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Work supported by the National Institute of Health, Public Health
Service, Grant No. GM-12868-04.
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ON SOME SELECTION PBOCEDU.RES IN TWO-WAY LAYOUTS
by
PRANAB KUMAR SEN AND MADAN LAL PURl
University of North Carolina, Chapel Hill
and Indiana University, Bloomington
Institute of Statistics Eimeo Series No. 630
June, 1969
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ON SOME SELECTION PROCEDURES IN TWO-WAY LAYOUTS*
BY PRANAB KUMAR SEN AND MADAN LAL PURl
University of North Carolina, Chapel Hill
and Indiana University, Bloomington
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~.
In the one-way layout problem, Puri and Puri (1969) have considered some
selection procedures which are nonparametric analogues of some earlier procedures
by Bechhofer (1954), Gupta and Sobel (1958), and Paulson (1952), among others.
the present paper, some of these procedures are extended to the two-way layout
problem in the parametric as well as nonparametric setup.
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Allied efficiency results
are also studied.
1.
For a two-factor complete block design with one observation per
~.
cell, we express the observable
(1.1)
where
X.
~QI
~
= ~
.
+
~
QI
is the mean-effect,
+
'J".
~
~andom
+ s.
w
~l""'~n
variables XiQl (i=l, ""
c
, L:
i=l
1". =
~
c; QI=l, ••• , n) as
0 ,
are the block effects (nuisance parameters
fer the fixed effects model or random variables for the mixed effects.mode1),
~l""'~c
I
In
are the treatment effects, and the siQI are the error components.
assumed that
S
'"'-Q'
= (e
1QI
, ••• ,e
CQl
It is
), QI=l, ••. ,n are independent and identically distrib-
uted stochastic vectors with a continuous cumulative distribution function (cdf)
F~),~ER
c
(the real c space), where
F~)
is symmetric in its c arguments, that is,
for any ~ERc and any permutation (i , ••• ,i ) of (l, •.• ,c)
1
c
(1.2)
F (e 1 ' ••• , e ) = F (e. , ... ,e. ) .
.
c
~1
~c
[Note that if all the nc errors are independent and identically distributed, (1.2)
holds, but the converse is not necessarily true.]
Our purpose is to study some
(parametric as well as nonparametric) multiple decision procedures for the following
three problems:
(i) selection of the best t treatments without regard to order,
(ii) selection of the best t treatments with regard to order, and (iii) selection
*
Work supported by the National Institute of Health, Public Health Service,
Grant No. GM-12868-04.
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of all the treatments which are as good as or better than a standard treatment.
The bestness of the treatments is judged by the largeness of the
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~
In the parametric case, Bechhofer (1954) has studied the first problem under
the assumption that the errors are independent and normally distributed.
It is
shown here that if the errors are jointly (within each block) normally distributed
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~.,
and are equally correlated then his procedure remains valid.
tion (1.2) which may arise often in mixed effects model.
This covers the situa-
It is also shown that if
(1.2) holds and F admits of the existence of second order moments, then the Bechhofer
procedure remains valid for large samples, even if F is not normal.
In the nonparametric setup, Puri and Puri (1969) have considered these selection procedures for the
on~-way
layout problems, and their procedures are based on
a class of rank order statistics.
the rank order estimators of
Here, instead of these statistics, we consider
{~.}
by Puri and Sen (1967) to provide asymptotically
~
distribution-free selection procedures for the
two-wa~
layout problems.
The asymp-
totic relative performances and efficiencies of the parametric as well as nonparametric procedures are studied.
The cases of paired comparisons designs as well as
one-way layout problems are also briefly presented.
2.
~.
are not known).
Let ~[l] ~ ... ~ ~[c] denote the (actual) ranked ~'s (which
Let then
=
=
-1 n
Z . = X .-X , X .= n
~ X. , i=l, ... ,c; X =c
n~
n~
n
n~
a=l ~a
n
(2.1)
We denote the ordered values of the Zni by Zn[l] <
the statistic associated with
~[i]'
i=l, ..• ,c.
-1 c _
~
i=l
X .
n~
< Zn[c]' and let Zn(i) be
The parametric procedures involve
the statistics Zn[ lJ ' ... 'Zn[ c] .
For the nonparametric procedures, to be considered later on, we let
*~ J ,a=X.W -X.Ja , e ~..J ,a=e.W -e.Ja , a=l, ..• ,n; b. 1...J =~.-~.,
X..
for
~
J
(2.2)
We denote the marginal cdf of e ..
N
~J,~
G(x-b. .. ).
1.J
l~i<j~c
.
*
by G(e), so that the marginal cdf of X..
~J,a
is
It may be noted that by virtue of (1.2), G(e) is symmetric about 0, so
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that the distribution of X..
Also, note that even if the block effects
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about~ ..
1J
~l""'~
for all
and
l~i<j~c
,'(
n
are random, the X..
1J,~
~=l,
... ,n.
are in-
We write X.. =(X *.. l""'X"*
), and consider the
-1J
1J ,
1J , n
dependent of these variables.
statistic
n
hQS~.)=n-lL:E
(2.3)
1J
n
where Z
n,~
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is symmetric
1J,~
=1 if the
~=
1
n,~
Z
n,~
~-th smallest observation among Ix~.
11 , ... ,lx~. I is from a
1J,
1J,n
positive X..
*1J,f.JQ' and Zn,~ =0, otherwise, for
the
~-th
~=l,
... ,n; E
n,~
is the expected value of
smallest observation of a sample of size n from the distribution
~
,'(
(x),
given by
~*(x) = {'l'(x) - 'l'(-x), X:20,
(2.4)
0,
_
X<O.
We assume that
~(x)
satisfies the following assumptions:
Assumption 1.
~(x)
is symmetric about zero, i.e.,
Assumption II.
n
Assumption III.
-1 n
L: [E
~=l
~
*-1
(~/n+l)JZ
+ 'l'(-x) = 1, for all x.
-~
n,~
= 0 (n 2).
p
J(u)='l'-l(u) (0 < u < 1) is absolutely continuous, and
I J(i) (u)1
(2.5)
n,~
-
~(x)
=
I diJ(u)/duil
~ K[u(l-u)J- i - W , i=O,l
where K(>0) is fini te and 6>0.
(It may be noted that the Wilcoxon signed rank statistic and the one sample normal
scores statistic are special cases of (2.2), see for example [5,6J).
Let 1 =(1, ... ,1),
-n
and note that by definition h QS .. -tl ) is non-increasing in t (-00 < t < <Xl).
n 1J -n
in Puri and Sen (1967), we consider as an estimate of
~
As
.. , the statistic
1J
(l~i<j~c)
(2.6)
where
" (n)
.. -tl ) > ~ },
~ij,l = Sup[t: h n (e 1J
-n
n
(2.7)
~ (n)
(2.8)
1J, 2
u ••
=
n
and
~
n
= (2n) -1 L: E
~=l n,~
Inf[t: h (X .. -tl ) < ~ },
n -1J -n
n
'I,
I
= ~E'l'[IXij,~}'
"
(n)_
Conventionally, we let ~ii - ~ii = 0, i=l, ••• ,c.
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Also, as in Puri and Sen (1967), we define
y~n) = c- 1 ~ A~~)
(2.9)
Let then
~
y[(~j
< ... <
Yf~~
statistic associated with
(n)
on y[ 1] ,
3.
. ..
, i=l, ... ,c
~J
j=l
be the ordered values of the
~[i]'
i=l, ... ,c.
y~n),
and let
y~~~
be the
The nonparametric procedures are based
(n)
, Y[c]
.
Exact as well as large sample
arametric) solutions to problem 1.
In the case of
independent and normally distributed errors, Bechhofer (1954) considered the solution based on the order statistics associated with X l""'X .
n
nc
His technique is
not directly applicable when the errors are not independent (as in (1.2)).
For
this reason, we consider the following modification of his procedure:
Extended Bechhofer procedure (to be referred in the sequel as the B*-procedure).
Select the t best treatments which are associated with
(3.1)
Zn[ c-t+l] , •.. ,Zn[ c] .
Our basic problem is to determine the sample size n is such a way that for any pre-
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assigned y (O<y<l), the probability of correct selection of the t best population
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(3.2)
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is
~
y, where ~[c-t] and ~[c-t+l] are subject to the condition that
~
[c-t+l]
-
~
[c-t]
~ ~
~,
, being the smallest worth detecting difference.
to the practical considerations.
Note that the choice of , is left
We denote the condition in (3.2) by a sequence
[, } where m is a positive integer and' ~ 0 as m ~
m
m
sequence of conditions by {(A )}.
m
00,
we denote the corresponding
In the context of one-way layout (which also
extends to two-way layouts for independent errors), Bechhofer (1954) has shown that
for the parent distribution being normal,
(3.3)
P{correct selection of the t best treatments} = y
when the following least favorable configuration holds:
(3.4)
("[ 1] = •• ~ = ~[c-t] = ~[c-t+l]
l ~[c-t+l]
... = ~[c]'
-
,
,
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We denote by
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the configuration in (3.4), and note that if
~
be replaced by
{~m}' the corresponding sequence will be denoted by {L(c,t;C )}.
m
L(c,t;~)
We first show that
is the least favorable configuration for the entire
class of symmetric dependent multinormal distributions.
This basic result will be
used throughout the paper.
Theorem 3.1.
Let (Wl"",W ) have jointly a multinormal distribution with mean vectbr
c
(T[ lJ ' ... 'T[ cJ) and dispersion matrix ~ = cr [(l-lP't! + p2:J, (c? > 0, -1/ (c-l)S:p<l,
2
~
.-
,
is the identity matrix of order c and J = 1 1).
-
-c-c
Then for given y, under (A),
(3.3) holds when (3.4) holds.
Proof.
Denote by
(3.5)
and note that {U .. , lS:iS:c-t, c-t+1S:jS:c} have jointly a (singular) multinormal distri~J
bution with null mean vector and dispersion matrix with the elements
~,
. .' ,
i=i' , J=J
;
1 '
,
i=i', j;':j or iri , j=j ,
0,
i;':i', j;':j' •
l'
(3.6)
Cov (U .. , U.,.,) =
~J
~ J
~
Now, by the procedure in (3.1), the probability of a correct selection is
(3.7)
P{Max[W , .. · ,W _t ] < Min[Wc _t + l ' ... , Wc ]}
c
l
= p{W.-W.<O, i=l, ••. ,c-t, j=c-t+l, •.. , c}
~
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J
k
= P{UijS:(T[j]-T[i]) I [20 2 (l-p)J\ i=l, ... ,c-t, j=c-t+l, ••. , c}.
Now, in view of (3.2), the T[i] satisfy
."
(3.8)
T[ lJ s: .•• s: T[ c-t] s: T[ c-t+1J
-
,..-'
"".-'
b ~ T[c-t+2] ~ ... ~ T[cJ •
Since the right hand side. of (3.7) agrees with the corresponding expression in
Bechhofer (1954, pp. 20-21), [with the only change that his
a2
(J2
is replaced by our
(l-p)], the least favorable configuration for which (3.3) holds can easily be
shown (as in [1, pp. 20-21J) to be (3.4).
Hence the theorem.
It may be noted that for the Z ., defined by (2.1),
n~
be replaced by
a2 In,
a2
in theorem 3.1 has to
and as in Sen (1968), it follows that the mean square due to
error consistently estimates cr2 (l-p), for all parent distribution having finite
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second moments.
While proceeding to the large sample solutions to the problem, we
note that by virtue of the consistency of the estimates Z ., i=l, ... ,c, for any
n~
fixed
~
in (3.2), the probability of correct decision will tend to unity as n
~
00.
Hence, to avoid this limiting degeneracy, we conceive of a sequence of worth-detecting differences which also converge to zero as n
with a Y, strictly less than 1.
~ 00,
in such a way that (3.3) holds
The justification of the use of such a sequence
can be made in the same light as the use of the well-known Pitman-efficiency is justified in nonparametric inference procedures.
[X. ,
the single sequence
i=l, .•. ,c],
~a
Theoretically speaking, we replace
.
~a
~
n
_
~a
~l}, where for the x~n~ the corresponding
wi th , replaced by
(n)
a=l, •.. ,n, i=l, ... ,c} by a double sequence [[X. , a-I, ... ,n,
, and ,
n
~'o
as n
~ 00.
(n)
~[i](say ~[i])
satisfy (3.2)
On the other hand, to practicing people,
the solution can be made justified for small values of
small enough to make the
~;
applied approximations to be valid and reasonable.
We now drop the assumption of multinormality of the errors, and, as in (1.2),
we let F to be arbitrary.
For the parametric procedures based on sample means, we
require that F possesses second order moments.
Theorem 3.2.
Suppose in (3.3) Y is fixed, and in (3.2),
~
is replaced by
~.
n
Then
for a y - probability of correct selection, we have
(3.9)
, P is the conunon correlation of e , ei~ (~k), and
ia
ia
6 is determined by the condition:
where a2 is the variance of e
(3.10)
y
= tQ
c-
1 (6 /./2 , . . . , 6/./2,
c-t times
°,toOl. times
..,
0),
Qc-l being the cumulative distribution function of a normally distributed (c-l)vector (U , .•• , U
, W t+2' ••. , W ), with EU.=EW.=O, Cov[U. ,U. ,]=~(l+6 .. ,),
c-t
cc
---~
J
~
~
~1
l
Cov[W.,W.,J=~(l+6
,) and Cov[U.,W.]=-~ for i,i'=l, ... ,c-t, j,j'=c-t+2, ... ,c;
J J
jj
1
J
-
6
rs
=0,1 according as r~s or r=s.
--The proof follows as a special case of (5.4) [with simplified (5.l)J, and hence
is omitted.
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holds [subject to (3.2), with
~
and we wish to determine n such that (3.3)
~*
replaced by
~*].
Then theorem 3.2 provides the
following large sample solution:
(3.11)
2
and as s , the mean square due to error estimates
°2 (l-p)
consistently, we have
also asymptotically
n
(3.12)
02 s 2 /(~*)2;
~
= n
X
= (n c )
~.
4.
-1
X,
s
n
c
2
1
= (n-l) (c-l) I
(Xia. -X,~. -X .a.+X •• ) ,
I
a.=l i=l
2
n
Ea.=l Xia.' i =l, ••• ,e,
c
n
1 Xia.
Ea.=l E,.
~=
-1
X = c
.a.
-1
c
E,~= 1 Xia.' a. = 1, ••• ,n, and
.
ased on rank order estimates.
Here, we select the t populations
associated with
(4.1)
The basic theorem of this section is the following.
Theorem 4.1.
Under (1.1) and (1. 2) and the assumptions I, II and III of Section 2,
, it~ng
' d '~str~' b
'
ut~on
t h e I ~m
0
f n 1/2[y(n)
(i) - yen)
(j) -
uA
ij ;
l<i<
,
__c-t, c-t+1<-1<
~_c ] ~s
multivariate normal with null means, and
°02 ,
1 2
2°0'
(4.2)
if i=i' ,j=j ,
if i=i',j;l:j' or i;l:i' ,j=j',
0, i f i:fi' ,j;l:j' ,
where
2
2
2
(4.3)
00 = [A +(c-2)A (F)]/(cB ),
(4.4)
A
(4.5)
J
2 =
AJ(F)
f
1
o
J 2 (u)du,
B~
f
00
(d/dx)J[G(x)]dG(x);
_00
=f
00
_00
f
00
_00
J[G(x)]J[G(x)]J[G(y)]dG*(x,y),
J(u) = ~-l(u),
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G*(x,y) is the joint cdf of (e!j,a' e!j' ,a) (j#j'), whose margina1s are
and
both G(x).
The proof of the theorem when the errors e. are mutually independent follows
1a
from theorem 3.2 of Puri and Sen (1967).
However, the multivariate approach of
Sen and Puri (1967) along with lemma 2.1 of Sen (1968) extends the proof to the
case of symmetric dependent errors.
For intended brevity, the details are omitted.
Now, the probability of correct selection of t best populations is given by
(n)
(n)
.
(n)
(n) }
P{max[Y(l)""'Y(c_t)] < m1n[Y(c_t+1)""'Y(c)]
(4.6)
= p{(....!L)1/2[Y(n)_Y(n.)_A .. ] < (....!L)1/2 A i l
.
+1
}
20~
(i) (J) L11J
20~
L1 ji ,
= , ••• ,c-t,J=c-t , ••• ,c
Thus, as in the asymptotic parametric case, we replace b,ji by a sequence
such that
n1/2b,J~)
+
A (real and finite) as n
ji
+
00.
{b,J~)}
Then, by using theorem 4.1,
we conclude that the right hand side of (4.6) is asymptotically equal to
p{U
(4.7)
ij
<
(n/20~)1/2 b,(~~, i=l, ••• ,c-t,j=c-t+1, ••• ,c} ,
where the U have a mu1tinorma1 distribution with null mean vector and covariance
ij
matrix, given by (4.2).
Since, this mu1tinorma1 distribution satisfies the condition
of theorem 3.1, we can again check easily that the least favorable configuration
turns out to be (3.4) with
~
replaced by
~.
Moreover, by the same technique as
0
as n +
n
in theorem 3.2, it follows that
In1/2~ _ 00
(4.8)
n
where 0
0
0
I+
is defined by (4.3) and 6 by (3.10).
00,
Thus, a large sample solution to the
sample size needed for the probability of correct selection being equal to y is
given by
(4.9)
where
~*
is a given small worth-detecting difference.
2
0
involves the two
"" as in Sen (1966), while
B can be estimated (by B)
unknown parameters Band AJ(F).
AJ(F) can be estimated (by LJ(F»
(4.10)
Now, 0
2
as in Puri and Sen (1967).
2
""2
n ~ 0 [A +(c-2)L (F)]/cB •
J
Hence, we have
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-9Now, using the same notion of asymptotic relative efficiency (ARE) as in
Puri and Puri (1969), it follows from (3.11) and (4.10) that the ARE of the rank
order procedure (based on
(4.11)
e(~,B*)
~-scores)
with respect to the B*-procedure is
= °2 (l-p)/OO2
22
= {[202
(l-p)]B }{c/2[A +(C-2)A (F)}
J
Now, B relates to the cdf G [cf. (4.4)] whose variance is 202(1-p).
Hence, the
first factor on the right hand side of (4.11) is the A.R.E. of the one-sample
rank order tests (for location) with respect to the Student t-test when the parent
e~,B*(G).
distribution is G(x) [cf. Puri and Sen (1969)], and we denote it by
Further, it has been shown in Puri and Sen (1967) that AJ(F) ~
21 A2 ,
where the
equality sign holds iff J[F(x)] is linear in x, with probability one.
As
such,
we have
e(~,B*) ~ e~,B*(G),
(4.12)
where the equality sign holds iff J[F(x)]
various known bounds for
example, if we use
~
e~,B*(G)
= a+bx,
with probability one.
can be used to provide bounds to
as the standard normal distribution,
e~,B*(G)
below by 1, where the lower bound is attained iff G is normal.
Now,
e(~,B*).
For
is bounded
Thus, the procedure
based on the normal Scores estimators is asymptotically at least as efficient as
the extended Bechhofer procedure.
follows that
e(~,B*)
If we use the Wilcoxon-Scores estimator, it
is bounded below by 0.864 (though not attainable) for all
F( G), while the same can be greater than unity for many non-normal F.
normal F, it is bounded below by 3/rr
for all
c(~2),
For
while it can be as high as
0.98.
5.
To study the relative performance of the
~-score
procedure with respect to the
B*-procedure, we consider any sequence of parameter points satisfying
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l 2
l 2
. 1 , ••• ,c- t ,c- t+2 , ••• ,c,
(n)
- T[(n.;)]
T[c-t+l]
... = U~i(n) = n- / Si +o(n- / ), ~=
(5.1)
where not all the Sl, ••• ,8 _ are equal to 8, and/or not all the 8 _ + , ••• ,8
c t
c t 2
c
are equal to zero, i.e. the least favorable configuration does not hold, but
T~~~ ~ T~;~
whenever
l~i~~c.
Then, for the B*-procedure, we have
lim P{correct selection of t best treatments}
(5.2)
n~
lim {
}
= n~ P man[Zn(l),···,Zn(c_t)] < min[Zn(c_t+l),···,Zn(c)]
lim
t
= n~ [E P{all the Zn(l)'··· ~Zn(c-t) < Z(c-t+,Q,) < min[Zn(i) ,i=.c-t+l, ••• ,c(;'c-t+,Q,)]}1
,Q,=l
=
t
L
,Q,=l
t
= E
,Q,=l
1.
[~m
Z ( )-z (,Q,) < 0, s=l, ••• ,c-t,
n
}]
Zn(,Q,)-Zn(r) < 0, r=c-t+l, ••• ,c(;'c-t+,Q,)
p{ n s
n+oo
lim P{U < ~(n) W < c(n) s=l, ••• ,c-t,
}]
[n~
s
~,Q"s'
r
~r,,Q,' r=c-t+l, ••• ,c(;'c-t+,Q,)
,
I_
where [Ul, ••• ,Uc_t,Wc_t+l' ••• 'W _l'Wi+l' ••• 'Wc ] has the multivariate normal
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(5.3)
distribution Qc- l' defined in theorem 3.2, and
the last identify in (5.2) is a direct consequence of the central limit theorem
as applied on the Zn(r).
Thus, from (5.1), (5.2) and (5.3), we obtain that (5.2)
is equal to
(5.4)
where
Si,j = Si- 8 j.
Consider now the rank procedures, where the assumptions I, II and III of
section 2 hold and the sequence in (5.1) also holds.
follows as in (5.2) that under (5.1), for the
lim
(5.5)
n~
~-scores
procedure
{
P correct solution of t best treatments}
c
=
Then, by theorem 4.1, it
L
=c-t+l
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-11where o~ is defined by (4.3).
we may conclude that the A.R.E. remains the same for (5.1), even when the least
favorable configuration may not hold.
6.
Selection of best treatments with re ard to order.
Here the Bechhofer procedure
consists in selecting the t best treatments associated with Zn[c-t+l]""'Zn[c]
respectively.
By virtue of our theorem 3.1, we can readily extend the original
procedure by Bechhofer (1954), and derive the following results.
[The details are
omitted for intended brevity.]
For a fixed
(6.1)
y(O<y<l) and under the condition that
T[i+l]-T[i]
~ ~n
(worth detecting distance), i=c-t, ••• ,c-l,
let n be determined so that (a) the following (least
favorabl~ configuration
holds:
(6.2)
~ (b)
(6.3)
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Thus, comparing (5.4) and (5.5) (in the Pitman sense),
max
}
P{ l<i<c-t Zn(i) < Zn(c-t+l)<"'< Zn(c) = y.
Then asymptotically,
Inl/2~ _ oo/l-pl
(6.4)
n
+
0, (as n+oo),
where a and p are defined as in theorem 3.2 and
(6.5)
(c-t)Q
c-
°
is determined by the condition
1(0, ••• ,0, 0/12, ••• ,0/12) = y ,
c-t-l times
t times
~ Qc-l i~~~e cdf of a normally distributed vector (Ul""'Uc-t-l,Wc_t""'Wc_l)
~atisfying EU i = EW j = 0, i=l, ••• ,c-t-l, j=c-t" •• ,c-l ~(i) Cov(Ui,U i ,) = ~ (l+oii')'
(ii) Cov(Ui,Wj ) = if j=c-t and 0, otherwise, and (iii) Cov(Wj,Wj ,) = 1, -~ £E_
t
° according as j=j',
Ij-j'l=l or Ij-j'l > 1, for i,i'=l, ••• ,c-t-l, j,j'=c-t, ••• ,c-l;
0ii' being the usual Kronecker delta.
Again by virtue of our theorems 3.1 and 4.1, it follows along the same line
as in [4] that for the rank scores procedure based on (4.1) (with regard to the
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-12~
order), the least favorable configuration turns out to be (6.2) (for small
n
),
and hence, for this procedure
(6.6)
where 0
0
is defined by (4.3) and 0 by (6.5).
Hence, the A.R.E. is the same as in (4.11) and (4.12).
section 5 can readily
Also, the results of
be extended in this situation and similar conclusions be
derived; for brevity the details are omitted.
7.
Selection of a subset of treatments better than a standard one.
Instead of
(1.1,), we consider the model
(7.1)
Xia =
~
+ Sa + Ti + E ia , i = O,l, ••• ,c; a = l, ••. ,n,
where the notations are all explained after (1.1) and TO is the standard treatment
effect.
We say that the ith treatment is better than the standard if
T. > TO +
(7.2)
~-
~;
~
= worth detecting difference.
For the one-way layout case with normally distributed errors, Gupta and
Sobel (1958) proposed the procedure:
Xni-X
> 0, where X
= n
nO
ni
-1 n
La=lX
ia
Select the subset of treatments for which
' i = O,l, ••• ,c.
An elegant solution for the
sample size (n) needed to achieve a y(O<y<l) probability of correct selection has
also been provided by them.
Noting that for (within-block) symmetric dependent
normally distributed errors, the
.
distr~bution
1/2 T
T
of [n
(Xn~.-Xn 0- ~.+ 0)'
•
~=l, •••
,c]
is also multinormal with null means and covariance matrix 02(1_p)[I +J ], (where
c c
o
2
and p are defined in theorem 3.2), it turns out that the only change needed in
Gupta-Sobel solution for the two-way layout problem is to replace their 0
o 2 (l-p).
2
by
By virtue of the central limit theorem, the same solution holds
asymptotically for the entire class of edf's with finite second moments.
For the rank scores procedure, we define the estimators
in (2.5) and (2.6).
A(n)
~iO
' i=l, ••• ,c as
Then, we select the subset of treatments for which
A(n)
~iO
> O.
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-13Along the same line as in theorem 4.1, it follows that
[nl/2(~~~)-Ti+To), i=l, ••• ,c]
have asymptotically a multinormal distribution with null means and covariance
matrix
O~[Ic+Jc]'
where
O~
is defined by (4.3).
Consequently, the Gupta-Sobel
solution also asymptotically holds for the rank scores procedure provided we
replace 0
2
by O~.
Hence, the A.R.E. of the rank scores procedure with respect
to the Gupta-Sobel procedure can again be measured by
with (4.11).
02(1-P)/0~, and it agrees
The details are therefore omitted. However, the results clearly
indicate the superiority of the normal scores procedure over the normal theory
procedure (for large samples).
8.
Few additional comments.
Unlike Puri and Puri (1969), we have considered
here the procedures based on the rank order estimates, not statistics.
The same
procedure can be suggested for the one-way layout problem as an alternative to
the procedures by Puri and Puri (1969).
Also, we note that the
procedure con-
sidered in this paper can readily be extended to incomplete block designs (such
as the paired comparisons designs, etc.), whereas the original procedures by
Puri and Puri face considerable difficulties.
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REFERENCES