OPTIMUM BLOCK DESIGNS FOR NEIGHBOR TYPE COVARIANCE STRUCTURES
by
John P. Morgan
A dissertation submitted to the faculty of
the University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in the Department of Statistics
Chapel Hill
1983
t\proved by
~~~
.. ~t
-.
Advisor
JOHN P. MORGAN.
Optimum Block Designs for Neighbor Type Covariance Structures.
(Under the direction of INDRA M. CHAKRAVARTI)
For a design with b blocks of k plots each, consecutively number the plots
within a block 1,2, ... ,k (i.e. number them according to their position).
The
covariance structure for the design is said to be an m-th-nearest neighbor
structure (NNm) if yields from plots i and j within the same block have correlation p for !i-j I = s, where p ~ 0 for s~m, and p = 0 for s>m, and if
s
s
s
yields from plots within different blocks are uncorrelated. This dissertation
explores the behavior of balanced incomplete block designs (BIBD's) with respect to the NNI and NN2 covariance structures, along with a few results concerning cyclic neighbor covariance structures.
Conditions are found for a
BIBD to be weakly universally optimum (w.u.o.) with respect to least squares
estimators for the NN2 structure, which give the NNI results of Kiefer and
Wynn (AnnaZs of Statistias, 9 (1981), 737-757) as a special case.
Necessary
conditions for the existence of these designs are found which are also useful
in determining if an optimum design is minimum optimum (for given k and v =
treatment number, a design is said to be minimum optimum if it has a minimum
b among the optimum designs).
It is shown that if k
~
0 (mod 4) then a min-
imum w.u.o. NN2 design is given by a k rowed semi-balanced array of strength 2
and index 1 whenever the array exists. Conditions are also found for which
transitive arrays give
minimum NNI and NN2 w.u.o. designs.
It is shown that for k=3, the NN2 and NNI conditions are equivalent. Two
algorithms for construction of these designs, depending on whether v is odd
or even, are given to solve this case.
A technique for developing ordinary BIBD's into NNI w.u.o. designs based
on Hamiltonian path decompositions of complete graphs is presented.
This tech-
nique gives several infinite series of minimum designs, and the case k=4 is
ii
solved for v
= 0,1,4,5,8,9
and 11 (mod 12).
A graph related procedure also
solves the construction of NN2 complete block designs.
Another technique for developing ordinary BIBD's with odd k is given
based on Hamiltonian cycle decompositions of complete graphs.
The developed
designs do not satisfy the conditions for weak universal optimality, but do
possess two important properties that make them useful: (i) the covariance matrix of the estimated treatment effects is constant on the diagonal and has
only two distinct values off the diagonal, hence elementary treatment contrasts
are estimated with two distinct variances, and (ii) for a given initial BIBD,
the developed design has fewer blocks than the w.u.o. design developed according to a Hamiltonian path decomposition.
Hence these designs can be considered
to serve some of the same purposes for the NNI structure as do the partially
balanced incomplete block designs for the case of uncorrelated errors.
The
2
difference in the two variances is usually a small fraction of PIa , and many
of these designs are shown to be minimax optimum for estimating elementary
treatment contrasts.
Construction of these designs depends upon appropriate
choice of a representative treatment for each block of the initial BIBD, whereby
an association scheme is induced on the treatment set.
In those cases for which
the association scheme is based upon a strongly regular graph, the initial BIBD
can be modified to give a PBIBD.
Hamiltonian cycle decompositions are also employed in the construction of
complete block designs with odd v.
These designs have less than half of the
number of blocks required for NNI weak universal optimality, are reasonably
variance balanced, and possess some interesting minimax properties as well.
Finally, an algorithm is given for constructing complete block designs
that are w.u.o. for an arbitrary degree of cyclic block correlation, for v
pn where p is an odd prime.
=
iii
ACKNOWLEDGEMENTS
I wish to thank my advisor, Dr. Indra M. Chakravarti, for the guidance
he has provided on this dissertation, and for the many hours in discussion
he has spent with me over the past two years.
I also wish to thank my committee members, Dr. Douglas G. Kelly,
Dr. Norman L. Johnson, Dr. Barry Margolin, and Dr. Ladnor D. Geissinger,
for the time and effort they have invested on my behalf.
And special thanks to Ms. Ruth Bahr, for her excellent job of typing,
and for tolerating the author.
iv
TABLE OF CONTENTS
CHAPTER I:
INTRODUCTION
CHAPTER II:
OPTIMALITY CONDITIONS FOR THE NN2 COVARIANCE
STRUCTURE
1
6
2.1
General notation and definitions
2.2
Optimality results
10
ARRAY CONSTRUCTIONS FOR OPTIMUM DESIGNS
20
3.1
Incomplete block designs from arrays
20
3.2
Optimum complete block designs
29
GRAPH THEORETIC CONSTRUCTIONS
39
4.1
Preliminaries
39
4.2
NNI designs from graphs
41
PARTIALLY VARIANCE BALANCED DESIGNS
49
5.1
Introduction
49
5.2
Incomplete block designs with partial
variance balance
50
5.3
Optimality results
64
5.4
Complete block designs with partial
variance balance
73
CHAPTER III:
CHAPTER IV:
CHAPTER V:
CHAPTER VI:
6
OPTIMUM DESIGNS FOR CYCLIC NEIGHBOR COVARIANCE
STRUCTURES
82
6.1
Introduction
82
6.2
Cyclic complete block designs
83
BIBLIOGRAPHY
101
CHAPTER I
INTRODUCTION
The theory for estimation of the parameter vector
model X =X,§ + f,
var(f,) =
in the literature.
0
2
~
in the linear
1, is well known and has been widely treated
In particular, consider the case ]T =
(~~ ! ~~) where
§l is the vector of parameters of interest (treatment effects) and §2 is
a vector of parameters that while not of direct interest to the experimenter, must necessarily be included in the model because of the nature of the
experimental material (block effects).
The choice of X is usually made to
satisfy some combination of two general objectives: that "good" estimates
of §l are obtained, where "good" is expressed in terms of some specified
criteria and subject to the constraints imposed by the available experimental material, and that the analysis of the experiment is not overly complex.
Hence the classical designs such as complete blocks, Latin squares,
balanaced incomplete block designs (BIBD's), and Youden
sq~ares.
They pro-
vide for unbiased estimation of the treatment effects, variance balance of
elementary treatment contrasts, and a choice of number of blocks, number of
plots per block, and elimination of single or two-way heterogeneity.
Their
analyses are not difficult and can be found in a number of standard references,
along with a
~reat
many plans (e.g. Cochran and Cox (1957)).
More recently,
designs.
mu~h
work has been done in the area of optimality of
e'"1 ' X = V be the covariance matrix of
Let V
of) §l with design matrix X, and let
line.
Then a design
X~
is said to be
~
(some set of contrasts
be a functional mapping V to the real
~-optimal
if it minimizes
~(V)
for
2
XEX, a general class of designs.
optimum if it minimizes
(1)
~(V)
A design is said to be weakly universally
over XEX for every
~
satisfying
is convex
~
is invariant under row and column permutations of 0
(2)
~(D)
(3)
~(bD) ~ ~(D)
for all
b~l
For instance, Kiefer (1958) has shown that for the class of designs with
b blocks and k<v (=
#
treatments) plots per block, the BIBD's are the unique
weakly universally optimum designs.
Similar results exist for many of the
classical designs (e.g. Wald (1943), Kiefer (1975), Takeuchi (1961)).
2
2
When var (U = a C f a I however, the case is not so clear.
If C is
known, the Gauss-Markov approach can be taken to find the best unbiased estimates of
~l
difficult.
for a given X, but choice of a good X depends on C and can be
The problem becomes even more complex when C involves one or more
additional unknown parameters: good estimates of
~l
are not easily found,
and the choice of X depends on the unknown variance parameters.
One could
ignore the fact that C II and proceed in the usual least squares fashion, but
this approach is feasible only if there is some reasonable assurance that V
is not overly inflated . . Kiefer and Wynn (1981) suggest the following twostep approach:
(1)
assuming C=I, find the class of designs X* in X that are optimal
when using least squares estimates
(2)
now find the class of designs X** in X* that are optimal for the
least squares estimates but for C in some relevant class of symmetric, positive definite matrices.
The experimenter will choose some XEX**
and upon completion of the experi-
ment proceed with the standard least squares estimation.
3
Of concern here will be several classes of covariance structures of
the nearest neighbor type: units are assumed to be correlated according to
their relative positions, or "nearness as neighbors", in say time or space.
The classes of designs X* to be considered are the complete block designs
and the balanced incomplete block designs.
Chapter II develops needed no-
tation and gives strong optimality conditions for both types of designs and
the second order nearest neighbor (NN2) covariance structure.
As a particu-
lar case, these conditions yield the results obtained by Kiefer and Wynn
(1981) in their treatment of the first order nearest neighbor (NNl) structure.
Some useful necessary conditions for existence are found as well.
The optimality conditions found involve the number of times a pair of
treatments occurs as both nearest and second nearest neighbors,
of these counts alone is not sufficient.
but balancing
This does, however, give a good
indication of the relationship of the designs constructed herein to those
previously dealt with in the literature.
Most similar may be those of E.J.
Williams (1949, 1950) and the handcuff designs. Assuming uncorrelated errors
but a special type of linear model leads Williams to complete block designs
with the requirement that ordered pairs of treatments occur equally often
as first neighbors and as second neighbors.
In the work presented here only
unordered pairs need be considered, but a not too difficult modification
often produces the designs he seeks, as will be seen in Chapter III.
Handcuff designs are incomplete block designs which require that unordered pairs of treatments occur equally often as first neighbors.
These
designs are essentially different, however, in that they do not constrain the
treatment intersection numbers to be equal.
signs is that of Lawless (1974).
Among the work on handcuff de-
Hung and Mendelsohn (1977) have given a
complete solution for handcuff construction.
4
Also as regards incomplete block designs there is work on the repeated
measurement designs, for example Heyadat and Afsarinejad (1975) (who give
an extensive bibliography).
They require equality of nearest neighbor counts
for ordered pairs, but again do not require equality of the treatment intersection numbers.
Further discussion of the relevance of these designs may
be found in Kiefer and Wynn (1981).
Chapter III is devoted to design construction from special types of
arrays, including the semi balanced and transitive arrays which yield optimum designs for the NNl and NN2 covariance structures.
A complete solution
for designs in three plot blocks is found and a characterization given. The
complete block case is also solved for the NN2 structure.
In Chapter IV a graph theoretic approach to NNI design construction is
presented.
Several infinite series of designs are found, and a partial so-
lution is given for the case of four plot blocks.
Chapter V considers the construction of BIBD's which exhibit partial
variance balance for the NNI structure.
This approach may be thought of
analogously to that taken for the partially balanced incomplete block designs
in the i.i.d. case: parameter combinations impossible with full variance
ance can be obtained.
bal~
In fact, the constructed BIBD's are shown to have an
interesting relationship with partially
balanced incomplete block designs.
The construction procedure is graph theoretic, and some weaker optimality resuIts are obtained.
A series of partially variance balanced complete block
designs, which enjoy several optimality properties, is also presented.
In a similar vein, Kiefer and Wynn (1981) have proposed the use of EBIBD's,
that is, BIBD's for which treatment pairs occur equally often as first neighbors.
"Such designs", they say, "besides being A-optimum, make
all
the
var(t.) equal and can be expected to perform fairly efficiently for criteria
1
other than A-optimality".
EBIBD's do not in general exhibit full variance
balance and hence may be compared to those constructed here.
In Chapter VI, cyclic neighbor covariance structures are briefly considered.
It is shown that a complete block design in pn-l(v_l)/2 blocks bal-
anced for an arbitrary degree of cyclic neighbor correlation can be constructed for v=p
n
treatments, where p is any odd prime.
Here one may compare the
neighbor designs of the literature: sets of cycles for which every pair of
treatments occurs equally often as first neighbors without requiring that a
treatment is present at most once in any cycle.
These designs were intro-
duced by Rees (1967); the construction problem was later solved by Hwang and
Liu (1977).
Cheng (1983) has shown that a neighbor design that is also a
BIBD may be developed to give an EBIBD.
Mendelsohn (1977) has proven some
results for incomplete cyclic blocks with ordered neighbor balance.
be mentioned are the type III designs of R. M. Williams (1952).
Also to
These are
single cycles in which every ordered pair of treatments occurs equally frequently as first neighbors and as second neighbors.
CHAPTER II
OPTIMALITY CONDITIONS FOR THE NN2 COVARIANCE STRUCTURE
2.1
General notation and definitions
The notation developed here will be maintained throughout the disser-
tation.
Define a balanced incomplete block design (BIB design or simply BIBD)
to be an arrangement of v symbols (treatments) into b groups (blocks) of k
treatments each such that each symbol occurs r times, no symbol occurs more
than once in any block and any pair of distinct symbols occur together in
A blocks.
r is said to be the replicate number and A the treatment inter-
section number.
Necessary conditions for the existence of such a design are
bk
= vr
(2.1.1)
A(v-l) = rlk-l)
A BIBD with the above parameters will often be denoted BIBD(b,v,k,r,A).
I
Each block of a BIBD(b,v,k,r,A) is assumed to be composed of k plots,
or experimental units, to which the treatments are allocated.
Arbitrarily
labeling the blocks 1.2 •... ,b and numbering the plots withing a block 1,2, ... ,k,
plot s of block i can be associated with the ordered pair (i,s) for i =
... ,b and s =
l,~,
... ,k.
1~2,
Take the v treatment symbols to be O,I,2, •.. ,v-l.
Now define g(i,s) as the treatment symbol allocated to plot s of block i
and write A. = {i: g(i,s) = i for some s}.
1
containing treatment i, i=O,l, ... ,v-l.
That is, A. is the set of blocks
1
7
Let Y
be the observation from plot (£,s).
£,s
The Yo
N,S
arranged in the
lexicographic order (£,s) are the coordinates of the observation vector
In general, the numbering of the plots within a block of the design will
represent an ordering of those plots.
Hence
block £ may be pictured along
with the corresponding observations as
1
2
3
·
(2.1.2)
·
·
k
This is what often is referred to as a linear block (e.g. Hwang (1976)), and
will be the focus of most of the work here.
Some results will be obtained,
however, for which it is convenient to picture circular blocks; appropriate
notation will be introduced as needed.
Note that no ordering of the blocks
is implied by their labels.
Define
h(i,£) = s
iff
P£: kxv matrix (£=1,2, ... ,b)
g(£,s) = i
. h p£ .
W1t
S,l
=
(P 0)
N
. = 1 i f g(£,s) = i
S,l
= 0 otherwise
8
L
I
Dxbk = b
Qvxl:
~bXl:
vector of treatment effects
e. = #{t: g(t,l) = i
or g(t,k) = i}
f. = #{t: g(t,2) = i
or g(t,k-l) = i}
1
1
0
.T
lk
vector of block effects
e . . = #{t€A.nA.: g(t,l) = i
or
g(t,k) = i}
#{t€A.nA.: g(t,l) = j
or
g(t,k) = j}
. = #{t€A.nA.: g(t,2) = i
J
1
J
or
g(t,k-l) = iJ
#{t€A.nA.: g(t,2) = j
or
g(t,k-l) = j}
1, J
J
1
+
J
1
f.
1,
+
J
1
N~.
= #{t: get,s) = i, g(t,q) = j,
1)
~
T
E
= (eO,el,···,e _ )
v l
= (e .. )
F = (f .. )
1J
1J
Is-q!
= t},
t=1,2,
i~j
fT
= (fO,f ,··· ,f _ )
l
v l
t
Nt = (N t.. ) with N..
= O.
1l.
IJ
Some explanation of the above quantities may prove insightful.
Pt is
the plot/treatment intersection matrix for block t; hence pt . = 1 iff treatS,1
ment i appears on plot s of block t.
A, then, is the treatment/plot inter-
section matrix for the entire design, with the plots ordered as are the coordinates of
y.
L
is the block/plot intersection matrix, where (L)n)("m = 1 iff
th
the m coordinate of
1
is an observation from block t.
block intersection matrix; (N).1,)(,n
=1
N is the treatment/
iff treatment i occurs in block t.
Referring to (2.1.2), plots 1 and k of a block are denoted the end plots
for that block.
Similarly, plots 2 and k-l are the next-to-end plots. Hence
e. = the number of blocks for which treatment i occurs on an end plot
l.
f. = the number of blocks for which treatment i occurs on a next-tal.
end plot
e..
1J
= the number of blocks containing both treatments i and j for
which at least one of i
and j is on an end plot, where a
9
block is counted twice if both of i and j are on an end plot
f ..
1J
• the number of blocks containing both treatments i and j for
which at least one of i and j is on a next-to-end plot, where
a block is counted twice if both of i and j are on a next-toend plot.
Again referring to (2.1.2), two adjacent plots are said to be first
neighbors, plots separated by a single plot are second neighbors, and so on.
For instance, in (2.1.2) plots 1 and 2 are first neighbors while plots 1 and 3
are second neighbors.
t
Thus N..
1J
is the number of times treatments i and j
. hb ors.
occur as t th ne1g
t
The quantities e., f., e .. , f .. , and N.. are collectively referred to
1
1
1J
1J
1J
as neighbor parameters. Some simple lemmas relating the neighbor parameters
will be useful.
1
N..
Lemma 2.1
1J
Proof:
= 2r-e.
1
Counting two first neighbors of treatment i in each of the r
blocks in which it occurs, one neighbor too many is counted for each of the
o
e.1 blocks in which it occurs on an end plot.
Lemma 2.2
. 2
N..
1J
jh
I
Lemma 2.3
I
Hi
Proof:
e ..
1J
= 2r-e.1 -f.1
= (k-2)e.+2r
1
The e. blocks for which treatment i occurs on an end plot each
contribute k to
1
I
e .. : a contribution of 2 for the other end plot and k-2
1J
Jr
for the remaining
k-2 plots. The r-e.1 blocks for wh1ch treatment i does not
.
• J. •
1
occur on an end plot each contribute 2 to
= (k-2)e.+2r.
1
I
j#i
e ...
1J
Hence
L
• J..
Jr 1
e ..
1J
= ke.+2(r-e.)
1
1
o
L
Lemma 2.4
f.
10
= (k-2)f +2r
i
lJ
j;ei
0
Finally, a few standard definitions will be given to insure clarity.
Let B = (b. 0) and C = (c .. ) be pxq and sxt matrices, respectively. De-
lJ
lJ
fine the Kronecker or direct product B
B
~
C = (b
0
0
lJ
~
C to be the psxqt matrix given by
C).
Define
is:
sxl
vector of ones
J pxq =ip~L,
"""y
I
J
pxp
==
J
P
pxq matrix with entry (i,i)=l, i=1,2, ... ,min(p,q)
pxq
and all other entries zero
I
0
P
-
pxq
I
pxp
pxq matrix of zeros, 0
0
0
0
pxp - p, pXl -
~p
For any two integers a and b,
g.c.d.(a,b) = greatest common divisor of a and b .
.Q,. C. T:l.
(Lh)
= least common multiple of a an db.
For any real number y, y = int(y)+frac(y) where
int (y) = integer part of y
frac (Y) = fractional part of y'
.2.2
Optimality results
Proceeding as discussed in Chapter I, the class of designs to be consid-
ered here is the class
X for which v treatments are to be tested in b blocks
of at most k plots per block.
The optimal subclass
X* of X for uncorrelated,
equi-variable observations is (Kiefer (1958, 75)):
(i)
k<v
the balanced incomplete block designs with k plots per block,
whenever they exist
11
(ii)
k=v
the designs for which every treatment occurs once in each
block, that is, the complete block designs.
The problem is to find the optimal subclass X** of X* with respect to a
given class of covariance matrices and employing least squares estimates.
Con-
sidered here will be classes of covariance matrices of the form
coveY N,S
,Yo,
,)=0
n
N,S
2
£=£', s=s'
(2.2.1)
2
= 0 ps , s' £=£', s#s'
=0
£~£'
that is, yields from pairs of plots within a block are correlated according
to their relative positions; yields from plots in different blocks are uncorrelated.
The assumed model for the observation vector
1=
where
]J
T
1
is
T
~k + A Q: + L .§ + £bkxl
]J
is a constant and
£
is a random vector such that E(£) =
Q
and
Vkxk ' v* = (02p s, t) with ps,s = 1. Optimality results will
1
be based on the set of contrasts t = (I - -J ) a. This is a reasonable choice
'"
vVV'"
in that £T ! = £TQ: for any contrast vector £. The well known least squares
var(~)
= Ib
estimates
~
of! for a BIBD(b,v,k,r,A) are
A
! = k(AV)
-1
(A-k
-1
NL)y = k(AV)
-1
Q.
The following lemma was stated by Kiefer and Wynn (1981).
Lemma 2.5
Write W = (I -k
k
structure given in (2.2.1),
var(!)
In particular,
A
A
cov(t. , t.)
1
J
-1
J)
k
V*(I _k-lJ )
k
k
=
(w
st
). With the covariance
12
Proof:
var(Q) = (A-k-1NL)(Ib~V*)(AT_k-1LTNT)
1 T
= A[(Ib@V*) - k- L L(Ib@V*) - k-1(Ib~V*)LTL+ k2LTL(Ib~V*)LTL]AT
= A[(Ib~V*) - k-1(Ib~JkV*) - k-1(Ib~V*Jk) + k2(Ib~JkV*Jk)]AT
(since LTL= Ib~Jk)
= A[Ib~(Ikk-1Jk)V*(Ik-k-1Jk)]AT
b
=
L P~
since
WPR,
R,=1
Hence, cov(Q.,Q.) =
1
J
b
(L
R,=l
T
b
PR, WPR,) . . = L
1,J
v
=
R,=1
v
v
L L
t=l u=l
R,' R,
P t ,1·PU,J· W
t,u
v
R,
R,
Pt,iPu,j Wt,u
R,EA. nA. t=l u=l
L I
L
1
J
are both in block
=
L
R,EA.nA.
1
Wh(i,R,) ,h(j ,R,)
~
Q;
since both Pt,i and Pu,j
are nonzero iff i and j
~.
by definition of h
J
o
Let the second order nearest neighbor (NN2) covariance structure (a subclass of (2.2.1)) be given by
Ps , t
=
1
Is-tl=o
PI
Is-t\=l
P2
Is-tl=2
a
Is-tl>2
(2.2.2)
The first order nearest neighbor (NN1), or simply nearest neighbor, covariance
structure is obtained from (2.2.2) by setting P = o.
2
Lemma 2.6
With the NN2 structure, for
k~4,
13
A
A
2
-2
2
1
2
cov(t. ,t.) = a (Av) {-A[k+2P (k+l)+2P (k+2)] + k [PIN .. +P N.. ] + k(P 1+P )e ..
2
1
2
2 lJ
1 J
lJ
lJ
+k p f .. }
2 lJ
(ih)
Proof:
w
s,t
For the NN2 structure,
k
l.
2
Ps , t = 1 + PI + P2 + PI IIt + P2 1 t
(k~4)
I
2
Ps , t = 1 + PI + P2 + P1 IIs + P2 1 s
(k~4)
s=1
k
t=1
k
I
k
l
t=1 s=1
where II = 1
if t
t
= 0
Hence
Ps , t = k + 2(k-l)P 1 + 2(k-2)p 2
E
{2,3, •.. ,k-l}
I~
otherwise
Ws,t = k-2a2[k2ps,t - kP 1 (I~+I~) -
if t
=0
otherwise
kP2(I~+I~)
Thus by Lemma 2.5
var(t.) = (Av)
-2 2
a
1
'\
2
[k -k-2(P +2P )
1
2
tEA.
L.
1
= (AV)
-2 2
{3,4, ... ,k-2}
= 1
a {r[k(k-l)-2(P 1+2P 2 )]
E
- k - 2(P 1+2P 2 )].
14
=
(AV)-2a2{-A[k+2Pl(k+l)+2P2(k+2)]+k2[PlN~.+P2N~.]+k(P
1J
1J
1
+p )e .. +kp f .. }
2
1J
2 1J
o
Lemma 2.7
With the NN2 structure, for k=3,
var(t.)
1
and for i#j
cov (t. , t.)
1
Proof:
J
Proceeding as in the proof of Lemma 2.6,
k
k
Ips t= 1+Pl+P2+(Pl-P2)I(S=2)' I Ps t=1+P l +P 2+(P l +P 2 )I(t=2)
t=l '
s=l '
k
I
k
I
P
= 3 + 4P l + 2P
2
s=l t=l s,t
2 -2
=> k a
w
s,t
Hence,
2
A
-2
var(t i ) = a (Av) £Ei. (9P. h (i,£),h(i,£) -3-2Pl-4P2-6(PI-P2) I (h(i,£)=2))
1
=
2
a (Av)
-2
[r(6-2P -4P 2)-6(P -P ) (r-e i )]
l
l 2
and for i#j
1
1J
2
1J
Using the relationship N.. + N.. = A for k=3 gives the result.
0
The following theorem, due to Kiefer, will be needed.
Theorem 2.8
Let 0 be a class of experimental designs and for each dED
15
let V be the covariance matrix for the estimates of a given set of parameter
d
contrasts.
Suppose iTVd = aT for each dED, and that there exists d*ED such
that
(i)
(ii)
Vd * = al+8J
for some a,8
trace Vd* = min trace V
d
dED
Then d* is weakly universally optimum in D.
Proof:
o
Kiefer (1975), Kiefer and Wynn (1981)
A matrix satisfying condition (i) of Theorem 2.8 is said to be completely
Since Le. = Lf. = 2b, it is an easy consequence of the preceding
symmetric.
1.
1.
two lemmas that for given b, v, and k, trace var(!) is independent of the
choice of BlBD.
Hence
by Theorem 2.8, the weakly universally optimum BlBD's
for the NN2 covariance structure will be those for which var(!) is completely
symmetric.
Theorem 2.9
A BlBD(b,v,k,r,A) is weakly universally optimum for the NN2
model i f
(i)
(ii)
1
the quantities kN .. +e .. are all equal (iFj)
1.)
1.)
2
the quantities kN .. +e .. +f .. are all equal (iFj)
1.)
1)
1)
For k=3, the conditions (i) and (ii) are equivalent to equality of the
(i~j)
.
Proof:
By Theorem 2.8 and Lemma 2.6, for
k~4,
the BlBD will be weakly
universally optimum if
(1)
(2)
1
Z
the quantities k(PlN .. +P N.. )+(p +P )e .. +PZf .. are constant in
1 Z 1)
Z
1)
1)
1)
i,j
(i~j).
1
N ..
1.)
16
Writing P2=aPl and rearranging, this requires that
(1')
the quantities a(e.+f.)+e. are constant in i
(2')
the quantities a(kN .. +e .. +f .. )+kN .. +e .. are constant in i,j
111
2
1
1J
1J
1J
1J
1J
By Lemmas 2.1-2.4
2
I
1
[a(kN .. +e .. +£ .)+kN .. +e .. ]
1J
jfi
1J
1J
1J
1J
= a[k(2r-e.-f.)+(k-2)e.+2r+(k-2)f.+2r] + k(2r-e.)+(k-2)e.+2r
111
1
1
1
= 2r[a(k+2)+(k+I)] - 2a(e.+f.)-2e.
111
Hence (2') => (1') and the theorem is proved for
For k=3 it is easy
k~4.
to verify that
1
N.. +
1J
2
N..
= A,
1J
f ..
1J
=
1
1
N.. , and
= 2A.
+ e ..
N ..
1J
1J
1J
Hence
kN I.. +e ..
1J
1J
0
2A+(k-l)N~.
1J
2
and kN . . +e . . +f ..
1J
1J
1J
=
1
(k+2) A-kN ..
1J
o
and the result follows from Lemma 2.7.
The NNI results of Kiefer and Wynn (1981) are obtained by setting a=O
in the proof of Theorem 2.9:
Corollary 2.10
(Kiefer and Wynn)
A BIBD is weakly universally optimum
1
for the NNI covariance structure if the quantities e .. +kN .. are all equal
1J
1J
(ifj).
A BIBD that satisfies the conditions of Theorem 2.9 will be said to be
NN2 optimum, or simply an NN2 design. Likewise a BIBD satisfying the conditions of Corollary 2.10 will be said to be NNI optimum, or an NNI design.
Taking k=v and A=b in Theorem 2.9 gives the result for the complete
17
block
In fact, the additional symmetry of a complete block design can
c~se.
be used to make a slightly stronger statement.
Theorem 2. 11
For a complete block design with v treatments arranged in
bblocks, and the NN2 covariance structure, yare!) is completely symmetric
2
(i ~j) are all equal and the N.. (i~j) are all equal.
1.)
1.)
For the NNl structure, yare!) is completely symmetric iff the N~. (i#j) are
1.J
all equal.
i f and only i f the
Proof:
N~.
The NN2 proof will be given.
For a complete block design, e .. =e.+e.
1.) 1. )
and f .. =f.+f .. So by Lemmas 2.1 and 2.2, equality of the N~ . and of the
1.J 1. J
1.J
2
N.. implies equality of the e .. and of the f ... Hence yare!) is completely
1.)
1.J
1.J
symmetric.
A
Now suppose yare!) is completely symmetric.
b2i
Lemma 2.6 gives
A
---2-- yare!) = b[v(v-l)-2(P l +2P )-2v(P +P )] Iv + 2v(P l +P 2) diag(~)
2
l 2
(J
. .
2 1
2 2
+ 2VP2 d1.ag(f)+Plv N +P 2v N -b[v+2Pl (v+l)+2P 2 (v+2)](J -I)
v
+ v(P +P ) (E-2 diag(~)) + vP 2 (F-2 diag(!))
l 2
2
2 1
2 2
= bv I - b[v+2P (v+l)+2P (v+2)]J + PlY N +P v N
2
2
l
(2.2.3)
+P )E + vp F
122
+ v(p
be any vxl vector such that ~Tiv=O and
"'V
.T
.
T
·T
. fT
f .Lv
E = ~.:Lv+.:Lv~
an d F = '"
+ .:Lv
'" •
Let a
T
~ ~=l.
Noting that
(2.2.4)
Since complete sYmmetry of yare!) implies variance equality for all normalized treatment contrasts, (2.2.4) requires equality of the
equality of the N~. (i~j).
1.J
N~.
1.) (i#j) and
o
18
The final few results of this section give restrictions on the parameters of a BIBD or complete block design imposed by the optimality results,
and hence necessary conditions for the existence of the optimum designs.
Theorem 2.12
An NN2 optimum BIBD(b,v,k,r,A) satisfies k(k-l) 14A, and if
kFO (mod 4), k(k-l) !2A.
Proof:
L L (e .. +kN~.)
i jl i
1J
1J
= L(2r(k+l)-2e.) = 2[vr(k+l)-2b]
i
1
= 2b(k+2) (k-l) = 2vr(k-l) (k+2)/k = 2A(k+2)v(v-l)/k
1
So
(e .. +kN .. )
\'
'i'
1J
1J
l
l
(1)
i jl i
v v-
4'
=
1\
(2.2.5)
2A + -k
=> k14A, since e .. +kN~. are equal integers.
1J
1J
L
L (kN~.+e
.. +f .. )
i j#
1J 1J 1J
Also,
= L(2r(k+2)-2(e.+f.)) = 2[vr(k+2)-4b]
i l l
= 2b[(k-l) (k+4)-k] = 2v(v-l) A[(k-l)(k+4)-k]/k(k-l)
2
(kN .. +e .. +f .. )
SA
2A
So L L __l...z.J--,-=lJ"-,--_l.LJ_ = 2A + - - => (k-l) I 2A since kl 4A.
i jli
v(v-l)
k
k-l
Now
g.c.d. (k,k-l)=l with the conditions kl4A and (k-l)! 2A implies k(k-l)1 41..
If kFO (mod 4), kl4A => kl2A => k(k-l)1 21..
o
An immediate consequence of the proof of Theorem 2.12 is
Corollary 2.13 (Kiefer and Wynn) An NNI optimum BIBD(b,v,k,r,A) satisfies k14A.
Similar methods show that a consequence of Theorem 2.11 is
Corollary 2.14
An NN2 optimum complete block design with v treatments in
b blocks satisfies v(v-l)! 2b.
An NNI or NN2 optimum_BIBD is said to be minimum if for given v and k
it attains the smallest possible value of b.
The above results indicate
that NNI and NN2 BIBD's will generally have several times the number of
19
blocks of ordinary (minimum) BIBD's.
The restrictions on NN2 complete
block designs are even more severe. Hence the goal of optimum design construction will be to, whenever possible, construct minimum optimum designs,
and more generally to keep b reasonably small.
CHAPTER III
ARRAY CONSTRUCTIONS FOR OPTIMUM DESIGNS
3.1
Incomplete block designs from arrays
A
txN array of v symhols is said to be semi balanced of strength d
mId index i=N/(~) if for any choice of d rows the N columns contain each
v
of the Cd) unordered d-tuples of distinct symbols exactly ~ times. Such an
an array will be denoted SB(N,t,v,d).
A transitive array, TA(N,t,v,d), of
strength d and index i=(v-d)!N/v! is a txN array of v symbols such that for
any choice of d rows the N columns contain each of the v!/(v-d)! ordered
d-tuples of distinct symbols exactly i times.
of strength d and index
~
Clearly a transitive array
is a semi balanced array of strength d and index
~(d!). An orthogonal array, OA(N,t,v,d), of strength d and index ~=N/vd
is a txN array of v symbols such that for any choice of d rows the N columns
contain each of the possible v
Theorem 3.1
Proof:
d
d-tuples exactly
~ times.
_
~v(v-l)
~t(v-l) it(t-l)
SB(iv(v-l)/2,t,v,2) ->NN2-opt.BIBD(
2
,v,t,
2
'
2
).
Taking the columns of the semi balanced array as the blocks of a
design, it is easy to see that the indicated BIBD is obtained.
1
1J
N.. =
Corollary 3.2
If
~(t-l),
t~O
2
1J
N.. =
~(t-2),
e .. = f .. = n(t-l)
1.1
1J
Also,
o
(mod 4), the BIBD given by SB(v(v-l)/2,t,v,2) is a
minimum NN2 design.
Theorem 3.3
(i)
Suppose g.c.d.(k(k-l), v(v-l)) = 2.
SB(v(v-I)/2,k,v,2) is minimum NNl as well as minimum NN2 optimum.
21
(ii)
If v is even and k is odd, TA(v(v-l),k,v,2) is minimum NNI and
minimum NN2 optimum.
(i)
Proof:
b=Av(v-l)/k(k-l)
and g.c.d. (k(k-l) ,v(v-l)) = 2 => k(k-l) In
=> v(v-IJ\2b.
(ii) v is even and k is odd so g.c.d. (k(k-l) ,v(v-l)) = 2 ~ g.c.d(k-l,v-l) = 1.
Now r=A(v-l)/(k-l) => k-l!A.
Thus k(k-l)12A (from (i)) and k:odd~ k(k-l) IA
0
=> v(v-l) lb.
, Rao (1961) introduced semi balanced arrays as orthogonal arrays of type II,
but used the name semi balanced in a later (1973) paper.
he gives a construction for SB(v(v-l)/2,t,v,2) where v=p
odd prime p and
Theorem 3.4
1 1
1
(Mukhopadhyay)
s
where
~
= IT
. 1
1=
(~.~!),
1.
1
The existence of
is a power of an
SB(~.v.(v.-l)/2,t.,v. ,2)
11111
i=1,2, ... ,s, implies the existence of
1
n
This result was extended by Mukhopadhyay (1978):
t~v.
OA ( ~ ! v ~ , t! , v . , 2) ,
In the 1961 paper,
s
v = IT vi'
i=l
and
SB(~v(v-l)/2,t,v,2)
and t=min (t ' ... , t S , til, ... , t SI) •
l
.
Another result from the same paper is:
Theorem 3.5
(Mukhopadhyay)
v=20s+l or 20s+5,
Proof:
SB(v(v-l)/2,t,v,2) exists for
1~t~5
and
s~l.
This is a consequence of Theorem 2.2 of Mukhopadhyay (1978).
0
Presented next is an algorithm for semi balanced array construction
which, although it includes only a subset of the array parameters covered by
Theorem 3.4, is of interest for its simplicity.
Theorem 3.6
Let v be an odd integer and suppose c ' c ' ... , c are n distinct,
l 2
n
nonzero residues (mod v) such that
22
c
(i)
(ii)
is coprime with v, r=1,2, ... ,n
r
c -c is coprime with v, r,sE{1,2, ... ,n},
r s
Write c n +1
= a and
r~s
form a (n+l)xv(v-l)/2 array by putting
(i+c~
) (mod v)
in column i(v-l)/2 +j of row r, i=O,l, ... ,v-l; j=1,2, ... ,(v-l)/2; r=1,2,
... ,n+l.
The array is SB(v(v-l)/2,n+l,v,2).
Proof:
It is easy to see that each of O,l, ... ,v-l occurs (v-l)/2 times
in each row.
Now fix some symbol m and pick any two rows, say rand s. The
(v-l)/2 2-tuples with m in row rare
m, i+c j (mod v)
s
where i+c j - m (mod v)
r
and the (v-l)/2 2-tuples with m in row s are
m, i+c j (mod v)
r
where i+c j - m (mod v).
s
If a pair is repeated among those for which m is in row r then there exists
(il,jl) ~ (i 2 ,j2) with i ,i 2do,1, ... ,V-l} and jlj2dl,2, ... ,(v-l)/2}
l
that
such
If a pair for which m is in row r is the same as a pair for which m is in
row s, then there exists i ,i E{O,1, ... ,V-l} and jl,j2E{1,2, ... ,(v-l)/2}
l 2
such that
=> jl+j2 - 0 (mod v), a contradiction.
23
Hence the v-I pairs involving m given by rows rand s are all distinct and
o
and the array is semi balanaced.
For given v, the maximum number of rows for a semi balanced array constructed according to Theorem 3.6 is pv = smallest prime divisor of v (take
c.=i, i=I,2, ... ,p -1).
1
v
That SB(v(v-I)/2,p ,v,2) can be constructed by Theov
2
rem 3.4 follows from the existence of OA(v ,p ,v,2) for all v and the result
v
of Rao mentioned above.
It should be noted that an equivalent form of the
construction of Theorem 3.6 was given by Gassner (1965), who presented the
result as a series of BIBD's , apparently failing to realize the semi balanced property of the blocks considered as an array.
Semi balanced arrays of strength 2 and index 1 do not exist for an even
number of symbols.
The next theorem gives a construction for three-rowed
transitive arrays of strength 2 and index I when the number of symbols is
even and at least four.
Theorem 3.7
Let
v~4
be an even integer and form a 3 x v(v-l) array as follows.
For i=O,l, ... ,v-I and j=I,2, ... ,v-I put i in column i(v-I)+j of row land
put (i+j) (mod v) in column i(v-l)+j of row 2.
put
(t+~)(mod
For row 3, for t=l, ... ,v-2·
v) in column
(~-l)
(v-l)+t-hl
~(v-I)+t-~+v
~= 1,
2, ... , t
~=t + 1,
(3.1.1)
... , v-l
Also, put 1 in column v-I and put v-2 in column (V-l)2+ 1 .
Finally, for
i=l, ... ,(v-4)/2 put
2i (mod v)
2i+1 (mod v)
in column i (v-2) +v
in column i(v-2)+v-l
and for i=(v-2) /2, ... , v-2 put
(3.1.2)
24
2i (mod v)
2i + 1 (mod v)
in column i(v-2)+v-l
(3.1.3)
in column i(v-2)+y
The array is TA(v(v-l),3,v,2).
Proof:
Take the columns to be numbered by i(v-l)+j, i=O,l, ... ,v-l;
j=1,2, ... ,v-l.
With each column associate the pair (i,j) and call i its
group and j its position within the group.
A little manipulation on (3.1.1)
shows that for row 3, group i
t+i+l (mod v)
t+i (mod v)
is in position t-i, t=i+l, ... ,v-2
(3.1.4)
is in position v+t-i, t=1,2, ... ,i-l
First it will be shoen that every ordered pair of distinct symbols is given
by rows 2 and 3.
Choose some symbol S10 or v-I.
group i1S at position j where i+j
=s
(mod v).
It appears in row 2 in
Hence it appears in group i
at position
s-i
for
s>i
v+s-i
for
s<i
Suppose s appears in a column with number i(v-2)+v-l or i(v-2)+v for some
iE{1, ... ,v-2} (i.e. a column in (3.1.2) or (3.1.3)).
Then there exists i,j
with i+j _ s (mod v) such that
i+j=v-l
i(v-2)+v-l = i(V-l)+j}
a contradiction
=>
or
i(v-2)+v = i(v-l)+j
or i+j =V ,
Hence the ,columns containing s in row 2 have entries given by (3.1.4) in
row 3.
These entries are (for each i take t=s in (3.1.4))
s+i+l (mod v)
s+i (mod v)
i=O,l, ... ,s-l
i=s+1,s+2, ... ,v-l
which are just the residues (mod v) excluding s.
Now take s=o.
In row 2
2S
it appears in column i(v-l)+j where i+j ::: 0 (mod v) and
i(v-l)+j = i(v-l)+v-i = i(v-2)+v.
i~O
(i.e. i+j=v) =>
Hence the corresponding entries in row 3
are (from (3.1.2) and (3.1.3))
2i (mod v)
i=l, ... , (v-4)/2
i= (v-2) /2, ... , v-2
2i+l (mod v)
and also v-2 in column (v-l)2+ 1 = (v-l)(v-2)+v, which are all the nonzero
residues (mod v).
Proceeding similarly for s=v-l, the corresponding entries
in row 3 are
2i+1 (mod v)
2i (mod v)
i=l, ... , (v-4)/2
i= (v-2) /2, ... , v-2
and I in column v-I, which are all the residues (mod v) excluding v-I.
Next it will be shown that every ordered pair of distinct symbols is
given by rows 1 and 3.
Choose some Sf 0 or v-I.
umns s(v-l)+j, j=l, ... ,v-I.
It appears in row 1 in col-
The corresponding entires in row 3 are
(from
(3.1.4))
t+s+l (mod v)
t+s (mod v)
t=s+1, ... ,v-2
t=l, ... ,s-l
and since Sf 0 or v-I, a pair of entries (from (3.1.2) or (3.1.3)) 2s (mod v)
and 2s+1 (mod v).
s=O.
These are all the residues (mod v) excluding s.
In row 1 it is in column j, j=l, ... ,v-1.
Now let
The corresponding entries in
row 3 are (from (3.1.4)) t+l (mod v), t=l, ... ,v-2 and 1 in column v-I, which
are all the nonzero residues (mod v).
Similarly for s=v-l, the corresponding
entries in row 3 are t+v-l (mod v) t=1, ... ,v-2 and v-2 in column (v-l)2+ l ,
which are all the residues (mod v) except v-I.
Finally, it is easily seen that every ordered pair of distinct symhols
is given by rows land 2.
o
26
/
The existence of TA(v(v-l),3,v,2) for all v>3, v#6, follows from the
work of Bose, Shrikhande and Parker (1960) on pairs of mutually orthogonal
Latin squares.
But as a means of construction, their techniques are not
so easily implemented as the method presented here.
Examp 1e 3. 1. 5
v=6
in Theorem 3.7 gives TA(30,3,6,2)
() 0 0 0 0
11111
2 2 2 2 2
3 3 3 3 3
44444
5 5 5 5 5
1 234 5
2 345 0
3 4 5 0 1
4 5 012
5 0 1 2 3
o
2 3 4 5 1
4 5
o
20145
2 3 5 0 1
40123
032
1 453
1 234
This case is especially interesting in that the array is not resolvable,
and it cannot be extended by the addition of another row, for either would
imply the existence of two mutually orthogonal Latin squares of order six.
Corollary 3.8
NNl/NN2 minimum optimum BIBD construction has been solved
for k=3.
Proof:
The NNI and NN2 conditions are equivalent for k=3.
result is a corollary to Theorem 3.6 with Corollary 2.13.
For v:odd, the
For even v, r =
A(v-l)/(k-l) = A(v-l)/2 => 21A and Corollary 2.13 => 31A, hence 61A and the
result is an immediate consequence of Theorem 3.7.
0
A number of other corollaries to the results of this section can easily
be written down.
Corollary 3.9
A few of these are listed below.
For k=4 and 5, minimum NN2 BIBD construction has been
solved for v=1,5,7
and 11 (mod 12) and for v=21 and 45 (mod 60).
The
de~
signs \vi th v=ll (mod 12) are minimum for the NNI model, as are those with
k=5 and v=7 (mod 12).
27
Proof:
The NN2 minimali ty with k=4 follows from Lemma 4.9.
If k=S and v=2,14 or 18 (mod 20), TA(v(v-l),k,v,2) is
Corollary 3.10
a minimum optimum BIBD for the NNl and NN2 models, whenever it exists.
If k is odd and v=k(k-l)+2, TA(v(v-l),k,v,2) is a minimum
Corollary 3.11
optimum BIBD for the NNI and NN2 models whenever it exists.
If v=2
Corollary 3.12
n
and v-I is prime, TA(v(v-l),k,v,2) is a minimum
optimum BIBD for the NNl and NN2 models for k=4t-l, t=1,2, ... ,2
n-2
-1.
v=32 in Corollary 3.12 gives the minimum optimum designs
Example 3.1. 6
BIBD(992,32,k,31k,k(k-l)), k=3,7,11,IS,19,23 and 27.
Two NNI or NN2 optimum designs are said to be equivalent if they can
be made identical by relabeling of blocks and treatments and by reversing
the order of blocks.
This leads to a characterization of the minimum opti-
mum BIBD's for k=3.
Theorem 3.13
NNI/NN2 minimum optimum BIBD with k=3 is equivalent to a
An
semi balanced array.
Proof:
The result will be proven for even v, the similar proof for odd v
being even simpler.
With v even, the minimum optimum design must have b =
1
1J
. 2
Arrange the blocks side to side
1J
in any order to form a 3x v(v-l) array, arbitrarily orienting each block
v(v-l), :\=6, N.. =4, and N.. =2 for all ijij.
(e.g.
g or
the block
t
~
with
g(~,l)=a, g(~,2)=b,
and
g(~,3)=c
one orientation being the reverse of the other).
may be taken
as
For any column of
the array, call the pair formed by rows 1 and 2 an upper pair, and the pa i r
formed by rows 2 and 3 a lower pair.
Let N~~
be the number of times symhols
lu
i and j occur as an upper pair, and let U = #{(i,j): i<j and N..
= 3} +
1J
1J
28
2x #{(i,j}: i<j and N~~=4}.
1J
If U=O, the array is semi balanced.
Suppose
U>O; it will be shown that U can always be reduced by 1.
Choose any pair i,j such that
N~~
1J
> 2 and reverse one of the columns
in which i and j occur as an upper pair. Either U is reduced by 1, or
Iu
Ni'j' is increased by I to 3 or 4 for some pair i', j' in which case
LJ
is unchanged.
In the latter case, choose another column in which i'
and j' occur as an upper pair and reverse it.
or
Again either
Iu
N."." is increased by 1 to 3 or 4 for some pair
1
J
'" ,
1
j" .
is reduced
U
If again the
latter occurs, choose another column in which i" and j" occur as an upper
pair and reverse it, being sure not to reverse any previously reversed
column.
Continue reversing columns in this manner, subject to the con-
straint that no column is reversed more than once, until either
duced by I or no column is available for reversing.
U
is re-
It will now be shown
that the second possibility cannot occur.
Suppose the algorithm stops without reducing U.
column reversed and let i,j be the upper pair of c .
l
there are
s~2
Let c
be the last
I
lu
Then N.. > 2 and
1J
other columns containing i,j as an upper pair which by assump-
tion have all been previously reversed.
Since the algorithm did not stop
upon reversal of any of these s columns, there are s corresponding columns
for which i,j is a lower pair.
Hence there are at least
2s+I~5
I
containing i and j as first neighbors, which contradicts N.. =4.
1J
columns
0
That a minimal optimum BIBD with k=3 and even v need not be equivalent
to a transitive array is shown by the following example.
29
Example 3.1. 7
1
1
1
2
1
2
2
2
3
3
3
4
2
2
2
1
3
3
4
3
1
4
4
1
3
4
4
4
2
1
3
4
4
1
2
3
By Theorem 3.13, this array is equivalent to a semi balanced array.
But column reversals cannot cause the ordered pairs (1,2) and (2,1) to
occur once each as upper pairs and once each as lower pairs.
Hence the
array is not equivalent to a transitive array.
Reversing the third and fourth columns gives a semi balanced array.
3.2
Optimum complete block designs
Define the vxv array B as the array given by placing
v
(mod v)
in row i and column j, i=O,l, ... ,v-l; j=O,l, ..• ,v-l.
Lemma 3.14
Proof:
B is a Latin square.
v
Clearly each column contains every symbol.
Suppose some row consuch that
j +1
j +1
(_1)1 int(~)
(mod v)
j +1
j -j
=> (-1) 1 2 int (~) _
But for j
Lemma 3.15
Proof:
E
{O,l, ... ,v-U,
(mod v)
0
~
.
(j-+1)
1nt
2
~
0
~
int(j+l)
~
2
(v-l)/2
for odd v
v/2
for even v.
1
Taking rows of B as blocks, N..
=2
v
1J
0
for irj.
It is sufficient to show that the forward and backward differences
30
of adjacent symbols in any row are all the nonzero residues (mod v) twice.
These neighbor differences are
± (( -1 )
± (-1)
j+l
j+l
j+2.
j+2
int (-2-) - (-1)
1nt (-2-))
j+l.
j+l .
j+2
(lnt (-2-) +lnt (-2-))
(j+l)
= ±
j=O,1, ... ,v-2
j=O,1, ... ,v-2
"
j=O,1, ... ,v-2
o
± 1, ± 2, ... ,±(v-1)
=
(mod v)
be the column of B for which symbol t occurs in
i,t
v
row i, and assume v is even. For 0 ~ t ~ v/2
Lemma 3.16
Let J'
J. i,t
v
and for '2
~
t
j.1, t
~
= 2 (i-t)
i=t, t+1, ... ,~-1
= 2 (t-i)-l
i=O,l, ... ,t-l
v-I
2(v+i-t)
i = 0 , 1 , . . . , t -}-l
.
v
v 1
1=t- 2 ,· .. '2-
= 2(t-i)-1
j.
Proof:
Verify i+(-l) 1,t
+1
j.
int( 1 t
2
+1
) _ t
(mod v)
o
For even v, let D he the ~ x v array given by the first ~ rows of B '
v
v
Lemma 3.17
Proof:
1
Taking rows of D as blocks, N.. = 1 for ifj.
v
Let a symbol t be given.
1J
First suppose 0 ~ t ~ I-I.
Then t occurs
in column 0 but not column v-I, so by Lemma 3.16 its row first neighbors
are t+(-1)2 int (}) - t+1
and
. (1)2(i-t)+2.
1+
1nt (2(i-t)+2)
2
. (1)2(i-t).
1+
1nt (2(i-t))
2
}
v
i=t+l, t+2""'Z-1
31
i+(_1)2(t-i)+lint(2(t;i)+1)
. (1)2(t-i)-1.
1+
1nt (2(t-i)-1)
2
1
i=O,l, ... ,t-l
J
which are t+l and
i=O,1, ... ,t-l,t+l,t+2,,,·,I-1.
2i-t+l, 2i-t
v
The case
o
is shown in a similar fashion.
2~t~v-l
For optimality with the NNl covariance structure, a complete block
1
design must have equality of the N.. (Theorem 2.11).
I)
Since there are
v(v-l)/2 such quantities, and each block contributes v-I to
L.
1
L.>.N~.,
) 1 I)
Hence, by
it is necessary that vl2b if v is even and vlb if v is odd.
Lemmas 3.15 and 3.17, Band 0 are minimum NNI complete block designs for
v
v
odd and even v respectively.
This problem was solved by Kiefer and Wynn
(1981) (based on a method "first proposed by E. J. Williams (1949) and rediscovered by a number of authors") using a construction equivalent to
that given above.
The approach taken here is more convenient for the ex-
tension to NN2 optimality.
Lemma 3.18
2
and N"
I)
=
Proof:
2
Taking rows of B as blocks, N., = v-2 if i-j
v
I)
==
1 or -1 (mod v),
° otherwise.
It is sufficient to show that the set of differences generated hy
second neighbors in any rows is 1 and -1 (mod v), each occurring v-2 times.
These second neighbor differences are
'+1
± ((_1))
'+1
'+3
'+3
int(~)-(-l)) int(~))
j=O,1, ... ,v-3
j(.
= ± ( - l) I
n t (j+3).
--2- -lnt (j+l))
--2-
=±
(-l)j
=
1,-1
j=O,1, ... ,v-3
o
32
2
Lemma 3.19
v-2
Taking rows of D as blocks, N..
v
IJ
2
(mod v), and N.. = 0 otherwise.
2
if i - j - 1 or -1
IJ
Proof:
Let a symbol t be given.
v
Then t occurs in columns
Case (i): l<t<--l
- -2
neighbors are t+(-1)3 int
(I)
o and 1, where its row second
t+l, respectively.
- t-l and t-1+ (-1) \nt (!)
2 -
By Lemma 3. 17 , t also has second neighbors
· (1)2(i-t)+3.
1+
Int (2(i-t)+3)
2
- t-l
· (1)2(i-t)-1.
- t+l
1+
Int (2(i-t)-1)
2
and
i+(_1)2(t-i)+2 int (2(t;i)+2) - t+l
· ( - 1) 2 (t-i) -2.Int (2 (t-i)2)
1+
2
- t-l
v
Case (ii): 2+ 1 ~ t
~
v-I
i=t+l,t+2, ... ,~ - 1
}
i=0,1, ... ,t-2
1
J
The row second neighbors of tare
v
v-3
v-3
t - 2- 1 + (-1)
int(-2-) :: t+l
v
in row t - - - 1 (t in column v-2)
2
v
v-2.· v-2 _
t - -+ (-1)
Int ( - ) =t-l
2
2
v
in row t -2
(t in column v-I)
i=O, 1 , ... , t -
.
· (1)2(t-i)+2.
1+
~
Int (2(t-i)+2)
2
- t+l
Case (iii):
t=O
}
l=t-
I -2
v 1
v
2 + ""'2-
1
The row second neighbors of t are
0+ (-1) 2 int (}) - -1
in row 0
(0 in column 0)
v
v-3
v-3
--1+(-1)
int(-O) - 1
2
2
v
in row 2- 1
(0 in column v-2)
i+(_1)2i-l int (2i-l)
2
-
1
· ( - 1)2i+3.Ul t(2i+3)
.1.+
--2- ::-1
I
i=1,2'''''I- 2
33
v
The row second neighbors of tare
t =-
Case (iv):
2
(I in column v-I)
(_1)V-2 int (V-2) _ ~-l
2
2
in row 0
~1 + (-l)\nt(~) :: ~2 + 1
2
2
v - 1 (v
.
in row "2
"2 1n
column 1)
. (1)V-2i-2.
1+
1nt (V-2i-2)
2
-
~-l}
i=1,2,···,I- 2
. ( - 1)v-2i+2.1nt (V-2i+2)
- -+ 1
1+
2
2
Theorem 3.20
Let v be odd.
o
Then there exists a complete design with
b=v(v-l)/2 such that the N~. (itj) are all equal, and the N~. (itj) are
1)
1)
all equal.
Proof:
By construction.
Define a permutation p. as the map of column 0
1
of B into column i, i=O,l, ... ,v-l.
v
For a vxl vector £ of the v distinct
£
symbols in some order, construct the vxv array B as
v
c
Bv = (PO (.s), PI (£), P2 (£), ... ,Pv-l (£))
c
By Lemmas 3.15 and 3.18, taking the rows of B~ as blocks, the N~. (itj)
v
1)
arc all equal and N~.=v-2 if and only if i and j are first neighbors when
1J
£ is considered as a cycle.
Hence
if there is a set of (v-l)/2 cycles of
length v, say £1' £2'· ··'£v-l ' with the property that every unordered pair
-2-
of distinct symbols from {O,l, •.. ,v-l} occurs exactly once as first neighhors
c.
in the set, then the rows of B~1, i=1,2, ... ,V;1 will be the required design.
Such a set of cycles is given by any Hamiltonian cycle decomposition of the
o
complete graph on v vertices (see Lemma 4.2).
Theorem 3.21
Let v be even.
Then there exists a complete block design
with b = v(v-l)/2 such that the N~. (itj) are all equal, and the N~. (ilj)
1)
are all equal.
1J
34
Proof:
Proceeding as in the proof of Theorem 3.20, the blocks will be the
c.
"'1
.
rows of the arrays D , 1=1,2, ... ,v-l. The set of cycles {c., i=1,2, ... ,v-I}
v
"'1
is given by any Hamiltonian cycle decomposition of the graph with v vertices
and two undirected edges joining every distinct pair of vertices
~ee
Lemma 4.3).
o
. Example 3.2.1
NN2 complete block design with v=7 treatments
8
()
1
6
120
7
2
5
3
4
364
5
231
405
6
342
5
1
6
0
453
6
2
0
1
564
031
2
605
142
3
An appropriate set of cycles is
c
c
c
= (6,0,1,5,2,4,3)
1
= (6,1,2,0,3,5,4)
2
= (6,2,3,1,4,0,5)
3
Hence the rows of the following 3 arrays give the design
c
8
c
1
8
7
c
2
8
7
3
7
603
1
4
5
2
6
1
4
2
5
o
3
6
2
5
3
0
1
4
016
5
3
2
4
1
2
6
o
4
3
5
2
3
6
1
5
4
o
150
2
6
4
3
2
o
1
3
6
5
4
3
1
2
4
6
o
5
521
4
0
3
6
o
3
2
5
1
4
6
1
4
3
o
2
5
6
245
3
1
6
o
3
5
o
4
2
6
1
4
o
1
5
3
6
2
432
6
5
o
1
5
4
3
6
0
1
2
o
5
4
6
1
2
3
364
o
2
1
5
4
6
5
1
3
2
o
5
6
o
2
4
3
1
Example 3.2.2
NN2 complete block design with v=6 treatments
8
6
015
2
4
1
0
3
5
231
4
0
342
5
1
o
c
453
o
2
1
c
504
1
3
2
2
c
c
c
1
2
3
4
5
= (5,3,2,4,1,0)
= (5,4,3,0,2,1)
= (5,0,4,1,3,2)
= (5,1,0,2,4,3)
= (5,2,1,3,0,4)
35
The rows of the following 5 arrays form the design.
c
c
l
c
2
°6
°6
3
°6
5
3
0
2
1
4
5
4
1
3
2
0
5
0
2
4
3
1
3
2
5
4
0
1
4
3
5
0
1
2
0
4
5
1
2
3
2
4
3
1
5
0
3
0
4
2
5
1
4
1
0
3
5
2
c
c
4
5
°6
°6
5
1
3
0
4
2
5
2
4
1
0
3
1
0
5
2
3
4
2
1
5
3
4
0
0
2
1
4
5
3
1
3
2
0
5
4
Some
of the complete block designs constructed here can be modified
to give designs for pairs of residual effects.
E. J. Williams (1949, 1950)
looked at the experimental situation in which each unit receives in sequence
each of the treatments.
He assumed that observations resulting from these
applications were uncorrelated, but that there may be residual effects from
the treatments that could be accounted for in the linear model.
For instance,
any given treatment applied to a unit may leave some residual that is still
present upon application of the following treatment: hence a first order
residual effect is included.
If some residual of a treatment may be present
for two successive applications, a second order residual effect is included.
Let the sequence of treatment application for a given unit define an
ordered block. Then Williams' conditions for variance balance in the second
order case are
(i)
each treatment immediately precedes every other treatment in an
equal number of blocks,
(ii)
each treatment precedes every other treatment by two positions in
an equal number of blocks,
(iii)
each ordered pair of treatments occurs at the end of a block equally
36
often (in this context, the end of a block is distinguished from the
beginning).
In particular, it is sufficient that every ordered triple of distinct
treatments occurs equally often.
These conditions present obvious similari-
ties to the NN2 complete block optimality conditions.
Lemma 3.22
Let v be odd, and let t #t be a pair of symbols that occur
l 2
as row second neighbors in B ' The common first neighbors of t and t in
l
2
v
those rows for which they are second neighbors are all the symbols except
Proof:
hY
.
1.
J,t
•.. , v-3
Write t =t and t 2 = '1-1.
l
()j+l.
.
+ -1
1nt (j+l)
-2- =t, 1.e.
Note t occurs in column j at row i. t given
J,
i.
J,t
1
- t+(-1)jint(j+2 ).
Now, for j=O,2,
so common first neighbors of t and t-l are, for j=O,2, ... ,v-3
·+2
0+2
..
·+1
·+20
·+2
ij,t+(-l)J int(~)=t+(-l)Jmt(~)+(-l)J 1nt(~ - t+j+l
Also, for
j=3,S,~ •.
,v-2
so common first neighbors of t and t-l are, for j=3,S, ... ,v-2
This covers all v-2 occurences of t and t-l as row second neighbors, and
t+j+l,
j=O,2, •.. ,v-3
t-j,
j=3,S, ... ,v-2
37
o
are all residues (mod v) except t and t-l.
There is a design with v(v-l) blocks satisfying Williams'
Theorem 3.23
conditions if v is odd.
Adjoin to the design of Theorem 3.20 a copy of itself with the
Proof:
order of the blocks reversed.
By Lemma 3.22, the resulting design contains
every ordered neighbor triple exactly once.
Let v be even.
Lemma 3.24
The row first neighbor ordered pairs of Bv
given by the pairs of columns (0,1) and (v-2, v-I) are (i,i+l) and (i+~+l,
i +
f),
respectively, i=O, 1, ... 'I - 1.
Proof:
Column
Row
0
i
i+ (-l)int (~)
I
i
i+(-1)2 int (l)
v-2
i
i+ (-1)
v-I
i
v
i+(-I)vint(I) - i +2
Theorem 3.25
Let v be even.
Entry
v-I
i
-
i+l
v-I
int (-2-)
-
. + 2+
v I
1
0
If there is a Hamiltonian cycle decomposition
of the complete directed graph on v vertices, then there is a design with
v(v-l) blocks satisfying Williams' conditions.
Proof:
Use the cycles of the postulated decomposition to construct the de-
sign of Theorem 3.21.
of the blocks reversed.
satisfied.
Adjoin to this design a copy of itself with the order
Theorem 3.21 shows that conditions (i) and (ii) are
Since the graph was directed, every ordered pair of symbols occurs
as an adj acentpair exactly once in the directed cycles.
Condition (ii i)
0
now follows from Lemma 3.24.
-+
The Hamiltonian cycle decomposition of C for even v is unsolved in
v
38
general.
made.
It is known that for v=4,6 and 8 the decomposition cannot be
Mendelsohn (1968) has given a decomposition for v=22.
CHAPTER IV
GRAPH THEORETIC CONSTRUCTIONS
4.1
Preliminaries
This section briefly develops some needed ideas from graph theory.
A graph G = G(E,V) consists of a set V of elements called vertices, and
a set E of unordered pairs of elements of V called edges.
An edge (i,j)EE
is said to be incident with the vertices i and j; i and j are adjacent vertices.
If the pairs of vertices in E are taken to be ordered, G is said to
he a directed graph, and the edge (i,j)€E is directed from vertex i to vertex j.
For
Ivi
=
v, the vertices will be labeled O,l, ... ,v-l.
A path in a graph is an alternating sequence of vertices and edges, beginning and ending with vertices, such that no vertex is repeated and such
that every edge is incident with the vertices immediately preceding and succeeding it.
For a path in a directed graph, every edge must be directed from
the vertex preceding it to the vertex succeeding it.
a path is its length.
The number of edges in
Clearly, a path of length £ may be denoted by the ap-
propriate sequence of £+1 vertices (the edges being understood).
A cycle is a closed path, that is, a path for which the intial and final
vertices are the same.
A Hamiltonian path
in G is a path that includes every vertex in G.
Hamiltonian path decomposition
A
of G is a set of Hamiltonian paths in G that
collectively include every edge of G, but for which no two of the paths contain a common edge.
One may similarly define
a
Hamiltonian cycle
and
a
40
Hamiltonian cycle decomposition.
A graph is complete if the edge set contains each of the unordered pairs
of distinct vertices from
V exactly once.
A directed graph is complete if
the edge set contains each of the ordered pairs of distinct vertices from V
exactly once.
Lemma 4.1
Proof:
-+
For Ivl=v, these graphs will be denoted Cv and Cv ' respectively.
For even v there is a Hamiltonian path decomposition of Gv •
Write v=2n.
One such decomposition is
(i,i+l,i-l,i+2,1-2, ... ,i-n+l,i-n)
i=O,l, ... ,n-l
where the vertex labels are evaluated (mod v).
Lemma 4.2
For odd v there is a Hamiltonian cycle decomposition of C .
v
Proof:
Write v=2n+l.
One such decomposition is
(2n,i,i+l,i-l,i+2,i-2, ... ,i-n+l,i+n,2n)
i =0 • 1 , ••• , n-l
where the vertex labels except for 2n are evaluated (mod 2n).
Denote by C~ the graph with v vertices and two undirected edges joining
each distinct pair of vertices.
Lerona 4.3
There is a Hanil tonian path decomposition and a Har.:iltonian cycle
decomposition of C2 .
v
Proof:
Case (i):
v is even.
The path decomposition follows from Lemma 4.1.
Writing v=2n+2, a cycle decomposition is
(00 ,
i +1, i "'" 1, i +2, i - 2 , ... , i +n , i - n ,(0)
where the vertex labels except for
odd.
00
i=O ,1, ... ,2n
are evaluated (mod 2n+l). Case (ii): v is
The cycle decomposition follows from Lemma 4.2.
decomposition is given by deleting
00
Writing v=2n+l, a path
from the set of cycles in\Case (i).
0
41
It is worthy of note that the paths of Lemma 4.1 are the rows of the
array Dv of Section 3.2, and for odd v the paths of Lemma 4.3 are the rows
of the array B .
v
4.2
NN1 designs from graphs
Theorem 4.4
(i)
Let k be even.
The existence of a BIBD(b,v,k,r,A) implies
the existence of a NNI optimum BIBD with parameters b =kb/2, vl=v, k =k,
l
1
r =kr/2 and A =kA/2. If A=l the design is minimal.
l
l
(ii) Let k be odd. The existence of a BIBD(b,v,k,r,A) implies the existence
of a NNl optimum BIBD with parameters bl=kb, vl=v, kl=k, rl=kr and Al=kA. If
A=l the design is minimal.
Proof:
(i)
Let the treatments in a block of the given BIBD(b,v,k,r,A) with
even k be associated with the vertices of C . Decompose the graph into k/2
k
Hamiltonian paths and form an ordered block from each path, where the order
of the treatments within a block is given by the order of the vertices in the
corresponding path.
Since each edge occurs exactly once in the path decompo-
sition, each of the (~) pairs from the k treatments appears once
neighbors in the set of ordered blocks.
path and at an end of
~xactly
e .. =2 for this set of blocks.
1J
as
first
Since each vertex occurs in every
one path, each pair i,j of the k treatments has
Repeating this process for each block of the
1
1J
given BIBD produces the optimum design with neighbor parameters N.. =A
and
e .. =2A.
1J
To establish minimality for the case A=l, let A denote the smallest poss
sible treatment intersection number for attainment of the optimality conditions.
If k=2 (mod 4), Corollary 2.13 implies k/zlA , so A =A . If k=O (mod 4),
s
s I
Corollary 2.13 implies k/41A , so it may be possible to find an optimum design
s
with A =k/4.
s
By (2.2.5) the value of e .. +kN~. for such a design would be
1J
1J
1
2A +4k- l A =k/2 + 1, which requires N..
=0 for all
s
s
1J
Hence again A SA.
s
i~j
(N1
.. ~l =>1
e .. +kNk
.. ~k>-Z + 1)
1J
1J
1J
.
42
(ii)
The proof here is essentially the same as the proof of (i), except that
for odd k one must employ
C~ in order to obtain a path decomposition. The
neighbor parameters for the optimal design are N~.=2A and e .. =4A.
1J
1J
Minimality
o
follows directly from Corollary 2.13.
By virtue of the results of Section 3.2 and the comment at the end of
Section 4.1, the construction in Theorem 4.4 is equivalent to one given by
Cheng (1983).
However, he does not establish minimality of the designs for
even k and A=l.
The remainder of this section will be devoted to applications
of the theorem.
Some of these results are also listed by Cheng (1983), in-
eluding slightly weaker versions of the following two corollaries.
Corollary 4.5
Let s=pn be a prime power.
If p is even, there exists a
2
BIBO((S+1)(s2+ s +1), 5 +5+1, s+l, (5+1)2, 5+1) that is minimum optimum for the
If p is odd, there exists a BIBO((S;1)(S2+ s +l ), 5 2+5+1, 5+1,
NNl model.
(5+1)2/2, (5+1)/2) that is minimum optimum for the NNl model.
Proof:
The initial design is given by the finite projective plane PG(2,s)
of order 5, with lines as blocks and points as treatments.
Corollary 4.6
Let s=pn be a prime power.
o
If p is even, there exists a
BIBO(s2(s+1)/2, s2,s,S(S+1)/2, s/2) that is minimum optimum for the NNl model.
If p is odd, there exists a BIBO(s2(s+1), s2,s,S(S+1),S) that is minimum optimum for the NNl model.
Proof:
The intial design is given by the finite affine plane EG(2,s) of
order s, with lines as blocks and points as treatments.
Theorem 4.7
Let k=v-l.
b=v(v-l), r=(v-l)
2
o
If v is even there is a minimum NNl design with
and A=(v-l) (v-2).
If v is odd there is a NNl design with
b=v(v-1V2, r=(v-l)2/ 2 and A=(v-l) (v-2)/2 which is minimum if v~3 (mod 4).
43
Proof:
NNI designs for k=v-l can be developed according to Theorem 4.4 from
the initial set of (vI) blocks given by all subsets of size v-I from the
vv treatments. It remains to establish minimality.
r = A(v-l)/(k-l) = A(v-l)/(v-2) => v-2IA.
If'v is even then k is odd, so Corollary 2.13 => klA => v-llA
(v-I) (v-2) lA,
since l.c.m. (v-l,v-2) = (v-I) (v-2).
k=2 (mod 4) => k/2A => (V;l) IA => (V-l~(V-2)
Theorem 4.8
and
hence
If v=3 (mod 4) then
D
IA.
The existence of a Hadamard matrix of order 4m implies the
existence of a minimum NNI BIBD with parameters b=v(v-l)/2, v=4m-l, k=(v-l)/2,
2
r=(v-l) /4 and A=(v-l)(v-3)/8.
Proof:
It is known (e.g. Hall (1967)) that a Hadamard matrix of
orde~
4m
implies the existence of a BIBD(4m-l,4m-l,2m-l,2m-l,m-l), which can be used
as the initial design in Theorem 4.4 to give the required optimum design. Now,
r=A(v-l)/(k-l) = 2kA/(k-l) => k-112A.
mUltiple of k(k-l)/2 = (v-I) (v-3)/8.
Also, k:odd => kiA.
Hence A is a
D
Hadamard matrices of order 4m are known to exist for a wide range of m,
including all
4m~lOO.
Cheng (1983) has found the designs of Theorem 4.8 for
the case v=3 (mod 4) is an odd prime power, in which case a Hadamard matrix
always exists.
For k=4 and
Table 4.2.1.
1~A~6,
the BIBD parameters satisfying (2.1.1) are listed in
44
Table 4.2.1
BIBD parameters for k=4, 1$11.$6
A
v (mod 12)
1
1,4
2
1,4,7,10
-r
b
(v-l)/3
v(v-l)/12
3
0,1,4,5,8,9
2(v-l)/3
v-I
4
1,4,7,10
4(v-l)/3
v(v-l)/3
5
1,4
any v;::: 5
5(v-l)/3
5v(v-l)/12
6
v(v-l)/6
v(v-l)/4
2 (v-I)
v (v-I) /2
In this case, the relationship kl4A of Corollary 2.13 provides no further
restrictions on the BIBD parameters.
That the designs with parameters listed
here all exist is a result due to Hanani (1961).
A pair of treatments i and j may occur together in a block of four plots
in four distinct orientations (a,B,y,c):
a
B
y
i
i
i
c
i
j
j
j
j
1
orientation
1)
a
5
B
1
y
2
c
4
1)
(4.2.2)
1
For 11.=3 and k=4, equality of the e .. +kN .. is impossible.
Lemma 4.9
Proof:
contribution to e .. +kN ..
1)
1)
If the e .. +kN~. are all equal, the common value is (from (2.2.5))
211. (k+2)/k=9.
1)
1)
It follows from (4.2.2) that a pair of treatments must have
45
either orientations a, y, and y or orientations S, 0, and o.
Looking at all
v
possible pairs of treatments, a total of 2(2)=v(V-l) y and 0 orientations are
required.
But since each block provides one y and one 0 orientation, only
o
2b=v(v-l)/2 are available.
Theorem 4.10
There exist NNI BIBD's with k=4 and b and v as indicated in
Table 4.2.3 ..
Proof:
Except for v=ll (mod 12), the listed designs can be constructed by
applying Theorem 4.4 to the appropriate BIBD' s with parameters
Table 4.2.1.
listed.
in
For v=l1 (mod 12) the designs can be constructed by Theorem 3.6
(as can those for v=5 (mod 12)).
For v=0,5,8, or 9 (mod 12), minimality is a consequence of Lemma 4.9.
The designs for v=l and 4 (mod 12) are minimal because A=l is impossible under
NNl conditions (an immediate consequence of Theorem 4.4).
The designs for
0
v=ll (mod 12) are minimal because no smaller BIBD exists.
Table 4.2.3
NNI
BIBD's with k=4
v (mod 12)
b
minimum?
0
v(v-l)/2
yes
1
v(v-l)/6
yes
2
v (v-I)
3
v(v-l)
4
v(v-l)/6
yes
5
v(v-l)/2
yes
6
7
v (v-I)
v(v-l)/3
8
v(v-l)/2
yes
9
v (v-l)/2
yes
10
v(v-l)/3
11
v(v-l)/2
yes
46
For those cases in which minimality has not been established, the
possibility of a design with half the listed number of blocks remains open.
Kiefer and Wynn (1981) have shown that the design with v=7 in Table 4.2.3
They have also shown that there is an EBIBD (all N~. equal)
is minimum.
1J
for v=7 and b=v(v-l)/6.
For k=5, the smallest admissible values of A and b (subject to the constraints (2.1.1)) for given v are listed in Table 4.2.4.
Table 4.2.4
Minimum BIBD parameters for k=5 and given v
v (mod 20)
A
1,5
1
2
v (v-I) /20
v(v-l)/5
9,3,17
4
5
3,7,19
10
v(v-l)/2
2,4,8,12,14,18
20
v(v-l)
11,15
0,6,10,16
b
v(v-l)/IO
v(v-l)/4
All designs listed in Table 4.2.4 exist except v=15, A=2 (Hanani (1972)).
Now the constraint kl4A implies 51A, and is useful in establishing the minimality of optimum designs.
Theorem 4.11
There exist minimum optimum NNI BIBD's with k=5 and para-
meters listed in Table 4.2.5.
Proof:
The designs with v=0,1,5,6,10,11,15 and 16 (mod 20) are developed
according
4.2.4.
to Theorem 4.4 from the corresponding designs listed in Table
The designs with v=7,19,23,43,47 and 59 (mod 60) are constructed
as semi balanced arrays by Theorem 3.6, and correspond to those values of
v~o
(mod 3) in row 5 of Table 4.2.4.
The designs with v=8 and 88 (mod 120)
are transitive arrays corresponding to those v in row 6 of Table 4.2.4 with
47
2
minimum prime power factor larger than four (hence there exists OA(v ,6,v,2)
=> TA(v(v-l),5,v,2)).
For each design minimality follows from the constraint
D
51\ and the minimum \ of Table 4.2.4.
Minimum NNI BIBD's with k=5
Table 4.2.5
v
b
1,5 (mod 20)
v(v-l)/4
5
11,15 (mod 20)
v(v-l)/2
10
7,19,23,43,47,59 (mod 60)
v(v-l)/2
10
0,6,10,16 (mod 20)
v(v-l)
v(v-l)
20
8,88 (mod 120)
Let v=lO and k=5.
Example 4.2.6
NNl-optimality => k14\ => 51\.
(except v=15)
20
Then \(v-l)/(k-l) = 9\/4 => 41\.
Hence 201\ for an NNl design.
Also,
A
BIBD(18,10,5,9,4) is
1
0
0
1
1
1
4
0
1
0
3
0
0
0
0
2
2
0
2
3
1
2
2
3
6
4
4
2
5
1
1
2
2
6
4
5
3
5
7
5
5
4
7
2
3
5
3
3
4
1
7
3
4
3
8
8
7
6
4
6
8
6
6
5
8
6
4
7
6
6
5
5
9
9
8
8
7
7
9
7
9
8
9
9
9
9
7
8
9
8
2
Using this path decomposition of C :
5
(0, 1, 2, 3, 4)
(1, 3, 0, 4, 2)
(3, 4, 1, 2, 0)
(4, 2, 3, 0, 1)
(2, 0, 4, 1, 3)
An NNI design is given by adjoining to the initial BIBD the following
72 blocks
48
231
2
2
3
644
2
5
1
2
233
1
8
7
6
4
6
8
6
6
5
864
7
665
5
100
1
1
1
4
0
1
030
0
0
0
0
998
8
7
7
9
7
9
8
9
9
997
764
5
3
5
7
5
5
4
7
2
3
5
344
3
8
646
8
665
864
7
665
5
9988779
798
9
999
7
9
8
233
1
3
4
4
3
0
2
2
0
8
8
7
1
23122
364
4
2
5
1
764
5
4
7
235
1
5
1
2
5
3
5
7
001
1
1
401
030
0
8
997
898
8
8
7
7
9
7
9
764
5
3
5
7
5
547
235
344
3
8
8
6
46
8
6
6
5
6
4
7
6
6
5
5
1
001
1
1
4
0
1
030
0
0
0
2
2
0
2
3
251
1
2
2
3
3
1
8
9
2
998
7
9
0
2
898
1
2
2
3
6
4
4
764
5
3
5
7
5
547
2
3
5
3
4
4
3
100
1
1
1
4
0
1
0
3
0
0
0
0
2
2
0
998
8
7
7
9
7
9
8
9
9
9
9
7
898
23122
3
644
2
5
1
1
2
2
331
8
6
8
5
8
6
4
7
6
6
8
7
6
4
6
6
5
5
CHAPTER V
PARTIALLY VARIANCE BALANCED DESIGNS
5.1
Introduction
One of the important
properties of the NNl weakly universally optimum
designs is that of complete variance balance:
T'"
var(~
the choice of the normalized contrast vector a.
t) does not depend on
This property is also ex-
hibited by the ordinary BIBD's and complete block designs in the case of uncorrelated, equivariable errors.
For reasons such as cost, lack of experi-
mental material, or other factors, however, it may be that the number of
blocks necessary for complete variance balance is impracticable or unachievable.
A resonable alternative is to find a design with smaller b for which
as little balance as possible is forfeited.
In the context of the ordinary
BIBD's this has led to the use of the partially balanced incomplete block
designs (PBIBD's).
To define the PBIBD, first consider an incomplete block design for which
no treatment appears more than once in any block, and for which a pair of
treatments occurs in either AI' A , ••. , or Am blocks . For this design, say
2
.
th
.
. A.
1
common blocks, i=1,2,
two treatments are I
assocIates I. f t h ey occur In
I
.•. ,m.
Evidently any given treatment partitions the remaining treatments
into m mutually exclusive associate classes, the i
.
I.th assocIates
0
f t h e gIven
.
treatment.
th
class consisting of the
In this manner the design defines
an m-class association scheme on the treatment set.
An m-class association scheme is strongly regular if
50
(i)
(ii)
each treatment has exactly n. i
th
1
associates (i=1,2, ••. ,m)
given any two treatments which are i
th
associates, the number of
treatments common to the jth associates of the first and the kth
associates of the second is p~ k and is independent of the chosen
J,
pair of treatments (i,j,k=1,2, ... ,m).
A partially balanced incomplete block design with m associate classes,
denoted PBIBD(b,v,k,r,Al, ... ,A ), is an incomplete block design with v treatm
ments in b blocks, with constant block size k and replicate number r, which
defines a strongly regular m-class association scheme.
For a PBIBD with m-class
association scheme and
2
var(~)=cr
I, var(l) is
constant on the main diagonal and contains at most m distinct values off
the diagonal.
Hence elementary treatment contrasts, that is, contrasts of
the form t. -t., are estimated with at most m distinct variances.
1
J
It is in
this sense that the design exhibits partial variance balance, and it is in
the same spirit here that incomplete block designs will be constructed which
enjoy partial variance balance for the NNI covariance structure.
In addi-
tion, a series of partially variance balanced complete block designs is proposed for odd v, and several optimality results are obtained.
5.2
Incomplete block designs with partial variance balance
Given a BIBD(b,v,k,r,A) for which k is odd, a design will be constructed
in a manner analogous to that of Theorem 4.4, but with a resulting smaller
number of blocks.
The cost of this savings in experimental material will. be
the loss of complete variance balance of
1.
Arbitrarily give the name "end" to one of the vertices of C (the reason
k
for this will soon become evident).
Assign the k treatments of one of the
blocks of the BIBD to the vertices of the graph, and decompose
it
into
51
(k-l)/2 Hamiltonian cycles (Lemma 4.2). Form two paths of length k-l from
each cycle by deleting first one of the edges that is incident with the end
vertex, then replacing it and deleting the other.
Proceeding as in the
proof of Theorem 4.4, take each of these k-l paths as an ordered block.
Since the end vertex is adjacent to each other vertex in exactly one of the
cycles,
= k-l
e.
1
or
1
for this set of blocks, where the value
with the end vertex, and I otherwise.
is k-l if treatment i is associated
And since every other pair of vertices
is also adjacent in just one cycle, it is easily seen that
1
e .. + kN ..
1J
1J
= 2k
or
2k+2
where the value is 2k if one of i or j corresponds to the end vertex, and
2k+2 otherwise •
.
Repeating this process for each block of the initial BIBD, all the
e. 's will be equal if each treatment can be made to correspond to the end
1
vertex an equal number of times.
This is equivalent to choosing one treat-
ment from each block of the initial BIBD as representative of that block
in such a way that each treatment is a representative an equal number of
times.
For this, it is necessary and sufficient that b is an integer multi-
pIe of v (Hartley, Shrikhande and Taylor (1953)).
For ilj, let A.. be the number of blocks containing both treatments i
1J
and j for which either i or j is the chosen representative.
A choice of
block representatives for a BIBD such that each treatment is a representative
an equal number of times, and such that the numbers A.. take on only m dis1J
tinct values A ,A , ... ,Am will be called an m-c1ass representative scheme.
l 2
Theorem 5.1
Let k be odd.
If for a BIBD(b,v,k,r,A) there exists
an
52
m-class representative scheme, then there exists a BIBD(b(k-l),v,k,r(k-l),
A(k-l)) that is partially variance balanced for the NNI covariance structure,
in the sense that elementary treatment contrasts are estimated with only
m distinct variances.
If A=l, the design has b fewer blocks than the NNI
mhimum optimum design with the same v and k.
Proof:
Developing each block of the initial BIBD according to the graph
decomposition discussed above, with the block representatives corresponding
to the end vertex, gives a BIBD with the appropriate parameters.
For this
design,
b
e. =
1
V
b
b
(k-l) + (r--) =
(k-2) +r
v
V
1
e .. + kN .. = (A-A .. ) (2k+2) + A.. (2k) = 2A (k+l) - 2A ...
1J
1J
1J
1J
1J
(5.2.1)
From Lemma 2.6 (with P2=O), var(t ) depends only on e and cov(ti,t )
i
i
j
1
depends only on e .. +kN ..• Hence for the developed design, var(t.) does not
1
1J
1J
depend on i and cov(t.,t.) takes on m distinct values corresponding to the
A
1
J
m values AI' A , •.• ,A of A...
m
2
1J
The result for A=l follows from Theorem 4.4.
o
Designs developed according to Theorem 5.1 will be denoted NNl-PBD(m)
or NNl-PBD.
The combinatorial properties of a representative scheme and the nature
A
of the covariance matrix var(!) for the developed design both suggest a relationship between BIBD's with representative schemes and PBIBD's.
The com-
binatorial nature of this relationship can be examined by using the m-class
representative scheme to define an m-class association scheme on the treat.
ment set: a pa1r
0
f treatments are 1. th assoc1ates
.
.
1. f t h ey occur toget her 1n
\. blocks for which one or the other is the representative, i=1,2, ... ,m.
1.
53
The existence of a BlBD(b,v,k,r,A) with an m-class representa-
Theorem 5.2
tive scheme implies the existence of an incomplete block design with m-class
association scheme and parameters bO=b, vO=v, kO=k-l, rO=r-b/v.
If the assoc-
iation scheme induced by the representative scheme is strongly regular, the
design is a PBlBD.
replicate number
Conversely, if an incomplete block design with constant
based on an m-class association scheme can be modified to
give a balanced incomplete block design by the addition of one treatment to
each block, the resulting BlBD has an m-class representative scheme.
Proof:
Delete the representative from each block of the BlBD(b,v,k,r,A)
to give the incomplete block design with parameters bO=b, vO=v, kO=k-l, and
ro=r-b/v.
... , or A-A
the BlBD.
For this design, a pair of treatments occurs in either A-AI' A-A ,
2
m
blocks, depending on their relationship as representatives of
Hence the association scheme is that induced by the representa-
tive scheme.
The converse follows by taking the treatment adjoined to a block to be
o
that block's representative.
The main concern here will be for BlBD's with 2-class representative
schemes, as they possess the most variance balance among the designs under
consideration.
Theorem 5.2 suggests looking at the extensively studied 2-
associate class PBlBD's in order to find members of the class which can be
modified to BlBD's.
o 0
Let DO be a PBlBD(b,v,k ,r ,A ,A ) that can be extended
o O l 2
to a BlBD(b,v,k,r,A) with 2-class representative scheme by the addition of
one treatment to each block.
Then writing b=sv,
k=k +1
r = sk = rO+s = s(ko+l)
A = r(k-l) =
v-I
s(ko+l)k
v-I
O
o
=
54
Hence the PBIBD must satisfy
vlb
(5.2.2)
and
(5.2.3)
If Al and A are the parameters of the representative scheme, then
2
Al+A~
=
A2+A~
= A
so the design must also satisfy
(5.2.4)
Clatworthy (1973) has given an extensive listing of 2-associate class
PBIBD's with
2~k~10
and
2~r~10.
Ignoring the cases k =v-2 and k =2 (which
O
O
will be treated later), his list includes only 2 designs with even k which
O
satisfy (5.2.2)-(5.2.4) and for which development according to Theorem 5.1
would yield a BIBD with less blocks than a corresponding NN1 optimum design
already constructed.
Using his notation, these are the designs T61 and LSIOI.
T61 is a PBIBD(15,15,6,6,3,1) with triangular association scheme formed
by taking the second associates of treatment i to form block i, i=1,2, ... ,15.
Adjoining i to block i for each i gives a BIBD(15,15,7,7,3).
Hence Theorem 5.1
can be used to construct an NN1-PBD(2) with parameters b=90, v=15, k=7, r=42,
A=18, Al=O and A =2.
2
kl4A => 71A.
For v=15 and k=7, an NNI optimum design must satisfy
Also, r=A(v-l)/(k-l) = 7A/3 => 31A => 211A.
An NNI optimum
BIBD(105,15,7,49,21) does exist (Theorem 4.4), hence the design constructed
here saves 105 experimental units compared to the minimum optimum design.
LSI01 is a PBIBD(25,25,8,8,3,2).
Inspection shows that a first associ-
ate of treatment 1 is in every block, hence if it is extendable to a BIBD,
the result must have
A~4.
But (5.2.3) => A=9(8)/24 = 3, so the extension
55
cannot be made.
This lack of success with 2-associate class PBIBD's seems to indicate
that they are generally too restrictive.
Of course, a representative scheme
inducing a strongly regular association scheme is not necessary; any representative scheme will do.
The next theorem gives a simple method for finding
a representative scheme for an arbitrary BIBD with v\b.
Theorem 5.3
A BIBD with treatment intersection number A for which vlb has
a representative scheme with at most min(1+A,1+2b/v) classes.
Proof:
Since vlb, a set of representatives can be chosen which includes
every treatment b/v times.
b~ocks
So a pair of treatments occurs in at most 2b/v
for which one is the representative.
And as the pair occurs in ex-
actly A blocks, they occur together in at most A blocks for which one is
o
the representative.
Corollary 5.4
A BIBD with A=l for which vlb has a 2-c1ass representative
scheme.
Corollary 5.5
A symmetric (b=v) BIBD has a representative scheme with at
most 3 classes.
Theorem 5.1 with Corollary 5.4 can be used to generate a great number
of partially variance balanced designs which have fewer blocks than the minimum optimum NN1 designs with the same v and k (and which will later be shown
to possess significant optimality properties of their own).
if s=2
n
For instance,
is an even prime power, the BIBD series with parameters
2
b=v=s +s+l, k=r=s+l, A=l
based on points and lines of the projective plane of order s, can be developed
56
to give the NNI-PBD series with
2
2
b=s(s +s+l), v=s +s+l, k=s+l, r=s(s+l), A=S, Al=O, A =1
2
(compare Corollary 4.5).
Another method for finding representative schemes is based on BIBD's
cyclically developed from a set of blocks.
The initial block (a ,a ,
l 2
,a )
k
developed cyclically gives the v blocks (a1+j, a +j, ... ,a +j), j=1,2,
2
k
where each a.+j is reduced (mod v).
A necessary and sufficient condition
1
that (a
il
,a
i2
, ... ,a
ik
,v
) i=1,2, ... ,u are a set of u initial
blocks
for
a
BIBD(uv,v,k,uk,A) is that the numbers
± (a .. -a.n)(mod v)
I)
i=1,2, ... ,u;
1JV
j ,5/,=1,2, ... ,k; j#5I,
are every nonzero residue (mod v) exactly A times (Bose (1939)).
Theorem 5.6
Suppose that for a BIBD(uv,v,k,uk,A) cyclically generated from
i=1,2, ... ,u, there are elements a ..
I).
1
i=1,2, ... ,u (called initial representatives) such that the numbers
i=1,2, ... ,u;
± (a .. -a. n) (mod v)
I) .
1
1JV
5/,=1,2, ... ,k
are every nonzero residue (mod v) either Al or A times.
2
and
5/,~j.
1
(5.2.5)
Then there is a
2-class representative scheme for the BIBD with parameters Al and A . In
Z
general, if the numbers (5.2.5) are the nonzero residues occurring with m
distinct frequencies, then there is an m-class representative scheme for the
BIBD.
Proof:
---
Write B. 8 = (a· +8,a. +8, ... ,a. k+8) (mod v), and let a .. +8 (mod v)
1,
1 l
12
1
I) .
1
be the representative of block B.
1,8
.
the representative of the blocks B. t
1,
Then any given treatment symbol t is
-a ..
I)
' i=1,2, ... ,u.
i
The treatments
57
occurring in these blocks are
a. n+t-a ..
Ix,
i=1,2, ... ,u; £=1,2, ... ,k.
IJ.
1
The symbol t also occurs in the blocks
i=1,2, ... ,u; £=1,2, ... ,k and £;tj.1
B.1 ,t-a.Ix,n
which have representatives
i=1,2, ... ,m; £=1,2, ... ,k and £;ij ..
a .. +t-a. £ :: t+(a .. -a. £)
1Ji
1
1Ji
1
1
Hence in the set of symbols
t ± (a .. -a. £) (mod v)
IJ
i
i=1,2, ... ,m; £=1,2, ... ,k
1
the number of times any symbol s;tt appears is just the number of blocks containing both s and t for which one of s and t is the representative. Each of
O,l,
,v-l (except t) occurs either Alar A times if and only if each of
2
1,2,
,v-l occurs either Alar A times among (5.2.5).
2
0
The extension to m-class representative schemes is obvious.
For a cyclic BIBD developed from the initial block (a ,a , ... ,a ), Corolk
l 2
lary 5.5
implies that anyone of the a. as choice of initial representative
1
yields a representative scheme with at most 3 classes.
For instance (com-
pare Theorem 4.8):
Corollary 5.7
If v:: 3 (mod 4) is a prime power, there is a BIBD with
parameters b=v, r=k=(v-l)/2, A=(v-3)/4 which has a 3-class representative
scheme, and hence an NNI-PBD(3) with parameters b=v(v-3)/2, r=(v-l)(v-3)/4,
2
k=(v-l)/2, A=(v-3) /8
which has v(v-l)/2 fewer plots than a minimum optimum
NNI design with the same v and k.
Proof:
(1967) .
Cyclic generation of these BIBD's is discussed, e.g., in Hall
o
58
Similar results may be written down for other families of symmetric,
cyclic BIBD's.
Unfortunately, aside from the case k=v-l, no general results
have been found for guaranteeing a 2-class representative scheme for these
designs.
One example with 2-class representative scheme is presented next.
Example 5.2.6
B = [0,6,8,9,11,15,25,32,33].
O
yields a BIBD(37,37,9,9,2).
o
Since no two elements of B sum to 0 (mod 37),
O
may be taken as the initial representative.
for C
9
Developing B (mod 37)
O
A set of Hamiltonian cycles
is
(0,1,7,2,6,3,5,4,8,0)
(0,5,7,6,8,2,3,1,4,0)
(0,3,7,4,6,5,8,1,2,0)
(0,7,8,3,4,2,5,1,6,0)
which, taking 0 as the end vertex, yields the paths
(1,7,2,6,3,5,4,8,0)
(5,7,6,8,2,3,1,4,0)
(0,1,7,2,6,3,5,4,8)
(0,5,7,6,8,2,3,1,4)
(3,7,4,6,5,8,1,2,0)
(7,8,3,4,2,5,1,6,0)
(0,3,7,4,6,5,8,1,2)
(0,7,8,3,4,2,5,1,6)
Hence a set of initial blocks for an NNI-PBD (2) with parameters b=296, v=37,
[6,32,8,25,9,15,11,33,0]
[15,32,25,33,8,9,6,11,0]
[0,6,32,8,25,9,15,11,33]
[0,15,32,25,33,8,9,6,11]
[9,32,11,25,15,33,6,8,OJ
[32,33,9,11,8,15,6,25,0]
[0,9,32,11,25,15,33,6,8]
[O,32,33,9,11,8,15,6,25J
Developing these ordered blocks (mod 37) gives the design, for which e i =16,
1
c .. +kN .. = 40 or 38.
I
J
1J
The minimum optimum BIBD for v=37 and k=9 has b=333.
More generally, consider a BlBD cyclically developed from u>l initial
blocks.
There are 2u(k-l) symmetric differences with respect to any choice
59
of initial representatives, out of the total of uk(k-l) symmetric differences
Since
within the blocks which give each nonzero residues A times.
2u(k-I)A/uk(k-l) = 2A/k, one would think it possible, in many cases, to
choose initial representatives yielding a representative scheme with less
classes than the bound of Theorem 5.3, particularly if 1<A<k/2.
Again, no
general results are offered, but for u=2 and 3 trial and error experience
shows that initial representatives yielding 3-class representative schemes
are usually easily found.
For k=v-l, Theorem 5.6 gives a representative scheme with 2 associate
classes.
Consider the block
[O,1,2, ... ,v-2]
The set of symmetric differences with respect to
a
are
±l, ±2, ... ,±(v-2)
which reduced (mod v) give the residues 2,3, ... ,v-2 twice each, and the rcsidues I and v-I once each.
It is easily verified that the given block is an
initial block for the BIBO with b=v, r=k=v-I, A=v-2, and hence
priate choice for initial representative.
a
is an appro-
For this scheme, call two treat-
ments first associates if they occur together in just one block for which
one of the two is the representative.
Then for i=O,l, ... ,v-l the first assoc-
iates of treatment i are (i-I) and (i+l) (mod v); all other treatments arc
second associates.
2
It follows that Pll = I for a pair (i,j)
min((i-j) (mod v), (j-i) (mod v)) = 2, and P~l =
a
otherwise.
such
that
Thus the assoc-
iation scheme induced by the representative scheme is not strongly regular.
An alternative choice of representatives can be made such that the corresponding association scheme is strongly regular.
a design 0
1
Assuming v is even, form
with v =v/2 treatments in blocks of size (v-2)/2 by taking each of
l
60
the v/Z subsets of size (v-Z)/Z from the treatment set twice; 01 is a
BIBO(v,v/Z,(v-Z)/Z,v-Z,v-4). Labeling the treatments of 01 o,l, ... ,I-l, form
a design 0z with v treatments in blocks of size v-Z by replacing treatment i
,f-
of 01 with the treatment pair (il,i ), i=O,l, ...
1. For 0Z' the treatz
ment pair (il,i z) occurs in v-Z blocks: they are first associates. Any other
pair, say (is,jt)' i#j, s,tE{l,Z}, occurs in v-4 blocks: they are second
associates.
It is seen that
= 1, n = v-Z
2
l
1
1
1
Pll = 0, PIZ = 0, PIZ = v-Z
n
Z
Z
Z
Pll = 0, PIZ = 1, PZZ = v-4
So the association scheme is strongly regular (of group divisible type), and
02 is a PBIBO(v,v,v-Z,v-Z,v-2,v-4).
By the construction, each pair of iden-
tical blocks in 0z lacks a unique first associate pair of treatments.
Ex-
tend 02 to a design 03 by adjoining these two treatments to the two blocks,
one treatment to each.
It now follows that 03 is a BIBO(v,v,v-l,v-l,v-Z).
Hence by Theorem 5.Z, the adjoined treatments may be taken as representatives
in 03' and the representative scheme induces a strongly regular Z-class association scheme.
Theorem 5.8
For even v there is a BIBO(v,v,v-l,v-Z,v-2) with Z-class rep-
resentative scheme, and hence an NNl-PBO(Z) with parameters
b=v(v-Z), v, k=v-l, r=(v-l)(v-Z), A=(V-Z)
Z
which has v less blocks than the corresponding NNl minimum optimum design.
The representatives may be chosen so as to induce a strongly regular association scheme, but need not be.
61
Proof:
o
The above discussion and Theorem 4.7.
Example 5.2.7
v=6, k=5
a
0
a a
1
1
The columns of
1
1
2
2
2
2
Underlined treatments give the cyclic represent-
3 -3
4 4
3
3
4
4
ative scheme.
5
5
5
5
PBIBD(6,6,4,4,4,2) of group divisible type.
2
1
4
a
3
5
Finally, consider the case k=3.
thi~
array are BIBD(6,6,5,5,4).
Deleting the last row gives the
If v is odd, the NNI minimum optimum
designs have parameters b=v(v-l)/2, r=3(v-l)/2, and A=3.
Since k-l=2, the
method of Theorem 5.1 can be used to construct an NNI-PBD with smaller b
only if the initial BIBD satisfies vlb and A=l.
Hence one must find repre-
sentative schemes for BIBD's in the series
b = v(v-l)/6, r = (v-l)/2, k=3, A=l, v = 1 (mod 6)
(5.2.8)
This problem is solved by Corollary 5.4, and the parameters of the represent-
If k=3 and v is even, NNI minimum optimum designs have
b=v(v-l), r=3(v-l), and A=6.
parameters
Theorem 5.1 will produce a design with b<v(v-l)
for initial BIBD's such that vlb and A=2, that is, for the series
b = v(v-l)/3, r = (v-I), k=3, A=2, v =4 (mod 6)
(5.2.9)
One construction of this series is given by developing the following set of
initial blocks mod (6t+3),
t~l
(oo,O,3t+l)
(O,i,2t+l-i)
i=1,2, ... ,t"
(O,2i,3t+l+i)
i=1,2, ... ,t
(O,2t+l,4t+2)
of period 2t+l
(5.2.10)
62
Here
00
represents an element that is unchanged by the cyclic development.
Theorem 5.9
There is a 2-class representative scheme with A =1 and A =2
1
2
for the design given by the set of initial blocks (5.2.10).
Proof:
By construction.
For the initial blocks (O,i,2t+l-i)
take () as the initial representative.
i=l, ... ,t
The resulting blocks containing a
given symbol s as representative also contain the symbols
s+i, s+2t+l-i
i=l, ... , t
(5.2.11)
and in the blocks containing s for which it is not the representative, the
representatives are
s-i, s-2t-l+i
i=l, •.. ,t
(5.2.12)
Por the purposes of this proof, let La,b] = a, a+l, a+2, ... ,b where each
symbol is reduced (mod 6t+3).
[s+l, s+2t]
Then (5.2.11) and (5.2.12) are, respectively,
and
(5.2.13)
ls-2t, s-l]
For the initial blocks (O,2i,3t+l+i)
i=l, ... ,t-l, take 0 as the initial
representative .. Then the symbols occurring in the resulting blocks which
have s as representative, or which are representatives in blocks containing
S,
are
s+2i, s+3t+l+i, s-2i, s-3t-l-i
i=l, ... ,t
which are
[s-3t-l,s-2t~3],
[s+2t+3,s+3t+l], and s+2i,s-2i
i=l, ... ,t-l
(5.2.14)
Take 4t+l as the initial representative in (O,2t,4t+l).
For this set of
blocks, the symbols in blocks for which s is a representative or which are
representatives of blocks containing s are
s-2t-2, s+2t+l, s+2t+2, s-2t-l
(5.2.15)
63
Combining (5.2.13), (5.2.14), and (5.2.15) gives
[s+1,s+3t+l], [s-3t-l,s-l], and s+2i, s-2i
i=1,2, ... ,t-l
(5.2.16)
For the blocks (00,j,3t+l+j) let 00 be the representative for j=O,l, ... ,2t
and let j be the representative for j=2t+l, 2t+2, ... ,6t+2.
For the blocks
(j,2t+l+j, 4t+2+j) let j be the representative for j=O,1, ... ,2t.
For these
last two sets of blocks, it can be verified that a symbol s is in blocks
with the following symbols for which one of s and the listed symbol is the
representative:
00, s-3t-l, s+2t+l, s-2t-l
O~s~2t;
2t+l
~
s
~
3t: 00, s-3t-l, s+3t+l, s-2t-l
3t+l~s~4t+l:
4t+2
~
s
~
00 , 00 , s+3t+l, s-2t-l
(5.2.17)
5t+l: 00,00, s+3t+l, s+2t+l
5t+2~s~6t+2:
00, s+-3t+l, s-3t-l, s+2t+l
Combining (5.2.16) and (5.2.17) shows that every symbol occurs in either
one or two blocks with every other symbol, for which one of the two is representative.
Also, it can be checked that each symbol is the representative
of 2t+1 blocks.
It is easily shown that the underlying association scheme
is not strongly regular.
The parameters of the representative scheme are
o
Example 5.2.18
BIBD(30,10,3,9,2).
The representatives are underlined.
0
1
2
-3 4
5
6
7
8
0
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
0
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
5
6
7
8
0
1
2
3
4
00
00
00
00
00
00
00
00
00
0
1
2
0
1
2
3
4
5
6
7
8
3
4
5
4
5
6
7
8
0
1
2
3
6
7
8
-
64
5.3
Optimality results
Starting with a BIBD(sv,v,k,sk,A) with an m-class representative scheme,
development according to Theorem 5.1 yields a BIBD DO with parameters
b
O
= (k-l)sv,
r
o=
(5.3.1)
1
ror DO' the values of e.+e. - (kN .. +e .. ) are
1 J
1J 1J
(A(2k+2)-2A) = 4s(k-l)-2A(k+l)+2A
t
t= 1,2,
t
... , m
(5.3.2)
where A ;A , ... ,A m are the parameters of the representative scheme.
l 2
If the
1
e.+e.-(kN .. +e .. ) were all equal, the common value would be (from (2.2.5))
1 J
1J 1J
4b
- vo -
2b O(k-l) (k+2)
v(v-l)
.
= 4s (k-l) -
[
4A '
0 k J
O
2A +-·1
4A
= 4s (k-l) -2A (k+l) +k
(5.3.3)
The differences between the "optimum value' (5.3.3) and the attained values
(5.3.2) are
t= 1,2,
... , m
(5.3.4)
In particular, a 2-class representative scheme must satisfy A < 2A/k and
l
A > 2A/k.
2
A design is type II optimum (Takeuchi (1961)) if
~mong
its competitors
it minimizes the maximum variance of estimates of normalized treatment contrasts ~T~ for which exactly two .coordinates of ~ are nonzero.
A type II
optimum BIBD for the NNI model, with parameters (5.3.1), must minimize
1
A
A
2
0
max var( - (t.-t.)) = max ---{kA OV+2 P (k+l) (Ao-rO)+kP[e1·+eJ'
• .J.' ,2 2
dj
12 1 J
1r-J I\ V
O
1
-(kN .. +e .. )]}
1J 1J
or, equivalently,
max
itj
1
P[e . +e . - (kN. .+e. .)]
1 J
1J 1J
(5.3.5)
6S
Hence (5.3.4) will be useful in checking for type II optimality for the designs of Section 5.2.
Theorem 5.10
More precisely,
Consider the design DO with parameters (5.3.1) given by de-
veloping a BIBD(sv,v,k,sk,A) with m-class representative scheme (Al, ... ,A ),
m
and suppose k 7- 0 (mod 4A).
(i)
If 4A - 2A > -1
k
t
with P ~
(ii)
t=1,2, ... ,m, DO is type II optimum for the NNI model
o.
If 4A - 2A < 1
k
t
t=1,2, ... ,m, DO is type II optimum for the NNI
model with p::;; O.
Proof: .
Since
k~4A
1
and the quantities e.+e.-(kN .. +e .. ) are integral,
1
J
1J
1J
1
(5.3.3) implies the e.+e.-(kN .. +e .. ) are not all equal.
1 J
1J 1J
Hence, since they
sum to a constant,
max(e.+e.-(kN~.+e.. )) ~ 4S(k-l)-2A(k+l)+int(4kA) + 1
ih
1
J
1J
1J
and
min(e.+e.-(kN~ .+e .. ))::;; 4S(k-l)-2A(k+l)+int(4 A)
i::f. j
1
J
1J
k
1J
o
The result follows from (5.3.4) and (5.3.5)
The purpose of this section is to prove type II optimality for several
of the series of designs constructed in Section 5.2.
First, two hard working
lemmas.
Lemma 5.11
A BIBD for which vlb and
min (e.+e.-(kN 1.. +e .. ))
i::f.j
1
J
1J
1J
must have equality of the e. IS.
1
~
4b
v
int(2b(k-l)(k+2)) _ 1
v(v-l)
(5.3.6)
66
Proof:
For a BIBD with vlb and equality of the e. 's, permute the treat1
ments within blocks so that for the resulting design the e. 's are not all
1
equal, say e I = 2b/v - p,
p>O.
Suppose that for this permuted design,
. ( e. +e. - (kN
1.. +e .. )) >- 4b _.1nt (2b(k-l)(k+2))
_ I.
mm
(1)
'..J'
1
J
1J
1J
V
V
v1r]
1
Write n .. = kN .. +e .. for this design; n .. = n.· for the unpermuted design.
1J
1J 1J
1J
1J
Recall that
'i
2p.
n.·
1J
'..J'
Jr 1
Now,
j=2, ... ,v
2p
:0;
(2~ _p) (v-I) +(2b- 2~ + p)
4b(v-l) + (v-I)
v
- (2r(k+l)_4b) + (v-I) int(2b(k-l)(k+2))
v
v(v-1)
4b
=> vp
p
:0; - -
v
:0;
~{4b-2bk(k+I)+V(V-I)+V(V-l) int(2b(~(~~i~+2))}
v
. (2b (k-l) (k+2))}
< v-I
P - .{- 2b (k-l) (k+2) + 1 + 1nt
v
p
:0;
Lemma 5.12
v(v-I)
v(v-I)
V~l {I - frac(2b(~(~~g+2))} < I => p=O
o
A BIBD for which vlb and
max
i:j.j
(e.+e.-(kN~.+e .. ))
1 J
1J 1)
:0;
4b _ int(2b(~-1)}k+2))
V
v v-I
(5.3.7)
must have equality of the e. 's.
1
Proof:
Similar to proof of Lemma 5.11, but use p<O.
o
67
Theorem 5.13
The series of NNl-PBD's with b =v(v-l)/3, ro=v-l, k=3,
O
v=6t+l, A =2, Al=O, A =1,
2
o
t~l,
given by developing the series (5.2.8) accord-
ing to Theorem 5.1, is type II optimum for the NN2 model.
Proof:
The values of (5.3.4) are
4A
1
3
and
k
Thus Theorem 5.10 gives the result for
p~O.
A design will perform better for
p<O only if it satisfies the condition (5.3.6), so that the e. 's must all
1
be equal.
Hence the design must satisfy
e .. +kN~ . ~ int(
1J
1J
2b 0 (k-l) (k+2)
v(v-l)
) + 1 = 7
1
1
1
Since k=3, e .. = 2A - N.. => e .. +kN .. = 4+2N ..
0
1J
1J
1J
1J
1J
1
2v(v-l)
N.. = b (k-l) =
o
3
i<j 1J
series is optimum for p< O.
But
l l
~
1
7 => N..
1J
~
1
so N..
> 1 for some ifj .
1J
1
ifj.
Thus the given
0
The series of NNl-PBD's with b =2v(v-l)/3, r =2(v-l), k=3,
O
O
Theorem 5.14
v=6t+4, A =4, Al=l, A2=2,
O
t~l,
given by developing the series (5.2.10) with
representative scheme constructed in Theorem 5.9, is type II optimum for
the NNI model.
Proof:
The values of (5.3.4) are
4A
and
k
4A
k
Thus Theorem 5.10 gives the result for
2A
2
=-4
p~O.
3
A design will
for p>O only if (5.3.7) is satisfied, in which case
1
e .. +kN .. ~ int(
1J
1J
2b (k-l) (k+2)
O
v(v-l)
) = 13.
perform better
68
1
1J
13 => N..
=>
N~. ~ 3v(v-l)/2.
II
i< j
Lemma 5.15
l I
But
1J
i< j
3
~
N~. = bO(k-l) = 4v(v-l)/3
< 3v(v-l)/2
1J
o
A BIBD with parameters b=s(k-l)v, v=sk(k-l)+l, r=sk(k-l),
A=k-l and k:odd for which all the e. 's are equal, satisfies e .. +kN~ .~2k
1
1J
1J
for at least one pair i#j.
Proof:
Suppose e .. +kN~ . ~ 2k+1 for all i#j.
By contradiction.
1J
1J
Since
A=k-1,
a
1
Thus N..
1J
~
e .. ~ 2(k-l)
1J
~
1 for all i#j
and
a
~
1
N..
~
1J
k-l.
and
1
N.. = 1 => e .. ~ k+ 1
1J
1J
Also, equality of the e. 's => e.=2b/v= 2s(k-l) =>
1
1
Let some treatment be given, say treatment 1.
a
Then
l
Nij~ 2 ,
a
v-I = sk(k-1)
+ a
=
2
v
and
a
1
N~. = 2r-e. = 2s(k-l)2
1J
1
Let
1
= # {j : N = 1 , j E{2, ... , v}}
lj
a 2 = # {j :
1
l
. J.'
Jr 1
S
I
1
N . = 2s(k-l)
j=2 lJ
+
2a
~
2s(k-1).
2
jE{2, ... ,v}}
2
Hence there are at least 2s(k-1) treatments j such that e
.~
1J
k+l.
e
Let Al be the set of blocks containing treatment 1, A1 the subset of Al
e
for which 1 is on an end plot, and Am = A - Ai· Then
1
1
#A
1
m
= sk(k-l), #A~ = 2s(k-l), #A = s (k-1) (k-2) .
1
69
Let a set VI of 2s(k-l) treatments j with N~j=l be given.
These treatments
If treatment j occurs on an end plot of A~,
take up 2s(k-l)2 plots in AI.
there is a contribution of 2 to e
lj
, and if it occurs on any other plot of
A~, there is a contribution of 1 to e
lj
.
Occurrence of j
on
one
of
the
2s(k-l)(k-2) end plots of A7 contributes 1 to e
; occurrence on any other
lj
Hence by counting over the 2s(k-l)2 plots,
plot of A7 makes no contribution.
2
L
je:V1
e . ~ 2[2s(k-l)] + [2s(k-l) -2s(k-l)] = 2sk(k-l)
l
J
But since N~j = 1 => elj~ (k+l)
I
je:V
1
it must hold that
o
e · ~ 2s (k-l) (k+l) => 2sk(k-l) ~ 2s(k+l) (k-l)
lJ
Developing the BIBD(sv,v,k,sk,l) with odd k by the method
Theorem 5.16
of Theorem 5.1 produces an NNI-PBD(2) with parameters
b
O
= s(k-l)v, r O = sk(k-l), AO=k-l, Al=O, A2 =1
that is type II optimum for the NNI model.
Proof:
If k=3, the result is given by Theorem 5.13.
4A
k - 2A l
so for
k~5,
=
4A
k
4
k and
the result for
Now,
4
2A 2 = k - 2
P~O
is given by Theorem 5.10.
1
The values of e .. +kN .. for a member of this series are
1J
2A(k+l) - 2A
1
1J
= 2(k+l)
and
2A(k+l)-2A
2
= 2k.
For a design to perform better than the given design for P>O, it must satisfy
(5.3.7).
Hence it must have
min (e .. +kN 1.. ) ~ int(
1J
1J
i!j
2b (k-l) (k+2)
O
v(v-l)
)
= 2k+l
(k~5)
70
o
which contradicts Lemma 5.15.
The remainder of this section will be devoted to evaluating and comparing the two series of NNI-PBD's with k=v-l
tion 5.2.
and even v constructed in Sec-
The series with cyclic representative scheme after the method
of Theorem 5.6 will be called the C series.
The other series, with repre-
sentative scheme which induces a group divisible association scheme, will be
called the G series.
Recall that for the C series, Al = 1 and A = 2, and for
2
the G series, Al = 0 and A = 2.
2
Theorem 5.17
The C and G series of BIBD's with k=v-l and even v are each
type II optimum for the NNI model with
p~O.
If p<O, the C series is super-
ior with respect to the type II optimality criterion.
Proof:
For the C series
4A _ 2A
k
2
=
-4
k
while for the G series
4
_!k
and
4A
k
2A
2
=
-4
k
The case k=3 follows from Theorem 5.14, and
p> 0:
k~5
is an application of
Theorem 5.10.
4A
max var(t. -t.) is determined by max(-k - 2A ) and hence the C series
t
..
1
J
t
A
P <0:
1,J
is superior.
It should be noted that the C series has less pairs of treatments that
attain the type II bound for p>O, and so may be considered superior in that
sense.
o
The two series may also be compared in terms of the E-optimality criterion.
A design is said to be E-optimum among its competitors if it minimizes
71
the maximum eigenvalue of var(!).
The following lemma, stated in a slightly
different form by Takeuchi (1961), will be useful.
Let V
Lemma 5.18
vxv
be a nonnegative definite, symmetric matrix of rank
v-I with zero row and column sums.
W = a V + SI + yv
has eigenvalues
W
o
J
Then the eigenvalues of V are
= S+y, w ,w ' ... ,w _ .
l 2
v l
i=1,2, ... ,v-l.
eo=O, e.=(w.-S)/a,
1
-1
Suppose
1
1
·22
The values of e .. +kN .. for the C series are 2(k -2) and 2(k -3). This
1J
1J
information in Lemma 2.6 (with P =O, pl=p) and some straightforward manipu2
lation yields var(!) = V
c
as
4
2 -2
2
3 2
(k-l) (k+l) a -V = [k(k-l) (k+l)-2p(k -k -k-l)]I
c
- [k(k-l)2_2P(k 2-2k-l)]J
where S
v
=
(s .. )
= 1 if
1J
Ii-j I
v
v + 2kp Sv
= 1 or v-I
= 0 otherwise.
S
v
is the sum of two circulants; its eigenvalues are
.
2 COS(21Tj/V)
v
J=1,2, ... '2- 1
2 and -2
each with multiplicity two
with multiplicity one
4
2
2
-1
2
3 2
Setting a = (k-l) (k+l) /(2kpa) and S = -(2kp) [k(k-l) (k+l)-2p(k -k -k-l)]
in Lemma 5.18, the eigenvalues of V
4kpa
c
2
cos (21Tj/V) •
are
~
2
3 2
2 [k(k-l)2(k+l)_2P(k _k _k_l)]
(k-l) (k+l)
j=1,2, ... ,~- 1 each with multiplicity two, j =
multiplicity one;
andO.
I with
72
Similarly, the values of e .. +kN:. for the G series are 2(k 2 -l) and
1)
2
A
2(k -3), so that var(!)
1)
= VG is
4
2 -2
2
3 2
(k-l) (k+l) a VG = [k(k-l) (k+l)-2p(k -k -k-l)]I
v
- [k(k-l)2_ 2P (k 2 _2k_l)]J + 4kpD
v
v
The eigenvalues of Dare 1 and -1, each with
v
mUltiplicity v/2.
Hence the eigenvalues of V are
G
2
4kpa
(k_l)4(k+l)2
--":---"... +
with mUltiplicity (v-2)/2
-4kpa
2
+
and
----:---~
(k_l)4(k+l)2
with multiplicity v/2,
Theorem 5.19
and O.
If p>O, the series C is superior to the series G in terms
of E-optimality. If
p~O,
the two series are equivalent with respect to E-
optimali ty.
theorems 5.17 and 5.19 taken together give a fairly strong argUment in
favor of the C series.
It should be noted, however, that the structure of
D provides for a useful set of mutually orthogonal contrasts.
v
One may take
a set of v/2 contrasts cpmparing first associates (e.g. of the form (1,-1,0,
... ,0)) and another set of (v-2)/2 comparing sums of first associates (e.g. of
the form (1,1,-1,-1,0, ..• ,0)), all of which are orthogonal eigenvectors of
V .
G
The two types are estimated with only two distinct variances, and the
estimates are uncorrelated.
73
5.4
Complete block designs with partial variance balance
It has been seen in Section 3.2 that
~~mplete
block designs with odd v
must have a mUltiple of v blocks to attain NNl optimality, and the minimum
designs in v blocks have N~ .=2 for all ifj. Since the N~ .'s determine the
1J
1J
balance here (Theorem 2.11), an experimenter desiring a design in fewer
blocks may ask how to assign the treatments in such a manner that the symmetry
in the N~ .'s is disturbed as little as possible. This may also be thought
1J
of as distributing the variance imbalance as evenly as possible throughout
the treatment set.
Such an approach motivates the following construction
v-I
of complete block designs with b = -2- blocks and odd v.
Decompose C into (v-l)/2 edge disjoint Hamiltonian cycles.
v
Form a
path from each cycle by deleting an edge in such a manner that among the
vertices previously incident with the set of (v-l) /2 deleted edges, no one
appears more than once.
For instance, for the set of cycles given in Lem-
. (n-l)) f or
. n+l 1---2-ma 4 . 2 , one may d e 1ete the e d ges ( 1+--2-'
. n
. n) f or even n, 1=
. 0 , 1 , ... ,n-l; V= 2n+ 1 .
( 1+2,1-2
0
dd n or the e d ges
Taking the paths as ordered
blocks, the vertices as treatments, and the pairs of vertices corresponding
to deleted edges as (0,1), (2,3), ... ,(v-3,v-2) gives
1
=
J1
N ..
N~ . = 0
1J
= 1
e. = 1
1
e
0
Write V = yare!).
j=i+l; i=0,2,4, ... ,v-3
otherwise
(5.4.1)
i=I,2, ... ,V-l
= 0
For this design and the NNI covariance structure
(set p 2=0 and Pl=P in(2.2.3)):
(V_l)2 v 2
V2 (V_l)
v(v-l)
V =
2
I - [2
+ p(v+l) (v-l)]J + pv
4i .
J
--~:~t-~:!
~
. T
1 v- 1
I'
I
I
0
}
74
2
v (v-I) I + [2pv+p _ V(V-I)]J
2
2
(5.4.2)
I v- 1
2
0
V
-
- pv
J2
:1v- I
0
I
t
-----------~---I
. T
~v+2
lv-I
where y
Lemma 5.20
=
The eigenvalues of the matrix V given by (5.4.2) are
eigenvalue of V
20
2
2
multiplici ty
(v-I) 12
I (v-I)
222
2
- 4po Iv(v-I)
20 I(v-I) - 4po I(v-I)
20
I
222
I (v-I) - 8po I (v-I)
(v-3)/2
o
Proof:
2pv+p-v(v·I)/2
I
First the eigenvalues of Wwill be found.
[I v _I
IW-AII
-2-
0 -
2
PV J 2] + yJ V_I - >..I v _I : (y-pv) iv-I
I
--------------------------------+----------T
=
(y-pv)
I
ly->..-pv(v+2)
i
~-l
I
Let
[(J-I)v_l
-2=
[I _
v l
2
0
G ] + [(J-I)v_l
2
0
yJ 2]
0
yJ 2]
say.
2
Then
IW->"II =
(y->..-pv(v+2)) lEv_II + (pv-y) (v-I) IFv_11
75
where F
is obtained by deleting the last column of E I and adjoining
v-I
v(y-pv) iv-I'
Now,
Ev _3
=
IYiv-3, (y-pv)iv_3
---------~---------------,
2
J
. ;y-pv -A y-pv
2
y 2x (v-3) I
Iy_pv
y-pv
0 3' (y-pv)i 3
Ev- 3- YJv-,
3: ""V""'1-----~----+--------------I
o
2x (v-3)
2
:-pv -A
I
2
-pv
y-pv
y-pv
Ev_3-yJv_3:2v_3(y-pv)iv_3
=
----------~---~----------I
o
2
;-pv -y
2x (v-3),
A
=
=
y-pv
a
A(pv-y) IE _3 - YJ _3 I
v
v
Hence,
IW-AII
= (y-A-pv(v+2)) lEv_II
2
+ A(v-l) (pv-y) IE v _3-YJ v _3 1
= (y-A-pv (v+2))1 (rv_1 0G2) +[(J- I) v-I ~J 2] I + A(v-l) (pv-y) 2 \I v - 30 (G2-yJ 2) I
-2-
=
-2-
-2-
v-3
v-3
(y-A-pv(v+2)) IG -YJ ,-2-!G +(V;3) YJ21 + A(v-l) (pv_y)2 IG -YJ 1-22
Z 2
2 2
Note that
IG2-yJ 2 I
so
=
I-pv 2J 2-AI 2 I
=
A(A+2pv 2)
v-I
v-3
2
2
IW-AII = A--(A+2pv )-2- [(y-A-pv(v+2)) (A-(v-l)y+2Pv 2)+(v-l) (pv_y)2]
=
v-I
v-3
-22 -22 (v 1)
-A
(A+2pv)
(A+v
(A+pv(v+l))
-T)
2
Therefore, the eigenvalues of Ware _v (v_l)/2, -pv(v+l), 0 with multiplicity
2
(v-l)/2, and _2Pv with multiplicity (v-3)/2. The eigenvalues of V follow
.2222
from Lemma 5.18 wlth ex = (v-I) v /40 and B = -v (v-l)/2.
0
76
For an arbitrary complete block design with (v-l)/2 blocks, and for any
a such that
~
T
~ ~ = 1
an d
T. = 0 ,
(2.2.4) =>
~.l,
2p
I I
i<j
a.a. N~.].
1J
1J
In particular, for an elementary treatment contrast,
1
A
var ( - (t. - t . ) )
12
1
J
so that the conditions for type II optimality are
p>O:
maximize min N..
1
1J
(5.4.3)
p<O:
minimize max N..
1
1J
(5.4.4)
i1j
• J. •
1r]
In this case,
I. <'I N~.
1J
1
= b(v-l)
=
2
(v-I) /2, so there are at least v(v-l)/2 -
J
1
(v-l)2/2 = (v-l)/2 unordered pairs (i,j) with N.. = O.
1J
Therefore, (5.4.3)
is satisfied by all competitors, and (5.4.4) is satisfied by any design with
N~.
1J
$
1 for all i#j.
So while eliminating obviously imbalanced designs with
some N~. ~ 2, the type II criterion is not exceedingly sharp.
1J
For example,
arbitrarily deleting an edge from each cycle of a Hamiltonian cycle decomposition of Cv will produce a type II optimum design. Say a design is type III
TA
optimum if it minimizes max var(~ 1) over all normalized contrast vectors ~
with exactly three nonzero components.
Lemma 5.21
··
.. 0
•
Nl" 0 =
F or d 1st1nct
1,J,N wr1te
1JN
(N~.,N~o,N~o)
and assume i,j,R.
1J IN 1N
are so ordered that N~.~N~o~N~o. A complete block design in (v-l)/2 blocks
1J
JN
1N
(v is odd) is type III optimum for the NNl model if
1
NijR. = (1,1,1) or (1,1,0)
for every triple i,j,R. of distinct treatments.
Proof:
Let Xl' x , x be such that LX~=I, LX.=O, and x.10, i=1,2,3. Assume
2
3
111
77
(5.4.5)
1
For a given triple (i,j,t), the assumed ordering of N.. o maximizes the
1J'"
value of (5.4.5). By inspection of (5.4.5), a design that performs better
p>O:
1
than the postulated type of design must have every N.. o of the form (a,b,c)
1J'"
where a>O, b>O, and c~O, and if a=b=l, c=l. Let Yl= # triples (i,j,t) with
I
1
N £ = (1,1,1), and let Y2= # triples with N
= (a,b,c) for which a>l. Then
ij
ijt
for the superior design,
\
l
I
I
)
= l\ (N 1.. +N·o+N·
•
•
0
1J J'" 1",o
1,J,'"
~
v
3(Yl+Y2) = 3(3)
= v(v-l) (v-2)/2
where the sum is over all subsets of size 3 from the treatment set.
But for
any complete block design in (v-l)/2 blocks,
l
= (v-2)
LI N~.
i<j
= (v-2) (V-l)2/2.
1J
2
Since x x +x x = -x <0, (5.4.5) => a design that is superior to the
3
l 3 2 3
1
postulated type must have every N.. o of the form (a,O,O). This implies that
1J'"
every treatment is the neighbor of just one other treatment, which is imposp<O:
o
sible.
Theorem 5.22
For odd v and b=(v-l)/2, the constructed series of designs
with neighbor parameters (5.4.1) and covariance matrix (5.4.2) is type II
and type III optimum within the class of complete block designs for the NNI
model.
Proof:
If
p~O,
Type II and type III optimality follow from the discussion above
and Lemma 5.21.
which is
the series is E-optimum.
For
p~O,
2
the maximum eigenvalue of var(!) is 2a /(v-l),
also the maximum variance of an elementary treatment contrast for
a member of this series.
Hence type II optimality => E-optimality.
0
78
That the designs of Theorem 5.22 are not necessarily E-optimum for p<O
is shown by an alternative construction.
(v-I)/2 Hamiltonian cycles.
Again for odd v, decompose C into
v
Now form paths and hence blocks by deleting
from each cycle one of the edges incident with a fixed vertex (this is
equivalent to adjoining an additional fixed treatment to the end of each
block of a minimum optimum complete block design with v-I treatments). Write
VI ={O,1, ... ,(v-3)/2}, V ={(v-l)/2, ... ,v-2}.
2
Then with appropriate labeling
of the treatments
1
N.. = 0
i=v-l, je:V 2 or j=v-l, ie:V 2 ·
1J
= I
otherwise
e. = 0
ie:V
= 1
ie:V
= (v-l)/2
i=v-l
1
For this design
(pv)
-1
(5.4.6)
I
2
and the NNI model, let V0 = var(;tl
and
let
W
o
=
2 2
2
2
(v-I)
l(v-l) v /4aV o - (v-I)v /2 I + -2-(v+2p(v+I))J] + vI - (v+I)J
. T + v I - ( v+ I ) J) •
( = vNI +!U.T +.u:,
Lemma 5.23
The eigenvalues of V are
o
eigenvalue
multiplicity
2
2
2a /(v-I) _ 4pa /(v_I)2 + 2pa2(1+/v+I)/v(v-I)
2a
2
2a 2/(v-l) _ 4pa 2/(v_l)2
v-3
222
/(v-I) - 4pa /(v-l) + 2pa (l-/v+l)/v(v-l)
I
o
Proof:
I
I
First the eigenvalues of W will be found.
o
79
o
J
v-I
2
,
v-I
v-I:-z-lv-l
0
2:
J
+
o
"'V-I
2
or
or
lv-]! ....v-l
-z-: -z-
1
v-lor
:v+I.r :
-2-1v-I :-Z-lv-l~
o
IV-3
v-I
0
: 2 lV-1 + vI - (v+I)J
2
---------~--------~--------
--------
0
I
v-I:
I~
'2J
:
v-I
Z
v-I
Z
v-I
-z--
-2-
-J
-Z-
---------,--------f--l-----
(J-I)v-l
.
: 2 lv-I
2'
2
I
2
--------f---------~---------
=
o
'J
: -v-l .
v-I'
v-I
, 2 lV-1
2 I
2
I
2
------ __ l
~--------v-3 or I-v-l or
-2- ~-l: -2-lV-I: v-2
- I
-
_
I
-2- 1
-2-'
I
3v-1
=>
IW-~II = (v-Z-~) I-JV-I-~Iv-11 IJv-l-AIv-11 + (V;l) (V;l) \EI!+(-l)
-2-
-z-
-2-
2
x
-z-
(v-I) (v-3) IE I
2
Z
Z
where E is given by deleting row v and column v-I of
I
by deleting row v and column (v-I)/2 of WO-~I.
Now
-J
v-I
-
-AI
2
I
I
v-I,
-,
2,
0
v-I v-3
-x22
WO-~I,
and E2 is given
:v-3
,
- ,.
I
2 Ny-I
:
2
-----------~-------------+---------,
,
ov-I
-
:CJ-AI) v-I v-3''-(v+l).-,
I
-x-,
2 Ny-I
Z
22;
:
-Z-
-J
v-I
-~I
-2-
v-I
v-3
= (-1~ ~-2- (v+l)
2
-J
v-I
-~I
-2-
v-I
-2-
v-I
-2-
-~I
I
!_(V+I~
v-I x v-3:
2 V-II
2
z
2
80
and
-J
v-I v-3
22
-AI
0
I
v-I v-3:
22:
-x-~-
=
: v-3 .
v - I ' '2_
lv-I
_
2
2
I
-------------------+-----------~----------
1:
0V_l v-3
z-><-2-
; J v- I-AI v- ,- (v+l).
-ly
-2-2-:
2
2
2
I
v-I
= (-1)
2
J
-J
-AI
v-I
v-I
-2-2-
-AI
:V-3 .
v-I v-3
v-I v-3, 2 lV-l
-=--=-x-=--=-X-I
2
2
2
2,
2
v-3
=
=>
(-1) v-2 A 2
IWO-AI
I
(v-3) J
-AI
2
v-I v-I
-2- -2-
= (v-2-A)
- J
V;l
-AI
v-I
v-3
+ (V+l)2 (v-I) (_1)-2- A 2
8
V;l
I-
I
J
. -AI
ViI
V;l
JV_1-AIv_1j
2
3v+l
v-3
2 (v-I) (V_3)2 A 2
+ (-1)
8
Easily shown is
and
J
-AI
v-I
v-I
\ -2-2-
I
-J
I::
-AI
v-I
v-I
-2-2-
I
I
2
I
J v-I -AI v-I
-2-2-
v-I v-3
(_1)-2- 1..-2- (A- V- 1 ),
2
v-I v-3
-2 -2
v-I!
= (-1)
A
(1..+-)
2
v-3
v-I
v-I
WO-AI = A
[(v-2-A) (1..---)(1..+-2-)
2
I
I
v+ 1
v+ 1
r-:-:;and the eigenvalues of W are -1, -2-(1+(v+l), -2-(1-vV+l), and 0 with
o
81
mUltiplicity v-3.
The eigenvalues of Va follow as an application of Lemma
2
5.18 with a = (v-I)·v/4pcr
2
and B = v(I-(v-I)/2p).
0
2
2
For p<O, the maximum eigenvalue of V is 2cr I (v-I) - 8Pcr I (v-I) 2 and
.
2
2
2
2
the maximum eigenvalue of Va 1.S 2cr I(v-l) - 4pcr I(v-I) + 2pcr (l-v'v+I)/v(v-l).
The difference is
max e.v. V - max e.v. Va =
2Pcr
2
~ + 1- 3v}
2 {(v-I)vv+I
v(v-I)
which is positive for v=5,7, and 9.
So for these values of v, the designs
of Theorem 5.22 are not E-optimal for p<O.
The series of designs with neighbor parameters (5.4.6) are interesting
in their own right.
Rather than distributing the variance imbalance through-
out the treatment set, as was done for the designs of Theorem 5.22, a member
of this series concentrates all the imbalance on one treatment.
With re-
spect to the other v-I treatments, the design is perfectly balanced.
This
type of design may be of interest in situations employing a control treatment,
comparisons with which are of less interest than those among the "true"
treatments.
CHAPTER VI
OPTIMUM DESIGNS FOR CYCLIC NEIGHBOR COVARIANCE STRUCTURES
6.1
Introduction
This chapter will be concerned with cyclic neighbor covariance struc-
tures.
th
Define the m order cyclic neighbor (CNNm) covariance structure as
that subset of (2.2.1) given by
s=t
=
min[s-t (modk), t-s (modk)]=q; q=1,2, ... ,m
otherwise
The number min[(s-t) (mod k), (t-s) (mod k)] is the circular distance separating
plots t and s in a block of size k.
So plots within a block are assumed to
be correlated according to their circular distance from one another.
A typi-
cal block for this type of structure may be conveniently pictured as in (6.1.1).
(6.1.1)
In (6.1.1), the distance from 0 to 5 is 5, and the distance from 0 to 6 is 4.
Let C~. = # {~: g(~,s) = i, g(~,t) = j, min[s-t (mod k), t-s (mod k)] = q}.
1J
q .
C.. 1S t h e numb er
1J
0
ft1mes
'
.
1. an d'J occur as q th nearest c1rcular
neighbors.
83
A BIBD is weakly universally optimum for the CNNm covariance
Theorem 6.1
structure
if for i;j
(k~2m+l)
(1)
1
the numbers C.. are all equal
(2)
2
the numbers C.. are all equal
(m)
the numbers C.. are all equal
1J
1J
m
1J
k
k
Proof:
I
k
Pst = k
t=l
=> k
2 -2
(J
k
m
k
=
=
k(I+2
I p.)
l.
I
Pst
Pst
i=l 1
t=l s=l
s=l
I
2
w
= k Pst
s,t
m
k(1+2
I P)
q=l q
So by Lemma 2.5,
=
cov(t. ,t.) =
J
1
2
(J
2
(J
(Av)
(Av)
m
_22
r[k -k(1+2 I P )]
q=l q
-2
[-Ak(1+2
2 m
m
I
P ) + k I p C~.]
q=l q 1J
q=l q
o
The result follows from Theorem 2.8.
6.2
Cyclic complete block designs
The purpose of this section is to prove:
N
Let v=p
Theorem 6.2
P
N-l
There
be an odd prime power.
exists
a
set
of
(v-l)/2 cycles constituting a weakly universally optimum complete block
design for the CNNm covariance structure,
N
For v=p , p an odd prime, let y.
J
m~(v-l)/2.
j=1,2, ... ,~(v) be the standard resi-
dues (mod v) that are coprime with v; RP
v
='{y., j=1,2, ... ,~(v)}.
J
Here ~ is
Euler's function for the number of positive integers coprime with and less
than a given integet.
Let 1=a,{a,2< •..<
a,~(v)
/2 be those residues between I and (v-I) /2 inclu-
84
sive that are coprime with v; RP+ = {a., j=1,2, ... ,~(v)/2}.
v
J
Form the cycles (O,li,2i, ... ,(v-l)i,O)
i=al,a2, .•.
,a~(v)/2
where the
coordinates are reduced (mod v), and where the cycle given by a = 1 is
l
called the initial cycle. Choose some arbitrary element t in the initial
cycle.
First consider those elements that occur at circular distances that
+
v
are coprime-v from t in the set of cycles.
_
-1
ment such that a. t.= t, i. e. t.= a. t
J J
J
J
Let dERP , and let t. be the eleJ
Then the elements at distance d from
tare
{a.(t.±d), j=1, ... ,~(v)/2}=· {t+y., j=1,2, ...
J
J
J
Hence the elements at distance d from
,~(v)}.
t in the set of cycles are those at a
distance coprime-v from t in the initial cycle.
It follows that every element
at a distance coprime-v from t in the initial cycle is at every distance coprime-v from t in the set of cycles exactly once.
Let the expression "distance a mUltiple of pR-" (R-<N) mean only those
distances a multiple of pR- that are not multiples of pj for R-<j~N.
Hence
"
. 1e of pR- are
t he d lstances
a mu 1tlP
RRRRn-RRp , 2p , ... , (p-l)p , (p+l)p , ... , (p
-l)p
The general term here is y j ,N-R-P R- where Yj ,N-R- E RPPN- Nn . The number 0 f t h ese
N-RN-R-1
N-R--l
distances is ~(p
) = P
(l-p) = P
(p-l). (This is not quite consistent with the definition of distance given in Section 6.1.
For instance,
with v=33, R-=l, the list is 3,6,12,15,21,24, which lists the distances
3,6, and 12 (mod 27) twice each.
The two approaches are equivalent, however,
and \vill be used interchangeably as is convenient.)
RNow look at the elements at some fixed distance a mUltiple of p from
Rt in the set of cycles, say at a dis tance Ys, N-R-P for some fixed Ys,N-RE RP N-R-.
p
The set of all such elements in the set of cycles is
8S
. {a..
J
(t.±y
J
s,
N oP5/.,), j=l,2, ... ,¢(v)/2}= {t+y.p5/."
J
-J(,
j=l,2, ... ,¢(v)},
which are elements each of which was at some distance a multiple of p
from t in the initial cycle.
5/.,
In fact, each element at some distance a
multiple of p5/., from t in the initial cycle occurs at each possible distance
5/.,
5/.,
a multiple of p from t exactly p times in the set of cycles. This follows
from
5/., _ 5/., N-5/.,
P = P (p
+l)
==
5/.,
N-5/.,
P (2p
+l) ==
.
.
.
_ 5/.,
N-5/.,
_
= p (sp
+l) = . . .
_ 5/.,
5/.,
N~5/.,
N
= P ((p -l)p
+l)(mod p )
and in general
5/.,_ 5/.,( N-5/.,
) _ 5/.,(2 N-5/.,
)_
_ 5/.,( N-5/.,
)_
Yj,N-5/.,P = P P
+Y j ,N-5/., =p
P
+Yj,N-5/., = ... =p sp
+Yj,N-5/., =...
_ 5/.,
5/.,
N-5/.,
N
= P ((p -l)p
+Y j ,N-5/.,) (modp )
where
o~s~p
.
{sp
5/.,
N-5/.,
-1, and from the fact that
N
5/.,.
N-5/.,
+Yj,N_5/.,(mod p), s=O,l, ... ,p -1; pl,2, ... ,¢(p
)} = RP N
P
Thus in the initial set of ¢(v)/2 cycles, any given pair at a distance
a multiple of p5/., in anyone of the cycles is at a distance a multiple of p5/.,
in every cycle, and occurs at each possible distance a mUltiple of p5/., exactly
p5/., times, 5/.,=0,1, ... ,N-l.
Suppose there is a set of (pN_l)/(p_l) permutations
of the initial cycle so that in the set of permuted cycles each pair occur·s
at a distance a mUltiple of p
5/.,
exactly p
N-5/.,-l
times.
Applying this set of
N N
permutations to the initial set of cycles produces a set of ¢(p )(p -l)/2(p-l)
N-l
= P
(v-l)/2 cycles with the property that every pair of symbols occurs at
each possible distance exactly p
N-l
times.
Hence Construction of this set of
permutations will prove Theorem 6 • 2.
For v=p
N
such a set of permutations will be generated from EG(N,p).
It
is well known that EG(N,p) can be decomposed into (pN_l)/(p_l) parallel bundles
of p
N-l
I-flats each.
Let the ordered linear forms generating
a
paticular
86
T
N
bundle be given by aox (= I a. ox.), i=1,2, ... ,N-l.
Then think of a
l
l.
JO=l l.J J
T
T
.
T
as generating a set of p (N-l),flats (by alx=O, alx=l, ... ,alx=p-l), and in
general think of a ,a , •.. ,a as generating a set of pj (N-j)-flats by
l 2
j
T
T
T
alx=c l , a 2x=c 2 , ... ,ajx=c as c l 'c 2 , ... ,c j take on all possible values
j
O,l, •.. ,p-l: it is in this sense that the linear forms of a bundle are ordered.
For the given bundle, let F . .
.
be the I-flat given by
l.l ' l. 2 ' ... , l.N -1
Identify each of the pN points of EG(N,p)
with one of the symbols in the initial cycle.
A permutation of the initial
cycle is obtained from a parallel bundle of order one in the following mannero
Choose a point from FO,O, ... ,O and let this be the starting point of
the cycle.
Choose the second point from FO,O, ... ,O,l' the third from
F
and so on until the pN-l_ th is chosen from F 1
p- ,p- 1 , ... ,p- 1
th
.
In general, the i
of the first PN-l pol.nts
l.S c h osen f r oFm . .
0,0, ... ,0,2
0
.
I N- l ,J N- 2 ,·· ·,J l
N-2
N-l
members of the permuted cycle,
= L\ Ps.J s + 1 + 1 • To get the next p
s=O
repeat this procedure of choosing points from I-flats with the restriction
n
N
h
were
that no point already chosen can be chosen again.
After p repetitions all
pN points have been chosen and hence a cycle obtained.
It is now easily verified that for a given bundle of l~flats, the cycle
generated from it satisfies:
(i)
any two points lying on the same I-flat of the bundle are at a
distance a multiple of p
(ii)
N-l
in the cycle.
any two points lying on a (N-j)-flat of the form
a~x=cl' a~x=c2' ... ,
aIx=c j that do not lie on any (N-j-l)-flat of the form
a~x=cl,a;x=c2'
... , a}x=c , a}+lx=c + are at a distance a multiple of pj in the
j
j l
cycle, j=1,2, ... ,N-2.
(iii)
any two points lying on different (N-I)-flats of the pencil generated
87
a~x are at a distance a multiple of PO=l in the
by the linear form
cycle.
Hence the problem will be solved if it can be shown that the equations
of the bundles can be chosen in such a way that
(1)
every pair of points occurs in different (N-l)-flats
for exactly p
(2)
N-l
bundles,
every pair of points occurs in the same (N-l)-flat
but in different (N-2)-flats for exactly pN-2 bundles,
(6.2.1)
(j)
every pair of points occurs in the same (N-j+l)-flat
but in different (N-j)-flats for exactly pN-j bundles,
(N)
every pair of points occurs in the same I-flat for
exactly one bundle.
Here, as explained above, an (N-j)-flat for a given bundle is given by the
.
. , ...
equat10ns
aT
x=1
l
l
T . 'for some 1. ,
h
T a Tx, ... ,a T_ x
,1 " .. ,1. , were
alx,
1 2
2
j
N1
,ajx=~,
are the ordered linear forms generating the bundle.
The case N=3
3
2
To find equations for (p -l)/(p-l) = P +p+l bundles such that
properties (1)-(3) of (6.2.1) are satisfied.
points in PG(2,p).
Note that there are (p3_ 1)/(p-l)
Let each of these points (in the standard Galois field
representation) be the "a " for generating the (N-l)-flats of a particular
1
bundle of EG(3,p).
Now the "a " must be chosen for each "a " such that a
l
and a are independent, and the vector spaces generated by the (a ,a ) pairs
2
l 2
are all unique.
2
1
(Such a choice guarantees that every line occurs once, and
every plane occurs once.
Every line occurs once => (3).
Any two points are
contained in p+l 2-flats, so every plane occurs once => every pair occurs in
88
a 2-flat without being in a generated I-flat (p+l)-l=p times => (2).
. occurs 1n
. dOff
.
pa1r
1 erent 2- f lats p 2+p+l-(p+l)=p 2 t1mes
=> (1).)
So a
This is
equivalent to finding a line through each point of PG(2,p) such that no two
of the lines are the same.
That such a choice of points and lines can be
made is guaranteed by the theory of distinct representatives; here an explicit method is given.
Choose an arbitrary line of PG(2,p) (the line at infin-
ity) and consider the other p2+ p lines to be grouped into p+l pencils of p
lines each, according to their having a common point at infinity.
Arbitrar-
ily label the pencils P ,P , ... ,P + and arbitrarily label the lines in penl 2
p l
cil i L.1, l' L1, 2"'" Lo1,p . For i;p+l let the point with which line L.1,J. is assoc0
iated be given by the intersection of L. . and L 1 . (i=1,2, ... ,p; j=1,2,
1,J
p+ ,1
... ,p-l) and let the point with which L.
is associated be given by the
1,p
intersection of L.1,p with the line at infinity (i=1,2, ... ,p).
Let the point
with which L 1 . is associated be given by the intersection of L 1 . with
p+ , J
p+ , J
Lo
(j=1,2, ... ,p). Finally, let the line at infinity be associated with the
J,P
point given by its intersection with P
p+l
This will be treated with an inductive/constructive proof,
The general case
by showing that the desired construction in EG(N-l,p) gives a construction in
PG(N-l,p) from which the desired construction in EG(N,p) can be obtained.
The PG(N-l,p)
+
EG(N,p) link is considered first.
For each of the
(pN_l)/(p_l) (N-2)-flats contained in PG(N-l,p), select a set of N-l indeIf a ,a , ... ,a _ are the corresponding
Nl
l 2
column vectors, then associate with each set a parallel bundle of I-flats
pendent points that generate it.
T
T
T
in EG(N,p) thus: let the bundle be given by alx=c , a2x=c2, ... ,aN_lx=cN_l
l
over all choices of c 'c , .•. ,cN_lEGF(p).
I 2
tinct.
Clearly the bundles are all dis-
Next, from each set of N-l points, delete one point in such a way that
89
the resulting sets of N-2 points each generate a distinct (N-3)-flat in
PG(N-l,p).
Note that each of these (N-3)-flats is contained in the same
number of (N-2)-flats.
If a ,a , ... ,a _ are the remaining N-2 points (as
N2
l 2
column vectors) of a given set, associate with them the parallel bundle of
T T
T
2-flats in EG(N,p) given by the linear forms a x,a x, ... ,a _ x.
N 2
2
l
Since each
of the chosen (N-3)-flats in PG(N-l,p) is contained in the same number of
(N-2)-flats, each I-flat of EG(N,p) is contained in an equal number of the
constructed 2-flats.
Continue in this manner, removing one point from each (N-j)-flat of
PG(N-l,p) in such a way to give distinct (N-j-l)-flats, and form the corresponding bundle of j-flats in EG(N,p) (j=1,2, ... ,N-2).
By the construction,
at each stage each I-flat of EG(N,p) is contained in an equal number of the
newly formed j-flats.
The order of the deletion of the points corresponds
to the ordering of the linear forms of the corresponding bundle; aN . is the
-J
.th
d
.
J
delete pOlnt.
More specifically, write
<I> (N, m, s)
= (sN+l_l)(sN_ l ).
m+l
m
(s
-1) (s -1).
. (sN-m+l_ l )
. (s-l)
= the number of m-flats in PG(N,s), where s is any
prime power.
Then each (N-j-l)-flat
of PG(N-l,p) is contained in
<I>(j-l,j-2,p) = (pj-l)/(p-l)
(N-2)-flats.
So a line formed in EG(N,p) by the process described in the
two previous paragraphs is contained in (pj-l)/(p-l) of the j-flats so formed.
Hence each line of EG(N,p) is contained in
90
pj-l
p-l
_ (pj-l_l) = j-l
P
p-l
of the constructed j-flats for which it is not in a corresponding (j-l)-flat.
Also, since there are (pN_I)/(p_l) bundles of order 1 in EG(N,p), every pair
of points occurs in different (N-I)-flats for
N
(pN-l_ l ) = N-I
P
p-l
P -1
p-l
of the bundles.
This verifies (l)-(N) of (6.2.1).
N
Hence the construction in EG(N,p) can be found from a set of (p -l)/(p-l)
descending chains of flat spaces in PG(N-I,p), each chain beginning with an
(N-2)-flat and ending with an O-flat, with the property that all the flat
spaces of a given order are distinct.
Note that this has been done for N=3.
Next, to show that the construction in EG(N,p) gives the descending
chains of flat spaces in PG(N,p).
For a given (N-I)-flat of EG(N,p), say
T
T
alx=c l , construct an (N-I)-flat of PG(N,p) by clxO+alx=O.
Doing this for each
(N-I)-flat of EG(N,p) and adding xO=O gives all the (N-I)-flats of PG(N,p).
Now to find an (N-2)-flat in each (N-I)-flat of PG(N,p) such that they are
all distinct.
Take the (N-2)-flat in PG(N,p) lying within the (N-I)-flat
T 0 and c x +aT x=O, were
h
T
.
alx=c
an d a T
clxO+a Tx=O to b e clxO+alx=
2 O 2
2x=c 2 1S one
l
of the (N-2)-flats of the assumed construction is EG(N,p) (c is arbitrary).
2
Are all of these (N-2)-flats distinct?
(a)
Suppose not;
two cases:
two (N-2)-flats from the same bundle yield the same (N-2)-flat in
PG(N,p) => 3 a ,a : independent N-vectors and (c ,c )I(c ,c ) such
3 4
l 2
l 2
T
T
T
that (cl,a ) and (c ,a ) form the same vector space as (c ,a ) and
3 l
l
2 2
T
(c ,a ).
4 2
(b)
This is impossible.
two (N-2)-flats from different bundles yield the same (N-2)-flat in
PG(N,p) => 3 a ,a and b ,b which generate distinct vector spaces,
l 2
1 2
91
and c ,c ,c ,c E: GF(p) (not necessarily distinct) such that (cI,a~)
l Z 3 4
T
T
T
and (CZ,a z) generate the same vector space as (c ,b ) and (c ,b )'
4 Z
3 l
which is contradictory.
Of course, it must also be shown that there is an (N-2)-flat at infinity distinct from all those just constructed, but this is trivial, as none of them
is at infinity.
In general, employ the same method to form (N-j)-flats in PG(N,p) from
the (N-j)-flats of EG(N,p) (j=I,Z, ... ,N-I).
distinct is done as the case j=Z above.
Showing that the flats are all
N+I
This gives a set of (p
-l)/(p-l)
chains of flat spaces, each chain beginning with an (N-l)-flat and ending
with a l~flat, such that all (pN+I_I)/(p_l) (N-j)-flats in the chains are distinct, j=I,Z, ... ,N-I.
It remains
to find an a-flat (point) in each I-flat
N+I
such that these (p-l)/(p-l) points are all the points of the geometry.
Noting that the (N-I)-flats of PG(N,p) form a symmetric BIBD with points
as treatments, one point can be chosen in each (N-I)-flat to represent that
flat in such a manner that each point is chosen exactly once.
Designate one
of the (N-I)-flats as the flat at infinity and consider the other (N-I)-flats
N
to be grouped into (p -l)/(p-I) parallel pencils of p (N-I)-flats each.
In
the construction above, each of the lines in the chains of flats in the same
pencil passes through a unique point at infinity.
For all but one of the pen-
cils, choose a finite point to represent (p-I) of the (N-I)-flats, letting
•
the common infinite point represent the remaining (N-I)-flat.
pencil, take only finite points as representatives.
For the final
Take the infinite point
determined by the chains of this final pencil as representative of the (N-I)flat at infinity.
T T
T
Let alx,aZx, ... ,aN_lx be the linear forms of one of the parallel bundles
of I-flats in EG(N,p).
In the above construction, a descending
chain
of
92
T
T
flats in PG(N,p) is formed by clxO+alx=O and cixO+aix=O
where c 2 '· .. ,c N_I are chosen arbitrarily for given c .
l
i=2, ... ,N-I
Now add the re-
striction that c. is so chosen that c.xo+a:x=o is satisfied by the chosen
1 1 1
T
representative of the (N-I)-flat clxO+alx=o.
If the representative is a
finite point, c. is determined by the representative and a., i=2, ... ,N-1.
1
1
If the representative is an infinite point, the c. 's may still be chosen
1
arbitrarily.
Constructing the chains in this manner, the I-fIat in each
chain will contain the point which is the representative of the (N-I)-flat
of that chain, and hence there is a distinct point to end each chain.
The proof is now complete if it can be shown that the representatives
can be so chosen.
The (N-I)-flats of PG(N,p) are given by the equations
where x = (x ,x '
l 2
... ,x ), the pN finite points of the geometry being those for which xOfo.
N
F
=
hxo+aTx=o: i=O,I, ... ,p-l; a€ PG(N-I,p)} and xO=O
Consider as (N-I)-flats of the embedded geometry EG(N,p) those (N-I)-flats
Then this is the set F = {aTx=i: i=l, ... ,p-l~
a €
o
PG(N-I,p)}, a set of pN_ I (N-I)-flats of EG(N,p) which collectively contain
of F for which VO.
N
N-I .
each of the p -1 nonzero points of EG(N,p) exactly p
t1mes.
Hence each
of these nonzero points may be taken as a distinct representative of exactly
one of these flats.
Now looking at the entire geometry PG(N,p), a represcnt-
ative which is a finite point has been chosen for p-l of the (N-I)-flats of
each parallel pencil, and every finite point except (1,0, ... ,0) has been
chosen as a representative.
Choose any infinte point to represent the (N-I)-
flat xo=O, and let (1,0, ... ,0) represent the remaining (N-I)-flat of the pencil whose chains determined the chosen infinit'e point. This leaves a unique
infinite point to represent each of the (N-I)-flats for which a representative has not yet been chosen.
Example 6.2.2
v=ll is a pure prime, so the "initial set" of cycles gives
•
93
the design
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, 0)
(0, 2, 4, 6, 8,10, 1, 3, 5, 7, 9, 0)
(0, 3, 6, 9, 1, 4, 7,10, 2, 5, 8, 0)
(0, 4, 8, 1, 5, 9, 2, 6,10, 3, 7, 0)
(0, 5,10, 4, 9, 3, 8, 2, 7, 1, 6, 0)
Example 6.2.3
v=9.
A set of cycles with complete distance balance is found for
The initial cycles are
(0, 1, 2, 3, 4, 5, 6, 7, 8, 0)
CI.
=1
1
(0, 2, 4, 6, 8, 1, 3, 5, 7, 0)
CI.
=2
2
(0, 4, 8, 3, 7, 2, 6, 1, 5, 0)
CI.
3
=4
Each pair at distance 3 in the first cycle is at distance 3 in each of the
cycles; every other pair occurs at distances 1, 2, and 4.
The geometry EG(2.3) may be represented by the following BIBD:
[0, 1, 2],
[3, 4, 5],
[6, 7, 8]
[0, 3, 6],
[1, 5, 7],
[2, 4, 8]
[0, 4, 7],
[1, 3, 8],
[2, 5, 6]
[0, 5, 8],
[1, 4, 6],
[2, 3, 7]
Here treatments are points, blocks are lines, and rows are pencils.
One
possible set of permutations of [0, 1, 2, 3, 4, 5, 6, 7, 8] generated by this
geometry is
0, 3, 6, 1, 4, 7, 2, 5, 8
0, 1, . 2, 3, 5, 4, 6, 7, 8
..
0, 1, 2, 4, 3, 5, 7, 8, 6
0, 1, 2, 5, 4, 3, 8, 6, 7
Hence the 3(9-1)
2
= 12 cycles are
94
(0,3,6,1,4,7,2,5,8,0)
(0,1,2,4,3,5,7,8,6,0)
(0, 6, 4, 2, 8, 3, 1, 7, 5, 0)
(0, 2, 3, 7, 6, 1, 4; 5, 8, 0)
(0,4,8,1,5,6,2,3,7,0)
(0, 3, 6, 4, 8, 2, 7, 1, 5, 0)
(0, 1, 2, 3, 5, 4, 6, 7, 8, 0)
(0,1,2,5,4,3,8,6,7,0)
(0, 2, 5, 6, 8, 1, 3, 4, 7, 0)
(0, 2, 4, 8, 7, 1, 5, 3, 6, 0)
(0,5,8, 3, 7, 2,6, 1,4,0)
(0,4,7,5,6,2,8,1,3,0)
6.2.4
~xamp1e
for v=33
A set of permutations of the initial cycle is constructed
Pairs of points which generate the 13 lines of PG(2,3) are
(0, 0, 1),
(1, 0, 0)
(1, 1, 0),
(0, 0, 1)
(0,1,0),
(0, 0, 1)
(1,1,1),
(0, 1, 2)
(0,1,1),
(1, 0, 0)
(1, 1, 2),
(1, 2, 0)
(0, 1, 2),
(1, 0, 0)
(1,2,0),
(0, 0, 1)
(1, 0, 0),
(0, 1, 0)
(1,2,1),
(1, 0, 2)
(1,0,1),
(0, 1, 0)
(1, 2, 2),
(1, 1, 0)
(1, 0, 2),
(0, 1, 0)
(6.2.5)
Taking collectively the first members of the 13 pairs gives the 13 points
of the geometry, so the first member of a pair may be taken as the representative of the line generated by the pair.
Now an ordered bundle of I-flats in
EG(3,3) is constructed from each pair by taking the representative to generate
2-flats and the other point to generate I-flats within the 2-flats.
That is,
the linear forms for the bundle corresponding to a line are a~x and a~x where
a
l
is the representative of the line and a 2 is the listed other point on the
line.
For instance, the pair (0, 0, 1), (1, 0, 0) yields the bundle of flats
F ,i given by x = i and x = i . A permutation of the initial cycle is
2
3
1
l
il 2
ohtained by ordering the lines F. . in the lexicographic order (i ,i ),
1 ,1
l 2
1
2
and then choosing points successively from the ordered lines.
Identifying
the point (x ,x ,x ) with the number 9x +3x +x , it is easily verified that
2 3
1
1 2 3
the ordered lines of F.1 ,1. are given by the triples of points
1
2
"
95
(0,3,6), (1,4,7), (2,5,8), (9,12,15), (10,13,16), (11,14,17), (18,21,24),
(19,22,25), (20,23,26)
(For example, 0=(0,0,0), 3=(0,1,0), and 6=(0,2,0) are the three points on
the line FO,O: x =0 and x =0.)
1
3
ordered lines F.
1
.
1
,1
A permutation of 0,1,2, ... ,26 based on the
is
2
0,1,2,9,10,11,18,19,20,3,4,5,12,13,14,21,22,23,6,7,8,15,16,17,24,25.,26
Similarly, one may order the lines generated by each of the pairs of (6.2.5).
These ordered sets of lines are listed in Table 6.2.6, and a corresponding
set of permutations in Table 6.2.7.
Table 6.2.6
Ordered bundles of I-flats in EG(3,3).
and columns within rows correspond to lines.
Rows correspond to bundles
The order of the columns is the
order of the lines.
0, 3, 6
1, 4, 7
2, 5, 8
9, 2,15 10,13,16 11,14,17 18,21,24 19,22,25 20,23,26
0, 9,18
3,12,21
6,15,24
1,10,19
0, 5, 7
1, 3, 8
2, 4, 6
9,14,16 10,12,17 11,13,15 18,23,25 19,21,26 20,22,24
0, 4, 8
2, 3, 7
1, 5, 6
9,13,17 11,12,16 10,14,15 18,22,26 20,21,25 19,23,24
0, 1, 2
9,10,11
8,19,20
3, 4, 5 12,13,14 21,22,23
0,11,19
1, 9,20
2,10,18
3,14,22
4,12,23
5,13,21
6,17,25
7,15,26
8,16,24
0,10,20
2, 9,19
1,11,18
3,13,23
5,12,22
4,14,21
6,16,26
8,15,25
7,17,24
0,15,21
3, 9,24
6,12,18
1,16,22
4,10,25
7,13,19
2,17,23
5,11,26
8,14,20
0,13,26
8, 9,22
4,17,18
7,11,21
3,16,20
2,12,25
5,15,19
1,14,24
6,10,23
0,14,25
2,13,24
1,12,26
8,10,26
7, 9,23
6,11,22
4,15,20
3,17,19
5,16,18
0,12,24
6, 9,21
3,15,18
1,13,25
7,10,22
4,16,19
2,14,26
8,11,23
5,17,20
0,16,23
6,13,20
3,10,26
8,12,19
5, 9,25
2,15,22
4,11,24
1,17,21
7,14,18
0,17,22
2,16,21
1,15,23
5,10,24
4, 9,26
3,11,25
7,12,20
6,14,19
8,13,18
4,13,22
7,16,25
2,11,20
5,14,23
8,17,26
6, 7, 8 15,16,17 24,25,26
96
Table 6.2.7
Permutations of the cycle (0,1,2, ... ,26,0).
0, 1, 2, 9,10,11,18,19,20, 3, 4, 5,12,13,14,21,22,23, 6, 7, 8,15,16,17,24,25,26, 0
0, 3, 6, 1, 4, 7, 2, 5, 8, 9,12,15,10,13,16,11,14,17,18,21,24,19,22,25,20,23,26, 0
0, 1, 2, 9,10,11,18,19,20, 5, 3, 4,14,12,13,23,21,22, 7, 8, 6,16,17,15,25,26,24, 0
0, 2, 1, 9,11,10,18,20,19, 4, 3, 5,13,12,14,22,21,23, 8, 7, 6,17,16,15,26,25,24, 0
0, 9,18, 3,12,21, 6,15,24, 1,10,19, 4,13,22, 7,16,25, 2,11,20, 5,14,23, 8,17,26, 0
0, 1, 2, 3, 4, 5, 6, 7, 8,11, 9,10,14,12,13,17,15,16,19,20,18,22,23,21,25,26,24, 0
0, 2, 1, 3, 5, 4, 6, 8, 7,10, 9,11,13,12,14,16,15,17,20,19,18,23,22,21,26,25,24, 0
0, 3, 6, 1, 4, 7, 2, 5, 8,15,19,12,16,10,13,17,11,14,21,24,18,22,25,19,23,26,20, 0
0, 8, 4,11, 3, 2, 5, 1, 6,13,22,17,21,16,12,15,14,10,26, 9,18, 7,20,25,19,24,23, 0
0, 2, 1, 8, 7, 6, 4, 3, 5,14,13,12,10, 9,11,15,17,16,25,24,26,21,23,22,20,19,18, 0
0, 6, 3, 1, 7, 4, 2, 8, 5,12, 9,15,13,10,16,14,11,17,24,21,18,25,22,19,26,23,20, 0
0, 6, 3, 8, 5, 2, 4, 1, 7,16,13,10,12, 9,15,11,17,14,23,20,26,19,25,22,24,21,18, 0
0, 2, 1, 5, 4, 3, 7, 6, 8,17,16,15,10, 9,11,12,14,13,22,21,23,24,26,25,20,19,18, 0
Example 6.2.8
A maximal set of distinct maximal length chains of flat spaces
is found in PG(3,3).
The 2-flats of PG(3,3) (aside from xo=O) are found from
the first members of the pairs of points (6.2.5).
For instance, (1,2,2) gives
the three 2-flats
x +x +2x +2x =0
o 1 2 3
(6.2.9)
A representative point is chosen for each of the constructed 2-flats (as discussed on page 92), then a I-flat within each 2-flat is obtained using the
second member of the pairs (6.2.5), subject to the constFaint that the I-flat
contains the representative point.
So if the representatives of the 2-flats
(6.2.9) are (0,1,2,2), (1,1,1,1), and (1,2,2,2) respectively, the pair (1,2,2),
(1,1,0) from (6.2.5) gives the I-flats
(6.2.10)
The O-flats for the constructed I-flats are just the representative points
97
already chosen.
For the I-flats (6.2.10),
these may be written as
x +2x +2x =0
2
1
3
x +x +2x +2x =0
2
O l
3
2x +x +2x +2x =0
o 1 2 3
x +x =0
l 2
x +x +x =0
o 1 2
2x +x +x =0
O l 2
x =0
0
x +2x =0
O 2
x +x =0
O 2
(6.2.11)
In general, any chain in PG (3,3) may be written as a series of three linear
equations, where the first equation gives the 2-flat, the first and second
equations give the I-flat, and the three equations together give the O-flat.
It remains to construct a chain beginning with the 2-flat (at infinity)
xO=O.
Take any infinite point as its representative, say (0,1,2,2).
Then
an appropriate chain is
x o=0
x +x =0
1 3
But (0,1,2,2) is also the point ending the first chain in (6.2.11). That chain
is reconstructed to end with (1,0,0,0): the only point that does not yet end
a chain.
Hence it becomes
x +2x +2x =0
123
•
x +x +x =0
123
The entire set of chains is listed below.
The three chains in each row
correspond to a common pair of points in (6.2.5).
is listed below it.
The point ending each chain
98
x =0
3
x +x =0
O 3
2x +x =0
3
x =0
1
x =0
1
x =0
1
x =0
x =0
2
x =0
2
(0,0,1,0)
(1,0,0,2)
(1,0,0,1)
x =0
2
x +x =0
O 2
2x +x =0
2
x =0
3
x =0
1
x =0
1
x =0
x =0
3
x =0
3
(0,1,0,0)
(1,0,2,0)
(1,0,1,0)
x +x =0
2 3
x +x +x =0
O 2 3
2x +x +x =0
023
x =0
1
x =0
1
x =0
1
x =0
x +2x =0
O 2
x +x =0
O 2
(0,0,1,2)
(I,0,1,1)
(1,0,2,2)
x +2x =0
2
3
x +X +2x =0
O 2
3
2x +x +2x =0
3
O 2
x =0
1
x =0
1
x =0
1
x =0
x +2x =0
O 2
x +x =0
O 2
(0,0,1,1)
(1,0,1,2)
(1,0,2,1)
°
°
°
°
°
°
.
x =0
1
x +x =0
1
2x +x =0
1
x =0
2
x =0
2
x =0
2
x =0
x =0
3
x =0
3
(0,0,0,1)
(1,2,0,0)
(1,1,0,0)
°
°
°
99
X +X =0
1 3
x +x +x =0
O 1 3
2x +x +x =0
O 1 3
x =0
2
x =0
2
x =0
2
x =0
0
x +2x =0
O 1
x +x =0
O 1
(0,1,0,2)
(1,1,0,1)
(1,2,0,2)
x +2x =0
3
1
x +x +2x =0
013
2x +x +2x =0
013
x =0
2
x =0
2
x =0
2
x =0
0
x +x =0
O 3
x +2x =0
o 3
(0,1,0,1)
(1,1,0,2)
(1,2,0,1)
x +x =0
1 2
x +x +x =0
O 1 2
2x +x +x =0
012
x =0
3
x =0
3
x =0
3
x =0
0
x +2x =0
O 1
x +x =0
O 1
(0,1,2,0)
(1,1,1,0)
(1,2,2,0)
x +x +x =0
1 2 3
x +x +x +x =0
O 1 2 3
2x +X +x +X =0
O 1 2 3
x +2x =0
2
3
x +x +2x =0
023
2x +x +2x =0
023
x =0
0
x +x =0
o 1
2x +x =0
o 1
(0,1,1,1)
(1,2,1,2)
(1,1,2,1)
x +x +2x =0
1 2
3
x +x +x +2x =0
012
3
2xO+x1+X2+2x3=0
x +2x =0
1
2
2x +x +2x =0
O 1
2
x +x +2x =0
2
O 1
x =0
0
x +x =0
O 1
2x +x =0
O 1
(0,1,1,2)
(1,2,1,1)
(1,1,2,2)
oJ
100
x +2x =0
1
2
x +x +2x =0
O 1
2
2x +x +2x =0
O 1
2
x =0
x =0
3
x =0
3
x =0
0
x +x =0
O 2
x +2x =0
O 2
(0,1,1,0)
(1,1,2,0)
(1,2,1,0)
X +2x +x =0
2 3
1
x +x +2x +x =0
O 1
2 3
2x +x +2x +x =0
0 123
x +2x =0
1
3
x +x +2x =0
013
2x +x +2x =0
O 1
3
x =0
0
x +2x =0
O 2
x +x =0
o 2
(0,1,2,1)
(1,1,1,2)
(1,2,2,1)
x +2x +2x =0
1
3
2
x +x +2x +2x =0
o 1 2 3
2x +x +2x +2x =0
2
3
O 1
xl +x 2=0
X +x +X =0
o 1 2
2x +x +x =0
O 1 2
x +x +x =0
1 2 3
x +2x =0
O 2
x +x =0
O 2
(1,0,0,0)
(1,1,1,1)
(1,2,2,2)
3
'.
x =0
o
x +x =0
1 2
x +x =0
1 3
(0,1,2,2)
.
101
BIBLIOGRAPHY
Bose, R. C. (1939) On the construction of balanced incomplete block designs,
Annals of Eugenics~ 9~ 353-399.
Bose, R. C. (1963) Strongly regular graphs, partial geometries and partially balanced designs, Pacific Journal of Mathematics~ 13~ 389-419.
Bose, R. C., Shrikhande, S. S. and Parker, E. T. (1960) Further results on
the construction of mutually orthogonal Latin squares and the falsity
of Euler's conjecture, Canadian Journal of Mathematics~ 12~ 189-203.
Cheng, C. S. (1983) Construction of optimal balanced incomplete block designs for correlated observations, Annals of statistics~ 11~ 240-246.
Clatworthy, W.H. (1973) Tables of Two-Associate-Class Partially Balanced
Designs, NBS Applied Mathematics Series 63.
Cochran, W. G. and Cox, G. M. (1950)
Wiley and Sons, Inc.
Experimental Designs, New York, John
Gassner, B. J. (1965) Equal difference BIB designs, Proceedings of the American Mathematical Society~ 16~ 378-380.
Gergely, Ervin (1974) A simple method for constructing doubly diagonalized
Latin squares, Journal of Combinatorial Theory (A)~ 16~ 266-272.
Hall, M. (1967)
Company.
Combinatorial Theory, Waltham, Mass., Blaisdell Publishing
Hall, P. (1935) On representatives of subsets, Journal of the London Mathematical Society~ 10~ 26-30.
Hanani, H. (1961) The existence and construction of balanced incomplete
block designs, Annals of Mathematical Statistics~ 32~ 361-386.
Hanani, H. (1972) On balanced incomplete block designs with blocks having
five elements, Journal of Combinatorial Theory (A)~ 12~ 184-201.
Hartley, H. 0., Shrikhande, S. S., and Taylor, W. B. (1953) A note on incomplete block designs with row balance, Annals of Mathematical Statistics~
24~ 123-126.
Heyadat, A. and Afsarinejad, K. (1975) Repeated measurement designs I, in
A Survey of Statistical Design and Linear Models (J. N. Srivastava, ed.),
229-242, Amsterdam, North-Holland.
102
Hung, H. Y. and Mendelsohn, N. S. (1977)
ematics~ 18~ 23-33.
Handcuffed designs, Discrete Math-
Linear neighbor designs, Proceedings of the 7th S.E.
Conference on CombinatoPics~ Graph Theory~ and Computing~ 335-340,
Washington, DC, American Statistical Association.
Hwang, F. K. (1976)
Hwang, F. K. and Lin, S. (1977)
Theory
A~
Neighbor designs, Journal of Combinatorial
302-313.
Kempthorne. Oscar (1952) The Design and Analysis of Experiments, New York,
John Wiley and Sons, Inc.
Kiefer, J. (1958) On the nonrandomized optimality and randomized nonoptimality of symmetric designs, Annals of Mathematical Statistics~ 29~
675-699.
Kiefer, J. (1975) Construction and optimality of generalized Youden designs,
in A Survey of Statistical Design and Linear Models (J. N. Srivastava, ed.)
333-353, Amsterdam, North-Holland.
Kieter, J. and Wynn, H. P. (1981) Optimum balanced block and Latin square
designs for correlated observations, Annals of Statistics~ 9~ 737-757.
On the construction of handcuffed designs, Journal
of Combinatorial Theory A~ 16~ 76-86.
Lawless, J. F. (1974)
Mendelsohn, N. S. (1966) Hamiltonian decomposition of the complete directed
n-graph, in Theory of Graphs, Proceedings of the Collo.~iu,!!.pl}~J~.at
T.thaI)L-lLunZ€!TY. (edited by P. Erdos and G. Katona), 237-241, Academic
Press, New York.
Mendelsohn, N. S. (1977)
63-68.
Perfect cyclic designs, Discrete
Mathematics~ 20~
Mukhopadhyay, A. C. (1978) Construction of BIBD's from OAl s and combinatorial arrangements analogous to OAl s , Calcutta Statistical Association Bulletin~ 21~ 45-50.
Rao, C. R. (1961) Combinatorial arrangements analogous to orthogonal
arrays, Sankhya~ 23~ 283-286.
Rao, C. R. (1973) Some combinatorial problems of arrays and apPlications
to design of experiments, in A Survey of Combinatorial Theory (J. N.
Srivastava, ed.), 349-359, Amsterdam, North-Holland.
Rees, D. H. (1967)
779-791.
Takeuchi, K. (1961)
Some designs for use in serology, BiometPics~ 2.3~
On the optimality of certain types of PBIB designs,
Reports of Statistical Application Research~ Union of Japanese Scientists
and Engineers~ 8~ 140-145.
103
On the efficient design of statistical investigations,
Annals of Mathematical Statistics, 14, 134-140.
Wald, A. (1943)
•
Williams, E. J. (1949) Experimental designs balanced for the estimation
of pairs of residual effects of treatments, Australian Journal of
Scientific Research, 2, 149-164 .
Williams, E. J. (1950) Experimental designs balanced for pairs of residual
effects, Australian Journal of ~cientific Research, 3, 351-363.
Williams, R. M. (1952) Experimental designs for serially correlated observations, Biometrika, 39, 151-167 .
•