UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Sceintific Research
Washington 25, D. C.
....
•
AFOSR Report No.
INFERENCE ON TREATMENT EFFECTS AND DESIGN
OF EXPERlMENTS IN RELATION TO SUCH INFERENCE
by
S. N. Roy
and
J. N. Shrivastava
January, 1961
Contract AF 49(638)-213
•
•
In this paper inference procedures and designs are discussed
that are more appropriate to those situations in ANOVA or
MANOVA where, against the customary or standard null hypothesis, we are interested in certain non-standard alternativ e,s or, in other words, we are interested in increasing
the discrimination along certain (non-standard) directions
of deviation, even at some cost to the discrimination along
other directions of deviation. It is indicated how,for several such situations,it is possible to improve upon both the
customary designs and the customary inference procedure.
Qualified requestors may obtain copies of this report from the
ASTIA Document Service Center, Arlington Hall Station, Arlington
12, Virginia. Department of Defense contractors must be established for ASTIA services, or have their "need-to-know" certified by the cognizant military agency of their project or contract.
Institute of Statistics
Mimeograph Series No. 274
li~FEHEi~CE Oil!
•
'l'REATl',iEN'l' EFF'EC'l'f: All::'; .;ESIGlIi*
OF EXPERIi'4E:N'I'S IN
RELATI01~
TO SUCR INFE;HEi,ICE
By S. N. Roy and J. N. Shrivustava
1.
Summary.
This paper starts from the customary model,
and the customary null hypothesis on any set of treatment
effects, both fo.c univariate and for multivariate response,
•
and consiuers such a null hypothesis not against the perfectly
General customary alternative, but against certain more specific
alternatives that mieht be more natural and meaningful in some
situations.
Such situations occur quite often in practice,
as, for example, when we are in possession of some prior information on the nature of the treatments either in terms of' one
or more characteristics other than the response we are looking
for, that puts the treatments into a natural relationship or
ordering, or even in terms of some very crude knowledge of this
response itself, an6 furthermore this prior information is such
as to focus our interest anJ attention on certain specific alternatives rather than the perfectly general one.
It is well known
that the usual F test for analysis of variance has some optimum
properties, and it has also been known for some time that each
"
..
of the several tests for multivariate analysis of variance has
certain good properties, the optimum and the good being all in
relation to the general alternative usually considered.
If,
however, one is interested in some more natural and specific
alternative, then, for any (jesignJ it will be indicated in this
This research was supported by the United States Air Force through the Air
Force Office of Scientific Research of the Air Research and Development Command,
under Contract No.AF 49(~38)-2l3. Reproduction in whole or in part is permitted
for any purpose of the United States Government.
*
•
2
paper how and in what sense an inference procedure based on a
finite set of unions and intersections would, in general, be
better than the usual F test for univariate problems or better
than the analogues of F for multivariate problems.
It will be
also indicated that if, at this point, one decides to stay with
this finite union-intersection procedure, then any customary
design can often be replaced by another design that would do
better.
Thus, one has a two-fold development, one in the direc-
tion of inference and one in the direction of design.
The same
kind of improvement is reflected in the confidence intervals when
we invert the test procedures to make confidence statements.
This paper discusses a sample of such specific alternatives
together with the corresponding inference procedures and designs,
that are by no means exhaustive but merely illustrative, and the
discussion will be expository rather than one involving full
proofs of the various statements made.
Furthermore, the state-
ments in this paper will be more concerned with the univariate
than with the multivariate aspects of the response.
The follow-up
papers will discuss the proofs of the statements made herein and
•
also more statements, together with proofs, of the multivariate
aspects.
2.
Notation and preliminaries. As usual b will stand for the
number of blocks,v for the number of treatments, k for the number of plots, ie, experimental units per block (assumed to be
the same for each block); r.for the number of replications of
J
3
•
the j-th treatment (j
=
1,2, ••• ,v) and
~
.. ,the number of
JJ
times the j-th and j I -th treatments (j#j';"1,2, ••• , v) occur
together (in a block).
With slight modification we take the
customary model I of ANOVA
(2.1) x
where x
ij
ij
=
l3 i +
T
j +
E
ij ,
is the observed (univariate) response from the j-th
treatment in the i-th block, 13
•
1
the (unknown) block contribution
to this response, o. the (unknown) treatment contribution and
J
the error, such that E •• I S are assumed to be independent
1J
1J
N(o,o2) (i = 1,2, ••• ,b; j= 1,2, ••• ,v). The corresponding model
E ••
I
of MANOVA is
(2.2) x' j
-1
pxl
= -1
13.
+ T + E. j ,
-j
-1
pxl pxl pxl
where everything is a p-dimensional vector and where
E •• I S
-lJ
are
~).
assumed to be independent N(Q,
pxl pxp
The customary null hypotheses on the treatment effects, for the
univariate and the multivariate case, are written respectively as
(2.3)
?8 (univariate):
T
1
=T
= •••
2
=T
v
and
(2.4)
'X (multivariate):
Let us now express the
= -v
T
.!l = .!2 =
pxl pXl
pxl
of (2.3) in four different alternative
= 0;
0; L c.
j
J
T
.
J
= 0,'
L c
,2
j
J
V
I:
j=l
c . . c,
J1J
= lJ
.= I)
J2 J
I
..
J r J2
(Kronecker 5)
4
where jl
i
= 1,2, ••• ,v-l
j2
and wherec j j'S (jl
= 1,2, ... ,v-l)
1
are associated with v-l specific orthonormal contrasts,
(iii)~
vn
;
=I
j
Z
j I =1
L-T = TjlJ
j
and
v-l (\ LT.
j=l
J
(iV)!f.
3
If we denote by
TI
= TV
J
... ,
the vector
T
vxl
v ,by~' the vector
the matrix whose elements are c
and by C
(v:-l)xv
.,
j lJ
then it will be convenient to rewrite (i) and (ii) respectively as
(v)}f.;
D
(vi)l1.;
n [ £'
T
= OJ
=0
[c
all Sand
C
T
CIJ
= OJ £1£...=
.J. = 0
j
C CI
~, (v-"'i)xv vxl (v:-~xl -vxl {v:-l)xl - -
The motivation behind expressing
lJ
= I(V-l)]
-
•
in the four different alternative
forms (i) - (iv) is that, for (i) we are looking for a test procedure that should, if possible, secure equal discrimination (in
terms of power) against all alternatives (or contrasts) !'
£,
for (ii) one that should, if possible, secure equal discrimination against the (v-l) specific contrasts C T , such that
v-Ixv vxl
this discrimination might presumably be sharper than in other
directions, ie, against other contrasts, for (1ii) one that should,
if possible, secure equal discrimination in the directions
T.
_
T
J
that might presumably be sharper than against other contrasts
and for (iv) one that should, if possible, secure equal discrimination in the direction
a standard treatment and
T. - T
J
j
v
(where v stands for some kind of
for any other treatment), that might
conceivably be sharper than in
other directions.
If the null
v
j ,
5
•
hypothesis
is expressed in forms 1 ike
It,
?
:If). 9f):>etc.,
'J
with particular ends in view (in terms of the power functions of
the respective test-procedures), then
Jtl ') 5f.
andJf.3 will be
J-
said to correspond to the structures 8 , 8 , 8 , etc. and each
1
2
will be said to be a structured hypothesis.
complete notation we shall use the symbols
3
For the sake of a
Ito and
8
to indicate
0
what from our viewpoint would be the unstructured case.
One
extension to multivariate response would follow if in (i)-(iv)
we replace
by
T.'S
T.'S
pxt
J
and if in (v)-(vi) we replace the
vector
T by the matrix T
• Other possible multivariate extenvxl
vxp
sions will also be discussed later.
Let us now set up the test procedures for the univariate
Jlfo:>?t l ')
J:t-~ andJt.
univariate
say
It is known that the usual F-test for the
has an acceptance region (at a level of significance,
which can be expressed in the form
Q
all
(2.6)
C
-
[(£'
<
!)2
2
'~nu
(0)
~o:
£'..J.
= 0,
where ~ is the customary least squares estimate of
£1£
1\
T,
and ~
0:
is the 100
%
'
2
the
nu
(J
-
1\
customary estimate of the variance of the estimate c'
numerator
=1]
T
in the
point of the F-distribution
with a degree of freedom (v-l) due to the hypothesis, and another
due to the error that depends upon the design used.
remarked that
1\
(£'!)
2
It may be
itself has the distribution of t
2
with
-;:T
(J
nu
the error degree of freedom of the F-distribution.
~(o)
0h
It should be
of (2.6) will be the 100
point of the
ex
2
t -distribution, when ex * > d... He shall let this region (2.6)
noticed that the
6
correspond to 8
andJ{ ana shall call i t ~D.
0
Corresponding
(Jf
to ~ , 8 ),
,82 ) and (~,83) let us set up test procedures
1
I
2:3
with the acceptance regions, say ~ G(\ torrespond ing to the
cVl)dJ2-
Tukey test
16])
(2 .7)
andJJ, given by
C?l
aU:
I
J
vA
L(
jtl
j=l
v
E
v
(\
j*j'=l
0
A
A 2 ~2
(1) v
c. j T.) /a
< ~ rv ; E c. .
Jl
J
nu ~
j=l J1J
A
2" 2
(T.-Tj,)/a
J
nu
)J
:.tjj':S~(;
2
=
0
,
and
where the
~'s
are given respectively by
(2.10) 1 - " =
It
PID,~I} p~I'pfJ= PL~~~
should be noticed that for the above purpose
:Ji:. If0'; Jf
l
-:: :!t-,2..;:: ~
We shall denote the power function of any test procedure
by
Wwith
suitable agruments to indicate dependence on parameters
and degrees of freedom and with suitable subscripts and superscripts,
etc. to indicate the test procedure it is based on.
In addition to
the structures and the associated test procedures already mentioned
a few other structures and the associated test procedures also
will be discussed in sections
4 and 5, with integers larger than
3 being used as subscripts to ~, 8 and~.
This is not a very
expressive and logical notation but would suffice for our present
purpose.
The different designs to be mentioned and discussed in
this paper will be sometimes called, for convenience of reference,
Do, D , D2 , etc. with the caution that no correspondence is necesl
sarily implied at this stage between the subscripts for the struc_
7
tured hypotheses and the subscripts for the designs.
Turning now to the case of multivariate response, we
propose for the multivariate analogues of cY(
to be
...
-1-
[J.I
)
called~ ~ '8f;
D ')
J62.' :9f.,
2;
n l
given by
It
(2.11)~o
~ ~ and)f ,
.3
tests with acceptance regions
'i
1-
(sl ~
a)2
I) 2.
1\
lxv Vxp pxl· /(a'E a)<
- -nu- -
all £. and a
I.L
(o)x
a:
£ I.J. = OJ £1 £=1 j
~ is non-null],
I.L
(O)x]
ex
,
1\
where E is the usual estimate of the dispersion matrix of the
nu
estimate c'
in the numerator· and S
is the "sample dispersion
-e
pxp
is the sample dispersion matrix
matrix due to the error", S
T
I\'P*P
due to the "hypothesis","
is the usual maximum likelihood
vxp
*
( ) is the 100 a:
estimate of T and I.L 0
point of the largest
v~
°z
a:
~
characteristic root, with appropriate degrees of freedom;
2
A
(1)*
!:) /(!.' E a) < 11
j
all !: l~l. v~ pxl·
-nu- --a
/\
A
I __ \ g£I.
n
A
!
A
1,,\ ["(T' - T
a
all!: l~ l~ pXl
I
,)
2
A
(2)*
/(!:' E a) < I.L
ja is
-nu - - a:
-
non-null?
<=->
1\ v
G\
r
L
jrj'=l
2
TJ'j'~
(2)*
I.L
a:
_7,
8
2
where T~jl is Rotelling's T with appropriate degrees of freedom,
the mean vectors being the maximum likelihood estimates of the two
vector responses .:!:j and 2:j
,
fr.QIlJi. the j-th
if, -th treatments
and the
error dispersion matrix being based on the overall residual;
<~>
v;{
I
J=l
•f
;-T~JV
-
<
-
~(3)*
7.
ex -
It should be noticed that ~(l)* ~(2)* and ~(3)* are defined by
ex
' a
a
(2.15)
with the understandin&as
3.
befor~
that for the above purpose we have
Remarks on the power properties (under three different designs) of
the test procedures appropriate to the four different structured hypotheses. With v treatments and r replications for each treatment (assumed
to be the same for all treatments) let us consider in the first instance
an RED and a comparable BIBD with b, k and A..
r < b.
the
~2
Notice that k < v and
Let us denote these two designs by Do and D .
l
of (2.1) depends on k and should be written as
Then for D ,
l
~k2 and likewise
2
E of (2.2) as E ; similarly for D we should write ~2 as ~ 2 and
k
o
r
2 2 2
E as Er (since k = r for Do)· Notice that,in general, ~k < ~r
We observe, furthermore, that, for Do and D , the respective degrees of
l
freedom due to the error are (rv - r - v+l) and (rv - b - v+l), and,
9
for both Do and D , the common degrees of freedom due to the hyl
pothesis~ (in the customary F-test) are (v-l).
;~u
For Do the
occurring in thec:JJ 0,J)1,JJ2 andJ)3 of (2.6)-(2.9) are given
respectively by
2
2
2
2
(i) Se(o)/r, (ii) s(o)/r, (iii) 8(0) /2r (iv) 6(0) /2r,
e
e'
e
and for D , the same are given by
l
(3.2)
(i)
s~1)2/~V,
Similarly, the
~nu
(i1)
6~1)2/~,
occurring
s~1)2/2~V,
(iii)
in~:,~~,JJ; and~;
(iv)
s~1)2/2~V .
of (2.11)-(2.14)
are given, for Do and D , by
l
and
We shall now consider in a symbolic form the power functions of the
critical regions of the different test procedures, ie.of the complementa
~0'~'.1l2 and~3
for Do and Dl , exhibiting, in particular,
the arguments involved in the power functions.
~
(3.5) (i)dJ
v
:w (a;
o
v-I; vr - v - r+l; r E (T.j=l J
0
v
where
T =E
j=l
For D these are
o
i)
2
/~
r
2
),
T./V,
J
(ii)~ 1: Wl(a;l,l, ... ,l; vr - v - r+l;
2
2
2
2
r~l/~r , ... ,r~v- l/~)'
r
10
v
where
E c .. ,r.,=
j'=l JJ J
v-l
E
j=l
that
~.
J
2
~.(
j=l,2, ••. ,v-l) and where, it should be remarked
J
v
_ 2
= E (T - T) of (i),
j=l j
2
2
2
2
(iii) ']) 2: ~2(a;1,l, ••. ,; vr - v - r+l; r~lv /rr r , .•• ,rn~~ 1 ,v/rr r )
and
(iv)
;J) 3:
2
/2
*3(a;1, .•. ,l;vr - v- r+l; r~lv2 /rr r 2 , ..• ,r~v1 ,v
rr r ).
The symbolic form (1,1, ..• ,1) in both (ii) and (iv) indicates that
the corresponding power functions are related to regions such that
each is the complement of the intersection of (v-l) acceptance regions
each based on a "t" with degrees of freedom vr - v - r+l; and in (iii)
this symbolic form indicates, instead of (v-I), (;). such acceptance
regions.
Further in both (iv) and (iii), ~j (j=1,2, •.. ,v-l)=
v
T.- T ;
J
v
in (iv) these are natural "deviation parameters," but in (iii) these
are not natural, in that any particular treatment, say Tv , can be
v
taken as the standard, and all (2) pairwise contrasts T.- T .• can be
J
J
expressed in terms of the (v-I) linearly independent contrast
T.- T
J
v
(J=1,2, ••• ,v-l), and thus the power functions can be expressed in
terms of these (V-I)
deviation parameters.
u
N~tice
that *2 and *3 are
entirely different functions.
For D , the corresponding power functions can be expressed as
l
11
t o (0;
AV v
v-ljvr-v-b+lj -- E
k
- 2
(T o- T)
01 J
/~
J=
C\ 3: t (0;1, ... , Ijvr-v+b+l'k
AV
(1V)-dJ
3
2 /
2
2
)
k;
AV
2
/
2
TJ lv O"k , •.• 'k ~-l,v ~k ).
By way of comparison between (i)-(iv) of (3.5) we shall now state
without proof a nwnbcr of results proved elsewhere.
v
liminary to that we recall that
E (T.j=l J
But as a pre-
_ 2
T)
of (i) can be expressed as
the sum of squares of any (v-l) orthonormal contrasts, including, in
particular, the specific set of (v-l) considered (ii), and also that
it can be expressed (though not as a sum of squares) in terms of
TJ~v's of (iii) or (iv). Furthermore, the specific contrast set of
(iii) can also be expressed in terms of TJo 's and vice versa, and,
JV
finally, all pairwise contrasts implicit in (iii) can be expressed
in terms of TJjv's.
With this remark we state that
(a) if in (ii) we put all nOJ 's
= 0,
(b) if in (iii) we put TJ jj , (5 T - T
j
j
except any particular one, then,
"
jfj')
f
0 for any particular
pair (j,j') and equate to zero all other pairwise contrasts that we could
se equate consistently with this particular pair being non-zero, then,
12
then ~3> ~2> *0'*1 (in general).
the directions
~'"I
JJ
IS
(JF v),
In other words, this means that in
2 is better than the others, in the
directions of a specific orthonormal set considered in (ii),
better than the others, while in other directions
or worse than the rest.
o
1 is
might be better
Indeed, there are directions in which
would be definitely better than the rest.
situation as among (i) - (iv) of (3.6).
o
We have an exactly similar
Next, as between (3.5) and
2
2
(3.6) we observe that, in general, ~k < ~r ' so that if the former
is sufficiently smaller than the latter (which, in fact, is achieved
in many experimental situations) then *i of (3.6) > *i (3.5) (i~0,1,2,3).
It is obvious, however, that the l.u.b. on ~k2/~r2 to accomplish this
would not be the same for i=0,1,2,3.
Another point to note is that
for both DO and D , or in other words,for all designs that are pairwise
l
symmetrical (with respect to all treatments),~o (or the F-test) is
equally sensitive with regard to all normalized
contrasts'~l equally
sensitive with regard to each of the specific set of (v-l) orthonormal contrastlj>
regard to
/J) 2 equally so with regard to all pairs anW 3 with
~.
I s (j=1,2, ••• ,v-l).
However, this equal sensitivity, as
JV
will be presently seen, will not carryover to designs that are not
pairwise symmetrical,as, for example, any PBIBD.
To illustrate this point let us take, for a given v and r, any
combinatorially possible PBIBD With, sa)) massociate classes, and let us
write for this design
(3. )
(i)~o :*0 (a;v-l;vr-b-v+l;T- 2* !/~k*2),
I
13
where
* 2 (depending
~
upon
the block size k* for this case) is what
k
occurs in (2.1), and
*
C* is such that C*j j ; r(k-l)/k and Cjj,(j~j');-Ae/k,
vxv
if the j-th and j'-th treatments are l-th associates.
in this case the power function
It is clear that
*o of the F-test already favors certain
directions of deviation more than other directions, and it might be mentioned here, without expressing *1'*2 and *3 in a symbolic manner as
in the previous cases, that, in general, this differential discrimination will be further sharpened under *1'*2 and *3 along the same general lines as indicated in the previous cases.
We shall not pursue
this in any detail here, since we are going to consider in the next
two sections certain other useful and interesting types of structured
hypotheses and the associated designs.
It may be remarked that in this
paper we shall not consider the general power properties and aspects of
the corresponding multivariate or multiresponse test precedures and
their possible impact upon designs.
This will be discussed in a
later communication.
4.
Some other types of structured hypotheses with the appropriate
test procedures (With power properties)
For concreteness of illustration, let us consider nine treatments
with hypothetical effects T , •.. ,T • On the general lines indicated
l
9
in section 1, assume that the treatments have been divided into two
groups (1, ••• ,5) and
(6, ..• ,9),
and that, although we would be in-
terested in all contrasts we might be much more interested in anyone
(and one alone) of the following types of contrasts or structured
14
hypotheses.
(84~Y;
= o;jh' =6, •.• ,cJ],
'! 4): (i)[iJ.-T J =O;jFj' =1,2, ••• ,2.7tlE-T
. .,
(4.1)
q
J-J
or
(ii)
5
9
~ CjT j = 2.7fL-n ~ c. Tj=
allc j=l
allcj=6 J
['/1
5
E c.
j=l J
(8
(4.2)
,CU 5):
5 df
2
=
9
E c.
2
j=6 J
97 with
5
9
E c.= ~ c.= 0 and
j=l J j=6 J
= 1.
)
T - T.,= 0; j=1,2, •• ·,5; j'= 6, ... ,9.
j
J
)
(8 , )(6): (i) (4.2) () (4.1)(i)
6
or (ii)
(87'~
(4.4)
(4.2)
n
(4. 1)( ii) ,;
): LT j - T = 0; j=1,2,3,!±7!/
5
T - T
5
Ef T9= OJ
j=6, 7,§.7
Y
= o.
9
Finally, with these nine treatments against a different background
(and without the division into two groups considered above) let us
consider a cyclically structured hypothesis
with the convention that T + = T •
v l
l
Before we discuss the associated test procedures, some remarks on the
above structures might be helpful.
9I 5,/f
It is easy to check that anyone of
6 orlt 9 constitutes the total hypothesis
J? (of equality of the
nine treatment effects), but this is not true Of?/4 or
If 7 or J¥ 8'
15
which suggests that to each of these we have to adjoin something else
(depedning upon the nature of our interest) so that the resulting structure constitutes the total hypothesis.
This means, for example, under
d¥8' that although we are more interested in
T
5
-
T
9
, we shall keep
open the possibility of studying other contrasts as well.
This also
means that by adjunction we shall be changing the nature of the original structure which, nevertheless, will still occupy a privileged
position in the altered structure.
at the proper points.
This adjunction we shall make
We next note that;)f 4 is related to "within
contrasts" of the two groups, expressed either in the form of all
one
contrasts within/group plus all contrasts within another group or in
the form of pairwise contrasts within one plus within another, de-
Likewise'~5 is related
pending upon the nature of our interest.
;)4- 6
to a modified version of "between contrasts,"
~7 is related to contrasts
standards'~8 to a contrast
is self-explained,
within each group against two respective
between just these two standards.
If 9 needs no further comments.
For the above structures we offer test procedures with the
following acceptance regions
n
n
~f1\I 9
jFj k 6
f"(T "To.)
j-
J
2 /~"2
nu
=t j J
O '
1 5 A
1 9" 2 "2
2
T - I. E To) /~
a t <
5 j =1 j ~ j=6 J
nu
-
~(- E
2 ~~
""
*(41)
ex
-
~
J7
~~(41
7;
Finally,
16
(\ ~() >( ~ c '~j)2/~2 ~
I 1(' j =6
r
(42)J or[F < 1J.(42)?7
llex2
2 0:)-
nu
J
all normalized contrasts
1 5 A
L{-5
~ T . 1 j
J=
~
f, \
(4.10)
~
.
_ A
j=l,
,5
Jk 6,
,9
n
9
6.1: / 1
1 9 A 2 A2
~ T j ) /a
If. 6
nu
J=
r.
L (T j -
2
It..
,~
j~j'=l JJ
A
Tj ,)
(61)
IJ.
0:
2 A2
/anu
3 t
2
< 1J.~42)
0:
2
=t j j ,
J.'
< 1J.(5)J.
ex
'
J,
which 1s precisely the same as:.1)2 of section 2;
iF -< 1J.(62)7/1
iF
C\)62 :L£l
ex - r I L£2
(4.11)GT\.
<: 1J.(62)Jn
-
ex
't~., -< 1J.(62)J
0:
'
LV JJ
where t 2jjl is the same as defined under (4.9);
(4.12)g[) 7:
LA(~j~5)2/~~u = t~5 ~ 1J.~7)L_7
J=J1
j
17
n
Notice that in (4.7), (4.8) and (4.12) the adjunction is
based on the hypothesis of zero contrast between the group means
and is in terms of a corresponding t-region and in (4.13) is
based on contrasts within the two groups against the respective
standards.
It may be remarked that in each case where adjunction
was used it would be qUite meaningful to use other kinds of adAs in section 2, the ~ 's in (4.9), (4.10), (4.11) and
a
(4.14) are given by
junction.
We next observe that in (4.7), (4.8) and (4.12), depending upon the
nature of our interest, it is open to us to take
*a= ~a
~
or keep the
two different in any prescribed manner that might be desired, but in
any case with the general overall requirement that
=1
However,
in
I( 4.13),
-
a .
*
the meaningful course seems to be to so choose ~a
as to make the discrimination with regard to (j5) and(j9) less sharp
than with regard to (59).
18
Turning now to the power properties of test procedures
offered for the different structured hypotheses we observe that
for any design, in general, and for any particular structured
hypothesis the suggested test procedure secures, in the sense indicated in section 3, better discrimination with regard to those
particular deviations than the other test procedures. In other words,
in this sense each procedure is appropriate to one structure.
Further-
more, given one structure and the test procedure claimed to be
appropriate to it (in that it is at least better than the others
in the class considered), a design will be offered in the next
section which would be appropriate to it, in that it would be at
least better, in the sense indicated in section 3, than others in
the class considered.
5.
The problem of design of experiments when the given set
of treatments has a structure.
We are now in a position to have a new look at the problem of
experimental designs.
suppose that the
Given a set of v treatments
set~ may
~,
we shall first
have no structure, i.e., we may not possess
any information regarding the nature of the treatments, or any
relationship in which they stand with respect to each other.
will, of course,happen rarely.
If
~has
This
no structure, we will proceed
in the usual way with the design of our expreiment, remembering the
three basic principles ~17 (i)
and (iii) local control.
replication, (ii) randomization,
The choice of the exact design to be used
19
will be determined by these three principles together with the
nature of our resources, especially in terms of the number of experimental units (n) available.
While analyzing the data of the
experiment so conducted, we will apply the usual over-all F-test for
testing the customary
hYPothesis~ of the equality of all treatment
effects.
Next, let us suppose that the set of
structure S.
treatments~has a
known
In this case we can introduce a fourth principle to
control the choice of the design viz. that the design, in a sense
to be presently explained, should correspond to the structure.
class of designs { D
J
The
which will be said to correspond to the struc-
ture S, is to be such that, if possible, it should control the relative variances of the treatment contrasts falling under S, in a
predetermined way.
After the class of design
i
D } corresponding
to S has been determined, an actual design will be selected from
this class by use of the three principles referred to in the last
paragraph, and the preassigned value of n.
Examples of many types
of structures are given in the earlier sections.
These structures,
theugh simple, are by no means the nnly ones to be found, nor are
they necessarily' the more general or more important ones.
The d1s-
cussion of other structures, because of their complexity, is
deferred to later communications.
As mentioned above, a class of designs
being
i 3
D
which corresponds
to a structure S controls the relative accuracies of the treatment
contrasts contained in S in the desired way both from the point of
20
view of ordinary point estimation and of ordinary testing.
Also,
in case we are interested in increasing from the testing point of
view the accuracies of any or all of the treatments contrasts
contained 1 in S, we can apply the corresponding intersection test as
exemplified in the earlier sections, instead of applying the overall F-test.
The relationship in which the various treatments stand, each to
the rest, vas recognized in case of factorial experiments from the
very beginning, and this information was utilized in a sense in the
construction of confounded factorial designs. However, in this case
also, the intersection tests were not used in any connection.
In the
case of experiments where the treatments are non-factorial in nature,
as, for example, when the treatments represent several varieties of
a crop, the information regarding structures, if any, was seldom
utilized and of course the intersection tests, as far as we are
sware, were never used.
We shall now consider classes of designs corresponding to
the structures discussed in the earlier sections.
the unstructured
ca~and
S represents
o
the BIB design or the randomized block
(RB) design can be utilized for this purpose.
The principle of
local control may sometimes necessitate the use of a PBIB design.
However, it must be rememberedthataPBIB design corresponds to
certain other structures and decreases the precision of certain
treatment contrasts while increasing the precision of certain
other treatment contrasts.
21
For 8 , a BIB
appears to be most reasonable. In 8 we
df~sign
2
3
increase the accuracy of the contrasts (T.- T ) (j=1,2, ••• ,v~1).
J
v
The class of design_ ~ D ~ which are consistent with 6 should ob-
want
t~
3
3
viously be
BU:.;r.. tha'~
A. jj , (j~j'~).
T:~e
A.
jV
(j=1,2, ••• ,v-I) are high compared to other
class of designs which are consistent with the
of the types 8 , 8 and 8 are the Intra and Inter Group
4 5
6
Balanced Incomplete Block Design (GBIB design, for short ). These
s~ructu~es
designs were first defined by K. R. Nair and C. R. Rao ~g7, and the
analysis of the designs was also given for the case where the number
of groups m = 2.
Recently, the authors independently arrived at
these designs (calling them generalized BIB or GBIB, for short)
from tnp.
~~~es~nt
Viewpoint, and obtained the complete analysis when
m is any positive integer.
The authors also introduced the inter-
section tests for tesUng!f, which was not done earlier by Nair and
Rae.
In the structures 8
(i=4,5,6,7,8), considered for the purpose
i
of a very simple illustration, the whole set of treatments has been
divided into two groups which may be denoted by Gl(T to T ) and
l
5
G (T to T ). In 8 , we want to increasethe accuracy of within
2 6
4
9
group comparisons, and so the corresponding class D is such that
4
i J
11. = ~jj,(j,j'E G ) and 11.
1
l
2
= A. jj .(j,j·EG2 )
are high compared to all
f
D ~ corresponding to 8 , 11.1 and 11.2 should
5
5
be low compared to other A.'s. The same considerations may eventually
other A.' s.
In the class
reduce the class
D
6
corresponding to 8 to the class of BIB designs.
6
22
Similarly,
in~D7~' ~\=
A. j5 (j
€
Gl ,
j~5)
and A.2- A. j9
G , j~9) should be high compared to other A.'s. The same
2
considerations indicate that in D8~ , A.
is high compared to
59
others, and in
A.= A.j,j +1' j=1,2, ••• ,9 is high.
9
For the sake of illustration one design D from each of the
i
classes D for i=4,5,7,8,9 is exhibited below. The values of the
i
parameters chosen are perfectly arbitrary, and thus the design may
(j
€
fD
J'
f
not be usable in a situation that involves another set of v, nand k.
Furthermore even if usable, no optimality
~s
claimed for any of these
designs among the general class to which such a design belongs.
However, it can be shown that the general pattern of any of these
designs is such that, for the
cor~esponding
structure of the hy-
pothesis, the design is better than any of the comparable customary
designs.
This, in fact, is the main point in exhibiting these designs.
The vector! = (r , •.. ,r ), where r = number of replications of
j
l
9
j-th treatment, j=1,2, ••• ,9 has been indicated below each design
beside the values of A.'s.
D4
A. =4,A. =3,other A.'s=l,
2
l
v=9, k=3, b=26,
!=(lO,lO,lO,lO,lO,6,6,6,6)
D
5
A. =A. =0, other A.'s= 1
1 2
v=9, k=2, b=20
r'= (4,4,4,4,4,5,5,5,5).
DESIQN
123
456
789
147
258
369
159
726
483
·'186 ':. 321
429
215
154
753
124
678
789
235
134
896
967
135
245
543
432
DEqIGN
16
17
18
19
26
27
28
29
36
37
38
39
46
47
48
49
56
57
58
59
23
D8
D
7
A = 1, j€G , A = 1, j€G
5j
1
9j
A = 1, other A'S
59
A = 7, A = 3, j€G or G
1
2
5j
59
2
=0
A = 3, j€G or G2
1
9j
v::9, k=2, b=8
E'=(3,3,3,3,7,3,3,3,7)
v=9" b=7, k==5
r'= (1,1,1,1,5,1,1,1,4)
DESIGN
DESIGN
15
25
35
45
1
69
79
89
59
2
3 5 9
14589
1
5 6 7 9
24569
25789
34579
35689
D
9
A
j, j+1(mod
9)
=2
A
=1
j, j+2(mod 9)
other AI s = 0
E'=(3,3,3,3,3,3,3,3,3)
DESIGN
123
234
345
456
567
678
789
891
912
6.
Concluding remarks.
The inference procedures offered or discussed in this paper are
based on a method extensively used by one of the authors and his
collaborators, being a particular application of a general principle
designated by them as the union-intersections principle
L3,4,27.
The
24
method used by them earlier and also in this paper consists essentially in expressing a complex (composite) hypothesis as the intersection of a number of more elementary (composite) components where
each such elementary component has a test procedure that has an
optimal property in a very natural and meaningful sense.
For the
complex hypothesis a test procedure is suggested that has an
acceptance region which is the intersection of the
acceptance
regions of the test procedures for the elementary hypotheses.
The chief motivation behind this method, only partially explained
in previous publications, is the following.
The expression of a
complex hypothesis as the intersection of a number of components is
by no means unique, and, when we express it as the intersection of
a particular class of components, we are looking for a test procedure that should have a large power against each of the associated
deviations, may be at the cost of being relatively poor against
other directions of deviation.
At this point of the process we
merely hope that this is precisely what would be achieved by the
procedure suggested.
For a wide class of specific problems in the
set up of univariate and multivariate
"norma~l
responses, it so turns
out that this 1s, 1n fact, achieved by the test procedures suggested,
in the sense that, in each case, the suggested procedure does better,
for its particular purpose, than the one ordinarily used or recommended.
At this point the further question arises.
Even assuming that, for
its particular purpose, the suggested procedure is better than the
one ordinarily recommended, is it in any sense optimal (again for
25
particular purpose)?
To answer this
~uestion
we have to define
carefully the criterion (or sense) under which we are looking for
optimality, and, in test procedures involving several parametric
functions, that go with the more complex problems (as opposed to
those like F, r, the two kinds of mUltiple correlation and Hotelling's
2
T , etc., that involve only one parametric function each, and go
with much simpler problems) any such criterion that might be laid
down would seem to be far less
"natura~1
and convincing than the
corresponding one for simpler types of problem.
the situation is this.
By and large,
In each case the procedure we have offered
can be shown to belong to a class that is good in a
"natura~'
sense;
and, among that class, the procedure can be shown to be optimal in
a sense that is far less "natural" and convincing. Such results on
the so-called "optimality' as are known to date on these specific
problems, and any light in this sense that
we may be able to
throw on the general method itself will be offered later.
But in
these complex situations, we must caution once more against any
hasty attempt to set up an optimality criterion, then obtain an
optimal procedure under that criterion, and finally stay happy
with that procedure.
Turning now to the design aspect of the total problem, we
recall that, once we lay down the objective, it has been possible,
at least for these problems, to suggest a test procedure that,
under any design, in general, is good in the sense already explained (at this point no optimality is claimed).
With this pro-
cedure at our disposal the further problem of choice of a good
26
design, at least for these problems, would be governed by the possibility of (i) reducing the
rr~ (a widely occurring empirical
phenomenon well known to the design of experiments and ANOVA
people, but apparently. almost totally unknown to other groups of
mathematical statisticians) and (ii) further increasing the discrimination in the preferred directions (already favored under the
test procedure) through the structure of the design sought for, if,
necessary, by introducing some kind of asymmetry.
The remarks made in
the previous paragraph about "good" and "optimal" in relation to a
test procedure would equally apply to the choice of a design.
As,
for example, in the above problems, it is' possible to choose a
design that would be good in the sense of being better than the
customary ones, but optimality is another matter.
For a general
theoretical treatment of optimality, in addition to the difficulties
already mentioned in relation to the test procedure, we encounter
the further difficUlty about defining a sufficiently wide and meaningful class of "good" designs among which the one sought for is to
be "optimal".
However, in the sense explained in section 5, we have
a very restricted treatment of optimality.
In other words, it is
broadly indicated there how it is possible, on the basis of the criteria and techniques (i) and (ii) mentioned just now, to pick out
an "optimal" design among a particular small class of designs each
of which is "good".
So far as tbe general treatment of optimality
is concerned, the little that has been done, for whatever it is worth,
will be discussed later.
Finally, we would like to add the same note
of caution as at the end of the last paragraph.
REFERENCES
1)
Fisher, R. A., The Design of Experiments, (Fifth Edition), Hafner
PUblishing Company, Inc., New York (1949).
2)
Nair, K. R., and Rao, C. R.,"Incomplete Block Designs for Experiments
3)
Involving Severa.l Groups of Varieties,"Science and Culture,VII,
(1942) .
Ramachandran, K. V., "On Certain Tests and The Monotonicity of
their Powers," Institute of Statistics, University of North
Carolina Mimeograph Series, No. 120 (1959).
4)
Roy, S. N., "On a Heuristic Method of Test Construction and Its
Use in Multivariate Analysis," Ann. Math. Stat., 24(1953) ,220-238.
5)
Roy, S. N., Some Aspects of Multivariate Analysis, John Wiley and
Sons, New York (1958).
6)
Tukey, J. W. "Allowances for Various Types of Error Rates,"
(Invited Address), Inst.Math.Stat., March, 1952.