lllllllllllflfllllllflllllllljlllljllfllllll ._

Michigan State
University

 

This is to certify that the

thesis entitled

On Structure Preserving Groups of Latin Squares
And Their Applications To Statistics

presented by

SH I N-SUN CHOW

has been accepted towards fulfillment
of the requirements for

Ph. D. Aegean Statistics and
Probability

 

 

54:2th get 0‘ (e .,
Major professor

Date August 17,1979 I

 

0-7639

 

 

ON STRUCTURE PRESERVING GROUPS OF LATIN SQUARES
AND THEIR APPLICATIONS TO STATISTICS

By
Shin-Sun Chow

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1979

ABSTRACT

ON STRUCTURE PRESERVING GROUPS OF LATIN SQUARES
AND THEIR APPLICATIONS TO STATISTICS

By
Shin-Sun Chow

We consider a symmetric property, invariance of probability
distribution under a group of transformations of the sample space, of
Latin square designs. The group of transformations of the sample
space will be called "the structure preserving group for a Latin
square design." We show that Latin square designs from the same
transformation set have isomorphic structure preserving groups. The
commutator algebras of the representation of the structure preserving
groups are then studied. The structure preserving group and commuta-
tor algebra are computed for one Latin square design from each trans-
formation set of Latin square designs of orders three, four and five.
Associated with the symmetric property, random assignment of treat-
ments (Latin square design as a fractional three factor design) to
subjects and randomization tests for Latin square designs are then

studied.

ACKNOWLEDGEMENTS

I would like to thank all the faculty members in the
Department of Statistics and Probability at Michigan State
University for their help during my four years (l975-1979) of
study here. I am especially grateful to my advisor, Dr. Esther
Seiden, for giving me encouragement during my frustrating days.

I must also thank my wife, Shu-In Huang and my parents
for giving me morale support.

I should also thank Emily Groen-White and Noralee

Burkhardt for their excellent typing job of my dissertation.

ii

TABLE OF CONTENTS

Page
INTRODUCTION ........................ l
Section
1 A SYMMETRIC PROPERTY OF LATIN SQUARE
DESIGNS AND SOME OF ITS RELATED CONCEPTS . . . 5
2 COMMUTATOR ALGEBRAS OF LATIN SQUARE
DESIGNS .................... l7
3 DIFFERENT SCHEMES FOR ASSIGNING
TREATMENTS TO SUBJECTS ............ 34
4 RANDOMIZATION TEST FOR THE LATIN
SQUARE DESIGN ANALYSIS ............ 46

BIBLIOGRAPHY ........................ 54

iii

INTRODUCTION

Latin square designs are used in agricultural experiments.
Suppose we wish to find out by experiments whether there is any signi-
ficant difference among yields of m different varieties v1....,vm.
The experimental field is subdivided into m2 plots laid out in m
rows and m columns and each plot is assigned to one of the m
varieties. If each variety appears once and only once in each row
and each column, we have a Latin square arrangement. Latin square
designs are also used in biological experiments to provide a method
of controlling individual differences among experimental units.
Another important use of Latin square designs is in the area of beha-
vioral sciences to counterbalance order effects in repeated measure-
ments plans. The dual balance, i.e. the varieties or treatments to
be compared are equally represented across each row and each column,
makes the statistical analysis for Latin square designs more precise
than those designs without such balance. Wald [l5] and Ehrenfeld [3]
studied the problem of testing linear hypotheses for a linear regres-
sion model. With respect to the problem, Wald [15] stated an optimality
criterion (called D-optimality by Kiefer [l0]) for designs in the
setting of two-way heterogeneity (m treatments are assigned to a
In><m array of plots in such a way that each plot receives one treatment)
and showed that Latin square designs are optimal among them. With re-

spect to the same problem, Ehrenfeld [3] stated another optimality

2

criterion (called E-optimality by Kiefer [10]) for designs in the
setting of two-way heterogeneity and showed that Latin square designs
are optimal among them. Their results thus enhance the superiority
of Latin square designs. Kiefer [l0] generalized their ideas and de-
fined several different optimality criteria for non-randomized designs,
with respect to the same problem of testing linear hypotheses for a
.linear regression model. He proved that balanced incomplete block
designs are optimal among designs in the setting of one-way hetero-
geneity (m treatments are assigned to b blocks, each of which con-
tains k plots). He also proved that Youden square designs are optimal
among designs in the setting of two-way heterogeneity (m treatments
are assigned to a k1 x k2 array of plots). So, with respect to the
problem of testing linear hypotheses for a linear regression model,
all Latin square designs of the same order are equally good.

James [6] studied the algebraic structures of randomized block
designs, Latin square designs and balanced incomplete block designs.
He introduced the concept of relationship algebra for those designs
and showed that the relationship algebras of Latin square designs are
always isomorphic to the algebra of all diagonal 5 x 5 matrices.
So, all Latin square designs of the same order are the same with re-
spect to the structure of their relationship algebras. Dubenko, Sysoev
and Shaikin ([l],[2],[lZ],[l3]) studied the symmetry properties of the
designs studied by James. They introduced the concepts of commutator
algebras, associated with the symmetry properties, for those designs.
They proved that the relationship algebra of any one of those designs

is always a subalgebra of the comnutator algebra of the design. For

Latin square designs, they stated that "It can be shown by direct Veri-
fication that the commutator algebra coincides with the relationship
algebra for all 2 x 2 and 3 x 3 squares and for 4 x 4 square of

the form Latin squares of larger dimensions have not yet

1 2 3 4
2 l 4 3
3 4 l 2
4 3 2 1
been investigated. But it is clear that the dimensions of the commu-
tator algebra, first of all, depend on the type of square, second,
increase with the order of the square."

In the first section of this dissertation, we study the symmetry
properties of designs in the setting of two-way heterogeneity (it in-
cludes Latin square designs in particular). We show that designs from
a transformation set (similar to the transformation set defined for
Latin square designs in Fisher and Yates [5]) have similar symmetry
property and similar commutator algebras associated with them. The
symmetry properties introduced by Dubenko, Sysoev and Shaikin thus
distinguish designs from different transformation sets. In the second
section, we compute the commutator algebras of Latin square designs,
from different transformation sets of orders three, four and five.
Commutator algebras, associated with designs from different transfor-
mation sets,are quite different for Latin square designs of orders
four and five. It also gives one counter example to the statement that
"the dimensions of the commutator algebra increase with the order of
the square." In the third section, we study different schemes to
assign treatments (Latin square design as a fractional three factor
design) to subjects and the impact of the different randomization

schemes on the analysis of Latin square designs. We show that a

particular scheme will justify the assumption about the covariance
matrix of the observed random vector. In the last section, we study
the randomization test (Fisher [4]) for Latin square designs. We show
that only those designs, from the same transformation set as the design

actually used, should be included in the randomization test.

51. A SYMMETRIC PROPERTY OF LATIN SQUARE DESIGNS
AND SOME OF ITS RELATED CONCEPTS

Many statistical experimental designs exhibit symmetries,
which provide natural restrictions to impose on the probability distri-
bution of the observed random variable. In this section, we study
symmetric properties of Latin square designs.

A Latin square design of order k can be represented by a
k x k matrix with elements from a finite set of k symbols, say
{l,2,...,k}, such that each of the k symbols appears once and only
once in each row and each column. Latin square designs are commonly
used in small scale pilot experiments to remove the heterogeneity of
experimental material in two directions. For example, a 4 x 4 matrix
3 4 represents a Latin square design of order 4. A Latin square

I 2
2 l
3 4
4 3

N-‘h
de

design can also be described as an incomplete 3-factor design (row
factor, column factor and treatment factor) which is balanced with re-
spect to main effects but only partially balanced with respect to
two-factor interactions (i.e. each row level occurs in combination
with each column level and each treatment level but does not occur in
combination with all possible pairs of column level and treatment level).
In the following, we discuss experimental designs which can be repre-
sented as a b x k matrix with elements from a finite set, say

{l,2,...,v}. The finite set {l,2,...,v} represents possible treatment

5

levels. Latin square designs are a specific type of designs out of
such designs. So all the results in this section are applicable to
Latin square designs in particular. The cell in the ith row and jth
column is ordered as the [(i - l)k + 33‘“ cell of a b x k matrix.
In a given design, there are three numbers (r ,c ,t ) associated with

XXX

each cell x, where r is the row level of the cell, cx is the

x
column level of the cell, and tx is the treatment level. A given

design thus can also be represented by tlt2 ......... tk , denoted

t(b_])k+]....t

bk
also by GDM (Given Design Matrix). In the Latin square design

, the following equality holds: (r9,c9,t9) = (3,l,3).

Different kinds of permutations of cells will be defined next.
Definition l.l: A row preserving permutation g of cells with respect
to a given design is a permutation of {l,2,...,bk} satisfying the

. . = - g f ’ =
condition that rx ry if and only if rg(x) rg(y) or x y
l,2,...,bk.

Definition 1.2: A column preserving permutation g of cells with re-

 

spect to a given design is a permutation of {l,2,...,bk} satisfying
the condition that cx = cy if and only if cg(x) = cg(y) for x,y =
l,2,...,bk.

Definition 1.3: A treatment preserving_permutation g of cells

 

with respect to a given design is a permutation of {l,2,...,bk}

satisfying the condition that tx = t if and only If tg(x) = t9(y)

Y
for x,y = l,2,...,bk.

Let

0
ll

r {gig is a row preserving permutation of cells}

(D
II

{gig is a column preserving permutation of cells}

CD
ll

{gig is a treatment preserving permutation of cells}.

It follows immediately that Gr’ G and Gt are subgroups of the

c
symmetric group Sbk’ i.e. the group of all permutations of
{l,2,...,bk} with its binary operation "composition of mappings."

A permutation belonging to the set of intersection of Gr’ Gc and

Gt (i.e. Gr n Gc n Gt) is called an admissible permutation of cells.
In the paper of (Dubenko, Sysoev and Shaikin 1976 [2]), they call the
group Gr n Gc rth "the symmetry group for a given design.“ To
avoid the confusion between symmetry group and symmetric group, let

us call Gr 0 Gc riGt "The structure preserving group for a given

design." It is also denoted by tlt2 ......... tk , i.e. K(GDM),

t(b-])k+]oooo
to emphasize its dependency on the given design GDM. Note that
Gr n Gc is a group which is independent of the given design. A per-

mutation g 6 Gr n Gc has the following representation:

g((b-l)k+l) ..... g(bk) (b-l)k+l ..... b
where L9 and R9 are two elementary matrices obtained from the
identity matrices Ib and Ik’ respectively, through permutation of

rows or columns of the identity matrices. Let us call

l 2 ......... k \ , the Cell Matrix, abbreviated as CM. Let us
............. ,1
\(b-l)k+l ..... bk}
call g(l) ............ g(k) , the Transformed Cell Matrix
g((b-l)k+l) ...... g(bk)

determined by g and abbreviated as TCMa. So the above representation
becomes TCMg = Lg-CM-Rg. The left multiplication of CM by Lg amounts
to a row permutation of CM. The right multiplication of CM by R

9
amounts to a column permutation of CM. Moreover, the representation
mentioned above is unique in the sense that if

(5(1) .......... g(k)\ l 2 ...... k l l 2 ...... k
............... = L -

g .......... jj-Rg g
\g((b-l)k¥D.u.. g(bk) (b4)k+l.. bk (b4)k+l..bk

then L9 = L6 and R9 = Ré, where L; and R; also are elementary
matrices. There are b! different elementary matrices which can be
obtained from Ib by permutation of rows or columns. From the above
representation, the order of the group Gr n Gc’ denoted by IGr n Gcl,
is equal to blkl. At this point, it is important to discuss the
following lemmas which will be used later.

Lemma l.l: Let h be a permutation of {l,2,...,bk} then

\
h(l) ......... h(k) hog(l) ............. hog(k) A
Lg- .............. -R9 = .................... 3
h((b-l)k+l)... h(bk) hog((b-l)k+l) ....... hog(bk)j
i.e. L oTCM -R = TCMhog, where L -CM-R = TCM and hog is the

9 h 9 9 9 9

composition of h and g with hog(i) = h(g(i)).
Proof: The mapping of CM + Lg-CM-R amounts to moving the element i

th th

9

from i cell to j cell where g(j) = i, and i = l,2,...,bk. So

the mapping TCMh‘+ L -TCMh-R amounts to moving the element h(i)

from ith cell to jth gell wheze hog(j) = h(i) and i = l,2,...,bk.
It follows that Lg-TCMh-Rg = TCMhog. Q.E.D.
Lemma l.2: If 91,92 6 Gr n GC then Lgl-ng = ngog], Rgz-Rg] =
R9209], Lgi] = La: and Rg;1 = Ré}.
Proof:

L91.L92.CM.R92.R91

= TCMgzog]

= ngogl CM R92°91

L9] L92 = ngog1 and Rgz'Rg] = R92°91 follows from the

uniqueness of representation.

L -L _]-CM-R _]-R

g1 g1 91 9l

L ~CM-R _1

9] °91 9] °91

-l -l
-L -CM-R -R
L91 91 91 91

-l -l
L = L and R = R
g]1 9l 911 91

follows from the uniqueness of

representation again. Q.E.D.

Let us define the set D = {Lg-GDM-RglgeGr n Gc} where GDM
is a given design. The number of elements in D (counting the possible
repetitions) is bIXkl. In other words, D is the collection of all

the possible designs which can be obtained from the given design by

ID

a row permutation and a column permutation.

Definition 1.4: Two designs Lg-GDM-Rg and Lg.-GDM~Rg., are defined

 

. , ' M'R N g
to be equ1valent Lg GD 9 L9

can be obtained from the other by a permutation of treatments.

oGDM-Rg., if and only if one of them

Let us define the set Da = {Lg-GDM-Rglg 6 Gr n Gc n Gt}' We
use the notation Da because an element of Gr FIGC riGt is called
an admissible permutation. It can also be defined as Da = {AIA E D
and A~GDM}. The number of elements in Da (counting the possible
repetitions) is equal to the order of the group Gr n Gen Gt’

[Gr rch rthl. Concerning the invariant property of the given design,
up to equivalence with respect to a row permutation and a column per—
mutation of the given design, the cardinality of Da is an indicator
for this invariance property. The equivalence relation defined above
partitions the set D into equivalence classes. It will be shown

that all the equivalence classes have the same number of elements, and
Da is one of the equivalence classes. If the number of equivalence
classes is given, then we know the order of the structure preserving

group, and vice versa.

Theorem l.l: The equivalence relation, which states that two designs

 

in the set D are equivalent to each other if one of them can be
obtained from the other by a permutation of treatments, partitions the
set D into a number of equivalence classes which have the same number

of elements as the structure preserving group does.

Proof: Let

~GDM-R .-GDM-R .
L90 90 ” L9 9

ll

then
L'I-L ~GDM-R -R-LML'1-L .-GDM-R .-R"
90 go 90 90 90 9 g 90
i.e. GDM m L _]-GDM0R _]
g'ogo g'oso
9'09;1 6 6r n cc n Gt

L .-GDMoR .‘= L -L -GDM-R ..R

for some 9. E G n G FIG .
9 9 90 91 91 90 i r c t

The equivalance class determined by L oGDM-R , denoted by D ,
go go go
is as follows:

= O O 0 OR
0 {L9 L9 eon Rg.

. 6 G G G
9o o i i 90'91 r n c n t}

= {Lg-GDM'Rglg 6 (Br n Gc n Gt)'90}

where (Gr n Gc rth)-gO is a coset of Gr riGc rth and the cardi-
nality of (Gr rch riGt)-go, denoted by |(Gr rch riGt)-gol, is equal

to [G n G FIG I. Note that D = (J D , and we thus have shown
r c t
gEGrnGc

that D is partitioned into a number of equivalence classes that have
the same number of elements as the structure preserving group. Q.E.D.

From the representation,

D
II

{Lg-GDM-Rglg E Gr rch}

{Lg- ..................... -Rg|g 6 Gr n Gc}.
t90((b-i)k+i)'tgo(bk)

Let us call tgo(1) ....... tgo(k) , the Transformed DeSign Matrix

t90((b-i)k+i)'tgo(bk)

determined by 90 and abbreviate it as TDMg where 90 e Gr n GC.

0

So if we start with a design TDMg from the set D, then the set of
0

all possible designs which can be obtained from this design by a row

permutation and a column permutation is still the set D.

Theorem 1.2: Let GDM be a given design and g 6 Gr n Gc' Then the

 

following equality holds:

K(TDMg) = g'1-K(GDM)-g.

Proof:
h €-K(TDMg)
. C C . .R ~ . .
iff Lh Lg GDM Rg h Lg GDM Rg
. -1- -1
iff Lg Lh Lg GDM Rg Rh Rg con
iff g . h o g“1 e K-(GDM)
iff h e g'1-K(GDM)-g Q.E.D.

Definition 1.5: A transformation set of designs is a set of designs

 

which can be generated from any one of its members by permutation of

rows, columns, and treatments.

Theorem 1.3: All designs from a tranformation set of designs have

 

isomorphic structure preserving groups.

l3

Eooof: From Theorem 1.2, it is proved that any two designs in a trans-
formation set that may be obtained from one another by permutation of
rows and columns have isomorphic structure preserving groups. Also,
any two designs which can be obtained from one another by permutation
of treatments have the identical structure preserving group, so the

theorem follows. Q.E.D.

Theorem 1.4: If we define an equivalence relation for designs in a

 

transformation set as in Definition 1.4, then the number of equivalence
classes in a transformation set is equal to lGr n GCI/lGr n Gc rust] =

(bEXk!)/|Gr n cc n th.

Proof: A transformation set of designs can also be represented as a
set of all designs which can be obtained by permutation of treatments
from those designs in D. Theorem 1.4 follows directly from Theorem
1.1. Q.E.D.
Relating to the concept of the structure preserving group, a
commutator algebra to represent the group is to be defined for later

discussion.

Definition 1.6: A commutator algebra C associated with a Latin

 

square design is defined as C = {XIX is a bk x bk matrix over the
T -

reals and Mg-z-Mg - z for all g e Gr n Gc rust} where M

bk x bk matrix, denotes the permutation g, i.e.

(9(1). 9(2).....g(bk))T = Mg-(i.2.....bk)T

pose operator on matrices.

g’a

and T denotes the trans-

T
= .. no a . . , to

If 2 (C13) then Mg 2 Mg (cg(1)g(j)) So the commuta r

algebra can also be characterized as C = {(cincij = ci'j' for all

l4

(i,j),(i',j') with g(i) = i' and g(j) = j' for some
9 6 Gr n Gc riGt}.

By using addition, scalar multiplication and multiplication
product for matrices, it can be shown that C is an associative
algebra over the reals in the usual algebraic sense. Since C is
defined according to a specific Gr DGc rth which depends on a
given design, C also depends on this design. So, C is also denoted
by C(GDM) to emphasize this dependency. Let 2i denote the observed
value in cell i of the given design and g is a permutation of the
set {l,2,...,bk}, (29(1)""’zg(bk))T = Mg'(zl""’zbk)T with M9

defined as in Definition 1.6. Let 2 be the covariance matrix for

(21’22""’zbk)' If we assume that (21’22’°"’zbk)'a"d

(29(1)’zg(2)""’zg(bk)) have the same probability distribution for

g e Gr n Gc riGt then Mg-Z-M; = 2 for all g E Gr 0 6c rth. In
other words, if we assume the invariance of probability distribution
of (21’22""’zbk) under the permutations from the structure preserving
group then the covariance matrix of (21’22""’Zbk) is an element of

the commutator algebra C.

Theorem 1.5: The dimension of the vector space C(GDM) is the same

 

as the dimension of C(TDMg), where g‘E Gr n GC.

fogxzfz Let us define two equivalence relations for ordered pair of
indices as follows:
(i.j) ~ (i'.j') if i' = h(i) and j' = h(i)
for some h 6 K(GDM)
(1.3) 5 Wu") if 1" = g"ohog(i) and j' = g"ohog(3)
for some h e K(GDM).

15

Then we have that
C(GDM) ={(c1j)lcij= ci'j' fOY‘ an (i,j),(I'sj') With (i,j)~(i.sj')}
and

C(TDMg) ={(CJ°1°J)IC = c .1j. for all (i,j) (1".5') with (i,j)§(i',.i')}.

The vector space dimension of C(GDM) is equal to the number of
different equivalence classes, corresponding to the equivalence rela-
tion "~", in the Cartesian product {l,2,...,bk} x{l,2,...,bk}.
Similarly, the dimension of C(TDMg) is equal to the number of
different equivalence classes, corresponding to the equivalence rela-

tion "5 ", in {l,2,...,bk} x {l,2,...,bk}. Let us denote the

equivalence class, corresponding to the equivalence relation ~ ,

containing the pair (i,j) by [(i,j)]. So we have
[(i,j)] = {(h(i),h(j))|h 6 K(GDM)}
Similarly, define
[(i,j)Jg =){(h(i h(i))lh 6 K(TDMg )}
From Theorem 1.2, we have that
[(i,j)Jg = {(9']°h°g(i)ag'1°h°9(i))lh. E K(GDM)}.

Given a permutation f of {l,2,...,bk}, define a function (f,f)
on {l,2,...,bk} x {l,2,...,bk} as follows:

(f,f): (iajl + (f(i).f(j))-

It is clear that the function (f,f) is a one-to-one function. Moreover,

we have the following equality:

16

(A) [(i.J')Jg = (9".9")<c(gm.g<am). where (9".9")(ug(i).gum)

is the image of [(g(i).9(]))] under (g'I,g']). Let m be the
dimension of the vector space C(GDM) and the set of different equi-
valence classes be {[(i],j])]. [(i2.j2)],....[(im.jm)]}. From (A),
it follows that the set of different equivalence classes, corresponding

to the equivalence relation "g" is as follows:

u“ “)(m ‘m (" “Inch “)1) (" “)(m on}
9 .9 1.31 . 9 .9 2.32 .....9 .9 I“.9"1 .

So the dimension of C(TDMg) is also equal to m. Q.E.D.

Theorem 1.5 shows that the dimension of the vector space
C(GDM) is the same for all the designs from the transformation set
generated from GDM. This is expected because the designs from a trans-

formation set have isomorphic structure preserving groups.

§2. COMMUTATOR ALGEBRAS OF LATIN SQUARE DESIGNS

From the discussion in section one, we know that Latin
square designs from the same transformation set have isomorphic
structure preserving groups and their associated commutator algebras
have similar structures. So, it is enough to consider one design
from each transformation set. In the following examples, the ele-
ments of structure preserving groups are found through computer
programming. The generators of structure preserving groups are

determined from their elements.

Examole 1: Latin Square Designs of Order Three
There is only one transformation set in this case. Let us

compute the structure preserving group and commutator algebra for the

design 1 2 3 . The structure preserving group of f1 2 1 is
2 3 112 3 1}
3 1 2 \3 1 2/

generated by the following permutations of {l,2,...,9}.

9 (12345678?
‘ 231564897
= 12345678m
92 }
546213879

9: C23456789
3 97831264&

17

18

The order of K (g g $) is equal to 18 and there are two equivalence

3 1 2
classes for the transformation set (Theorem 1.4). The general form

of a matrix in the commutator algebra is as follows:

F
l a b b c d e c e d

 

 

d e c e d c b b a
.. .J

The commutator algebra is a five dimensional vector space. The five

 

 

 

 

equivalence classes of index pairs are as follows:
{[(1,1)], [(1,2)]. [(1,4)], [(1,5)]. [(1,6)]} where
[(1.1)] = {(1.1).(2.2).(3.3).(4.4).(5.5).(6.6).(7.7).(8.8).(9.9)}

[(192)] = {(1’2)9(]’3)9(291)9(293)’(391)9(392)9(4’5),(4’6)9(594)9(596)9
(5.4).(6.5).(7.8).(7.9).(8.7).(8.9).(9.7).(9.8)}

[(195)] = {(1’5)9(]99)9(296)9(297)9(394)9(398)9(493)9(498)9(591)9(599)’
(5.2).(6.7).(7.2).(7.6).(8.3).(8.4).(9.1).(9.5)}

{(194)!(]’7)’(2’5)’(2’8)’(3’6)9(399)9(4,])9‘497)9(5,2)9(598)9
(5.3).(6.9).(7.1).(7.4).(8.2).(8.5).(9.3).(9.6)}

[(1.4)]

19

[(1.6)] = {(1.6).(1.8).(2.4).(2.9).(3.5).(3.7).(4.2).(4.9).(5.3).(5.7).
(6.1).(6.8).(7.3) .(7.5).(8.1).(8.6) .(9.2).(9.4)}

Example 2: Latin Square Designs of Order Four
There are two transformation sets in this case. Let us con-
sider one design from each transformation set.

(I) The structure preserving group of’ /1 2 3 4 is generated by

2 1 4 3
3 4 1 2
4 3 2 1
the following permutations: (Dubenko, Sysoev and Shaikin, 1976 [2])

‘
I

{12 3 4 5 6 7 8 910111213141516

g:
I (2 3 4114151613101112 9 6 7 8 5)

g =(12 3 4 5 6 7 8 910111213141516).
2 15141316 3 214 7 6 5 81110 912/

g (12345678910111213141516)
3

12 4 3 5 6 8 713141615 9101211

The order of f1 2 3 4\ is equal to 96 and there are six equivalence
’<{2 1 4 3
3 4 1 2
4 3 2 1

classes for the transformation set (Theorem 1.4). The general form

of a matrix in the commutator algebra is as follows.

20

 

 

 

m
0.
O
m
D.
(D
O
m
m
(D
n
O.
U’ U" U'
U"
Q

d e e c e d e c e e d c

— d

 

 

 

 

 

The commutator algebra is a fiVe dimensional vector space. Since any
matrix in the commutator algebra is symmetric, it follows that the
algebra is commutative. An algebra of square matrices which can be
generated by symmetric matrices is a semi-simple algebra (James,

1957 [6]). According to a theorem of Hedderburn, a semi-simple algebra
is isomorphic to a direct sum of complete matrix algebras. (Van Der
Naerden, 1950 [14], Chapter XVI). So, the commutator algebra is

isomorphic to the algebra of all diagonal 5 x 5 matrices.

(II) The structure preserving group of f; a 2'4\ is generated
: 1
i 3 4 l 2;
\4 1 2 3/

21

by the following permutations:

g :(12 3 4 5 6 7 8 910111213141516)1
1 .

412 3 8 5 6 712 9101116131415)
9 =(12 3 4 5 6 7 8 910111213141516\
2 1110 9127658321415141316)
g:12345678910111213141516)
3 13141516123456789101112

The order of is equal to 32 and there are eighteen

equivalence classes for the transformation set (Thoerem 1.4). The

general form of a matrix in the commutator algebra is as follows.

— ‘11
a b c b d e f g h i j i d g f e

b a b c g d e f i h i j e d g f
c b a b f g d e j i h i f e d g
b c b a e f g d i j i h g f e d

 

U"
DJ
(D
'h
(D
Q
.5
L;
—l
3"

(B) 9 f e d b c

 

3"
a.
‘—J
—l
O-
‘0
..h
m
D!
0"
O
U'
D.
(D
‘h
In

 

 

 

 

 

 

22

The commutator algebra is a ten dimensional vector space. It is also
a commutative algebra. The commutator algebra is isomorphic to the
algebra of all diagonal 10 x 10 matrices (follows again from the
Wedderburn theorem). Applying the results in section one, the commu-
tator algebra of a design which belongs to the same transformation set

\

as 1 will be examined.

dth

3 4
4 1
1 2
2 3

1 2 3 4 5 6 7 8 9
7 6 8 5 3 2 4 1 11
t

an element of he group Gr 0 Gc'

hum—-
\‘__..-

Let g 10 11 12 13 14 15 16 . which is
1012 915141613

\
((11 t2 t3 t4 \ /f
1 t5 t6 t7 t8

1 t9 t10t11t12
\t13t14t15t16

“#001“
Nd-Dw
DON—'4:-

1
2
3
4

“9(1) t9(2) t9(3) t9(4)\3‘ f4
t915) t9(6) t9(7) t9(8)

\\f9(9) t9(101t9(11)t9(12) 1

flIC~ a.“ ‘

M5...—

, and

N—‘OO

d-th
wN-kd
45de

t9(131t9(14)t9(15)t9(16)/ \

g“: 12 3 4 5 6 7 8 910111213141516).
8 6 5 7 4 2131210 91116141315;

The ten equivalence classes of ordered pairs of indices, for the design

\
\

, are as follows.

Nd-fiw
(AN—‘45

/4>wN-:o\
d-DwN

23

[(1.1)] = {(1.1).(2.2).(3.3).(4.4).(5.5).(6.6).(7.2).(8.8).(9.9).
(10,10),(11,11),(12,12),(13,13),(14,14),(15,15),(16,16)}

[(192)] = {(192),(]94)9(2,])9(293)9(3’2)9(394)9(49])9(493)9(5’6)9
(598)9(695)9(697)9(796)9(798)9(895)3(897)9(9910)9(9912)9
(10.9).(10.11).(11.10).(11.12).(12.9).(12.11).(13.l4).
(13,16),(14,13).(14,15),(15,14),(15,16),(16,13).(16,15)}

[(1,3)] = {(193)!(294)9(331)9(492)9(597)9(6’8)9(7’5)9(896)9(9911)9

(10,12),(11,9),(12,10),(13,15),(14,16),(15,13),(16,14)}

[(1,5)] = {(1,5),(1,13),(2,6),(2,14),(3,7),(3,15),(4,8),(4,16),(5,1),
(5,9),(6,2),(6,10),(7,3),(7,11),(8,4),(8,12),(9,5),(9,13),
(10,6),(10,14),(11,7),(11,15),(12,8),(12,16),(13,1),
(13,9),(14,2),(14,10),(15,3),(15,11),(16,4),(16,12)}

[(1,6)] = {(1 ,6).(1 ,16),(2,7),(2,13),(3,8),(3,14),(4,5),(4,15),(5,4),
(5,10),(6,1),(6,11),(7,2),(7,12),(8,3),(8,9),(9,8),(9,14),
(10,5),(10,15),(11,6),(11,16),(12,7),(12,13),(13,2),
(13,12),(14,3),(14,9),(15,4),(15,10),(16,1),(16,11)}

[(1.7)] = {(1,7).(1.15).(2,8),(2,16),(3,5),(3,13),(4,6),(4,14),(5,3),
(5,11),(6,4),(6,12),(7,1),(7,9),(8,2),(8,10),(9,7),(9,15),
(10,8),(10,16),(11,5),(11,13),(12,6),(12,l4),(13,3),
(13,11),(l4,4),(l4,12),(15,1),(15,9),(16,2),(16,10)}

{(1,8),(1,14),(2,5),(2,15),(3,6),(3,16),(4,7),(4,13),(5,2),
(5,12),(6,3),(6,9),(7,4),(7,10),(8,1),(8,11),(9,6),(9,16),
(10,7),(10,13),(11,8),(11,14),(12,5),(12,15),(13,4),
(13,10),(14,1),(14,11),(15,2),(15,12),(16,3),(16,9)}

[(1.8)]

24

{(1.9).(2.10).(3.11).( 4.12).(5.13).(5.14).(7.15).(8.16).
(9.1).(10.2).(11.3).(12.4).(13.5).(14.6).(15.7).(16.8)}

[(1,9)]

{(1,10),(1,12),(2,9),(2,11),(3,10),(3,12),(4,9),(4,11),
(5,14),(5,16),(6,13),(6,15),(7,14),(7,16),(8,13),(8,15),
(9,2),(9,4),(10,1),(10,3),(11,2),(11,4),(12,1),(12,3),
(13,6),(13,8),(14,5),(14,7),(15,6),(15,8),(16,5),(16,7)}

[(1.10)]

{(1.11).(2.12).(3.9).(4.10).(5.15).(5.16).(7.13).(8.14).
(9.3).(10.4).(11.1).(12.2).(13.7).(14.8).(15.5).(16.5)}

[(1.11)]

The images of the ten equivalence classes under the mapping (g'],g'])

are as follows:

(9".9'111um11) [(1.111

{(8.6) .(8.7) .(6.8).(6.5) .(5.6).(5.7).(7.8) .(7.5) .
(4.2).(4.3).(2.4).(2.l).(1.2).(1.3).(3.4).(3.1).
(12,10),(12,11),(1o,12),(1o,9),(9,10),(9,11),
(11,12),(11,9),(16,14),(16,15),(14,16),(l4,13),
(13,14),(13,15) ,(15,16) ,(15,13)}

(9".9")(t<1.211)

{(8.5).(6.7).(5.8).(7.6).(4.1).(2.3).(1.4).(3.2).
(12.9).(10.11).(9.12).(11.10).(16.13).(14.15).
(13,16),(15,14)}

(g".g“1<t<1.311)

{(8,4),(8,16),(6,2),(6,14),(5,1),(5,13),(7,3),
(7,15),(4,8),(4,12),(2,6),(2,10),(1,5),(1,9),
(3,7),(3,11),(12,4),(12,16),(10,2),(10,14),(9,1),
(9,13),(11,3),(11,15),(16,8),(16,12),(14,6),
(14,10),(13, 5),(13,9),(15,7),(15,11)}

(g".g“1<c(1.5)1)

25

{(8,2),(8,15),(6,1),(6,16),(5,3),(5,14),(7,4),
(7,13),(4,7),(4,10),(2,8),(2,9),(1,6),(1,11),
(3,5),(3,12),(12,3),(12,14),(10,4),(10,13),
(9,2),(9,15),(11,1),(11,16),(16,6),(16,11),
(14,5),(14,12),(13,7),(13,10),(15,8),(15,9)}

(9".9’111c1m11)

{(8,1),(8,13),(6,3),(6,15),(5,4),(5,16),(7,2),
(7,14),(4,5),(4,9),(2,7),(2,11),(1,8),(1,12),
(3,6),(3,10),(12,l),(12,13),(10,3),(10,15),
(9,4),(9,16),(11,2),(11,14),(16,5),(16,9),
(14,7),(14,11),(13,8),(13,12),(15,6),(15,10)}

(9" .9")(t(1 .711)

{(8.3).(8.14).(5.4).(6.13).(5.2).(5.15).(7.1).
(7.16).(4.5).(4.11).(2.5).(2.12).(1.7).(1.10).
(398)9(399) 3(1292),(12315)9(1091)’(]0’]6),

(9.3).(9.14).(11.4).(11.13).(16.7).(15.10).(14.8).
(14.9).(13.6).(13.11).(15.5).(15.12)}
{(8.12).(5.10).(5.9).(7.11).(4.16).(2.14).(1.13).

(3,15),(12,8),(10,6),(9,5),(11,7),(16,4),(14,2),
(13.1).(15.3)}

(9-].9-1)([(1.8)])

(9’1 .9")(1(1 .9)1)

{(8,10),(8,11),(6,12),(6,9),(5,10),(5,11),(7,12),
(7,9) ,(4,l4),(4,15),(2,16) ,(2,13) ,(1 ,14),(1 ,15) ,
(3,16) ,(3,l3),(12,6),(12,7),(10,8),(10,5).(9,6) ,
(9,7),(11,8),(11,5),(16,2),(16,3),(14,4),(14,1),
(13,2),(13,3),(15,4),(15,1)}

(9'1 .9")(t(1 .1o)1)

{(8.9).(6.11).(5.12).(7.10).(4.13).(2.15).(1.16).
(3.14).(12.5).(10.7).(9.8).(11.6).(16.l).(14.3).
(13.41.05.211

(9“.9‘111t1mnn

26

The general form of a matrix in the commutator algebra ;/4 3
i 3 2
C'14

2 1

hde

1
4
2
3

is as follows:

 

 

 

h i g h f e d j j e d f b c a b

 

i h h g e j f d e f j d c b b a

 

 

 

 

\
\ is still isomorphic to the algebra

2

1 l
3
4
of all diagonal 10 x 10 matrices.

Let us do more comparisons for Latin square designs from

different transformation sets. Assume the observed values from the

27

Latin square designs ([1 2 3 4\ and [l 2 3 4 are arranged
2 1 4 3 1 2 3 4 1
\3 4 1 2/ 3 4 1 2
43 2 1 4 l 2 3

correspondingly in a matrix 21 22 23 24‘\.

1
z5 z6 z7 Z8 l
29 Z10211212 j
213214215216/
The following fixed effect model is assumed for the analysis of the
design {[1 2 3 4\ .
g 2 1 4 3
1 3 4 1 2
\4 3 2 1

27 = u + a2 + 83 + Y4 + 87

Z8 ‘ u I “2 1 B4 1 Y3 I 88

Z9 = u + “3 * 81 + Y3 1 E9

210 “ T “3 I 82 1 Y4 T 810
Z11 ‘ u I “3 I 83 1 Y1 I 811
z12 “ “ I “3 1 B4 * Y2 T E12
213 = u I “4 + 81 I Y4 + 813
z14 ' u I “4 I 82 1 Y3 T 814
Z15 = u T “ 1 B3 1 Y2 T 815
216=U+a +B4+Yl+€16

where u is the overall effect, a. B. Y are main effects and

+
e - (6],..

28

"€16) is assumed to have (A) as its covariance matrix.

The estimators of elementary contrasts of treatment effects

are as follows.

A A

Y1 ' Y2 = (21 + Z6 + 211 1 216)/4 ‘ (22 + Z5 I 212 1 215”4

A A

Y] - Y3 = (21 + 26 + 2]] + 216)/4 - (23 + 28 + 29 + 214)/4

A A

«{1- Y4 =

(21 T 26 T 211 1 216)/4 ' (Z4 1 z7 * 210 I 213”4

The variances and covariances of the estimators are as

follows.

(2.1)

e = (2],..

following

(2.2)

Var(§i - §j) = (8a + 24d - 8(b + 2e + c))/16 for i a j

Cov(§.-§. ,§.-y. ) = (-4a-12d+4b+8e+4c)/16
J1 J2 J2 J3
for j],j2,j3 not equal

A

A

Cov(;. - y. ,;. - §. ) = 0 for j ,j ,j ,j not equal
J] 12 33 34 1 2 3 4

similar analysis of the design assuming

1 2 3 4
2 3 4 1
3 4 1 2
\4 1 2 3

"€16) to have (8) as its covariance matrix has the

results.
Var(v1 ' 12) = Var(vi - I4) = Var(12 - Y3) = Var(v§ - Y4)
= (8(a + 29 + j) - 8(b + d + i + f))/16
V6F<Y1 ‘Y3) = Var(v2-v4) = (8(a+29+j) ‘ 8(C+29+h))/16
Cour-1723243) = (8(b+f+i +d) -4(a+29+.1) -4(c+2e+h))/16

C0V(Y1 - Y2.Y2 - Y4) = (4(c + 28 + h) - 4(a + 29 + 311/16

29

and so forth.

From (2.1) and (2.2), the two designs are shown to have
different statistical properties.
Examole 3: Latin Square Designs of Order Five

There are two transformation sets in this case. Let us
consider one design from each transformation set.

(I) The structure preserving group of is generated

by the following permutations.

8 910111213141516171819202122232425“.
2 3 4 517 8 910 6121314151117181920162223242521)

’ .3
N
w
4:
01
01
\1

0‘
\l

8 910111213141516171819 20 2122 23 24 251

( 1 2 3 4 5
1215131114 2 5 3141720181619 710 8 6 922252321247)

93 =
( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 251
232425212234512891067131415111218192016177

The order of the structure preserving group is equal to 100 and there
are 144 equivalence classes for the transformation set. The general

form of a matrix in the commutator algebra is as follows.

30

 

 

 

 

 

 

 

 

 

rl-a-bb bcdefgcfdgecegdfcgfed
bab bgcdefecfdgfcegddcgfe
bba bfgc'degecfddfcegedcgf
bbb befgcddgecfgdfcefedcg
bbb adefgcfdgecegdfcgfedc
cgf dabbbbcdefgcfdgecegdf
dcg ebabbbgcdefecfdgfcegd
edc fbbabbfgcdegecfddfceg
fed gbbbabefgcddgecfgdfce
gfe cbbbbadefgcfdgecegdfc
ceg fcgfedabbbbcdefgcfdge
fc,e ddcgfebabbbgcdefe.cfdg
dfc gedcgfbbabbfgcdegecfd
gdf efedcgbbbabefgcddgecf
egd cgfedcbbbbadefgcfdgec
cfd ecegdfcgfedabbbbcdefg
ecf gfcegddcgfebabbbgcdef
gec ddfcegedcgfbbabbfgcde
dge fgdfcefedcgbbbabefigcd
fdg cegdfcgfedcbbbbadefgc
cde gcfdgecegdfcgfedabbbb
gcd fecfdgfcegddcgfebabbb
fgc egecfddfcegedcgfbbabb
efg ddgecfgdfcefedcgbb ab
ief cfdgecegdfcgfedcbbbb:

 

31

The corrmutator algebra is a 7-dimensional vector space. It is also

a commutative algebra. The commutator algebra is thus isomorphic to
4 5 is generated
3 4

5 2

1 3

2 l/

12 3 4 5 6 7 8 910111213141516171819202122232425)
14 2 3 51619171820 6 9 7 81011141213152124222325

the algebra of all diagonal 7 x 7 matrices.

(II) The structure preserving group of

01¢de
0001-th
hN-‘U‘Iw

by the following permutations.

(D
.4
II

1.0
N
I

(12 3 4 5 6 7 8 910111213141516171819202122232425\
15 4 3 2212524232216201918171115141312 610 9 8 7)

CD
0.)
ll

(12 3 4 5 6 7 8 910111213141516171819202122232425)
14 5 2 316192017182124252223 6 910 7 81114151213

The order of the structure preserving group is equal to 12 and there
are 1200 equivalence classes for the transformation set. The general

form of a matrix in the commutator algebra is as follows.

32

ML.

 

 

 

 

 

c oc oc oc Nm Nc mc cc mc Pc Nc mc mc cc Fc Nc cc mc mc pc cN mN mN mN

oc m mm cm cm mm om mm um Nc mm am pm on wc —m No no Nm mc mN 5N mN am mp
cc cm m mm cm Po Nm No no ac mm mm om mm mc mm on on —m wc mN om NN mN mp
cc mm cm m cm mm pm on mm mc pm mm Nm Nm mc mm mm mm om Nc mN mN om 5N m—
pm mm mm mm m mm mm mm mm mm mm mm mm mm mm mm mm mm Rm mm MN mN mN cN P—
om mm um om mc m cc cm mm cm No Fe Nm mm mc am mm om pm mc KN mN om mN mp
mc Nc mc cc —c cc c cc cc Nm cc Nc mc mc pc mc Nc cc mc pc mN cN mN mN N—
—m mm mm on mc mm cc m cm cm mm mm om mm mc mm Pm Nm Nm mc mN mN RN om mp
Nm pm me Nm mc cm cc mm m cm on wm pm mm wc um mm mm om mc om mN mN NN mp
mm mm mm mm mm mm Pm mm mm m mm mm um mm mm mm cm mm mm mm mN mN oN oN Pp
om mm mm mm Nc mm Pm mm on mc m mm cc cm cm No no Pm Nm mc NN mN mN om mp
Nm Nm Pm me ¢c mm om mm mm Nc cm m cc mm cm so mm am pm mc om RN mN mN mp
mc cc Nc mc Pc mc mc Nc cc pc oc oc c oc Nm cc mc Nc mc —c mN mN cN mN Np
—m cm mm mm mc mm Nm pm No mc mm cm cc m cm om Km mm om Nc mN om mN NN mp
um mm mm mm mm mm Rm om mm mm mm mm _m mm m mm mm mm um mm mN mN MN 0N pp
om mm mm mm Nc No Nm mm pm mc am om pm mm mc m cm mm cc cm NN om mN mN mp
—m mm on mm mc cm on um mm mc no Nm Nm pm mc mm m cm cc cm oN NN om mN mp
Nm me No Pm ac om Fm mm mm wc um cm on mm mc cm mm m cc cm om mN RN mN mp
mc mc cc Nc Pc cc mc mc Nc pc mc cc mc Nc pc oc oc oc c Nm mN mN mN cN N—
nm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm mm pm m cN cN mN mN pp
mp up up up mp mp 0N FN NN mp mp NN ON pN mp mp PN NN ON mp N cp cp cp op
0N mp NN FN mp n— op N— up mp PN mp ON NN mp NN mp PN oN mp c» N c_ cp o_
8 —N 3 NN m: NN ON 3. N E t 2 2 2 mp FN NN or 8 3 cp cp N c— 0—
ON NN N 3 2 PN ON NN 2 mp NN PN ON 2 2 2 2 2 3 mp 3 cp c_. N S
w m m m N m w a m N m m m m m m m m m m o c o m _

 

 

 

 

 

33

The elements corresponding to the same number are equal to
each other. The structure of this commutator algebra is completely
different from the one in (I). It is a 63 dimensional vector space
and a noncommutative algebra.

Romook: For Latin square designs of order k, the number of equivalence
classes in a transformation set is equal to the product of the number
of standard Latin squares in the transformation set and the number
(k-l)!. From the table in Fisher and Yates, 1938 [5], the number of
standard squares are listed for Latin square designs of order up to 6.
There are 22 transformation sets of Latin squares of order 6. Some
of them have equal number of standard squares, so they have the same
order for their structure preserving groups. Are the vector space
dimensions of the commutator algebras (representing structure pre-
serving groups of larger order) smaller than the dimensions of the
commutator algebras (representing structure preserving groups of
smaller order)? We need a computer to study these problems. A com-
puter program was written to compute the structure preserving groups
for 4 x 4 designs, however, it is quite expensive to run and further
work is needed to improve its efficiency. It is hoped that further
results for Latin square designs of order 6 will be obtained in the

future.

§3. DIFFERENT SCHEMES FOR ASSIGNING TREATMENTS TO SUBJECTS

Concerning the problem of estimation of covariance matrices
of Latin square designs, it is desirable to assume that the covariance
matrix of the observed values belongs to the commutator algebra asso-
ciated with the Latin square design (Dubenko, Sysoev and Shaikin,

1976 [2]). A special randomization scheme, depending on a group of
permutations of the cells, to assign treatments to subjects is to be
discussed for Latin square designs.

As mentioned in section one, a Latin square design can be con-
sidered as a 3-factor design. Let L be a given Latin square design

2

of order m, with m subjects corresponding to m2 cells in this

given design. The level of each of the three factors (i.e. row factor,
column factor and treatment factor) for a specific x is denoted by

(rx,cx,tx) according to section one. Let G be a subgroup of the

symmetric group S 2. The randomization scheme is described as follows.
m

First, one element (g) will be randomly chosen from G. Then the

h

condition (rx,cx,tg(x)) will be assigned to the xt experimental sub-

ject (rx is the row level associated with cell x, cx is the column

level associated with cell x and t is the treatment level

9(X)
associated with cell g(x) in the given design L). For example, if

35

\
12 3 4 5 6 7 8 910111213141516.
( 5 7 8 213 410 9111214131516)15randomly

,.

chosen from G, then the condition (r1,c],tg(x)) will be assigned to
the first subject, where (r],c],tg(1)) = (1,1,1). Similarly, the con-
dition (r2,c2,tg(2)) will be assigned to the second subject, where
(r2,c2,tg(2)) = (1,2,2), and so forth. With this randomization scheme,

th

the observed value for the x subject can be written as follows

(fixed effect model):

2 = u + a + B + y + (as) + e
x rx cx tg(x) rxcx x

where u is the overall effect; a,8,r are the main effects; (a8)

is the row-column interaction and ex is the random error which is
independent of the randomization scheme. The row-treatment interaction,
column-treatment interaction and three-way interaction are assumed to

be negligible. Let us rewrite zx as follows: 2x = ux + nx, where

u is not

U = u + a + B + (a8) + e and nx = Yt x

X T‘x CX Y‘xCx X g(x)

affected by the randomization scheme, but nx is affected by the ran-
m m m

domization scheme. It is assumed that X “i = 2 Bi = Z Yi =
i=1 i=1 i=1

m m
X (a8)ij = Z (“8)1j = 0. Under the above model, we have that
i=1 '=1

(3.1) sz = Eux + Enx
and ‘
(3.2) Cov(zx,zy) = Cov(ux,uy) + Cov(nx,ny).

It is interesting to study what the impact of the randomization
scheme is upon the njs. Let us introduce the following notations.

INX)(k) = P(t = k), g(x)(k) is the probability that the condition

9(X)

36

h subject under the randomization

= k'). n‘x”)(k.k')

(rx,cx,k) will be assigned to the xt

scheme. K(x’y)(k,k') = P(t = k and t

9(x) g(y)
is the joint probability that the condition (rx,cx,k) will be assigned

to the xth subject and the condition (r ,c ,k') will be assigned to the

Y y
yth subject. Note that.the probability comes from the random choice

of g from the group G and the following equalities hold.

(3.3) Enx = g(n‘*)<k> x yk)
(3.4) Var(nx) = 1(H(X)(k) x YE) - (Enxlz
(3.5) Cov(nx.ny) = kzk.(n(x:Y)(k.k'1 x vk x yk.) - (Enx)(Eny)

where summations are over all possible k and k'. Some theorems

concerning Enx, Var(nx) and Cov(nx,ny) will be shown next.

Lemma 3.1:

a) K(x)(k) = H(y)(k) for all k if x = g(y) for some 9 E G

b) K(X’Y)(k.k') = n‘x'11'11k.k') for all k.k'. if

(x.y) = (9(X').9(y')) for some 9 6 6.

Proof:

 

a) The probability distribution of K(x)(k) is completely dependent
on Gx = {g(x)lg 6 G}, the orbit of x under G. We also know
that x = g(Y) with g E G implies that Gx = Gy.

H(X)(k) = H(y)(k) for all k, if x = g(y) for some 9 6 G.

b) Similarly, the probability distribution of H(x’y)(k,k') is com-
pletely determined by the set {(g(x),g(y))|g 6 G}. Also, if
(x,y) = (g(x'),g(y')) for some 9 e G, then

37

{(9(X).9(y))|9 E G} = {(9(X').9(y'))|9 6 G}.

n‘x’Y)(k.k') = n(x"yi)(k.k') for all k,k' if

 

(x.y) = (9(X').9(y')) for some 9 5 G. Q-E-D~
Theorem 3.1:
a) If x = g(y) for some 9 E G then Enx = Eny and

Var(nx) = Var(ny).
b) If (x,y) = g(x'),g(y')) for some 9 E G then

Cov(nx,n ) = Cov(nx..ny.)-

Y

Proof: Theorem 3.1 follows directly from lemma 3.1. Q.E.D.

 

Theorem 3.2: If G is a subgroup of the symmetric group S 2 which
m

induces one and only one orbit on {l,2,...,m2} then Enx = 0 for

x = l,2,...,m2. (A group of transformations which induces one and

only one orbit on its domain is called a transitivegroup.)

Proof: From the assumption that G is transitive, we have that

{1.2.....m21 = {9(1119 e 61. Let H] = {919(1) = 1}.

-H2 = {g|g(l) = 21,...,H 2 = {g|g(l) = m2} and then H1 is a subgroup
m

G. Next, we prove the following equalities:

(*1 Hi = g-H1 for some 9 E G, i = 2,3,...,m2.

Proof of (*):

iff g(l) = 1
iff g(l) = 90(1) for some 90 E G
g 6 Hi
-1

iff g0 og(l) = l for some 90 6 G
iff g E gO-H1 for some 90 E G

38

(*) is thus proved.

From (*), we have that |H1| = [HZ] = ... = |H 2I, thus
m

n(‘)(1) = n(‘)(2) = ... = n(‘)(m) = l/m

En-I = er/m = O
k
En] = En2 = ....... = En 2 = O
m
follows from theorem 3.1. Q.E.D.

Theorem 3.3: If we choose G = Gr n Gc Fth, the structure preserving

 

group of the given Latin square design, and let '6 = (n1...-.n 2),
m
then Cov(fi) 6 C (where C is the commutator algebra of Definition

1.6).

Proof: From Theorem 3.1, Cov(ni,nj) = Cov(ni..nj.) if (i,j) =
(g(i'),g(j')) for some 9 e Gr n Gc n Gt and i,j,i',j' e

{l,2,...,m2}. :. Cov(6) e 6 follows from the definition of c. Q.E.D.

Let g e Gr 0 Gc and (9,9) be a function defined on
{l,2,...,m2} X {l,2,...,m2} as in the proof of Theorem 1.5. Let
[(i,j)] = {(9:9)(1.j)|g E Gr n Gc}’ it is the orbit of (i,j) under
the set of transformations {(g.g)lg G Gr n Gc}' We have that
' {l,2,...,m2} x {l,2,...,m2} = [(1,1)] 0 [(1,2)] U [(1,m+1)] u [(1,m+2)].
From Lemma 3.1, the next theorem completely specifies the value

2

n(x”)(k',k') for all x,y = l,2,...,m and k,k' = l,2,...,m.

Theorem 3.4: If we choose G = Gr n GC for our randomization scheme,

then

39

a) K(x)(l) = H(x)(2) = ... = K(x)(m) = l/m for x = l,2,...,m2

b) n("2)(i,j) = 1/m(m-1) for i e j and i,j = l,2,...,m

= 0 for i = j and i = l,2,...,m

c) 11“'""'”(i.i)

l/m(m-l) for i f j and i,j = l,2,...,m

0 for i = j and i = l,2,...,m

d) n("m+2)(i (m-2)/m(m-1)2

1/m(m-l) for i = j and i = l,2,...,m

for i f j and i,j l,2,...,m

.1)

Proof:

a) It follows from the proof in Theorem 3.2 and Lemma 3.1 (2’ Gr n Gc

is a transitive group)

b), c), d) Let

3
I

1 - {(i,j)li e j and i,j = l,2,...,m}

2 {(k.j)|i # j and i,j m+l,m+2,...,2m}

3
II

M = {(1.1111 f j and 1.3

N] = {(isj711fj and in].

21

(m-l)m+1,...,m

1,m+1 ,2m+1 ,. . . ,(m-1)m+1}

N2 = {(i,j)li # j and i,j = 2,m+2,2m+2,...,(m-l)m+2}

Nm = {(i,j)li f i and i,j = m,2m,...,m2}

R1 = {(1,j)|j a l,j f 2,...,j a m,j r 1,j # m+l,j r 2m+1,...,
j # (m-1)m+l}

R = {(2,j)|j f1,j as 2,...,j ,1 m,j ,1 2,j ,1 m+2,j ,1 2m+2....,

j? hrlMHZ}

4O

Rm=umnwr1dizaunrmnrzmiimhudrmh
Rm,1 = {(m+l..1')lj 1‘ m+l.3' 1‘ m+2.....3' 7‘ 2m..i 21.1 i‘ m+1.....

.1' 1‘ (m-l)m+l}

mm=HMJHirmLiimauuiimnimnrzmuuiimi

$121.3? m+1......1' 1‘ (m-1)m+1}

R 2 = {(m2.i)lj f (m—1)m+1.i r (m-l)m+2.....j i m2.j r m.i # 2m.
111
...,j #112}

where M1,...,Mm,N],...,Nm,R],R2,...,Rm2 are disaoint sets.

Note that

[(1,2)] = M1 u M2 u ... u Mm

[(l,m+l)] = N] U N2 U ... U Nm
[(l,m+2)] = R1 u R2 U ... U Rm2
and
111.1 - 11121 = -- lel = rum-11
1N1} = 1N2} = = lel = m(m'1)
181 181 =18 1=<m-1>2 -
1 2 2
m
l[(l,2)]| = m2(m-l), l[(l,m+l)]l = m2(nrl) and l[(1,m+2)]l = m2(m-l)2

41

LEt [(1,2)] = {(1:2)9(129j2)s(139j3)90009(i )}

m2(m-1)’jm2(m-l)
(1.2)}

I
ll

1 {919 E Gr 0 Gc and (9(1).9(2))

I
ll

2 {9|9 E G, n GC and (9(1).9(2)) = (12.32)}

H = {9|9 6 G, 0 GC and (9(1).9(2)) = (i

.1 )
m2(m-1) m2(m-l) m2(m-l)
then H1,H2,...,H 2 are disjoint sets and
m (m-l)
m2(m-1)
(1) Gr n Gc = i3] Hi‘

Moreover, H1 is a subgroup of Gr n GC. Next, we prove the following

equalities:

2
(2) Hk = 9 H1 for some 9 6 Gr 0 Gc’ k = 2,3,...,m (m-l).

Proof of (2):

 

iii (9111,9121) = (1k..ik)

iff (901.9(2)) (900159091) [0" 5°“ 90 6 Gr “ Ge

9 6 H
iff (galog(1),gBIog(2)) = (1,2) for some 90 E Gr n GC
iff g E gO-H.l for some 90 E G

(2) is thus proved.

[H | = |H I = ... = [H l follows from (2).

Note that

42
(3) {<t,.tj)1(i,i) 5 M11 = {(ti.tj)l(i.j) e 821 = ... =
{(t,.tj)l(1.j) e Mm}.
(1‘ In a Latin square design each treatment appears once in each row.)
Let (i0,j0) 5 M1 then
Ii(i.3)1(i.3)_e [(1.211 and (ti’tj) = (10.30111 = m (2' (3))

i.e. [{Hi|(tg(]),tg(2)) = (i0,j0),i = l,2,...,m2(m-l)}| = m

. H(1,2)(i . - m'1H11 m’lH'II '1

H3110 1 m2(m-l)|H1| m(m-1)

 

 

for (i0,j0) 6 M1 (1' (1)) b) is thus proved.

The proof of c) is completely similar to b), so we omit
it.

For d): Let

[19 2]={], 29 .'9.' 90°09 .' 9 .'
( m+ ) ( m+ ) (12 32) (1m2(m_1)2 sz

)2):

(m-l

01 = 191(911).g(m+2)) = (1.m+21 and 9 E Gr “ Gc}’

= 1, 2 = q a. }
Qm2(m-])2 {91(9( 7 g(m+ )) (1m2(m-])2 jm2(m-1)2)

Similar to the proof of (2) for {H],...,H 2( 1)1, we have
m m-

2 2
0k = 9°01 for some 9 6 G, 0 6c, k = 2,3,...,m (m-l)

q = Q = ... = Q .
1 11 1 21 1 m2(M21

43

Note that
m2(m-l)2
(4) Gr n Gc = 12] 01
(5) |{(t..tj)|(ti.tj) = (1.1) and (1.3) 6 R1}! = m - l
and

l
3

1
N

|{(t,.tj) (ti’t‘) = (1.k) and (1.3) 5 R111 -

J
for k = 2,3,...,m.

(1‘ structure of Latin square design.) Moreover,

(6) |{i|Ri has the same property as that of R1 in (5)}| = m.
Similarly,

(7) l{(t..tj)l(ti.tj) = (2.2) and (1.3) 6 R211 = m - l

and

[{(tl’tj)l(t1’tj) = (25k) and (1,j) 6 R211 m - 2

for k 6 {l,2,...,m}\{2}

 

and
(8) ({ilRi has the same property as that of R2 in (7)}| = m
and so forth.
m(m-2)|Q | _
n“i"‘*2)(i,j)= 1 = "'2 for iii
[Gr n Gcl m(m-l)

and i,j = l,2,...,m then follows from (4), (5), (6), (7), (8). and

Gr n GC m(m-l)

for i = j

 

and i,j = l,2,...,m. d) is thus proved. Q.E.D.

44

Discussion: In practice, Enx = 0 for all x are required for

(3.1) so that no bias is introduced through the randomization scheme.
Theorem 3.2 tells us that a transitive group G will fulfill the
requirement. The structure preserving groups for Latin square de-
signs of order 3, order 4 and the Latin square designs from the

transformation set containing are transitive groups.

U'l-fiWN-fl
~01th
NdUT-Dw
(JON-#0145
hWN-‘U‘I

S 2 and Gr 0 Gc are two other transitive groups. In case we would
m

like to assume that all the observed values zjs have equal variances,
the randomization scheme with a transitive group G will justify the
assumption. The reasoning is as follows:

Var(n1) = Var(n2) = ... = Var(nm2) if G is transitive (2‘ Theorem
3.1). Moreover, Var(zx) = Var(ux) + Var(nx) for x = l,2,...,m2
(1°(3.2)). If the original Var(ux)'s are not equal to each other,
then Var(nx)'s have the stabilization effect to make the assumption
"Var(zx)'s are all equal" justified. Similarly, if we would like to
assume that Cov(§) belongs to the commutator algebra associated with
the Latin square design. From (3.2) we have that Cov(§) =

Cov(U) + Cov(fi), Theorem 3.3 tells us that if G is the structure
preserving group then Cov(fi) belongs to the commutator algebra. So
when Cov(fi) is an element of the commutator algebra or very close to
an element of the commutator algebra, the randomization scheme using
the structure preserving group will have a similar stabilization

effect. The randomization scheme using Gr 0 Gc corresponds to the

usual Fisher and Yates randomization for Latin square designs. Under

45

the Fisher and Yates randomization scheme, Cov(n1,n2) is not equal

to Cov(n1,nm+2) in general. In fact,

C0V(n1.n2) = Z '(H(]’2)(K,K') x rk x rk.) = X

k,k 173
(1' Theorem 3.4c))

(rirj)/m(m-l)

 

 

Cov(n].nm+]) = 37E2117' 1.gjhirj) + mil—1)'§lr$ (2' Theorem 3.4d))
- m- 1:
= __E:Z_§_ 2 ( .r.) - 1 Z (r.r.) (2’ g r = 0)
m(m—l) ifj 1 3 m(m-l) ifj ‘ 3 i=1 1
= -1 o a .
m(m—i) i;j(r1rJ)

We thus introduce some bias into Cov(2) under the Fisher and Yates
randomization scheme. In this section, I attempt to understand more
about the impact of the randomization scheme on the statistical
analysis that follows it. I think that randomization changes the
probability distributions of the observed values. However, I doubt
that it will make the normal theory analysis more appropriate as is

claimed by some people.

§4. RANDOMIZATION TEST FOR THE LATIN SQUARE DESIGN ANALYSIS

From the computation in section two, it is seen that the in-
trinsic structures for Latin square designs from different transforma-
tion sets are quite different. However, Latin square designs from the
same transformation set share some common properties. Let us compare
different Latin square designs from the point of view of estimation
of the covariance matrix of the observation values. Among Latin
square designs of the same order, those with small vector space dimen-
sion for their associated commutator algebras should be preferred
(since less parameters need to be estimated). $0, for Latin square

designs of order 4, designs from the transformation set containing
1 2 3 4

2 l 4 3 are preferred to designs from the transformation set
3 4 l 2
4321/
1234\
containing 2 3 4 l 1 For Latin square designs of order 5, designs
3 4 1 2.
4123/!
l 2 3 4 5
the transformation set containing g 2 g $ g are preferred to
4 5 l 2 3
l 2 3 4/

designs from the transformation set containing In this

U'I-hWN—J
(3001th
Jam-401w
N-dU'lw-k
dwN-QU'I

section, we discuss the comparison for different Latin square designs

46

47

from the point of view of randomization.

The idea of randomization tests was originated by R.A. Fisher
(Fisher, 1935 [4]). Let us quote and reexamine an example of a ran-
domization test (Kempthorne, 1952 [7]).

Example 4.1: Suppose we have 8 experimental objects, a,b,c,d,e,f,g,h,

 

of which four a,b,c,d have received treatment 1 and the other four
treatment 2, and let the experimental results be:

Treatment 1: a, b, c, d
1813 317

Treatment 2: e f g h
9 l6 l7 17

There are 70 possible ways of assigning treatment 1 to four objects
.and treatment 2 to the other four objects. Using all the seventy
possible ways of assigning the eight observed values, the absolute
value of the difference between treatment averages was computed. The

frequency distribution of these seventy values was as follows.

Value (0 .5 1 2 2.5 3 4 4.5 5 6 6.5 7

(A)
Frequency [ 6 8 6 6 8 6 6 8 6 2 6 2

 

The computed value for the assignment actually used is 2."The signi-
ficance probability is 50/70 (i.e. approximately 71 percent). Suppose
the way of assigning treatments to objects was obtained through a ran-
domization scheme, each object receiving either treatment 1 or treat-
ment 2 according to the outcome of flipping a coin. Then the
randomization test should allow for those other ways of assigning
treatments to objects. For example, three objects receive one of the

two treatments and five objects receive the other treatment (call it

48

{3,5} ways of assigning treatments to objects) and so forth. Totally
there are 254 ways excluding the two ways which assign one of the two
treatments to all objects. The absolute value of the difference between
treatment averages is computed for each of the 254 ways. The frequency
distribution for such computed values are tabled as follows:

The frequency distribution of the computed values for {3,5}

ways of assigning treatments to objects:

Value I .4 .6 .93 1.20 1.46 1.73 2.26 2.53 2.80 3.06

 

FrequencyI 6 2 8 6 6 8 8 6 6 8

(B) . .
Value I3.6 3.86 4.40 4.6 4.93 5.2 5.73 6 6.53 7.06 8.6

Frequency I 6 2 6 6 2 8 6 2 6 2 2

The frequency distritution of the computed values for {2,6}

ways of assigning treatment to objects:

() Value I .3 1 1.6 2.3 3.6 4.3 5.0 5.6 7.6 10.3
c

 

FrequencyI 2 8 8 2 8 10 12 2 2 2

The frequency distribution of the computed values for {1,7}

ways of assigning treatments to objects:

Value I .8571 2.57 3.714 4.857 5.428 12.28

 

(0)
Frequency I 2 2 6 2 2 2

The mean values and standard deviations for the four distributions are
as follows:

(A) Mean Value: 3.07 Standard Deviation: 2.11

(8) Mean Value: 3.29 Standard Deviation: 2.00

(C) Mean Value: 3.69 Standard Deviation: 2.21

(0) Mean Value: 4.64 Standard Deviation: 3.27

49

A comparison between the four cases from another point of view is as
follows: Let 21’22""’28 denote the observed values for the experi-
ment and assume they are independently distributed with the same

2

variance 0 . The computed values in (A),(B),(C),(D) can be represented

in the following forms:

(A) (21] +21.2 +21-3 + 214)/4 - (2i +’z1 + 217 + 218)/4
(8) (21-1 + 21 + 213)/3 - (2i + + zi8)/5

(c1 (21] + 2121/2 - (21.3 + + 2181/6

(D) (21] - (21-2 + + 218)/7

The variances of the computed values are as follows:
2
o 8 2 8

(A) '2' , (3)1302, (C) 362, (0) 7o . This suggests that the randomi-
zation scheme according to the outcome of flipping a coin is inappro-
priate. In practice, the number of objects to receive treatment 1 and
the number of objects to receive treatment 2 should be predetermined
before doing the randomization procedure. In other words, randomiza-
tion should be performed among those designs with the same structure
or symmetric property. So in practice, a particular transformation
set of Latin square designs should be chosen according to some cri-‘
terion and a Latin square then randomly chosen from the transformation
set. A criterion for the choice between different transformation
sets is discussed below in connection with the randomization test.

A Latin square design of order k is randomly chosen from a

transformation set. The observed values from the Latin square design

50

experiment are arranged in a matrix 2 ........... Zk as befbre.

2(k-])k+]ooazk2
Under the null hypothesis that k treatment effects are all equal,

the observed values would have been the same had any other Latin square
design been used from the transformation set. If we superimpose the
observed values on all the possible Latin square designs in the trans-
formation set, then we can compute the mean treatment sum of squares,

*
mean error sum of squares and the F statistics as in the usual

= mean trt SS
mean error SS

using the actual design used, is the test statistic for the randomiza-

analysis of variance (F* ). The computed F* value,
tion test. The computed F* values, using designs from the same
equivalence class of a transformation set, have the same value.

Let me explain more clearly, from the point of view of analysis of
variance, why we should include only those designs in the same trans-
formation set as the design actually used. We know that two Latin
square designs from different transformation sets can not be obtained
from each other by permutations of rows and columns. In other words,
if we superimpose the observed values on designs from the other
transformation set, then the computed values for the sum of squares
due to rows and columns are vitiated in this case. The computed F*
value is thus inappropriate for comparison. For Latin square designs
of order 3, there is only one transformation set and it has two equi-
valence classes. The significance probability for testing the null
hypothesis with one replicate is either .5 or 1.0. For Latin

square designs of order 4, the attainable significance

51

l 2 3 4
probabilities are multiples of 1/6 for , g l I g and the attainable
4 3 2 l
l 2 3 4
significance probabilities are multiples of 1/18 for g 2 I g
4 1 2 3/
l 2 3 4 l 2 3 4
We say that g 2 I 6 is more precise than g l I g (or more
4 l 2 3 4 3 2 l

sensitive) with respect to the randomization test. The transformation

/1 2 3 4
set containing {\g 2 I 3 is preferred to the transformation set con-
4 1 2 3
l 2 3 4
taining g l I g in terms of the randomization test. Similarly,
4 3 2 1

applying the results of Example 3, Section two, the transformation set

1 2 3 4 5
2 l 5 3 4
containing 3 4 l 5 2 is preferred to the transformation set con-
4 5 2 1 3
5 3 4 2 l
1 2 3 4 5
2 3 4 5 1
taining 3 4 5 l 2 in terms of the randomization test. These are
4 5 1 2 3
5 l 2 3 4

contrary to the results in terms of estimation of covariance matrix.
It shows that the main interest of statistical study should be deter-
mined before an optimal design for an experiment can be found. Let
us look at two examples of randomization tests for a Latin square
design of order 4.

Example 4.2: The observed values from a randomly chosen Latin square

design are arranged correspondingly in a matrix

dawn)
bum—a
com—44>
N—‘Otw

52

21 8 17 9
10 3 12
20 10 15 21
4 15 3 9

U1

The analysis of variance table is as follows:

 

 

Source Sum of Squares d.f. Mean Square F ratio
Row 240.25 3 80.08

Column 28.25 3 9.42

Treatment 216.25 3 72.08 4.55
Error 95.00 6 15.83

Total 579.75 15

The significance probability 3 .07

The computed F* values, using designs from 18 equivalence classes
2 1 4 3

of the transformation set containing the design are as

‘90)
#00“)
com-4
,N—I-t:

follows:

(4.1) F*values: .096, .207, .231, .284, .318, .390, .422, .485,
.638, 1.051, 1.242, 2.07, 2.61, 2.75, 4.55, 4.62,
4.92

Mean value of F* = 1.53
Standard Deviation of F* values = 1.67
The computed F* values using designs from six equivalence

classes of the other transformation set are as follows:

(4.2) F*values: .128, .318, .441, .594, 4.29, 4.92
Mean Value of F* = 1.78, Standard Deviation of F* = 2.20

53

Example 4.3: The observed values from a randomly chosen Latin square

1.1 1.5 1.0 1.7

design are arranged in a matrix 1.4 1.9 1.6 1.5
2.8 2.2 2.7 2.1

.4 2.5 2.9 2.7

 

*
The computed F values, using designs from eighteen equivalence

classes of the transformation set containing the design I 3 Z I g
1 2 3

are as follows:

(4.3) F* values: .0905, .127, .155, .155, .205, .205, .3095, .332.
.378, 1.03, 1.19, 1.44, 1.67, 1.85, 2.85, 7.33.
10.76, 11.1

Mean Value of F* value = 2.29

Standard Deviation of F*va1ue = 3.58

*
The computed F values, using designs from six equivalence

classes of the other transformation set are as follows:

(4.4) F* values: .0905, .155, .205, .377, 5.95, 15.34
Mean Value of F* value = 3.67

Standard Deviation of F* value = 6.15

The computed F* values, using designs from different transfor-
mation sets in the above two examples suggest that distributions of
computed F* values for different transformation sets might be quite
different for Latin square designs of order greater than 4. It indi-
cates again, that the randomization test should be performed using only

those designs in the same transformation set as the design actually used.

BIBLIOGRAPHY

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

BIBLIOGRAPHY

Dubenko, T.I., Sysoev, L.P. and Shaikin, M.E. Symmetry proper-
ties and characterization of the covariance matrices in the
randomized-block experimental design problem with randomized
blocks. Automation and Remote Control (1976), 378-376.
Translated from Avtomatika i Telemekhanika, No. 3 (March 1976).
73-82.

Dubenko, T.I., Sysoev, L.P. and Shaikin, M.E. Sufficient
statistics and estimates of covariance matrices of specially
structured covariance matrices for two experimental design
models. Automation and Remote Control (1976), 492-500. Trans-
ggtgg from Avtomatika i Telemekhanika, No. 4 (April 1976).

Ehrenfeld, S. On the efficiency of experimental designs.
Ann. Math. Statist. Z§.(1953): 247-255.

Fisher, R.A. The Design of Experiments. Oliver and Boyd,
Edinburgh, 1935 (rev. ed., 1960).

Fisher, R.A. and Yates, F. Statistical Tables for Biological
Agricultural and Medical Research. Oliver and Boyd, Edinburgh,
1938 (rev. ed., 1957).

James, A.T. The relationship algebra of an experimental design.
Ann. Math. Statist. 2§_No. 4 (1957), 993-1002.

Kempthorne, O. The Design and Analysis of Experiments. John
Wiley & Sons, New York, 1952.

Kempthorne, O. The randomization theory of experimental
inference. J. Am. Statist. Assoc. §Q_(l955), 946-967.

Kempthorne, O. Inference from experiments and randomization.
A Survey of Statistical Design and Linear Models (J.N. Srivastava,
ed.), North-Holland Publishing Company, 1975, 303-331.

Kiefer, J. On the nonrandomized optimality and randomized
nonoptimality of symmetrical designs. Ann. Math. Statist. 29_
(1958), 675-699.

Kiefer, J. Optimum experimental designs. J. Roy. Statist. Soc.,
Ser. B, 21_(1959), 272-319.

54

[12]

[13]

[14]

[15]

55

Sysoev, L.P. and Shaikin, M.E. Algebraic methods of investi-
gating the correlation connections in incompletely balanced
block schemes for experiment design. I. Characterization of
covariance matrices in the case of asymmetric block schemes.
Automation and Remote Control (1976), 696-705. Translated
from Avtomatika i Telemekhanika, No. 5 (May 1976), 64-73.

Sysoev, L.P. and Shaikin, M.E. Algebraic methods of investi-
gating the correlation connections in incompletely balanced
block schemes for experiment design. II. Relationship algebras
and characterization of covariance matrices in the case of
symmetric block-schemes. Automation and Remote Control (1976),
1022-1031. Translated from Avtomatika i Telemekhanika, No. 7
(July 1976), 57-67.

Van Der Waerden, B.L. Modern Algebra, Vol. 2. ‘Springer,
Berlin, 1940. English translation, Ungar, New York, 1950.

Wald, A. On the efficient design of statistical investigations.
Ann. Math. Statist. 14_(1943), 134-140.

      

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