li-].
var:table of 71, for example, given S = 6. +
r,
II
(~:,~:
In thi s case, tIle cane..:.t .. -::mal
becomes
m
(7) Is
= 6.+ r) =
(T +
;!, (
I
V~)1()/(6.
+ T).
u=l
~ ViI);! are not mutually independent in the stochastic
u=1 u
sense lJ.nc1er the permutation cl.istribution due to the rand.onization in general, the
But, since -r'(b'1( and 1(1(
,_ ,
TIl
#
condit:i,:mal distribution lof ::..::.::.::::. 1(1( I: V)1( given -r'If)'1( = 0 is no longer asymptotically
6.0
u=l u
approxi:r:-iated (S) by the central clli-square distribution of degrees of freedom
v-I, but by certain weighted chi-square distribution whose weighing coefficients
(epencls an the matrix C given in tIle section 2 of the
preV:~'Jus
paper [2].
This
.-
causes a difficulty in pr~tical application, because it is difficult to express the
elements of the matrix C by the parameters of association and the design uncler
consideration.
In cOlli1ection with this, it should be noted that, in order to derive tIle
results of this section by using Theorem 3.2 of
[4], it is not necessary to state
thc asyr:lptotic norw.aUty of -;;11e variables Xl' X or X given by (3.11), but ,nerely
2
to obtain the variances of these variables.
However, if we intend to work out another
aSYI:lJ?totic distribution of tl'le .:.1on-null distribution':lf
F
OJ:
I
I
I
I
II
F by the meth:)d stated
ab:>ve, the asymptotic n:)rJ1W,J.it;:,r of Xl' X or X will be needed.
2
23
I
I
I
I
I
~
I
I
I
.e
The authors want to e::prcss their gratitude to P:toi'css:n' N. L. Jol1nson
f:J:' ·il:Ls encouragements.
T·jlOJ.1.1;:s
are also due to Mrs. KaJr Herring for her nice
I
I
I
I
I
I
-.
I
I
I
I
I
I
24
I
I
I
References
[lJ
[2J
J. Ogavla, S. Ikeda and. H. Ogasawara, liOn tIle null-6.istribution of the
F-statistic for testing a 'partial' null-hypothesis in a randomized
partially balancef. incomplete blocl\: design >.Tit}l :-il associate classes
under the N(:ytnan :cn0.el. II (35th session of the InternL Stat. Inst.
1965)
S. Il:eda, J. Ogawa an(1. 1,·-1. Ogasa\-Tara, "On the asymptotic distribution of
the F-statistic Ullder the null-hypothesis in a randomized PBIB design
vlith m associate classes under the Neyman madel, " Uni v. of North
Carolina, Inst. of stat. Mimeo Series, No. l~54 (1965)
[)J s. D\:eda, "On certain types of asymptotic equivalence of real probability
distributions, I, Definitions and some of their properties," Univ.
of North Carolina, Inst. of Stat. Mimeo Series, No 455 (1965)
[4J
S. Ikeda, "On certain types of asymptotic equivalence of real probability
distributions, II, Further results on the prJperties of type (5)
asymptotic eqc:ivalence in the casebf equal basic spaces, II Univ. of
North Carolina, Ins"(;. of Stat. Mimeo Series, ITJ. 1+65. (1966)
-.
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~
25
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