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Some new extended block designs and their analyses

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Some new extended block designs and their analyses
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Schreckengost, Jack Franklyn, 1944-
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ix, 9 leaves. : ; 28 cm.

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Thesis -- University of Florida.
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Includes bibliographical references (leaves 106-108).
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Typescript.
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Vita.

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Full Text
SOME NEW EXTENDED BLOCK DESIGNS AND THEIR ANALYSES
By
JACK FRANKLYN SCHRECKENGOST
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974


TO MY WIFE


ACKNOWLEDGMENTS
I would like to express my appreciation to Dr.
John A. Cornell for his guidance and assistance while di
recting this dissertation. My thanks, also, to the other
members of my advisory committee, Dr. F. W. Knapp, Dr. Frank
G. Martin, Dr. John G. Saw, and Dr. P. V. Rao, for their
helpful suggestions.
A belated thanks is expressed to Mr. Ronald E.
Boyer, a good teacher and a valued friend, who gave me much
encouragement from the very beginning.
Appreciation to my wife, Donna Rae, cannot be ex
pressed as deeply as is felt. I thank her for her patience
and understanding during my many hours of study, research,
and writing.
iii


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
ABSTRACT vii
CHAPTER
1 INTRODUCTION 1
1.1 Blocking Designs .... 2
1.2 Extended Complete Block Designs 2
1.3 Purpose of This Work 5
2 LITERATURE REVIEW 7
3 EXTENDED COMPLETE BLOCK DESIGNS WITH
CORRELATED OBSERVATIONS 16
3.1 Notation and Definitions 17
3.2 Intrablock Estimation of the
Treatment Effects 21
3.3 A Test for the Presence of Correlation 26
3.4 The Exact Distribution of SSR for p > 0 30
3.5 An Approximate Distribution of SSR
for p > 0 40
3.6 An Estimate of the Correlation 4 3
4 A PARTIALLY BALANCED GROUP DIVISIBLE ECBD ... 47
4.1 Definitions and Notation 50
IV


TABLE OF CONTENTS (Continued)
CHAPTER
4 (Continued) Page
4.2 Intrablock Estimation of the
Treatment Effects 53
4.3 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses 58
4.4 Mixed Model Analysis 62
5 A PARTIALLY BALANCED ECBD WITH THE L
ASSOCIATION SCHEME 7 66
5.1 Intrablock Analysis 68
5.2 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses 72
6 THE GENERAL PARTIALLY BALANCED EXTENDED
COMPLETE BLOCK DESIGN 77
6.1 Intrablock Analysis 78
7 CONCLUDING REMARKS AND A SENSORY TESTING
EXAMPLE 86
APPENDIX
1 THE EXACT DISTRIBUTION OF SSR FOR b = 2t
AND k = t+1 WHEN p>0 94
2 AN APPROXIMATE DISTRIBUTION OF SSR FOR
b = 2t AND k = t+1 WHEN p>0 98
BIBLIOGRAPHY 106
BIOGRAPHICAL SKETCH 109
V


LIST OF TABLES
Table Page
1 Intrablock Analysis of Variance for an
ECBD 25
2 Values of g and h for the Approximate
Distribution of SSR, I 42
3 Intrablock Analysis of Variance for Partially
Balanced Group Divisible Extended Complete
Block Designs 57
4 Mixed Model Analysis of Variance for Partially
Balanced Group Divisible Extended Complete
Block Designs 63
5 Intrablock Analysis of Variance for Partially
Balanced ECBD of the Association Scheme ... 73
Al Values of g and h for the Approximate
Distribution of SSR, II 99
A2 Comparison of the Exact Distribution and Two
Approximate Distributions of SSR 104
vi


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOME NEW EXTENDED BLOCK DESIGNS AND THEIR ANALYSES
By
Jack Franklyn Schreckengost
August, 1974
Chairman: Dr. John A. Cornell
Major Department: Statistics
An extended complete block design is a balanced
block design consisting of t treatments in b blocks each of
size k such that k varies between t and 2t. The balance
among the treatments is achieved by selecting duplicates of
some of the treatments for each block according to the scheme
followed when selecting blocks from the class of balanced
incomplete block designs. Under the assumption of an addi
tive model, it may be of interest to investigate the exist
ence of correlation between responses to the same treatment
in the same block. When a positive correlation between du
plicate observations is present, it has been previously shown
that k should be taken equal to t+1 for maximum efficiency
with the extended complete block designs when compared to
complete block designs.
A procedure for the test of the hypothesis of zero
correlation is presented as is a method for estimating the
vii


correlation if the hypothesis of zero correlation is rejected
in favor of the alternative hypothesis of positive correla
tion. Particular attention is given to the distribution of
the sum of squares for remainder, where remainder is defined
as residual minus duplication error, under the alternative
hypothesis of positive correlation. The distribution of the
sum of squares for remainder is necessary for calculating
the power of the test and for obtaining an approximation to
the distribution of the estimator of the correlation. A
specific formula for the distribution of the sum of squares
for remainder is given for the case t = b and k = t+1. The
exact distribution and an approximate distribution of the
sum of squares for remainder are also presented for the case
b = 2t and k = t+1.
The general partially balanced extended complete
block design is defined as a partially balanced block design
consisting of t treatments in b blocks each of size k greater
than t. The analyses of variance for the non-additive fixed
effects and mixed models are presented for the special class
of designs called partially balanced group divisible extended
complete block designs. The analysis of variance of the
additive fixed effects model is also presented for the class
of partially balanced extended complete block designs with
the (Latin Square) association scheme. The analysis of
variance of the non-additive mixed model for this class of
designs is mentioned briefly.
The intrablock analysis of variance for the
viii


additive model is developed for the general partially bal
anced extended complete block designs. Also, the recovery
of the interblock information and the combined intrablock
and interblock analysis for these general designs are men
tioned briefly.
The final chapter contains some comments about the
assumptions made and about directions for future study. A
numerical example of a taste testing experiment is also pre
sented with the resulting analysis for the balanced extended
complete block designs considered in this work.
ix


CHAPTER 1
INTRODUCTION
In many fields of experimentation, a distinction
that has long been implicit in the statistical literature is
the difference between experiments designed for the es
timating of absolute treatment effects and experiments of the
comparative type. In comparative experiments, the emphasis
is on performing comparisons between the effects of the dif
ferent treatments such as the effects of different doses of
a drug or the effects of different levels of nitrogen on the
average yield of soybeans. While the distinction between
comparative experiments and experiments designed to es
timate the absolute treatment effects individually is per
haps not always clearly defined, nevertheless, the idea of
a comparative type of experiment remains convenient and
useful.
For comparative experiments, it is clear that an
advantage is to be gained by comparing the treatments under
homogeneous conditions. To achieve this end, much of the
effort in choosing the homogeneous conditions is directed
toward the selection and use of block designs. Over the
years, both complete and incomplete block designs have been
discussed in detail. In this work, we shall be concerned
mainly with combinations of these block designs for use in
comparative type experiments.
1


2
1.1 Blocking Designs
In blocking experiments where the objective is the
comparison of different treatment effects, the number of
experimental units in each block may or may not equal the
number of treatments to be compared. When the size of the
block, where size refers to the number of experimental units
in each block, is equal to the number of different treatments
and each treatment is randomly assigned once with every other
treatment in each block, the design is known as a randomized
complete block design. If the size of the block is less than
the number of treatments, an incomplete block design may be
used. Incomplete block designs are common in applications
where either the number of treatments is large or the size
of the block must be kept small in order to ensure homogene
ity of the experimental units in each block.
Still another type of block design exists when the
size of the block exceeds the number of different treatments.
In this latter design, if each block contains first repli
cates of all of the treatments plus duplicates or second
replicates of some of the treatments, the design is called an
"extended complete block design." We now discuss such
designs.
1.2 Extended Complete Block Designs
In an attempt to increase the precision of the com
parisons between the effects of each of the treatments and
the effect of a control treatment, Pearce (1960) introduced


3
blocking experiments where in each block the control treat
ment was replicated. Later, Pearce (1964) considered possi
ble methods for designing experiments in which for a given
experiment the blocks are of varying sizes. The analysis
of a fertilizer experiment on strawberries in which an ex
tended complete block design was used is mentioned briefly
by Pearce (1963).
Extended complete block designs, as introduced by
John (1963), are block designs in which each block contains
a first replicate of all of the treatments plus a duplicate
or second replicate of some of the treatments. These second
replicates in each block comprise an incomplete block se
lected from the class of balanced incomplete block designs.
An example of an extended complete block design formed by
augmenting complete blocks of size three with balanced in
complete blocks of size two resulting in extended complete
blocks of size five is presented in Figure 1, where the three
treatments are denoted by A, B, and C.
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
complete block design
balanced incomplete block design
Figure 1. An extended complete block design
consisting of three blocks each of size five
experimental units containing treatments A,
B, and C.
Extended complete block designs can be used in a
variety of experimental situations. In sensory experiments


4
where the objective is the comparison of preferences for dif
ferent food samples (treatments) expressed by a panel of
judges (blocks), the number of food samples that a panelist
may effectively evaluate at a single sitting is limited but
may be more than the number of different samples to be eval
uated. Acquiring panelists for these sensory experiments is
often difficult and/or costly. Hence, if a panelist can ef
fectively evaluate all of the different samples plus repli
cates of some of the samples at a single sitting and if a
fixed number of observed values of each sample is necessary,
a smaller number of panelists would be required with the use
of an extended complete block design than if each panelist
could evaluate each of the samples only once. The use of a
smaller number of panelists would result in a savings in
terms of time and cost.
In an agricultural setting, an experimenter wishing
to compare the effects of different chemical sprays on cit
rus trees may have available more trees in a block than the
number of sprays to be tested. On the additional trees in
each of the blocks, second replicates of some of the dif
ferent chemical sprays could be applied. In an industrial
experiment, on a given day an experimenter may be able to ob
tain observed responses from each of the treatments as well
as responses from second replicates of some of the treat
ments. If he does not have enough time to observe the re
sponses from second replicates of all the treatments, an
extended complete block design could be used.


5
The analysis of a block design in which two treat
ments are applied to the experimental units in blocks of
size three was discussed by John (1962). The following year,
John (1963) introduced extended complete block designs and
presented their analysis. In his designs, the block size k
could vary between t and 2t, where t is the number of dif
ferent treatments used in the experiment. Trail and Weeks
(1973) generalized this latter work of John to include de
signs in which k is greater than 2t. In the papers by John
(1963) and Trail and Weeks (1973), the analysis of the fixed
effects model as well as the mixed model was presented in
detail.
The application of the extended complete block de
signs of John to the area of sensory evaluation was con
sidered by Cornell and Knapp (1972, 1974). Also considered
by Cornell (1974) was the efficiency of these designs com
pared to randomized complete block designs.
1.3 Purpose of This Work
The first part of this work will be concentrated
on extending the works by Cornell and Knapp (1972) and
Cornell (1974) with special emphasis on the area of sensory
evaluation. Specifically, we shall be interested in the
analysis of extended complete block designs where correlation
is present between duplicate responses to the same treatment
in the same block and the magnitude of the correlation is
constant over all treatments and blocks. In sensory


6
experiments for example, the presence of correlated observa
tions easily could arise as a result of using highly skilled
judges ( as the blocks). Therefore, we shall be interested
in testing whether there is any evidence of correlation pre
sent in the data. Futhermore, if there is sufficient evi
dence to indicate that correlation is present, we shall seek
to obtain an estimate of the magnitude of the correlation,
which is denoted by p. An approximate test on the treatment
effects in the presence of a value of p greater than zero
will be suggested, since an exact test on treatment effects
cannot be performed for this experimental situation.
In the second part of this work, we shall general
ize the work of Trail and Weeks (1973) to include extended
complete block designs generated by partially balanced in
complete block (PBIB) designs with two associate classes.


CHAPTER 2
LITERATURE REVIEW
The analysis of block designs in which the block
size k could vary between t and 2t, where t denotes the num
ber of treatments in the experiment, was first introduced by
John (1963). These designs, called extended complete block
(ECB) designs, contain in each of the b blocks first repli
cates of all of the treatments plus second replicates of k-t
of the treatments. The method taken by John of choosing the
k-t second replicates in each block was to use the class of
balanced incomplete block (BIB) designs of block size k-t.
Using formulae similar in structure to the formulae
used in the analysis of balanced incomplete block designs,
John discussed the intrablock analysis, the interblock anal
ysis, and the recovery of the interblock information. The
recovery of the interblock information was achieved by com
bining the two independent intrablock and interblock es
timates of the effects of each of the treatments.
In the intrablock analysis, in addition to ob
taining the treatment effects adjusted for blocks and the
unadjusted block analysis, John obtained estimates of both
the experimental error variation and the block x treatment
interaction. The measure of the interaction was obtained by
subtracting the experimental error variation from the
7


8
residual variation in the intrablock analysis of variance.
With the interblock analysis, however, an additive model was
assumed. That is, the block x treatment interaction variance
component was assumed to be zero. Using the assumption of
the additive model then, the combined estimate of each of
the treatment effects was formed using a linear combination
of the weighted intrablock and interblock estimates. The
weights used with the intrablock and interblock estimates
were the reciprocals of the estimates of their respective
variances. The special case where t = b and k = t+1 was
presented in detail.
Trail and Weeks (1973) considered the aforemen
tioned extended complete block designs (ECBD) as a special
case of the more general class of designs which they called
extended complete block designs generated by balanced in
complete block designs (BIBD). Their generalization of the
work of John (1963) included balanced block designs in which
the block size could exceed 2t. An example of this more
general design in which a balanced incomplete block design
is added to a double complete block design (CBD) is pre
sented in Figure 2. A second example of these more general
designs in which the complete block design is augmented by
two balanced incomplete block designs is presented in Fig
ure 3. In both of these figures, the three treatments are
denoted by the letters A, B, and C. It should be noted that
the treatments would be randomly assigned to the experimental
units within each block when the experiment is performed.


9
A
A
A
B
B
B
C
C
C
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
CBD
CBD
BIBD
Figure 2. An ECBD for three treatments generated
by a BIBD consisting of three blocks each of size
eight experimental units.
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
A
B
A
C
C
B
CBD
BIBD
BIBD
Figure 3. An ECBD for three treatments generated
by a BIBD consisting of three blocks each of size
seven experimental units.
In a block design in which t treatments are ar
ranged in b blocks, properties of the design can be obtained
by studying the elements of the incidence matrix of the de
sign. The incidence matrix N = (n^j) is a t x b matrix such
th
that n^j denotes the number of times the i treatment ap-
th
pears in the j block. The elements of the incidence matrix
for the extended complete block designs may be constructed
by appropriately summing the elements of the incidence matrix
of a balanced incomplete block design and the elements of
the incidence matrix of a complete block design.
For the extended complete block designs generated
by BIB designs, Trail and Weeks showed that the incidence


10
matrix N can be generated from the incidence matrix N* of
any balanced incomplete block design by using the equation
N = cQJ + (C;l-c0)N* (2.1)
where J is the incidence matrix of a complete block design,
that is, J is a t x b matrix of ones, and c^ and c^ are ele
ments of the set of positive integers. The model used by
the authors is
y = C^x + X23 + Y) +
(2.2)
where x is a t x i vector of treatment effects, 3 is a b x l
vector of block effects, y is a bt x i vector of interaction
effects, and e is a bk x l vector of independent random er
ror effects. Letting 1 denote a t x l vector of ones, I
i.U
denote the t x t identity matrix, and y. represent the &1"
1 X/
-H Vi
response to the i1" treatment in the j11 block, the vector y
and the matrices C, X^, and X2 in (2.2) are of the forms
y =
111
lln
11
211
tbn
tb
?i -
it
I
, x =
' it
1^
~t
' ~2
~t



- it
btxt
1



rt*
i
bkxl
btxb
and


11
:n
11
:n
21
C
. (2.3)
bkxbt
In our notation, 1 is an n~. x 1 vector of ones.
~n21 21
In addition to presenting the intrablock and
interblock analyses, Trail and Weeks expressed the formula
for calculating the combined estimates of the treatment ef
fects using the method presented by Seshadri (1963a) for
combining unbiased estimators. Trail and Weeks also dis
cussed how "best" designs might be obtained. They defined
the "best" design as that design for which the variance of
the difference between the intrablock estimates of the ef
fects of any two different treatments is a minimum for fixed
t and k. The minimum value of the variance of the difference
is achieved by minimizing the absolute value of the differ
ence Cq-c^, where Cq and c-^ are the magnitudes of the ele
ments in the incidence matrix N of the design.
An application of extended complete block designs
to sensory testing experiments was presented by Cornell and
Knapp (1972). Separate estimates of block x treatment


12
interaction and experimental error were obtained in their
analyses. Cornell and Knapp showed that the use of the
experimental error only as a measure of the within treatment
variability when comparing treatments results in a more ef
ficient test than when using the residual variation (the sum
of the experimental error and the interaction variation)
when some measure, however small, of interaction is present.
Replication of extended complete block designs was
also discussed by Cornell and Knapp (1974). Replication of
the designs was performed to achieve a balance between the
blocks and the treatments. By balance is meant, each and
every treatment appears in each block (is evaluated by each
panelist) the same number of times over the replications.
Hence, pairwise comparisons of the treatments in each block
can be made with equal precision.
With the replicated designs, the assumption of
negligible replication variation was made by the authors.
This assumption resulted in simpler expressions for the for
mulae for calculating estimates of the treatment effects as
well as the intrablock sums of squares when compared to the
formulae used with the unreplicated extended complete block
designs. An example of a replicated extended complete block
design is presented in Figure 4.
Using a non-additive model, Cornell (1974) dis
cussed the efficiency of extended complete block designs
compared to complete block designs for uncorrelated observa
tions. The efficiency of each design was defined as the


13
reciprocal of the variance of the difference between any
pair of treatment means with the respective design. To
illustrate, with the extended complete block design con
sisting of b blocks each of size k, the estimate of the
variance of the difference between any two treatment means
1 ? is
A
Var(x.-T.,
i i
ECB
2k(t-1) Q 2
b (k2-3k+2t) e
(2.4)
where cr* is the intrablock estimate of experimental error.
An estimate of the efficiency of the extended complete block
design would be the reciprocal of (2.4). (The author used k
for the size of the blocks in the balanced incomplete block
design used in the extension. In this work, k* denotes the
size of the blocks in the balanced incomplete block design
used in the extension, while k refers to the size of the
blocks in the extended complete block design.)
Replications
I
II
III
1
ABCAC
ABCBC
ABCAB
2
ABCBC
ABCAB
ABCAC
3
ABCAB
ABCAC
ABCBC
Figure 4. A replicated extended complete block
design consisting of three replicates of an ECBD.
In a complete block design with the same number of
replicates of each of the treatments, that is, with bk/t
complete blocks of size t, the estimate of the variance of
the difference between any two different treatment means is


14
Var(xi t,)cb bk jesidual (2.5)
where Residual ;''s t^ie residual mean square. From (2.4) and
(2.5) an estimate of the efficiency of the extended complete
block design compared to the complete block design is ob
tained using the ratio
Var(r-T,)rn
Ef f (ECB to CB) = 1 z
Var(xi-Ti,)ECB
t(k2-3k+2t) a2 ^
v residual x
l b J
k2(t-1) a*
To obtain estimated efficiency values for dif
ferent values of t and k, the value of the ratio of
^2 o .
aresidual to ae 1S re(2uired- For the value of this ratio,
Cornell used the ratio of the mean square for interaction to
the mean square for error, which is easily obtainable from
the analysis of variance table of the extended complete
block design. With this ratio, denoted by F, Cornell showed
that when the hypothesis of zero interaction effect is true,
resulting in F = 1, the extended complete block design is a
slightly less efficient design than the complete block de
sign with the same number of replicates of the treatments.
However, when F is greater than 1, the extended complete
block design is the more efficient design with the efficiency
increasing with increasing values of F.
Cornell (1974) also considered the situation where
a positive correlation p exists between the two responses to


15
a treatment in the same block. For an extended complete
block design having fixed balanced incomplete block size k*,
it was found that as P approaches one the efficiency of the
extended complete block design compared to the complete block
design decreases. In fact, the larger the value of k*
(k* + t) the faster the efficiency of the extended complete
block design approaches one-half that of the complete block
design with twice as many blocks. This implies that if one
suspects a positive correlation to be present between du
plicate treatment responses in the same block, one should
use k* equal to one for maximum efficiency if using an ex
tended complete block design.
Owing to the results previously found concerning
the effect of correlated observations on the efficiency of
extended complete block designs compared to complete block
designs, in the next chapter we shall investigate the for
mulation of the test of the hypothesis of zero correlation.
If there is evidence of correlation present between du
plicate treatment responses in the same block, we shall want
to estimate the correlation p. With an estimate of p, an
estimate of the variance of the difference between any two
intrablock estimates of the treatment effects can be cal
culated.


CHAPTER 3
EXTENDED COMPLETE BLOCK DESIGNS
WITH CORRELATED OBSERVATIONS
In the extended complete block designs discussed
to this point, we have observed that some of the treatments
in each block are duplicated. In the papers by John (1963),
Trail and Weeks (1973) Cornell and Knapp (1972) and Cornell
(1974), the responses to the duplicated treatments in each
block are assumed to be independent and are used to obtain
an estimate of the experimental error. Comments on the ef
ficiency of these designs when the duplicated observations
are not independent but rather are positively correlated
were made in the latter paper.
As mentioned previously in Section 1.3, in sensory
experiments correlated observations are a real possibility.
A panelist's response to a treatment might very likely be
positively correlated with his response to the duplicate of
the treatment, particularly if the panelist has previously
been trained for these experiments. The presence of posi
tive correlation between responses to the same treatment by
a panelist reflects a measure of the efficiency of the pan
elist. That is, the closer in magnitude the responses to
the same treatment by a panelist are, the more consistent
the panelist is in evaluating that treatment. Although the
correlation could be different for each treatment and/or
16


17
each panelist, we shall consider only the case where the
correlation is.assumed to be constant and equal for all pan
elists (blocks) and treatments.
3.1 Notation and Definitions
The parameters associated with an extended complete
block design are as follows:
t = the number of treatments,
b = the number of blocks,
k = the number of experimental units in each block
(block size),
r = the number of replications of each treatment in the
experiment,
X = the number of distinct pairs of experimental units
which receive any fixed pair of treatments while
appearing in the same block, and
N = (n..) = the incidence matrix, where n.. denotes the
ID ID
number of times the i^ treatment appears in the
j*"*1 block.
The following parameters are associated with the balanced
incomplete block design used to form the extended complete
block design:
t = the number of treatments,
b = the number of blocks,
k* = the block size,
r* = the number of replications of each treatment,


18
X* = the number of times over the b blocks each pair of
treatments appears in the same block, and
N* = (n* ) = the incidence matrix,
ij
The following identities involving the aforemen
tioned parameters are satisfied:
1. r = r*+b
2. k = k*+t
3. r*t = bk*
4. rt = bk
5. N = N*+J, where J is a t x b matrix of I's
6. X = 2r-b+X*
7. X*(t-1) = r*(k*-l)
8. X(t-1) = rk-3r+2b .
The model written in matrix notation is
y = C(yl + X.t + X 3) + e (3.1.1)
~ ~ ~Dt ~~
where all symbols are defined following (2.2) with the ex
ception of y and e. These parameters are y, the overall
mean, and e, a bk x 1 vector of random errors with the prop
erties
E 0
where E() denotes mathematical expectation and


19
E(eijieij
'pa2 i = i', j = j l jt l'
a2 i = i1 j = j 1 =5/' ,
0 otherwise
(3.1.2)
where a2 denotes the variance of the distribution from which
the errors are sampled and p denotes the correlation between
the duplicate observations. We shall assume that the values
of p lie in the interval 0 < p < 1.
Owing to the properties in (3.1.2) of the random
errors, then
E(y) = C(ylbt + X^r + X23)
and
(3.1.3)
Var(y) = E(ee') = V .
(3.1.4)
The matrix V consists of the following partitions corre
sponding to the form of the vector y in (2.3); on the main
diagonal of V are positioned the matrices
and a2 [ 1 ] ,
while there are zeros located in all other positions. Hence,
under the assumption of the normality of the random errors,
y ~ N( C(ylbt + X1t + X2B), V ) .
Before discussing the intrablock estimation of the
treatment effects, we illustrate the form of the matrix V by


20
referring to the extended complete block design presented in
Figure 1. If the vector of observations y is written as
Y =
All
A12
A21
A22
A31
Bll
B12
B21
B31
B32
Cll
C21
C22
C31
L y
C32 J
then the corresponding matrix V is
V =
1
P
1
P
1
P
1
P
1
P
1
P


21
On the main diagonal of the matrix V are positioned the
matrices
which correspond to the duplicate responses to a treatment
in the same block, and the matrices a2[ 1 ], which corre
spond to the response to only a single treatment in the
block.
We shall now discuss the intrablock estimation of
the treatment effects where both the treatments and the pan
elists are assumed to represent fixed effects in the model
in (3.1.1). The panelist effects represent fixed effects
either when it is desired to compare the specific panelists
used in the experiment or when the panelists chosen to eval
uate the treatments cannot realistically be assumed to rep
resent the general public. A case which comes to mind in
this latter situation is when trained panelists are used in
an attempt to enhance the efficiency of the comparisons be
tween the treatments.
3.2 Intrablock Estimation of
the Treatment Effects
To obtain the intrablock estimates of the effects
of the different treatments, we recall the form (3.1.1) of
the model
y = C(ylbt + X t + x23) + e ,


22
where the elements of the random error vector e have the
properties specified in (3.1.2). If the method of least
squares is used to obtain the intrablock estimates t of x,
the normal equations are
IbtSi
ibt?ibt
IbtHi
it?;
XjC'y
=
?12ibt
?i5?i
51552
-
- *2~bt
?255i
;j5;2
(3.2.1)
where the bt x bt matrix D = C'C and the hat (~) denotes
estimate. According to the definitions and parameter iden
tities specified in Section 3.1, the forms of the matrices
~1~~1' XiDX2' and X2DX2 n (3-2*1) are
~1~X1 = rit ?1??2 = i? and X'DX2 = klfa ,
and therefore the normal equations (3.2.1) are expressed
as
G
-
T
=
_ B
bk
ri;
ki
rl
rl
N
~t
~t
~
kl.
N'
kl.
~b
~b
y
A
T
A
L e J
(3.2.2)
where G denotes the grand total of the observations and T
and B are the t x 1 vector of treatment totals and the b x 1
vector of block totals, respectively.
For a solution to the normal equations, both sides
of the equality in (3.2.2) are premultiplied by the matrix


23
1/bk O O
O Ifc -N/k
O -N'/r I,_
~ ~b -J
A A
and the constraints 1' r = 0 and 1,' 3 = 0 are imposed on the
~t~ ~b~
parameter estimates. Corresponding to these particular con
straints imposed, the following relation results
G/bk
r /s _
y
kQ
=
A
Ax
rB N'T
(rkl N1N)3
~ JD ~ ~ ~
where kQ = kT NB and A = rkl^ NN'. Characteristic of
these designs, the matrix NN' = (rk-At)I + AJ. Hence, the
matrix A can be expressed in the simple form
(l/k)A = (At/k)[lt (l/t)j] .
From the equation kQ = At in (3.2.3), the t x 1 vector x of
intrablock estimates of the treatment effects is
x = kQ/At ,
(3.2.4)
where A = (rk-3r+2b)/(t-1). Furthermore, with the properties
J_ l- /N
of the vector e specified by (3.1.2), the i element x_^ of
the vector x is unbiased for x. since E(e)=0 and with l'x=0
1 -V, -V ~
Cov (x\ x ,)
(t-1)(Ak+2[A(k+t)-2rk]p)o2/(tA)2 i = i'
-Var(xi)/(t-l)
f i ^ i'
(3.2.5)


Since we are interested in the pairwise comparison of the
treatment effects, we also have
(3.2.6)
and
, 4{X (k+t)-2rk} 7
+ po-
tx2
(3.2.7)
In the formula (3.2.7), the quantity 2kcr2/tX on
the right-hand side of the equality is the variance of the
difference between the intrablock estimates of the treatment
effects t. and x., in the case of uncorrelated errors. Thus,
if correlation is present between responses to the same
treatment in the same block, the variance of the difference
between the intrablock estimates of any two treatments, over
all blocks, is greater than the variance between the same
two treatments when the observations are uncorrelated, since
the quantity [X (k+t) 2rk] is always positive.
The intrablock analysis of variance table is pre
sented in Table 1. It is clear from Table 1 that an exact
test does not exist for testing the hypothesis of equal
treatment effects when a non-zero correlation is present.
If we wish to test this hypothesis, an approximate test must
be performed. Before suggesting an approximate test for the
equality of the treatment effects when p > 0, we shall first
consider a procedure for testing for the presence of correla
tion. If correlation is present, we shall need to know how
this correlation affects the distributional properties of


TABLE 1
Intrablock Analysis of Variance for an ECBD
Source
df
Sum of Squares
EMS*
Treatments
(adjusted)
t-1
SST = (k/Xt) l Q?
A i 1
E(MST )
A
Blocks (unadjusted)
b-1
SSB = (1/k) l B2 -
j 3
(G2/bk)
Residual
bk'
-t-b+1
(by subtraction)
E(MSR )
e
Total
bk-1
TSS = l l l y? (G2/bk)
i j £ ^
*
E (MST ) = a2 {1
A
+ iP
Xk
[(k+t> _2rkR + k(t-i) J
CM -H
E(MSR ) = a2 +
1
, { (b-1) (t-1) d> bJ*'
-*> )pa2
= Xk'-fb-ry {X ~ 2(k+t) bk] + 4rk}


26
the sums of squares associated with the two sources, treat
ments and residual.
3.3 A Test for the Presence of Correlation
Although one of the initial steps in the analysis
of data arising from a comparative type experiment is a test
on the equality of the treatment effects, in this section we
shall first investigate the possible presence of correlation
between duplicate observations in the same block. The rea
son for this investigation is that if correlation is present,
an exact test of the hypothesis of equal treatment effects
cannot be performed and an approximate test must be derived.
Furthermore, if correlation is present, the formula for the
variance of the intrablock estimates of the treatment ef
fects contains p and an estimate of p is needed to estimate
this variance. The same is true of the formula for the dif
ference between the intrablock estimates of the effects of
two treatments.
To determine if there is evidence of correlation
in the data, we shall consider a test of the hypothesis
Hq: p = 0. If this hypothesis is rejected in favor of the
alternative hypothesis H : p > 0, we shall conclude that the
duplicate observations are not uncorrelated and insist on
finding an estimate of p. If, on the other hand, the hy
pothesis is not rejected, the inference made here shall be
that the duplicate observations are uncorrelated, or, if
they are correlated, there is not sufficient information in


27
the data to show that the magnitude of the correlation is
greater than zero.
In order to test the hypothesis Hq: p = 0, we
first need to derive the form of a test statistic. To this
end, recall from Table 1 that the source of variation termed
residual has bk-b-t+1 degrees of freedom. The residual var
iation is a composite of duplication variation as well as
another source of variation which we shall call remainder
variation. To see this, let d^j be the difference or range
th th
of the observations made on the i treatment in the j
block so that if n^j = 2, then d^j > 0, and if n^j = 1, then
dj-j = 0. If each of the d^j is squared and these squares
are summed over all treatments and blocks, then the resulting
quantity when multiplied by one-half is called the sum of
squares for duplication variation (SSDV). In summation no
tation, the sum of squares for duplication variation is
given by
t b
SSDV = h l l d[. .
i j
The sum of squares for remainder (SSR) is found by cal
culating the difference, sum of squares for residual SSDV.
To derive the form of a test statistic for testing
the hypothesis, we require the separate distributional prop
erties of the sum of squares for duplication variation and
the sum of squares for remainder. The distributional prop
erties of SSDV and SSR are most easily obtained by rewriting
SSDV and SSR as quadratic forms and then using our knowledge


28
of the distributional properties of quadratic forms. In ma
trix notation then, SSDV and SSR can be expressed in the
quadratic forms
SSDV = y' [i. -CD_1C'] y = y'A,y (3.3.1)
~ L~bk ~ J ~ ~ ~1~
and
SSR I' SC5'1 e 525 it (bt E
(bt e5;22>Js' X '
(3.3.2)
where both the matrices and A^ are real, symmetric, and
idempotent.
In the quadratic form (3.3.1) for SSDV, the matrix
A^ consists of the square matrices
and [ 0 ]
on the main diagonal and zeros in all other positions. This
partitioning corresponds identically to the partitioning of
the matrix V as defined and illustrated in Section 3.1.
Thus, by direct computation
A V = (l-p)a2A. (3.3.3)
~ -L~ ~
and since A^ is a real, symmetric, idempotent matrix, so
also is the matrix A V/{ (1-p) cr2 } The trace of the matrix
~ 1 ~
A^ is equal to b (k-t) and therefore under the assumption of
normality of the errors,


29
SSDV ~ (l-p)a2 X(k_t) / (3.3.4)
where x* denotes a random variable with a central chi square
distribution with v degrees of freedom. With E(*) denoting
mathematical expectation, then
E(SSDV) = b(k-t)(1-p)o2 (3.3.5)
and
E(MSDV) = (1-p)a2 (3.3.6)
where MSDV denotes the mean square for duplication variation.
(An alternate derivation of the distribution of SSDV is pre
sented in Section 3.4.)
In the quadratic form (3.3.2) for SSR, the matrix
A^ is real, symmetric, and idempotent. However, it can be
shown that if c is a scalar constant, the equality
AVA
~2~~2
is not true in general. (To see this would only require
working through the small example where t = b = 3 and
k = r = 4.) Hence, unlike SSDV, the random variable SSR
does not have an exact weighted chi square distribution when
p > 0. The exact distribution of SSR is discussed in Sec
tion 3.4.
The distributions of SSDV and SSR are independent,
since A VC = 0. The trace of the matrix A V is equal to
~ 1 ~ ~ Z ~
(b-1) (t-1) (l+ square for remainder is


30
E(MSR) = (l+p) cr2 (3.3.7)
where 4> is defined in Table 1.
When the hypothesis HQ: p = 0 is true, the random
variable SSR has a chi square distribution. Hence, to test
the hypothesis Hq: p = 0, the test statistic F^ = MSR/MSDV
is used. When p = 0, the test statistic has an F distri
bution with (b-1) (t-1) and b (k-t) degrees of freedom in the
numerator and denominator, respectively. Therefore, the hy
pothesis is rejected in favor of the alternative hypothesis
H : p > 0 for large values of F .
a p
A brief discussion of the power of this test is
reserved for a later section, since we must first consider
the distributional properties of SSR when p > 0.
3.4 The Exact Distribution of SSR for p > 0
In the previous section, it was shown that when
p > 0, the distribution of the sum of squares for remainder
does not in general have a weighted chi square distribution.
This is because the matrix A2 of the quadratic form SSR does
not necessarily satisfy the equality A2VA2 = CA2, w^ere c
some constant and V is the covariance matrix of the observa
tions. In this section we shall seek to find an expression
involving independent chi square distributed random vari
ables for which the moments of the distribution of SSR can
be found.
The approach we shall use to find the distribution
of SSR involves rewriting the model in (3.1.1) in the form


31
yij = P + T + gj + (l-p)?Szijjl + p^Ujlj (3.4.1)
i 1 / 2 f t / ^ 1 / 2/ Id ^ dnd = 1 / 2 / j ,
where the x^ are the treatment effects, the Bj are the block
effects, p is the magnitude of the correlation between du
plicate responses to the same treatment in the same block,
and zj_j£ and uij are independent, identically distributed
normal random variables each with mean zero and variance a2.
Let us now define the random variable SSR|u^j to
be the usual sum of squares for interaction (which we have
chosen to call remainder in our additive model) given the
u- Since the conditional distribution of SSR given u-
1J J
can be found, the form of the unconditional distribution of
SSR is obtained by taking the expectation of the random
variable SSR|u^j with respect to u^j.
The distribution of the random variable SSR|u^j is
given by
SSR | u^ j ~ a2 (1 p) X2(b_i) (t-1) ( 2 (-p) r2) (3.4.2)
where x2 (1) denotes a random variable with a non-central
chi square distribution with v degrees of freedom and non
centrality parameter X. In the noncentrality parameter of
the distribution in (3.4.2), R2 is of the form
>2 _
= min l Z x* B*)2 ,
(3.4.3)
T*, B* i j
where x* and B are the parameters in the conditional distri
bution corresponding to y+x^ and y+Bj, respectively, in the
unconditional distribution. In order that the distribution


32
in (3.4.2) be expressed in a simpler form, it is convenient
to write R2 in terms of the design parameters t, b, k, and
r. To this end, let D~^ denote the diagonal matrix of cell
frequencies associated with a t x b table in a two-way cross
classification, that is,
n
11
n
12
n
lb
n
21
Then (3.4.3) may be written in the form
R2 = min (u-y)'D*1(u-y) ,
y :F y = 0
where
(3.4.4)
y' (yxl, ..., ylb, v21, ytb)'
u' (ui;l, ..., ulb, U21, utb) '
and F is a matrix of constraints for additivity in a t x b
cross classification. The form of the matrix F is given
shortly.
The quantity R2 equals the minimum value of the


33
quadratic form in (3.4.4). To find this minimum value, we
write
Q = (u-y)'D*1(u-y) + 2H'F'y
where II is a (b-1) (t-1) x l vector of Lagrange multipliers.
Differentiating Q with respect to y and setting the result
equal to zero, the minimum value in (3.4.4) is
R2
u'F(F'D*F)
u .
(3.4.5)
An expression for (3.4.5) involving the design
parameters t, b, k, and r for our problem requires the la
tent roots of the matrix F(F'D*F) ^F'. If these roots are
denoted by 0, then 0 are the solutions to the equation
I ?(r?*!,)~V ei(b-l)(t-l) I = 0 (3.4.6)
Using the following identity,
6(b-1) (t-1)
F' F'D*F
Qb+t-lj 0F.D*F p.F|
F'D*F
01
(b-1) (t-1)
- F(F'D
tF) 1fi
we find that b+t-1 roots of (3.4.6) are zero while the re
maining (b-1)(t-1) roots are positive. These latter
(b-1)(t-1) positive roots can be found by solving for 0' in
the equation
F'D*F
0 1 F F | = 0
(3.4.7)


34
and setting 9 = 1/9'. Since the 9' are non-zero, the frac
tion 1/9' is not undefined.
We should like to express equation (3.4.7) in a
simpler form to find the values of 9'. To this end, the ma
trix of constraints F is written as the direct product of
two other matrices. This direct product is
F = F(t-l) F(b-l) ,
where the two matrices are defined by
and
F(t-l)
14-i
-I
t-1
tx(t-1)
F(b-l) =
l-i
-I
b-1 J
bx(b-1)
Then
F'F
1
-b-1
f
where
L = I + J (3.4.8)
~a ~a
That is, L is an a x a matrix with 2's on the main diagonal
~a
and l's in all other positions. We now make use of the fol
lowing theorem and corollary to find the values of 9' satis
fying (3.4.7) .
Theorem. Let W be an m x m matrix with the distinct latent


35
roots w_. with respective multiplicities m., j = 1, 2. Let
R be an m x s matrix satisfying R'R = I Then the roots of
~ ~ ~ ~s
the matrix R'WR are the values of 0' satisfying the equation
R'WR 6'I | = 0. These values are
9' = w10" + w2(l-0") ,
where 0" are the solutions of | R'MR 0"I | =0 with
V v M = (W w2)
Proof; The proof follows directly by replacing M with
(W-w^Im)/(w^-w2) in the determinant | R'MR 0"I | = 0 and
simplifying.
Corollary. If W is defined as in the theorem, and F is an
m x s matrix of full rank s < m, then the solutions 0' of
the equation | F'WF 0'F'F | = 0 are
0 = w-^0 + w2 (1-0" ) ,
where 0" are the solutions of | F'MF 0"F'F | =0 with M
defined in the theorem.
Proof: There exists a matrix K such that K'F'FK = Ig. Let
r' = k'F' and apply the theorem.
In our problem, W is the matrix D*. Therefore
from the theorem, M is a diagonal matrix of ones and zeros.
Referring to the corollary to obtain the values of 0' satis
fying (3.4.7), we now need only to find the solutions 0" of


36
| F'MF 6"F'F |=0. (3.4.9)
Because the structure of the matrix F'F depends on the
matrices L^._^ and Lb_^ as defined in (3.4.8), a forward
Doolittle procedure is performed on La to find that the val
ues of 0" satisfying (3.4.9) are the same as the values of
0" satisfying
I e"i(b-l) (t-l) I = 0 > (3.4.10)
where F* = H(t) 0 H(b) with the matrix H(a) defined as the
first a-1 columns of the a x a Helmertz orthogonal matrix.
Since M = M* and M'M = M, then F;MF* = (MF*)'MF*, and the
positive values of 0" satisfying (3.4.10) are the positive
solutions 0" satisfying
| MF*F;M 0"Ibt |=0. (3.4.11)
Since we may write
f*f; = (Gt 0 Gb)/bt ,
where Ga is an a x a matrix with a-1 on the main diagonal
and -1 in the other positions, then the positive solutions
0" of (3.4.11) are functions of the positive solutions 0*
satisfying the equation
| M(Gt 0 Gb)M 0*Ibt |=0, (3.4.12)
where 0* = bt0".
At this stage, an expression for the random vari
able SSR is presently untenable for general t, b, k, and r.


37
In the remainder of this section, we shall derive an expres
sion for the random variable SSR when p > 0 for the special
case t = b and r = k = t+1. A similar expression for SSR
when p > 0 for the case b = 2t, k = t+1, and r = 2 (t+1) is
presented in Appendix 1.
For the special case considered in this section,
the matrix M(G 0 G, )M in (3.4.12) has the non-zero partition
~ ~t ~b ~
{(t-1)+ J /
for which the positive latent roots are t(t-2) and t(t-l)
with multiplicities t-1 and 1, respectively. Hence, since
these roots are simple multiples of the solutions 0" of
(3.4.9), the 0" are
0"
(t-2)/t with multiplicity t-1
(t-l)/t 1
0 (t-l)2-t
With the use of the corollary where w^ = % and w^ = 1/ the
values of 0' satisfying (3.4.7) are
0'
(t+2)/2t with multiplicity t-1
(t+1)/2t 1
1 (t-l)2-t
and the values of 0 satisfying (3.4.6) are
0 =
2t/(t+2) with multiplicity
2t/(t+1)
1 11 11
t-1
1
(t-l)2-t


38
In the distribution of the conditional random
variable SSR given u^ in (3.4.2), we may now express R2 as
R2 =
2t X2
2t 2
t+2 At-1 t+1
xi + x(t-D2-t
Furthermore, upon taking the expectation of SSR|u^_. with re
spect to u,, and using the notation SSR = E(SSR|u,.), the
lj 1 lj
sum of squares for remainder when p > 0 is distributed as
SSR/a:
al Xvx + a2 xv2 + a3 Xv3 '
(3.4.13)
where
and
al = 1 + p TT
a2 1 + P t+1 '
a3 1 ,
V1 = t-1 ,
v2 = 1 ,
v3 = (t-1)2 t
An approximate distribution to (3.4.13) will be obtained in
Section 3.5.
The conditional distribution of the usual sum of
squares for error given u^ is
SSE|uj ~ a2(1-p) X(k_t)(0) ,
and the random variable SSE|u. is independent of u_^
Hence, we have


39
SSDV ~ a2(1-p) x(k-t)(0) '
where SSDV denotes the expectation of the random variable
SSE|u^j with respect to j. Since the duplication var
iation sum of squares random variable is also independent of
the random variable SSR|u^j, then the random variable SSDV
is independent of the random variable SSR. By independent
random variables is meant, the distributions of the random
variables are independent. (The independence of the distri
butions of SSDV and SSR was established previously in Sec
tion 3.3 through the use of quadratic forms.)
To this point, it has been shown that for the
special case t = b and r = k = t+1, the random variable SSR
is distributed as a sum of weighted independent chi square
random variables when p > 0. We still do not have the exact
form of the density of the random variable SSR which is nec
essary in order to specify the distribution of the random
variable SSR/SSDV. The distribution of SSR/SSDV is also
necessary in order that we may calculate the power of the
test of the hypothesis HQ: p = 0 for non-zero values of p.
Since an exact form of the density of SSR would likely re
quire an excessive amount of work and since a simpler form
of an approximating distribution of SSR would suffice for
our problem in a majority of cases, an approximate distri
bution of the random variable SSR will now be considered.
A check on the accuracy of the approximate distribution when
compared to the exact distribution of SSR when p > 0 is pre
sented in Table A2 of Appendix 2.


40
3.5 An Approximate Distribution of SSR for p > 0
In this section we shall consider an approximation
of the distribution of the sum of squares for remainder when
p > 0 for the special case t = b and r = k = t+1. An ap
proximate distribution of SSR when p > 0 for the special
case b = 2t, k = t+1, and r = 2(t+1) is given in Appendix 2.
There are numerous approaches that could be used
to approximate the distribution of SSR. The approach used
in this section (and also used in Appendix 2) was introduced
by Box (1954). The rationale in selecting Box's ap
proximation lies not only in its relative ease of appli
cation but also in the fact that it was shown by Box that
the approximate distribution compared to the exact distri
bution of a quadratic form is fairly good except when small
differences in probability are to be examined. We now state
the theorem in his paper which we shall use.
Theorem. The quadratic form
is distributed approximately as where
g
(3.5.1)
and
h
(3.5.2)
In both of the expressions for the scale constant g and the


41
degrees of freedom h, the are scalars and the are the
degrees of freedom of the respective chi square random vari
ables that are summed to form Q, j = 1, 2, ..., p.
In our problem we seek to approximate the distri
bution of the random variable SSR, where
SSR/a2 ~ ax + a2 xv + a3 X*
v.
with a^ and vj, j = 1, 2, and 3, defined following (3.4.13).
From the theorem by Box previously stated then, if the aj
and Vj are substituted into (3.5.1) and (3.5.2) to find g
and h, respectively, we may say that SSR is approximately
distributed as a scaled chi square random variable with h
degrees of freedom. A tabulation of the values of g and h
corresponding to the integer values 3, 4, 5, 6, and 7 of t
and to some values of p in the interval between zero and one
is presented in Table 2, where h has been rounded to the
nearest integer and g has been rounded to four decimal
places.
The approximate distribution of SSR may be used,
when testing the null hypothesis Hq: p = 0 against the gen
eral alternative hypothesis H : p > 0, to compute the power
Cl
of the test under the alternative hypothesis for values of p
greater than zero but less than one. Since the distribution
of SSR is independent of the distribution of SSDV, then
under the alternative hypothesis we approximate the distri
bution of the statistic


TABLE 2
Values of g and h for the Approximate Distribution of SSR, I
p
t
V,
a.
a
g
h
1
2
3
1
2
3
0.1
3
2
1
1
1.025
1.05
1
1.0253
4
0.3
1.075
1.15
1.0776
4
0.5
1.125
1.25
1.1319
4
0.7
1.175
1.35
1.1880
4
0.9
1.225
1.45
1.2457
4
0.1
4
3
1
5
1.04
1.06
1.0205
9
0.3
1.12
1.18
1.0645
9
0.5
1.2
1.3
1.1121
9
0.7
1.28
1.42
1.1629
9
0.9
1.36
1.54
1.2166
9
0.1
5
4
1
11
1.05
1.0667
1.0173
16
0.3
1.15
1.2
1.0554
16
0.5
1.25
1.3333
1.0978
16
0.7
1.35
1.4667
1.1441
16
0.9
1.45
1.6
1.1940
15
0.1
6
5
1
19
1.0571
1.0714
1.0149
25
0.3
1.1714
1.2143
1.0485
25
0.5
1.2857
1.3571
1.0867
25
0.7
1.4
1.5
1.1291
24
0.9
1.5143
1.6429
1.1754
24
0.1
7
6
1
29
1.0625
1.075
1.0131
36
0.3
1.1875
1.225
1.0431
36
0.5
1.3125
1.375
1.0778
35
0.7
1.4375
1.525
1.1168
35
0.9
1.5625
1.675
1.1599
35


43
MSR
Fp MSDV
with a weighted F distribution with h and b(k-t) degrees of
freedom, where the weight is given by g/(l-p). That is,
1-P
g
' b (k-t)
(3.5.3)
approximately. The probability that Fp exceeds some value
Fq is approximately equal to the probability that the random
variable F^^-t) exceec^s (l-p)Fo/g*
A method for estimating the magnitude of the
correlation between duplicate responses observed with the
same treatment in the same block will be discussed in the
following section.
3.6 An Estimate of the Correlation
In Section 3.3 a procedure for testing the hy
pothesis Hq: p = 0 was outlined in detail. As mentioned at
the beginning of Section 3.3, the test of the hypothesis of
zero correlation is normally the first action to be taken
during the analysis of the experimental data. If the hy
pothesis is rejected, we should then want an estimate of p.
The estimate of p would be used when estimating the variance
th
of the intrablock estimate of the effect of the i treat
ment as shown in (3.2.5) and/or when estimating the variance
of the difference between the intrablock estimates of the
effects of two treatments as shown in (3.2.7). Still anoth
er use for the estimate of p would be when estimating the


44
efficiency of the extended complete block design compared to
the complete block design as shown in the paper by Cornell
(1974). The value of the relative efficiency of the two de
signs could be very useful when considering designs for sub
sequent experimentation, particularly in a sensory exper
iment where the same panelists are to be used in additional
experiments.
Referring to the formulae (3.3.6) and (3.3.7), we
see that the expectations of the mean squares for duplica
tion variation and for remainder variation are
E(MSDV) = (l-p)CT2 (3.6.1)
and
E(MSR) = (l+4>p) CT2 (3.6.2)
where is defined in Table 1. If a linear combination of
these mean squares and a ratio of two linear combinations of
them are considered, we can express the correlation p in the
form
p = {E (MSR) E (MSDV) }/{E (MSR) + c¡>E (MSDV) } (3.6.3)
As an estimate of p then, the expectations in (3.6.3) are
replaced by their respective mean squares resulting in the
formula
p = MSR ~ MSDV (3.6.4)
M MSR + <{>MSDV


45
Similarly from (3.6.1) and (3.6.2), an estimate of a2 may be
obtained as
^2 MSR + MSDV
1 + 4
Since the calculated value of MSR is always greater
A
than or equal to zero, then from (3.6.4) p > -l/cj>. Further
more, since the calculated value of MSDV is always greater
A
than or equal to zero, then p < 1. If these extremes are
considered as the endpoints of the range for the values of
A A
the estimate p, then -l/ < p < 1. However, since we are
interested only in the values of p in the interval between
A
zero and one, any negative value of p calculated is con
sidered meaningless and is set equal to zero in this case.
(Setting a negative estimate of a non-negative parameter e-
qual to zero is a procedure practiced when estimating vari
ance components in random and mixed models.)
A
The distribution of the random variable p depends
on the forms of the distributions of the random variables
SSR and SSDV. It was shown in Section 3.4 that the random
variable SSR is distributed as a weighted sum of independent
chi square random variables. Although an approximate distri
bution of SSR was given in Section 3.5, an approximation to
\
the distribution of p is at present untenable. Nevertheless,
A
the first two moments of the distribution of p could be ap
proximated using a Taylor series. That is, the formula in
A
(3.6.4) for p may be expressed in a Taylor series and the
A
mean and the variance of the distribution of p could be


46
approximated with a finite number of terms in the series by
taking the appropriate expectations.


CHAPTER 4
A PARTIALLY BALANCED
GROUP DIVISIBLE ECBD
In the extended complete block designs presented
thus far, the class of balanced incomplete block designs was
only considered in the extended portion of the b blocks.
That is, in making the extended complete blocks of size k,
we have considered in combination with the complete blocks
of size t only balanced incomplete blocks of size k-t. By
restricting attention to the use of balanced incomplete block
designs only in the extended portion, the extended complete
block designs retain the property of balance among the treat
ments. By balance is meant, the off-diagonal elements in
the matrix A (or NN') in (3.2.3) are all equal, resulting in
a single value of the variance for all pairwise treatment
comparisons. Hence, all pairwise treatment comparisons could
be made with the same precision.
When it is not necessary to have equal precision
for all pairwise treatment comparisons or when to achieve
balance the use of balanced incomplete block designs requires
a large number of replications of the treatments or possibly
too many blocks, a partially balanced incomplete block design
(PBIBD) might be used in the extended portion of an extended
complete block design. To illustrate this point, consider
an extended complete block design consisting of six treatments
47


48
in blocks of size nine. If a BIBD were used in the extended
portion of the.blocks, the balanced design would require ten
extended blocks supporting fifteen replicates of each of the
six treatments. On the other hand, if a PBIBD were used in
the extended portion, only six blocks supporting nine repli
cations of each treatment would be required.
In this and subsequent chapters then, the use of
PBIB designs in the extended portion of the extended complete
block design will be considered. Specifically, we shall
limit our attention to the use of PBIB designs with two asso
ciate classes. By relaxing the requirement of balance, in
most cases we do not sacrifice that much precision when con
sidering PBIB designs with two associate classes where in
stead two variances are required for making all pairwise
comparisons of the treatments. The two variances arise be
cause with each treatment a subset of the t-1 other treat
ments are first associates while the remaining other treat
ments are second associates. One variance is used for pair
wise comparisons among the treatments that are first asso
ciates while the second variance is used among treatments
that are second associates. The generalization to partially
balanced incomplete block designs with more than two asso
ciate classes should be straightforward.
For the balanced extended complete block designs,
the case where responses to the same treatment in the same
block were positively correlated was presented in detail.
The general theory developed in Section 3.4 on the exact


49
distribution of SSR when p > 0 with the additive model could
be used with partially balanced extended complete block de
signs. Hence, since the theory is general, the analysis of
partially balanced ECB designs with correlated observations
will not be presented. Our discussion will be limited to the
additive and non-additive models when all observations are
uncorrelated.
The group divisible association scheme for t = mn
treatments where m and n are integers is derived by parti
tioning the treatments into m groups of n treatments each
with those in the same group being first associates and those
in different groups being second associates. For example,
with six treatments (denoted by the numbers 1, 2, 3, 4, 5,
and 6) a group divisible association scheme for three groups
of two treatments each would be given by the 3x2 rectangu
lar array
1 2
3 4
5 6 .
In the array, treatments in the same row (1 and 2, 3 and 4,
5 and 6) are first associates. Treatments not in the same
row as a specified treatment are second associates of that
treatment. For example, the set of second associates of
treatment 1 consists of the treatments 3, 4, 5, and 6.
For a PBIBD with the group divisible association
scheme and incidence matrix N*, the matrix N*N*' may be


50
arranged in a particular pattern that will be described in
detail in Section 4.2. The particular pattern of the matrix
N*N* facilatates finding a solution of the normal equations
for the intrablock estimates of the treatment effects. The
pattern of N*N*' carries over to the matrix NN', where N is
the incidence matrix of the extended complete block design.
Extended complete block designs generated by the class of
PBIB designs with the group divisible association scheme
will be called "partially balanced group divisible extended
complete block designs."
4.1 Definitions and Notation
An extended complete block design generated by a
PBIBD is defined as a connected, two-way classification with
the following properties:
1. Each treatment is applied either Cq or c-^ times in
a block, c. >0, i = 0, 1.
i
2. In the incidence matrix of the design, replacement
of Cq by zero and c^ by unity results in the inci
dence matrix N* of a PBIBD (with two associate
classes).
It follows from this definition that the incidence
matrix N of such a design can be generated from the incidence
matrix of any PBIBD. That is, given a PBIBD with the inci
dence matrix N* and denoting by J a matrix of l's (the inci
dence matrix of a complete block design), the incidence
matrix of an extended complete block design (ECBD) generated


51
by a PBIBD is
N = cQJ + (C;l-c0)N* (4.1.1)
This equation is identical to the equation (2.1) for the
incidence matrix of a balanced ECBD except that N* is now
the incidence matrix of the generating PBIBD.
The parameters associated with an ECBD generated
by a PBIBD are as follows:
t = the number of treatments,
b = the number of blocks,
k = the block size,
r = the number of replications of each treatment in the
experiment,
= the number of distinct pairs of experimental units
which receive any fixed pair of i^ associates
while appearing in the same block, i = 1, 2,
th
n^ = the number of i associates of each treatment,
i = 1, 2, and
4-Vi
p., = the number of treatments that are both j asso-
th
ciates of treatment a and k associates of treat
. .th
ment 3 given that a and 3 are i associates.
The corresponding design parameters of the generating PBIBD
will be denoted by t, b, k*, r*, A*, n^, and Pl
owing to the definitions given above for the param
eters of the generating PBIBD as well as the ECBD generated
by a PBIBD, the following relationships are satisfied:


52
1.
r =
clr* + (b-r*)cQ = cQb + (crc0)r*
(4.1.2)
2.
k =
gxk* + (t-k*)cQ = cQt + (Gl-c0)k*
(4.1.3)
3.
r*t
= bk*
(4.1.4)
4.
rt
= bk
(4.1.5)
5.
Xi
= (cl"c0)2Xi + c0(2r-bcQ) i = 1, 2
(4.1.6)
6.
rk
- (cQ+c^)r + CQC^b = n-^X^ + n2X2 *
(4.1.7)
In
matrix notation, the non-additive model
is
y = + ?11 + + Y) + e (4.1.8)
where y is the overall mean effect, x is a t x 1 vector of
treatment effects, 8 is a b x 1 vector of block effects and y
is a bt x l vector of block x treatment interaction effects.
Letting lt denote a t x l vector of l's, It denote the t x t
th
identity matrix, and Yj_j£ represent the l response to the
th
i treatment in the j block, the vector y and the matrices
C, Xj, and X2 in (4.1.8) are of the forms
y =
ym
Ylln
Y211
11
Ytbn
tb
' ?1 =
it
it
'it
r Xr)
btxt
it
it J
btxb
bkxl
and


53
1
~n
11
C
bkxbt
4.2 Intrablock Estimation of
the Treatment Effects
Consider the model in (4.1.8) where the interaction
effects are all zero. This additive model is written as
y = C(ylb + xr + X28) + e (4.2.1)
Setting up the normal equations exactly as detailed in Sec
tion 3.2 results in the equality
G/bk
r a
y
kQ
=
/A
Ax
rB N'T
(rkl, N'N) §
~ D ~ ~ ~ J
where all symbols are defined in Section 3.2.
The solution of kQ = Ax in (4.2.2) depends upon
the form of the matrix
A
rkl NN'
t
(4.2.3)


54
which in turn depends upon the form of the matrix N* from
(4.1.1). For a PBIBD with the group divisible association
scheme, the matrix N*N* may be arranged in a particular
pattern as follows. For an association scheme consisting of
m groups each containing n treatments (so that t = mn), let
the treatments in the first group be labeled 1 through n; in
the second group, the treatments are labeled n+1 through 2n;
th
and so on, so that in the m group, the treatments are la
beled (m-l)ntl through mn. Now, with the corresponding
blocking plan and the labeled treatments listed in numerical
order in the incidence matrix N*, the matrix N*N*' becomes
N*N*' = (r*-A*) (Im In) + (Asj'-A*) (Im Jn) + X* (Jm Jn) ,
(4.2.4)
where denotes the direct (Kronecker) product of two matri
ces and A| and A| are the number of distinct pairs of exper
imental units over all blocks which receive any fixed pair
of first and second associates, respectively, in the same
block in the generating PBIBD.
Since the incidence matrix of the ECBD generated
by a PBIBD equals cgJ + (c]_-cq)N*, then
NN' = (c-l-Cq) 2N*N*' + cQ (2r-cQb) (Jm Jn) (4.2.5)
By substituting (4.2.4) into (4.2.5) and simplifying, we
obtain
NN* = [(c0+c1)r-c0c1b-X1] (Im In) + (XrX2) (Im Jn) +
(4.2.6)
X0(J J) .
2 ~m ~n


55
To facilitate finding a solution of kQ = At for x
in (4.2.2), an-expression for the matrix A may now be found
by substituting (4.2.6) for NN' into (4.2.3). By further
simplification using the identity (4.1.7) involving X-^ and
X2, the result is
kQ = [(nXi+njjkj) (Im 0 In) (X1-X2)(Im 0 Jn) -
x2h '
th
of which the i element is given by
kQi = (nX1+n2X2)xi (^i~^2^G^Ti^ X2T*
(4.2.7)
where G(x^) denotes the sum of all the estimated treatment
. th
effects of the treatments m the group containing the 1
treatment. In other words, G(x^) is the sum of the effect
th
of the 1 treatment plus the effects of all first associates
th
of the i treatment. Also in (4.2.7), x. denotes the sum
of all the estimated treatment effects. However, one of the
restrictions used on the treatment effects to obtain (4.2.2)
was to set x. equal to zero. Therefore, if G(kQ^) denotes
the sum of the kQ^ in (4.2.7) plus the kQ^1, i ^ i', corre-
. th
spondmg to all the first associates of the 1 treatment,
we obtain
G(kQ^) (nX j+n2 X 2) G (x ^) n (X^-X2) G (x^)
= tX2G(xi) (4.2.8)
By substituting G(Xj_) from (4.2.8) back into (4.2.7), the
th
intrablock estimate of the effect of the i treatment is


56
found by solving for x^ in the expression
X -X
(nX +n X )x = kQ. + G (kQ. ) .
1 2 2 i i i
(4.2.9)
The difference between the intrablock estimates of
the effects of the treatments i and i', i ^ i', can be writ
ten as
A A
T -X .
1 1
t
= MQ.-Q. ,) +
i x'
VX2
tX0
[g (kQ ) -G (kQ ) ] .
i l
Under the assumption that the errors are normally distributed
, 2
with mean zero and variance structure o I, the difference
e~bk
A A
x^-x^, has the properties
E(t.-t,
i i
T -T .
1 1
i
and
Var(x.-T.
i i
,)
2ka2
r i & i' are 1st associates
nXi+n2X2
2ka2
e
nXi+n2X2
(1 +
i & i' are 2nd
associates .
The intrablock analysis of variance table for the
partially balanced group divisible extended complete block
design is presented in Table 3. In the sum of squares ex
pressions in Table 3,
CM
_1
bk
l l
i j
l
a
i j Z
r


TABLE 3
Intrablock Analysis of Variance for Partially Balanced
Group Divisible Extended Complete Block Designs
Source
df
Treatments (adjusted) t-1 SSTA = (n~~Tn~~~r H^kQi)2 + t\ ^ (kQi)G(kQi)]
Sum of Squares
1' **22' i tX2
Blocks (unadjusted) b-1
SSBjj = | l B? CM
j
Remainder
(t-1) (b-1) SSR =11^- R?j "
1 D ID
SSTa SSB0 CM
Error
b(k-t) sse -III Yin 11 ht: RL
j H
i D ID
Total
bk-1
TSS
yij£
i j £
CM
* E (MSTa)
= ae + k(t-1) t'At E(MSR) = a* + y'Dy and E(MSE) =
- rr 2
a where
A = rklfc NN' and D = C'C
EMS*
E (MSTa)
E (MSR)
E (MSE)
cn
-J


58
As can be seen from Table 3, the ratio of
block, and
treatment
the mean square for remainder to the mean square for error
(previously called the mean square for duplication variation)
provides a statistic for a test on the validity of the addi
tive model. If the hypothesis is not rejected, a test of
the hypothesis of equal treatment effects could be performed
using the statistic the ratio of the mean square for treat
ments (adjusted) to the mean square resulting from pooling
the mean square for remainder with the mean square for error.
If the hypothesis is rejected, then we might wish to consider
a non-additive model. With the non-additive model, we could
concern ourselves with the estimation of the block x treat
ment interaction effects or concentrate on testing the hy
pothesis of equal treatment effects in the presence of inter
action effects.
The next section gives validity to both of the
above mentioned tests of hypotheses. As will be shown, the
sums of squares in Table 3 are each distributed as weighted
chi square random variables and each sum of squares is dis
tributed independently of the others.
4.3 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses
In order to validate the tests mentioned at the
end of Section 4.2, we must first obtain the distributions


59
of the sums of squares in Table 3. The approach that will
be used to derive the distributions of these sums of squares
is to express them as quadratic forms and use our knowledge
of the distributions of quadratic forms.
The sums of squares in Table 3 can be written in
the quadratic forms
SETA = X'dlX = X :,..,+n2y S'ibt k ;2525>?1
H5 X '
SSBu X'*2X = X' BE tbcx2xc J) y ,
SSR = y'A3y = y' (CD_1C' £ CX^C' A-^ y
SSE = y'A4y = y' (Ibk CD 1C') y ,
(4.3.1)
(4.3.2)
(4.3.3)
(4.3.4)
and
TSS = y'A5y = y' (Ibk ££ J) y (4.3.5)
where in (4.3.1), F = Im IR.
In the quadratic forms (4.3.1) through (4.3.5),
each of the matrices A^, A2, A^, A^, and A^ is real, sym
metric, and idempotent, and
4
I ip = *5
P=1 P
Also, the ranks of the five matrices, where r(Ap) denotes
the rank of the matrix A are
~ IT
riA^) = t-1 ,


60
r(A2) = b-1 ,
r(A3) = (b-1)(t-1) ,
r(A4) = b(k-t) ,
r(Ac) = bk-1 = l r(A ) .
P=i ~p
Hence, by applying Theorem 5 in Searle (1971) on the dis
tribution of quadratic forms, it is found that when
y
N( y,
aeibk
/
then
y'A Y ~ CTe xr(A ) ( y'^p!y2a ) (4.3.6)
P
for p = 1, 2, 3, 4 and the y'A y are mutually independent.
~ ~ IT ~
The distributional forms in (4.3.6) will be used in con
structing the aforementioned tests of hypotheses.
The test of the assumption concerning the validity
of the additive model corresponds to the test of the hy
pothesis of zero interaction effects when the non-additive
model is considered. To test the validity of the additive
model, the test statistic used is the ratio of the mean
square for remainder to the mean square for error. If the
additive model holds, then the test statistic possesses an
F distribution with the appropriate degrees of freedom, and
the hypothesis concerning the validity of the model is re
jected for large values of this ratio. If the additive
model assumption is valid, a test of the hypothesis of equal


61
treatment effects would be performed using the ratio of the
mean square for treatments (adjusted) to either the mean
square for error or the pooled mean square for remainder
plus error. Under the hypothesis of equal treatment effects,
this ratio possesses an F distribution. On the other hand,
if there is evidence to reject the assumption of the addi
tive model in favor of the non-additive model, then an ap
proximate test on the treatments could be performed if de
sired.
The intrablock analysis given in this section and
in the previous section is, of course, an analysis of a model
in which the treatment, block, and interaction effects are
considered as fixed effects. In comparative type experi
ments, usually we seek to draw inferences about the effects
of the specific treatments used in the experiment and hence
the assumption of fixed treatment effects presents little
argument. Now the block effects, on the other hand, may be
fixed or random. In this latter case the emphasis may be on
drawing inferences about the magnitude of the variance of
the population from which the sample of block effects was
assumed to be drawn. A model in which the treatment effects
are fixed while the block effects are random is called a
mixed model.
Since the partially balanced group divisible ex
tended complete block designs would frequently be used with
random block effects, we shall now present the analysis of
a mixed model for these designs.


62
4.4 Mixed Model Analysis
In the mixed model analysis, all symbols in the
model are defined exactly as in (4.1.8) with the exception
of 8 and y. In the mixed model, the parameters 8 and y are
assumed to be independently distributed normal random vari
ables with
8 ~ N( 0, oIb ) ,
I ~ N< btibt > '
and each distributed independently of the random errors e.
Under these assumptions, the expectation of the vector of
observations is
E (y) = c(ylbt + x-jt) = y
and the variance of the observations is
(4.4.1)
Var(y) = e[c(X28 + y) + e] [c (X28 + y) + e] '
= ?X2XC-a + CC'at + a^Ibk = V
(4.4.2)
The analysis of variance table for the mixed model
is presented in Table 4. The differences between the en
tries in Table 4 and Table 3 for the fixed effects model are
the replacement of the source of variation called blocks
(adjusted) for the source of variation called blocks (un
adjusted) and the expected mean square expressions. Of
course other differences exist in the use of the two tables,
namely in the interpretation of the tests of hypotheses.
The expected mean squares in Table 4 were obtained using the
identity


TABLE 4
Mixed Model Analysis of Variance for Partially Balanced
Group Divisible Extended Complete Block Designs
Source
df
Treatments (adjusted) t-1
Blocks (adjusted) b-1
Interaction (t-1)(b-1)
Error
b (k-t)
SS*
SST
A
SSB,
SSI
SSE
Expected Mean Squares**
1
e + t-1 * abt + k (t-1)
r 'Ax
e + (si- 7 + hT<>* 7F (r-k) s,}at2
b-1 V1 r 2' b b-11^ rk
ae + (b-i)Tt-l) {S1 k S2 ^*}abt
2J bt
Total
bk-1
TSS
* SSBa = SSTa + SSBy SSTy and the other SS are defined in Table 3 with SSI = SSR .
p
** A = rklt NN' s = l l ni- for p = 1, 2, and 3, X** = (cq+c-^Aj. CqCjT and
** = bk(n>^2A2) '1 + ) (kslS2-2slS3+S|) + (X1s22X**s1) } .

OJ


64
E(y'Apy) = tr(ApV) + u'Apu
involving the expectation of a quadratic form with
y ~ N( y, y )
and where tr(A V) denotes the trace of the matrix A V.
~ IT ~ ir ~
To test the hypothesis HQ: = 0, the statistic
takes the form of the ratio of MSI to MSE, where MSI equals
SSI/(b-l)(t-1) and MSE equals SSE/b(k-t) with SSI and SSE
defined in Table 4. The hypothesis is rejected for values
of this ratio larger than the appropriate tabular F value.
/s _
If the hypothesis is rejected, an estimate of could
be found using the analysis of variance approach. An es-
timate of the random blocks component of variance may
also be obtained using the analysis of variance approach.
When the hypothesis HQ: = 0 is rejected, we
may still wish to test the hypothesis that the treatment
effects are small relative to the magnitude of the inter
action variation. For this test the statistic is given by
s-,^- (1/k) s2~(j)5
(b-1) (t-lT~
MSTa -
t-1
MSI
s1~ (1/k) s2-(f)*
(b-1)(t-1)
t-1
MSE
where expressions for calculating the values of s^, s2, and
<})* are presented at the bottom of Table 4. The statistic FT
can be shown to possess an approximate F distribution with
f and b(k-t) degrees of freedom in the numerator and denom
inator, respectively, where the number f is computed by the


65
procedure given by Satterthwaite (1946) for approximating
the distribution of the estimate of a variance component.
If the hypothesis HQ: cr£t = 0 is not rejected, a
simpler test on the treatment effects may be performed. For
this case, the statistic is the ratio of the mean square for
treatments (adjusted) to the pooled mean square consisting
of the mean squares for interaction and error.


CHAPTER 5
A PARTIALLY BALANCED ECBD WITH
THE L2 ASSOCIATION SCHEME
The partially balanced group divisible extended
complete block designs presented in Chapter 4 comprise a
large class of partially balanced ECB designs. However,
because of its general applicability, still another class of
partially balanced ECB designs to be considered is the class
of partially balanced extended complete block designs with
the L2 (Latin Square) association scheme. The L2 asso
ciation scheme for t = n2 treatments is characterized from
the arrangement of the treatments in a square array. The
classification of the treatments to one another in the L2
association scheme is such that the treatments in the same
row or same column are first associates and the treatments
not in the same row or same column are second associates.
For example, with sixteen treatments (denoted by the numbers
1 through 16) an 1>2 association scheme would be determined
from the square array
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The first associates of treatment 1 are the treatments 2, 3,
66


67
4, 5, 9, and 13, while the second associates of treatment 1
are the treatments 6, 7, 8, 10, 11, 12, 14, 15, and 16.
In this chapter, we shall present only the intra
block analysis of the partially balanced extended complete
block designs with the L2 association scheme. The mixed
model analysis follows the procedure outlined in Section
4.4. In fact, the only difference in the final analysis of
the mixed model with the two association schemes is that with
the partially balanced ECB designs with the L2 association
scheme, the expected mean square expressions are slightly
more complicated than the corresponding expressions with the
partially balanced group divisible ECB designs.
In the intrablock analysis, we shall make use of
row sum and column sum operators which we denote by RS()
and CS () respectively. For the square array containing the
treatment effects x^ through Xg
the row sums RS(x^), RS(Xg), and RS(Xg) are given by
RSx^ = ti+t2+t3 = RS(x2) = RS(x3) ,
RS(x5) = t4+x5+t6 ,
and
RS(x 9)
t7+t8+t9


68
The column sums for the same treatment effects are given by
CS(t1) = T1+T4+T7 = cs(t4) = CS(x7) ,
cs = t2+t5+t8 ,
and
CS(t9) = t3+t6+t9 .
Note that for the i1" treatment effect,
RS (T) + CS(Ti) = s1(Ti) + 2xi ,
where S^(t^) denotes the sum of the effects of all treatments
that are first associates of the i*" treatment.
For a PBIBD with the association scheme, the ma
trix N*N*' may be arranged in a particular pattern which will
be described in Section 5.1. The particular pattern of the
matrix N*N*1 facilitates solving the normal equations to ob
tain the intrablock estimates of the treatment effects. As
expected, the pattern of N*N*' is reflected in the matrix
NN', where N is the incidence matrix of the ECB design.
5.1 Intrablock Analysis
The definitions and notation presented in Section
4.1 will be used again in this section with the exception
that N* now denotes the incidence matrix of the generating
PBIBD with the association scheme.
Let us consider an additive model consisting of
the overall mean parameter, a treatment parameter, a block


69
parameter, and a random error term. Since the interaction
effects are all zero, the model in (4.1.8) may be written as
y = C(ulbt + Xjj + X28) + e (5.1.1)
Using the same form of the normal equations as detailed in
Sections 3.2 and 4.1, we have
G/bk
r a -
y
A
kQ
=
At
A
rB N'T
(rklb N'N)B
where all symbols are defined in Section 3.2.
A
The solution of kQ = At in (5.1.2) depends upon
the form of the matrix NN', since A = rklt NN'. For a
PBIBD with the L^ association scheme, the matrix N*N* may
be arranged in a particular pattern as follows. In the
association scheme, let us label or number the treatments
from 1 to n2. Then, if the treatments are listed in numeri
cal order in the incidence matrix according to the particu
lar blocking plan used, the matrix N*N* is of the form
N*N*' = (r*-2A*+A*)(I 0 I ) + (A*-A*)(I 0 J )
- ~ 2 ~n ~n 1 2 ~n ~n
+ (A*-A*)(Jn 0 In) + A*(Jn 0 Jn) (5.1.3)
J. ^
where 0 denotes the direct product of two matrices and where
over all blocks in the generating PBIBD, A* and A* are the
number of distinct pairs of experimental units which receive
any fixed pair of first and second associates, respectively,
in the same block.


70
Since the matrix N can be expressed in the form
N = CqJ + (c-l-CqJN* ,
then
NN* = (C-Cq)2N*N*' + c0(2r-c0b)(Jn 0 Jn) (5.1.4)
Now if the form (5.1.3) of the matrix N*N* is substituted
into NN' in (5.1.4), the resulting expression for NN' after
simplifying is
NN = [(c1+c0)r-c0c1b+X2-2A1] (In 0 IR)
+ (Xl-X2)[(Jn In)+(In Jn)] +X2(Jn0Jn) .
(5.1.5)
This expression for NN' can now be substituted into
A = rklfc NN' ,
so that the intrablock estimates of the treatment effects
are obtained by solving the equation
kg = (n[2Aj+(n-2)X2] (Jn In> i.\1~X2) [(Jn Jn) + (Jn Jn>]
- X2 th
of which the i element is
kQi = n[2X1+(n-2)X2]xi (Xx-X2) [RS (t) +CS (t) ] (5.1.6)
since -A2t. = 0. In (5.1.6), RS(t^) and CS(t^) are the row
th
sum and column sum, respectively, of the estimated i treat
ment effect.


71
To obtain the expression for from (5.1.6), we
need the row and column sums of kQ^ in (5.1.6). These sums
are respectively
RS(kQ) = [nX1+n(n-l)X2]RS(Ti) (5.1.7)
and
CS(kQ^) = [nX-^+n (n-1) X2] CS (x^) .
(5.1.8)
Replacing RS(x^) and CS(x^) in (5.1.6) by their respective
equivalent expressions in (5.1.7) and (5.1.8), then kQ^ in
(5.1.6) may be rewritten as
*>i = n[2X1+(n-2)X2lT1-nX1+Mn-l)X2 [RS (kQi)+CS (kQl) ] ,
th
and hence the intrablock estimate of the effect of the i
treatment is given by the equation
n [2X^+(n-2)X2]Ti = +
A1 A2
nX^+n(n-1)X^
[s1(kQi)+2kQi]
(5.1.9)
where Sj(kQ^) is the sum of kQ^ in (5.1.6) plus the kQ^,,
th
i ^ i', corresponding to the first associates of the i
treatment.
The difference between the estimated effects of
treatments i and i' can be written as
Ti~Ti' =
X X
k (QiQi>) + nX +n(n-1)X
i.
[S1(kQi)-S1(kQi,)+2k(Qi-Qi,)] .
Under the assumption that the random errors in (5.1.1) are


72
normally distributed with mean zero and variance structure
A A
eIbk' the difference has the properties
A A
and
A A
Var(Ti-xi,)=
Kl> + X.+n (n-T)'rf i & i' are lSt associates
X1 X2
K[l + nx (n-)X 1' i & i' are 2nC^ associates,
where K = kcr* /n[2X-^ + (n-^)^] and n = /t*
The intrablock analysis of variance table for the
partially balanced extended complete block designs with the
association scheme is presented in Table 5. All symbols
in Table 5 are defined as they were defined in Table 3 with
the exception of S^(kQ^) which has been defined in this
section.
5.2 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses
As in Section 4.2, before considering any relevant
tests of hypotheses, we shall obtain the distributions of the
sums of squares in Table 5. Resorting once again to the the
ory of the distributional properties of quadratic forms, the
sums of squares formulae in Table 5 are expressed as quad
ratic forms in the following matrix notation


TABLE 5
Intrablock Analysis of Variance for Partially
Balanced ECBD of the Association Scheme
Source
df
Sum of
Squares
EMS*
X.
-X,
Treatments (adjusted)
t-1
ssta =
2nX-
k
L+n (
n-2)X2
Kq? +
i
1
nX-^+n
2
(n-l)X2
[Qisi
(Q) +
2Q?]}
E (MST
Blocks (unadjusted)
b-1
SSBu
1
k
l
j
B2
D
- CM

Remainder (t-
1)(b-1)
SSR =
l
l
1
. R? -
SST -
SSB 7
- CM
E (MSR)
i
j
nij
ID
A
U
Error b
(k-t)
SSE =
I
i
1
j
I y
a
2 _
i j a
l l r
i j I]
R2 .
ID
E(MSE)
Total
bk-1
TSS =
I
i
l
j
C
2 _
i j £
CM
* E (MST ) = a2 + x1 At E(MSR) = a2 + y' Dy and E(MSE) = a2 where
A e k(t-l) ~ ~~ e ~ ~~ e
A = rkl NN' and D = C'C .
co


74
SST = y'A y = y'
A ~ ~1~ ~
C(I J. X X D) X,
2nA2_+n (n-2) X2 ~ ~bt k ~2~2~ ~1
A -A
IX + 1 2 h] X' (I 1 DX X' ) C' y (5.2.1)
(n-1)A o ~ ~1 ~bt k ~~2~2 ~
'~t nAi+n
SSB = y' A y = y' I(CX X'C I J) y ,
U ~ ~2~ ~ k ~~2~2~ b ~ ~
(5.2.2)
SSR = y'A y = y' (CD-1C' -ij-A -A)y, (5.2.3)
~ ~3~ ~ ~~ ~ bk ~ ~1 ~2
SSE = y'A y = y' (I CD_1C') y ,
~ ~4~ ~ ~bk ~~
(5.2.4)
and
TSS = y'A y = y' (I 1 J) y ,
~ ~5~ ~ ~bk bk ~
(5.2.5)
where in (5.2.1), the matrix H is given by
H = (I J ) (J I ) .
~n ~n ~n ~n
The matrices A., A., A-, A., and Ac are each real,
~1 ~2 ~3 ~4 ~5
symmetric, and idempotent, and
l A = A_ .
L. ~p ~5
p=l
Furthermore, the ranks of the five matrices, where r(A ) de-
~ hr
notes the rank of the matrix A are
~P
r(^l}
t-1 ,
b-1 ,
(b-1)(t-1) ,
b(k-t) ,
and
r(A ) = bk-1 = V r (A )
~ R _L, ~ D
p=l


75
Hence, by again referring to Theorem 5 in Searle (1971) on
the distribution of quadratic forms, we find that when
y ~ N( H' aeibk } '
then
yApy ~ e xr(A )( H'^pH/2ae ) (5.2.6)
for p = 1, 2, 3, and 4 and the y'A y are mutually independ-
~ ~ IT ~
ent. The distributional forms (5.2.6) can now be used to
construct statistics for the tests of hypotheses in the
usual manner.
The test of the assumption concerning the validity
of the additive model corresponds to the test of the hy
pothesis of zero interaction effects when the non-additive
model is considered. To test the validity of the additive
model, the test statistic used is the ratio of the mean
square for remainder to the mean square for error. If the
additive model assumption holds, then the test statistic
possesses an F distribution with the appropriate degrees of
freedom, and the hypothesis concerning the validity of the
model is rejected for large values of this ratio. If the
additive model assumption is valid, a test of the hypothesis
of equal treatment effects would be performed using the ratio
of the mean square for treatments (adjusted) to either the
mean square for error or the pooled mean square for remainder
plus error. Under the hypothesis of equal treatment effects,
this ratio possesses an F distribution. On the other hand,


76
if there is evidence to reject the assumption of the addi
tive model in favor of the non-additive model, then an ap
proximate test on the treatment effects could be performed
if desired.


CHAPTER 6
THE GENERAL PARTIALLY BALANCED
EXTENDED COMPLETE BLOCK DESIGN
In Chapter 4 the analysis of the fixed effects
model as well as the analysis of the mixed model for the
class of partially balanced group divisible extended com
plete block designs was presented in detail. In Chapter 5
the analysis of the fixed effects model was presented in de
tail and the analysis of the mixed model was mentioned for
the class of partially balanced extended complete block de
signs with the I12 association scheme. These two special
cases of the general partially balanced (GPB) extended com
plete block designs were presented in detail, not only be
cause of their general applicability, but also because the
constants (containing the parameters of the designs) were of
the same form for both special cases.
In this chapter, we shall present the analysis of
the GPB extended complete block designs. For this general
class of designs, it will be necessary to introduce new con
stants to aid in simplifying the forms of the necessary cal
culating formulae. The introduction of these new constants
stems from the desire to conform to the use of the standard
notation for the analysis of general partially balanced in
complete block designs. In particular, the new constants
are d., d and A which correspond to the constants c c ,
12s 12
77


78
and A as defined and used by Bose and Shimamoto (1952) in
the analysis of PBIB designs.
We now present the intrablock analysis of variance
for the GPB extended complete block designs.
6.1 Intrablock Analysis
Let us consider the model in (4.1.8) where again
the interaction effects are all zero. The additive model is
written as
y = C(ylbt + XjT + X23) + e (6.1.1)
where the symbols y, C, y, lbt, X^, t, X2, 3, and e are de
fined following (4.1.8). The normal equations are set up
exactly as presented in Sections 3.2 and 5.1, resulting in
G/bk
1
< 2-
\
kQ
--
A
At
A
rB N'T
(rklb -N'N)3
where G is the grand total of the observations, Q is the
vector of adjusted treatment totals, B and T are respec
tively vectors of the unadjusted totals of the blocks and
treatments, N is the incidence matrix of the design, r is
the number of replications of each treatment in the exper
iment, and k is the block size. As in the previous sections
where the intrablock analysis was discussed, the matrix A is
given by
A = rklt NN' ,


79
A A A
and y, x, and 3 are respectively the estimates of the param
eters y, x, and g in (6.1.1).
To find an expression for the intrablock estimate
th
of the effect of the i treatment, we note from the t x 1
a th
vector of equations kQ = Ax in (6.1.2) that the i element
can be written as
b t
kQi = rkTi J I nijnhjTh ,
(6.1.3)
th
where is the adjusted total for the i treatment and n^
and n^j are elements of the incidence matrix N of the de
sign. The quantities
1 l ijVTh
3 h J
can be expressed in terms of the parameters of the design as
follows:
l 1
j h
nijnhjTh
l nij*i
3
l xiTi
1
_ A
l Vi
i = h
s t
i f h, i & h are 1 associates
i ^ h, i & h are 2n<^ associates
(rk-n-^X j-n2X2^ Ti
XjS^(x^) ( i / h
^2^2 i = h
s t
i & h are 1 associates,
nd
i & h are 2 associates
where S-^ix^) is the sum of the estimated treatment effects
of all treatments (n-^ in number) that are first associates
th a
of the i treatment, and likewise, S2(x£) is the sum of the


80
estimated treatment effects of all treatments (n^ in number)
J_ T_
that are second associates of the iT'n treatment. By re
placing the quantities
£ l n. .n t
j h x3 h3 h
in equation (6.
,1.3) with their equivalent expressions in-
/\
A
volving S^(t_^) and S2(t^), we maY wr;i-te equation (6.1.3) in
the form
A A Z\
kQi = ^ni^l+n2^2^Ti ^lSl^Ti^ ^2S2^Ti^ (6.1.4)
Equation (6.1.4) is now summed over the first asso
ciates of the x treatment resulting in the expression
kS1(Qi) =
"hVi + Sl(i> (nlXl+n2A21lPl'i2p2)
+ S2(.)(-X1p^1-X2p2) (6.1.5)
Summing (6.1.4) over the second associates of the i1" treat
ment results in
kS2(Q.) =
X2n2il + S2(il> (Vl+n2X2'XlP12X2P22)
+ S1(.)(-X1p>2-X2P2) (6.1.6)
In (6.1.5) and
(6.1.6) px is the number of treatments that
3k
are both a jth
+ Vi
associate of treatment a and a associate
of treatment 6
given that treatments a and 8 are ix asso-
ciates. As in
Bose and Shimamoto (1952), we write equations
(6.1.5) and (6.
.1.6) in the forms
kS1(Qi) =
= -n1X1i + a11S1(?l) + a12S2(.) (6.1.7)
and


81
kS2(Qi) n2A2Ti
(6.1.8)
all nlXl + n2X2 Xlpl X2P12
a
12 Xlpl X2P2
a
21 "Xlp2 X2P22
and
X1P2 X2P22 '
In order to express the intrablock estimate x^ as
a function of Q^, S^(Q^), and S2 (Q) having arrived at the
equations (6.1.7) and (6.1.8), we interrupt the development
A A
briefly to see how the sums S^ix^) and S2(x^) can be re
placed in (6.1.7) and (6.1.8) with the quantities S^(Q^)
and S2(Q^). To this end, consider the linear combination
L = k2Q + d1kS1(Qi) + d2kS2(Qi)
(6.1.9)
1^1vvi
involving only the Qi and parameters of the design with d^
and d2 being constants consisting of linear functions of the
aij, i < j = 1, 2. If both (6.1.7) and (6.1.8) are substi
tuted into (6.1.9) for kS^Q-^) and kS2(Q^), respectively,
the resulting expression for is
= [k (njX^+n2X2)-djn^X-^-d2n2A2] x^ + (d^a^j+d2a22_~kA^) S^ (Q^)
(6.1.10)
+ (d1a12+d2a22-kX2)S2(Q^) .
The quantities d-^ and d2 in (6.1.10) are now chosen so that


82
upon equating the right-hand side of (6.1.9) to the right-
hand side of (6.1.10), the equation (6.1.10) becomes
k2Qi + d1kS1(Qi) + d2kS2(Qi) = k (n^+n^) (6.1.11)
A
That is, equation (6.1.11) expresses the estimate as a
function of the quantities S-^iQ^), and s2 (Q^) .
To obtain the values for d-^ and d2 so that equation
(6.1.11) is as shown, we require the identities
k^i = d^ (a-j^+n^A-^) + d2 (a2^+n2X2) (6.1.12)
and
kX2 d^ ^2 ^a22+n2^2^ (6.1.13)
Solving equations (6.1.12) and (6.1.13) simultaneously by
the use of determinants, we have d-^ = D-^/D and d2 = D2/D,
where D, D-^, and D2 are given by
D (a^^tn^X^) (a22+n2A2) (a^2"^~^2_^i^ (a2*^tn2X2) ^ (6.1.14)
~ kX^(a22tn2X2) kX2 (a2i+n2'*''2^ t (6.1.15)
and
D2 kA2(a^*^~^"^2_^}_^ """* kXi (a-^2+n^X^) (6.1.16)
Substituting for a^, a^2, a2^, and a22 in (6.1.14), (6.1.15),
and (6.1.16), simplifying, and writing D = k2A the fol-
b
lowing equations for d-^ and d2 are obtained
kAgd-j_ = A1 (n1X1+n2X2 + X2) + (X-^X2) (^ 2P2 ^ lp 2 ^
and


83
k^s^2 ^2 l"^^2^2^^ 1^ **" ^2^ ^2P12^lp12^ f (6.1.18)
where
k2A
s
^nl^l+n2^2+^l^ ^nl^l+n2^2+^2^
+ (X1-X2) [ (niAi+n2A2) (P3_2"P2^ + X2P12 ^lp12^ *
The expressions (6.1.17) and (6.1.18) for the values of d-^
and d2, respectively, are now substituted back into (6.1.11).
th
The intrablock estimate of the effect of the i
treatment is
Ti k(n1A1+n2A2) tk2Qi + ^i^Qd.) + d2S2(kQi)] (6.1.19)
which has the alternate forms
Ti k(n1A1+n2X2) t(k dl)kQi + {dl d2)s1(kQi)] (6.1.20)
and
- = 1
Ti k(n^X^+n2X2)
[Oc-a2)kQ. + (a2-a1)s2(kQ.)]
(6.1.21)
The use of one of the alternate forms would probably be more
convenient since only one sum, either S-^(kQ^) or S2(kQ^),
for each treatment need be calculated.
/\ /v
The difference of the estimated effects of
treatments i and i', i ^ i', using the alternate form in
(6.1.20) above is
Ti"Ti* =
k (n.X^+nX.) [(k"di)k(Qi Q,)
11 2 2
+ (d1-d2)k{S1(Q.)-Si(Q.I)}]
Under the assumption that the random errors in (6.1.1) are


84
independent normally distributed random variables with mean
r.
zero and variance a^r the difference has the properties
and
A A
Var(xi-xi,)
f2(k"dl)ae st
-:rr; i & i' are 1 associates
nl^l+n2^2
2(k-a2)a a
-;rrr 1 & 1 are 2 associates .
[nlh+n2X2
The analysis of variance table for the general
partially balanced extended complete block design is exactly
of the same form as Table 5 except that the calculating
formula for the sum of squares for treatments (adjusted) is
now given by
SSTa
1
k2(n]_^]_+n2^2)
I
[ (k-d1) (kQi) 2+ (d1-d2) (kQi) S1 (kQi) ] .
Also, the tests of hypotheses usually performed are con
ducted in the manner described in Section 5.2.
The recovery of the intrablock information and a
combined estimate of the intrablock and interblock treatment
effects can also be obtained with the straightforward appli
cation of the maximum likelihood method of Rao (1947).
The utility of the general partially balanced ex
tended complete block designs is at present limited. This
limitation is imposed by the complexity of the formulae for


85
the estimates of the treatment effects, in that the con
stants and d^ must be calculated for any design used. A
similar difficulty is encountered in the analysis of PBIB
designs, unless one has reference to the extensive listing
of PBIB designs (with two associate classes) and their asso
ciated constants necessary for an analysis as given by
Clatworthy (1973).


CHAPTER 7
CONCLUDING REMARKS AND A
SENSORY TESTING EXAMPLE
Throughout the development and presentation of
this work, it has been necessary to make certain assumptions
concerning the model as well as the type of correlation
present in the data in order to formulate our methods of
analysis. In Chapter 3 for example, in the development con
cerning the possible presence of the correlation p between
duplicate responses to the same treatment in the same block,
the additive model only was assumed. Without making the
assumption that the interaction effects are all zero, the
estimate of the magnitude of the correlation would be con
founded with the estimates of the interaction effects. In
this case, an estimate of the correlation free of the inter
action effects would not have been possible by the method
we used. Although the assumption of additivity for many
realistic situations is somewhat restrictive, the additivity
assumption was made as a matter of necessity and to be in
line with the assumption employed in the analysis of ran
domized complete block designs (of which our designs are
just extensions).
The assumption that the correlation between du
plicate responses to the same treatment in the same block is
constant and equal for all treatments and blocks may also
86


87
appear to be somewhat restrictive. It would seem more appro
priate perhaps to assume the correlation is not constant but
rather varies over the treatments and blocks. That is, for
many practical applications it may be more realistic to con
sider the correlations p^j, i = 1, 2, ..., t and j =1, 2,
..., b. However, this non-equality of the correlations
would give rise to the difficulty of having to estimate a
larger number of parameters than the number of observations
present in the experiment. Also, all the correlations p^j
could not be estimated in a given experiment since not all
block-treatment combinations would have duplicate responses.
A simplification of the problem of non-equality of
the correlations would be to consider the correlations pj,
j = 1, 2, ..., b, that is, a different correlation is asso
ciated with each block. Such a case might arise when pan
elists of varying degrees of proficiency are used in sensory
testing experiments. The estimation of the Pj and the sub
sequent test on the treatment effects is being considered
for future work.
The methods presented for testing the hypothesis
of zero correlation and for estimating p may appear to be
somewhat intuitive. First attempts in finding a likelihood
ratio test of the hypothsis of zero correlation and a maxi
mum likelihood estimator of p resulted in complex expres
sions which did not seem to simplify. These likelihood pro
cedures could be investigated in future work. At that time,
it may be of interest to compare the likelihood results with
the results contained in this work.


88
The possible lack of utility of the general par
tially balanced extended complete block designs developed in
Chapter 6 was mentioned at the end of that chapter. Inves
tigations into the possibility of expressing our solutions
in terms of the parameters and constants already tabulated
for PBIB designs by Clatworthy (1973) could be considered.
Hopefully, this would enhance the utility of the GPB ex
tended complete block designs with respect to the calcula
tions involved in the analysis.
As mentioned at the end of Section 3.2, we shall
now suggest a test of the hypothesis of equal treatment ef
fects in the presence of a non-zero correlation. The test
procedure is just a suggestion since the properties of the
procedure have not been studied in detail at this time and
remain for future consideration.
The quadratic forms for the sums of squares for
treatments (adjusted) in Table 1 of Section 3.2 and for re
mainder in (3.3.2) are not independently distributed in gen
eral. However, the quadratic forms for SST^ and the sum of
squares for duplication variation in (3.3.1) are independ
ently distributed. In fact, we have that
SSTa ~ a2{1 + [(k+t)X 2rk]} x^-i
when the hypothesis of equal treatment effects is true, that
SSDV ~ o2(1P) X(k_t) ,
and that these random variables are distributed independently


of one another. Thus, to test the hypothesis under con
sideration, we may use the test statistic
89
MST /{I + |£ [ (k+t)A 2rk]}
pi A AK
t MSDV/(1 p)
which possesses a central F distribution with t-1 and b(k-t)
degrees of freedom in the numerator and denominator, respec
tively. When the hypothesis is not true, the test statistic
F has a non-central F distribution which depends upon the
true value of p as well as the unknown value of the ratio of
£ x2 to ct2. Thus, the power of the test could be calculated
for various values of p and the ratio J x2/a2.
The value of the test statistic F depends upon
the true value of p which is usually unknown. The suggested
A
procedure, therefore, is to replace p with p resulting in
the approximate test statistic F* given as
MST /{I + ||. [(k+t) A 2rk] }
p* A AK
t MSDV/(1-p)
The distribution of the approximate test statistic
F* depends upon the unknown distribution of p. At this
point, complications arise in arriving at the exact or an
approximate distribution of F* since p is calculated using
the value of SSR which, as a random variable, is not inde
pendent of the random variable SST Hence, until more
A
investigation may be made into the distribution of the es
timator p, the distribution of the test statistic F* might
be approximated by an F distribution with t-1 and b(k-t)


90
degrees of freedom (the distribution of F ). The closeness
of this approximation to the exact distribution is being
considered and will hopefully be reported in later work.
The following is a numerical example of a taste
testing experiment. The objectives of the experiment were
twofold. First, it was of interest to compare the degree of
preference for the treatments by the specific panelists used.
Second, it was suspected that correlation would be present
in the data and hence a test for its presence was to be per
formed.
Each of the trained panelists (denoted by the num
bers 1 through 10) was asked to evaluate five different
treatments (denoted by the letters A, B, C, D, and E) by as
signing a numerical value of 1 through 9 according to his or
her degree of preference for the treatments. The lower end
of this hedonic scale reflects an extreme non-preference
while the upper end reflects an extreme preference for the
treatments. Since there was an interest in measuring the
consistency of the panelists used and since each panelist
could evaluate six food samples effectively at one sitting,
each panelist was asked to evaluate each of the treatments
plus a replicate of one of the treatments. The data with
some calculations is


91
Treatments
where the kQ. are
i
calculated using the formula
which follows
formula in (3
tions are
kQ. = kT. l n..B
13 3
(3.2.3) and the are calculated using the
2.4). For treatment A, the necessary calcula-
kQ = (6) (89) {(2) (38+19) +(1) (34+27+23+36+31+42+30+22)}
A
= 175
and
£ 175 o c
ta JT4TT5T ~ *b *
The sums of squares for treatments (adjusted) and
blocks (unadjusted), the total sum of squares, and the sum
of squares for residual are calculated using the formulae in
Table 1. The results of these calculations are


Full Text
SOME NEW EXTENDED BLOCK DESIGNS AND THEIR ANALYSES
By
JACK FRANKLYN SCHRECKENGOST
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974

TO MY WIFE

ACKNOWLEDGMENTS
I would like to express my appreciation to Dr.
John A. Cornell for his guidance and assistance while di
recting this dissertation. My thanks, also, to the other
members of my advisory committee, Dr. F. W. Knapp, Dr. Frank
G. Martin, Dr. John G. Saw, and Dr. P. V. Rao, for their
helpful suggestions.
A belated thanks is expressed to Mr. Ronald E.
Boyer, a good teacher and a valued friend, who gave me much
encouragement from the very beginning.
Appreciation to my wife, Donna Rae, cannot be ex
pressed as deeply as is felt. I thank her for her patience
and understanding during my many hours of study, research,
and writing.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
ABSTRACT vii
CHAPTER
1 INTRODUCTION 1
1.1 Blocking Designs .... 2
1.2 Extended Complete Block Designs 2
1.3 Purpose of This Work 5
2 LITERATURE REVIEW 7
3 EXTENDED COMPLETE BLOCK DESIGNS WITH
CORRELATED OBSERVATIONS 16
3.1 Notation and Definitions 17
3.2 Intrablock Estimation of the
Treatment Effects 21
3.3 A Test for the Presence of Correlation 26
3.4 The Exact Distribution of SSR for p > 0 30
3.5 An Approximate Distribution of SSR
for p > 0 40
3.6 An Estimate of the Correlation 4 3
4 A PARTIALLY BALANCED GROUP DIVISIBLE ECBD ... 47
4.1 Definitions and Notation 50
IV

TABLE OF CONTENTS (Continued)
CHAPTER
4 (Continued) Page
4.2 Intrablock Estimation of the
Treatment Effects 53
4.3 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses 58
4.4 Mixed Model Analysis 62
5 A PARTIALLY BALANCED ECBD WITH THE L
ASSOCIATION SCHEME 7 66
5.1 Intrablock Analysis 68
5.2 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses 72
6 THE GENERAL PARTIALLY BALANCED EXTENDED
COMPLETE BLOCK DESIGN 77
6.1 Intrablock Analysis 78
7 CONCLUDING REMARKS AND A SENSORY TESTING
EXAMPLE 86
APPENDIX
1 THE EXACT DISTRIBUTION OF SSR FOR b = 2t
AND k = t+1 WHEN p>0 94
2 AN APPROXIMATE DISTRIBUTION OF SSR FOR
b = 2t AND k = t+1 WHEN p>0 98
BIBLIOGRAPHY 106
BIOGRAPHICAL SKETCH 109
V

LIST OF TABLES
Table Page
1 Intrablock Analysis of Variance for an
ECBD 25
2 Values of g and h for the Approximate
Distribution of SSR, I 42
3 Intrablock Analysis of Variance for Partially
Balanced Group Divisible Extended Complete
Block Designs 57
4 Mixed Model Analysis of Variance for Partially
Balanced Group Divisible Extended Complete
Block Designs 63
5 Intrablock Analysis of Variance for Partially
Balanced ECBD of the Association Scheme ... 73
Al Values of g and h for the Approximate
Distribution of SSR, II 99
A2 Comparison of the Exact Distribution and Two
Approximate Distributions of SSR 104
vi

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOME NEW EXTENDED BLOCK DESIGNS AND THEIR ANALYSES
By
Jack Franklyn Schreckengost
August, 1974
Chairman: Dr. John A. Cornell
Major Department: Statistics
An extended complete block design is a balanced
block design consisting of t treatments in b blocks each of
size k such that k varies between t and 2t. The balance
among the treatments is achieved by selecting duplicates of
some of the treatments for each block according to the scheme
followed when selecting blocks from the class of balanced
incomplete block designs. Under the assumption of an addi
tive model, it may be of interest to investigate the exist
ence of correlation between responses to the same treatment
in the same block. When a positive correlation between du
plicate observations is present, it has been previously shown
that k should be taken equal to t+1 for maximum efficiency
with the extended complete block designs when compared to
complete block designs.
A procedure for the test of the hypothesis of zero
correlation is presented as is a method for estimating the
vii

correlation if the hypothesis of zero correlation is rejected
in favor of the alternative hypothesis of positive correla
tion. Particular attention is given to the distribution of
the sum of squares for remainder, where remainder is defined
as residual minus duplication error, under the alternative
hypothesis of positive correlation. The distribution of the
sum of squares for remainder is necessary for calculating
the power of the test and for obtaining an approximation to
the distribution of the estimator of the correlation. A
specific formula for the distribution of the sum of squares
for remainder is given for the case t = b and k = t+1. The
exact distribution and an approximate distribution of the
sum of squares for remainder are also presented for the case
b = 2t and k = t+1.
The general partially balanced extended complete
block design is defined as a partially balanced block design
consisting of t treatments in b blocks each of size k greater
than t. The analyses of variance for the non-additive fixed
effects and mixed models are presented for the special class
of designs called partially balanced group divisible extended
complete block designs. The analysis of variance of the
additive fixed effects model is also presented for the class
of partially balanced extended complete block designs with
the (Latin Square) association scheme. The analysis of
variance of the non-additive mixed model for this class of
designs is mentioned briefly.
The intrablock analysis of variance for the
viii

additive model is developed for the general partially bal
anced extended complete block designs. Also, the recovery
of the interblock information and the combined intrablock
and interblock analysis for these general designs are men
tioned briefly.
The final chapter contains some comments about the
assumptions made and about directions for future study. A
numerical example of a taste testing experiment is also pre
sented with the resulting analysis for the balanced extended
complete block designs considered in this work.
ix

CHAPTER 1
INTRODUCTION
In many fields of experimentation, a distinction
that has long been implicit in the statistical literature is
the difference between experiments designed for the es
timating of absolute treatment effects and experiments of the
comparative type. In comparative experiments, the emphasis
is on performing comparisons between the effects of the dif
ferent treatments such as the effects of different doses of
a drug or the effects of different levels of nitrogen on the
average yield of soybeans. While the distinction between
comparative experiments and experiments designed to es
timate the absolute treatment effects individually is per
haps not always clearly defined, nevertheless, the idea of
a comparative type of experiment remains convenient and
useful.
For comparative experiments, it is clear that an
advantage is to be gained by comparing the treatments under
homogeneous conditions. To achieve this end, much of the
effort in choosing the homogeneous conditions is directed
toward the selection and use of block designs. Over the
years, both complete and incomplete block designs have been
discussed in detail. In this work, we shall be concerned
mainly with combinations of these block designs for use in
comparative type experiments.
1

2
1.1 Blocking Designs
In blocking experiments where the objective is the
comparison of different treatment effects, the number of
experimental units in each block may or may not equal the
number of treatments to be compared. When the size of the
block, where size refers to the number of experimental units
in each block, is equal to the number of different treatments
and each treatment is randomly assigned once with every other
treatment in each block, the design is known as a randomized
complete block design. If the size of the block is less than
the number of treatments, an incomplete block design may be
used. Incomplete block designs are common in applications
where either the number of treatments is large or the size
of the block must be kept small in order to ensure homogene
ity of the experimental units in each block.
Still another type of block design exists when the
size of the block exceeds the number of different treatments.
In this latter design, if each block contains first repli
cates of all of the treatments plus duplicates or second
replicates of some of the treatments, the design is called an
"extended complete block design." We now discuss such
designs.
1.2 Extended Complete Block Designs
In an attempt to increase the precision of the com
parisons between the effects of each of the treatments and
the effect of a control treatment, Pearce (1960) introduced

3
blocking experiments where in each block the control treat
ment was replicated. Later, Pearce (1964) considered possi
ble methods for designing experiments in which for a given
experiment the blocks are of varying sizes. The analysis
of a fertilizer experiment on strawberries in which an ex
tended complete block design was used is mentioned briefly
by Pearce (1963).
Extended complete block designs, as introduced by
John (1963), are block designs in which each block contains
a first replicate of all of the treatments plus a duplicate
or second replicate of some of the treatments. These second
replicates in each block comprise an incomplete block se
lected from the class of balanced incomplete block designs.
An example of an extended complete block design formed by
augmenting complete blocks of size three with balanced in
complete blocks of size two resulting in extended complete
blocks of size five is presented in Figure 1, where the three
treatments are denoted by A, B, and C.
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
complete block design
balanced incomplete block design
Figure 1. An extended complete block design
consisting of three blocks each of size five
experimental units containing treatments A,
B, and C.
Extended complete block designs can be used in a
variety of experimental situations. In sensory experiments

4
where the objective is the comparison of preferences for dif
ferent food samples (treatments) expressed by a panel of
judges (blocks), the number of food samples that a panelist
may effectively evaluate at a single sitting is limited but
may be more than the number of different samples to be eval
uated. Acquiring panelists for these sensory experiments is
often difficult and/or costly. Hence, if a panelist can ef
fectively evaluate all of the different samples plus repli
cates of some of the samples at a single sitting and if a
fixed number of observed values of each sample is necessary,
a smaller number of panelists would be required with the use
of an extended complete block design than if each panelist
could evaluate each of the samples only once. The use of a
smaller number of panelists would result in a savings in
terms of time and cost.
In an agricultural setting, an experimenter wishing
to compare the effects of different chemical sprays on cit
rus trees may have available more trees in a block than the
number of sprays to be tested. On the additional trees in
each of the blocks, second replicates of some of the dif
ferent chemical sprays could be applied. In an industrial
experiment, on a given day an experimenter may be able to ob
tain observed responses from each of the treatments as well
as responses from second replicates of some of the treat
ments. If he does not have enough time to observe the re
sponses from second replicates of all the treatments, an
extended complete block design could be used.

5
The analysis of a block design in which two treat
ments are applied to the experimental units in blocks of
size three was discussed by John (1962). The following year,
John (1963) introduced extended complete block designs and
presented their analysis. In his designs, the block size k
could vary between t and 2t, where t is the number of dif
ferent treatments used in the experiment. Trail and Weeks
(1973) generalized this latter work of John to include de
signs in which k is greater than 2t. In the papers by John
(1963) and Trail and Weeks (1973), the analysis of the fixed
effects model as well as the mixed model was presented in
detail.
The application of the extended complete block de
signs of John to the area of sensory evaluation was con
sidered by Cornell and Knapp (1972, 1974). Also considered
by Cornell (1974) was the efficiency of these designs com
pared to randomized complete block designs.
1.3 Purpose of This Work
The first part of this work will be concentrated
on extending the works by Cornell and Knapp (1972) and
Cornell (1974) with special emphasis on the area of sensory
evaluation. Specifically, we shall be interested in the
analysis of extended complete block designs where correlation
is present between duplicate responses to the same treatment
in the same block and the magnitude of the correlation is
constant over all treatments and blocks. In sensory

6
experiments for example, the presence of correlated observa
tions easily could arise as a result of using highly skilled
judges ( as the blocks). Therefore, we shall be interested
in testing whether there is any evidence of correlation pre
sent in the data. Futhermore, if there is sufficient evi
dence to indicate that correlation is present, we shall seek
to obtain an estimate of the magnitude of the correlation,
which is denoted by p. An approximate test on the treatment
effects in the presence of a value of p greater than zero
will be suggested, since an exact test on treatment effects
cannot be performed for this experimental situation.
In the second part of this work, we shall general
ize the work of Trail and Weeks (1973) to include extended
complete block designs generated by partially balanced in
complete block (PBIB) designs with two associate classes.

CHAPTER 2
LITERATURE REVIEW
The analysis of block designs in which the block
size k could vary between t and 2t, where t denotes the num
ber of treatments in the experiment, was first introduced by
John (1963). These designs, called extended complete block
(ECB) designs, contain in each of the b blocks first repli
cates of all of the treatments plus second replicates of k-t
of the treatments. The method taken by John of choosing the
k-t second replicates in each block was to use the class of
balanced incomplete block (BIB) designs of block size k-t.
Using formulae similar in structure to the formulae
used in the analysis of balanced incomplete block designs,
John discussed the intrablock analysis, the interblock anal
ysis, and the recovery of the interblock information. The
recovery of the interblock information was achieved by com
bining the two independent intrablock and interblock es
timates of the effects of each of the treatments.
In the intrablock analysis, in addition to ob
taining the treatment effects adjusted for blocks and the
unadjusted block analysis, John obtained estimates of both
the experimental error variation and the block x treatment
interaction. The measure of the interaction was obtained by
subtracting the experimental error variation from the
7

8
residual variation in the intrablock analysis of variance.
With the interblock analysis, however, an additive model was
assumed. That is, the block x treatment interaction variance
component was assumed to be zero. Using the assumption of
the additive model then, the combined estimate of each of
the treatment effects was formed using a linear combination
of the weighted intrablock and interblock estimates. The
weights used with the intrablock and interblock estimates
were the reciprocals of the estimates of their respective
variances. The special case where t = b and k = t+1 was
presented in detail.
Trail and Weeks (1973) considered the aforemen
tioned extended complete block designs (ECBD) as a special
case of the more general class of designs which they called
extended complete block designs generated by balanced in
complete block designs (BIBD). Their generalization of the
work of John (1963) included balanced block designs in which
the block size could exceed 2t. An example of this more
general design in which a balanced incomplete block design
is added to a double complete block design (CBD) is pre
sented in Figure 2. A second example of these more general
designs in which the complete block design is augmented by
two balanced incomplete block designs is presented in Fig
ure 3. In both of these figures, the three treatments are
denoted by the letters A, B, and C. It should be noted that
the treatments would be randomly assigned to the experimental
units within each block when the experiment is performed.

9
A
A
A
B
B
B
C
C
C
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
CBD
CBD
BIBD
Figure 2. An ECBD for three treatments generated
by a BIBD consisting of three blocks each of size
eight experimental units.
A
A
A
B
B
B
C
C
C
A
A
B
B
C
C
A
B
A
C
C
B
CBD
BIBD
BIBD
Figure 3. An ECBD for three treatments generated
by a BIBD consisting of three blocks each of size
seven experimental units.
In a block design in which t treatments are ar
ranged in b blocks, properties of the design can be obtained
by studying the elements of the incidence matrix of the de
sign. The incidence matrix N = (n^j) is a t x b matrix such
th
that n^j denotes the number of times the i treatment ap-
th
pears in the j block. The elements of the incidence matrix
for the extended complete block designs may be constructed
by appropriately summing the elements of the incidence matrix
of a balanced incomplete block design and the elements of
the incidence matrix of a complete block design.
For the extended complete block designs generated
by BIB designs, Trail and Weeks showed that the incidence

10
matrix N can be generated from the incidence matrix N* of
any balanced incomplete block design by using the equation
N = cQJ + (C;l-c0)N* (2.1)
where J is the incidence matrix of a complete block design,
that is, J is a t x b matrix of ones, and c^ and c^ are ele
ments of the set of positive integers. The model used by
the authors is
y = C^x + X23 + Y) +
(2.2)
where x is a t x i vector of treatment effects, 3 is a b x l
vector of block effects, y is a bt x i vector of interaction
effects, and e is a bk x l vector of independent random er
ror effects. Letting 1 denote a t x l vector of ones, I
i.U
denote the t x t identity matrix, and y. represent the &1"
1 X/
-H Vi
response to the i1" treatment in the j11 block, the vector y
and the matrices C, X^, and X2 in (2.2) are of the forms
y =
111
lln
11
211
tbn
tb
?i -
it
I
, x =
' it
1^
~t
' ~2
~t



- it
btxt
1



rt*
i
bkxl
btxb
and

11
:n
11
:n
21
C
. (2.3)
bkxbt
In our notation, 1 is an n~. x 1 vector of ones.
~n21 21
In addition to presenting the intrablock and
interblock analyses, Trail and Weeks expressed the formula
for calculating the combined estimates of the treatment ef
fects using the method presented by Seshadri (1963a) for
combining unbiased estimators. Trail and Weeks also dis
cussed how "best" designs might be obtained. They defined
the "best" design as that design for which the variance of
the difference between the intrablock estimates of the ef
fects of any two different treatments is a minimum for fixed
t and k. The minimum value of the variance of the difference
is achieved by minimizing the absolute value of the differ
ence Cq-c^, where Cq and c-^ are the magnitudes of the ele
ments in the incidence matrix N of the design.
An application of extended complete block designs
to sensory testing experiments was presented by Cornell and
Knapp (1972). Separate estimates of block x treatment

12
interaction and experimental error were obtained in their
analyses. Cornell and Knapp showed that the use of the
experimental error only as a measure of the within treatment
variability when comparing treatments results in a more ef
ficient test than when using the residual variation (the sum
of the experimental error and the interaction variation)
when some measure, however small, of interaction is present.
Replication of extended complete block designs was
also discussed by Cornell and Knapp (1974). Replication of
the designs was performed to achieve a balance between the
blocks and the treatments. By balance is meant, each and
every treatment appears in each block (is evaluated by each
panelist) the same number of times over the replications.
Hence, pairwise comparisons of the treatments in each block
can be made with equal precision.
With the replicated designs, the assumption of
negligible replication variation was made by the authors.
This assumption resulted in simpler expressions for the for
mulae for calculating estimates of the treatment effects as
well as the intrablock sums of squares when compared to the
formulae used with the unreplicated extended complete block
designs. An example of a replicated extended complete block
design is presented in Figure 4.
Using a non-additive model, Cornell (1974) dis
cussed the efficiency of extended complete block designs
compared to complete block designs for uncorrelated observa
tions. The efficiency of each design was defined as the

13
reciprocal of the variance of the difference between any
pair of treatment means with the respective design. To
illustrate, with the extended complete block design con
sisting of b blocks each of size k, the estimate of the
variance of the difference between any two treatment means
1 ? is
A
Var(x.-T.,
i i
ECB
2k(t-1) Q 2
b (k2-3k+2t) e
(2.4)
where cr* is the intrablock estimate of experimental error.
An estimate of the efficiency of the extended complete block
design would be the reciprocal of (2.4). (The author used k
for the size of the blocks in the balanced incomplete block
design used in the extension. In this work, k* denotes the
size of the blocks in the balanced incomplete block design
used in the extension, while k refers to the size of the
blocks in the extended complete block design.)
Replications
I
II
III
1
ABCAC
ABCBC
ABCAB
2
ABCBC
ABCAB
ABCAC
3
ABCAB
ABCAC
ABCBC
Figure 4. A replicated extended complete block
design consisting of three replicates of an ECBD.
In a complete block design with the same number of
replicates of each of the treatments, that is, with bk/t
complete blocks of size t, the estimate of the variance of
the difference between any two different treatment means is

14
Var(xi t,)cb bk jesidual (2.5)
where Residual ;''s t^ie residual mean square. From (2.4) and
(2.5) an estimate of the efficiency of the extended complete
block design compared to the complete block design is ob
tained using the ratio
Var(r-T,)rn
Ef f (ECB to CB) = 1 z
Var(xi-Ti,)ECB
t(k2-3k+2t) a2 ^
v residual x
l b J
k2(t-1) a*
To obtain estimated efficiency values for dif
ferent values of t and k, the value of the ratio of
^2 o .
aresidual to ae 1S re(2uired- For the value of this ratio,
Cornell used the ratio of the mean square for interaction to
the mean square for error, which is easily obtainable from
the analysis of variance table of the extended complete
block design. With this ratio, denoted by F, Cornell showed
that when the hypothesis of zero interaction effect is true,
resulting in F = 1, the extended complete block design is a
slightly less efficient design than the complete block de
sign with the same number of replicates of the treatments.
However, when F is greater than 1, the extended complete
block design is the more efficient design with the efficiency
increasing with increasing values of F.
Cornell (1974) also considered the situation where
a positive correlation p exists between the two responses to

15
a treatment in the same block. For an extended complete
block design having fixed balanced incomplete block size k*,
it was found that as P approaches one the efficiency of the
extended complete block design compared to the complete block
design decreases. In fact, the larger the value of k*
(k* + t) the faster the efficiency of the extended complete
block design approaches one-half that of the complete block
design with twice as many blocks. This implies that if one
suspects a positive correlation to be present between du
plicate treatment responses in the same block, one should
use k* equal to one for maximum efficiency if using an ex
tended complete block design.
Owing to the results previously found concerning
the effect of correlated observations on the efficiency of
extended complete block designs compared to complete block
designs, in the next chapter we shall investigate the for
mulation of the test of the hypothesis of zero correlation.
If there is evidence of correlation present between du
plicate treatment responses in the same block, we shall want
to estimate the correlation p. With an estimate of p, an
estimate of the variance of the difference between any two
intrablock estimates of the treatment effects can be cal
culated.

CHAPTER 3
EXTENDED COMPLETE BLOCK DESIGNS
WITH CORRELATED OBSERVATIONS
In the extended complete block designs discussed
to this point, we have observed that some of the treatments
in each block are duplicated. In the papers by John (1963),
Trail and Weeks (1973) Cornell and Knapp (1972) and Cornell
(1974), the responses to the duplicated treatments in each
block are assumed to be independent and are used to obtain
an estimate of the experimental error. Comments on the ef
ficiency of these designs when the duplicated observations
are not independent but rather are positively correlated
were made in the latter paper.
As mentioned previously in Section 1.3, in sensory
experiments correlated observations are a real possibility.
A panelist's response to a treatment might very likely be
positively correlated with his response to the duplicate of
the treatment, particularly if the panelist has previously
been trained for these experiments. The presence of posi
tive correlation between responses to the same treatment by
a panelist reflects a measure of the efficiency of the pan
elist. That is, the closer in magnitude the responses to
the same treatment by a panelist are, the more consistent
the panelist is in evaluating that treatment. Although the
correlation could be different for each treatment and/or
16

17
each panelist, we shall consider only the case where the
correlation is.assumed to be constant and equal for all pan
elists (blocks) and treatments.
3.1 Notation and Definitions
The parameters associated with an extended complete
block design are as follows:
t = the number of treatments,
b = the number of blocks,
k = the number of experimental units in each block
(block size),
r = the number of replications of each treatment in the
experiment,
X = the number of distinct pairs of experimental units
which receive any fixed pair of treatments while
appearing in the same block, and
N = (n..) = the incidence matrix, where n.. denotes the
ID ID
number of times the i^ treatment appears in the
j*"*1 block.
The following parameters are associated with the balanced
incomplete block design used to form the extended complete
block design:
t = the number of treatments,
b = the number of blocks,
k* = the block size,
r* = the number of replications of each treatment,

18
X* = the number of times over the b blocks each pair of
treatments appears in the same block, and
N* = (n* ) = the incidence matrix,
ij
The following identities involving the aforemen
tioned parameters are satisfied:
1. r = r*+b
2. k = k*+t
3. r*t = bk*
4. rt = bk
5. N = N*+J, where J is a t x b matrix of I's
6. X = 2r-b+X*
7. X*(t-1) = r*(k*-l)
8. X(t-1) = rk-3r+2b .
The model written in matrix notation is
y = C(yl + X.t + X 3) + e (3.1.1)
~ ~ ~Dt ~~
where all symbols are defined following (2.2) with the ex
ception of y and e. These parameters are y, the overall
mean, and e, a bk x 1 vector of random errors with the prop
erties
E 0
where E() denotes mathematical expectation and

19
E(eijieij
'pa2 i = i', j = j l jt l'
a2 i = i1 j = j 1 =5/' ,
0 otherwise
(3.1.2)
where a2 denotes the variance of the distribution from which
the errors are sampled and p denotes the correlation between
the duplicate observations. We shall assume that the values
of p lie in the interval 0 < p < 1.
Owing to the properties in (3.1.2) of the random
errors, then
E(y) = C(ylbt + X^r + X23)
and
(3.1.3)
Var(y) = E(ee') = V .
(3.1.4)
The matrix V consists of the following partitions corre
sponding to the form of the vector y in (2.3); on the main
diagonal of V are positioned the matrices
and a2 [ 1 ] ,
while there are zeros located in all other positions. Hence,
under the assumption of the normality of the random errors,
y ~ N( C(ylbt + X1t + X2B), V ) .
Before discussing the intrablock estimation of the
treatment effects, we illustrate the form of the matrix V by

20
referring to the extended complete block design presented in
Figure 1. If the vector of observations y is written as
Y =
All
A12
A21
A22
A31
Bll
B12
B21
B31
B32
Cll
C21
C22
C31
L y
C32 J
then the corresponding matrix V is
V =
1
P
1
P
1
P
1
P
1
P
1
P

21
On the main diagonal of the matrix V are positioned the
matrices
which correspond to the duplicate responses to a treatment
in the same block, and the matrices a2[ 1 ], which corre
spond to the response to only a single treatment in the
block.
We shall now discuss the intrablock estimation of
the treatment effects where both the treatments and the pan
elists are assumed to represent fixed effects in the model
in (3.1.1). The panelist effects represent fixed effects
either when it is desired to compare the specific panelists
used in the experiment or when the panelists chosen to eval
uate the treatments cannot realistically be assumed to rep
resent the general public. A case which comes to mind in
this latter situation is when trained panelists are used in
an attempt to enhance the efficiency of the comparisons be
tween the treatments.
3.2 Intrablock Estimation of
the Treatment Effects
To obtain the intrablock estimates of the effects
of the different treatments, we recall the form (3.1.1) of
the model
y = C(ylbt + X t + x23) + e ,

22
where the elements of the random error vector e have the
properties specified in (3.1.2). If the method of least
squares is used to obtain the intrablock estimates t of x,
the normal equations are
IbtSi
ibt?ibt
IbtHi
it?;
XjC'y
=
?12ibt
?i5?i
51552
-
- *2~bt
?255i
;j5;2
(3.2.1)
where the bt x bt matrix D = C'C and the hat (~) denotes
estimate. According to the definitions and parameter iden
tities specified in Section 3.1, the forms of the matrices
~1~~1' XiDX2' and X2DX2 n (3-2*1) are
~1~X1 = rit ?1??2 = i? and X'DX2 = klfa ,
and therefore the normal equations (3.2.1) are expressed
as
G
-
T
=
_ B
bk
ri;
ki
rl
rl
N
~t
~t
~
kl.
N'
kl.
~b
~b
y
A
T
A
L e J
(3.2.2)
where G denotes the grand total of the observations and T
and B are the t x 1 vector of treatment totals and the b x 1
vector of block totals, respectively.
For a solution to the normal equations, both sides
of the equality in (3.2.2) are premultiplied by the matrix

23
1/bk O O
O Ifc -N/k
O -N'/r I,_
~ ~b -J
A A
and the constraints 1' r = 0 and 1,' 3 = 0 are imposed on the
~t~ ~b~
parameter estimates. Corresponding to these particular con
straints imposed, the following relation results
G/bk
r /s _
y
kQ
=
A
Ax
rB N'T
(rkl N1N)3
~ JD ~ ~ ~
where kQ = kT NB and A = rkl^ NN'. Characteristic of
these designs, the matrix NN' = (rk-At)I + AJ. Hence, the
matrix A can be expressed in the simple form
(l/k)A = (At/k)[lt (l/t)j] .
From the equation kQ = At in (3.2.3), the t x 1 vector x of
intrablock estimates of the treatment effects is
x = kQ/At ,
(3.2.4)
where A = (rk-3r+2b)/(t-1). Furthermore, with the properties
J_ l- /N
of the vector e specified by (3.1.2), the i element x_^ of
the vector x is unbiased for x. since E(e)=0 and with l'x=0
1 -V, -V ~
Cov (x\ x ,)
(t-1)(Ak+2[A(k+t)-2rk]p)o2/(tA)2 i = i'
-Var(xi)/(t-l)
f i ^ i'
(3.2.5)

Since we are interested in the pairwise comparison of the
treatment effects, we also have
(3.2.6)
and
, 4{X (k+t)-2rk} 7
+ po-
tx2
(3.2.7)
In the formula (3.2.7), the quantity 2kcr2/tX on
the right-hand side of the equality is the variance of the
difference between the intrablock estimates of the treatment
effects t. and x., in the case of uncorrelated errors. Thus,
if correlation is present between responses to the same
treatment in the same block, the variance of the difference
between the intrablock estimates of any two treatments, over
all blocks, is greater than the variance between the same
two treatments when the observations are uncorrelated, since
the quantity [X (k+t) 2rk] is always positive.
The intrablock analysis of variance table is pre
sented in Table 1. It is clear from Table 1 that an exact
test does not exist for testing the hypothesis of equal
treatment effects when a non-zero correlation is present.
If we wish to test this hypothesis, an approximate test must
be performed. Before suggesting an approximate test for the
equality of the treatment effects when p > 0, we shall first
consider a procedure for testing for the presence of correla
tion. If correlation is present, we shall need to know how
this correlation affects the distributional properties of

TABLE 1
Intrablock Analysis of Variance for an ECBD
Source
df
Sum of Squares
EMS*
Treatments
(adjusted)
t-1
SST = (k/Xt) l Q?
A i 1
E(MST )
A
Blocks (unadjusted)
b-1
SSB = (1/k) l B2 -
j 3
(G2/bk)
Residual
bk'
-t-b+1
(by subtraction)
E(MSR )
e
Total
bk-1
TSS = l l l y? (G2/bk)
i j £ ^
*
E (MST ) = a2 {1
A
+ iP
Xk
[(k+t> _2rkR + k(t-i) J
CM -H
E(MSR ) = a2 +
1
, { (b-1) (t-1) d> bJ*'
-*> )pa2
= Xk'-fb-ry {X ~ 2(k+t) bk] + 4rk}

26
the sums of squares associated with the two sources, treat
ments and residual.
3.3 A Test for the Presence of Correlation
Although one of the initial steps in the analysis
of data arising from a comparative type experiment is a test
on the equality of the treatment effects, in this section we
shall first investigate the possible presence of correlation
between duplicate observations in the same block. The rea
son for this investigation is that if correlation is present,
an exact test of the hypothesis of equal treatment effects
cannot be performed and an approximate test must be derived.
Furthermore, if correlation is present, the formula for the
variance of the intrablock estimates of the treatment ef
fects contains p and an estimate of p is needed to estimate
this variance. The same is true of the formula for the dif
ference between the intrablock estimates of the effects of
two treatments.
To determine if there is evidence of correlation
in the data, we shall consider a test of the hypothesis
Hq: p = 0. If this hypothesis is rejected in favor of the
alternative hypothesis H : p > 0, we shall conclude that the
duplicate observations are not uncorrelated and insist on
finding an estimate of p. If, on the other hand, the hy
pothesis is not rejected, the inference made here shall be
that the duplicate observations are uncorrelated, or, if
they are correlated, there is not sufficient information in

27
the data to show that the magnitude of the correlation is
greater than zero.
In order to test the hypothesis Hq: p = 0, we
first need to derive the form of a test statistic. To this
end, recall from Table 1 that the source of variation termed
residual has bk-b-t+1 degrees of freedom. The residual var
iation is a composite of duplication variation as well as
another source of variation which we shall call remainder
variation. To see this, let d^j be the difference or range
th th
of the observations made on the i treatment in the j
block so that if n^j = 2, then d^j > 0, and if n^j = 1, then
dj-j = 0. If each of the d^j is squared and these squares
are summed over all treatments and blocks, then the resulting
quantity when multiplied by one-half is called the sum of
squares for duplication variation (SSDV). In summation no
tation, the sum of squares for duplication variation is
given by
t b
SSDV = h l l d[. .
i j
The sum of squares for remainder (SSR) is found by cal
culating the difference, sum of squares for residual SSDV.
To derive the form of a test statistic for testing
the hypothesis, we require the separate distributional prop
erties of the sum of squares for duplication variation and
the sum of squares for remainder. The distributional prop
erties of SSDV and SSR are most easily obtained by rewriting
SSDV and SSR as quadratic forms and then using our knowledge

28
of the distributional properties of quadratic forms. In ma
trix notation then, SSDV and SSR can be expressed in the
quadratic forms
SSDV = y' [i. -CD_1C'] y = y'A,y (3.3.1)
~ L~bk ~ J ~ ~ ~1~
and
SSR I' SC5'1 e 525 it (bt E
(bt e5;22>Js' X '
(3.3.2)
where both the matrices and A^ are real, symmetric, and
idempotent.
In the quadratic form (3.3.1) for SSDV, the matrix
A^ consists of the square matrices
and [ 0 ]
on the main diagonal and zeros in all other positions. This
partitioning corresponds identically to the partitioning of
the matrix V as defined and illustrated in Section 3.1.
Thus, by direct computation
A V = (l-p)a2A. (3.3.3)
~ -L~ ~
and since A^ is a real, symmetric, idempotent matrix, so
also is the matrix A V/{ (1-p) cr2 } The trace of the matrix
~ 1 ~
A^ is equal to b (k-t) and therefore under the assumption of
normality of the errors,

29
SSDV ~ (l-p)a2 X(k_t) / (3.3.4)
where x* denotes a random variable with a central chi square
distribution with v degrees of freedom. With E(*) denoting
mathematical expectation, then
E(SSDV) = b(k-t)(1-p)o2 (3.3.5)
and
E(MSDV) = (1-p)a2 (3.3.6)
where MSDV denotes the mean square for duplication variation.
(An alternate derivation of the distribution of SSDV is pre
sented in Section 3.4.)
In the quadratic form (3.3.2) for SSR, the matrix
A^ is real, symmetric, and idempotent. However, it can be
shown that if c is a scalar constant, the equality
AVA
~2~~2
is not true in general. (To see this would only require
working through the small example where t = b = 3 and
k = r = 4.) Hence, unlike SSDV, the random variable SSR
does not have an exact weighted chi square distribution when
p > 0. The exact distribution of SSR is discussed in Sec
tion 3.4.
The distributions of SSDV and SSR are independent,
since A VC = 0. The trace of the matrix A V is equal to
~ 1 ~ ~ Z ~
(b-1) (t-1) (l+ square for remainder is

30
E(MSR) = (l+p) cr2 (3.3.7)
where 4> is defined in Table 1.
When the hypothesis HQ: p = 0 is true, the random
variable SSR has a chi square distribution. Hence, to test
the hypothesis Hq: p = 0, the test statistic F^ = MSR/MSDV
is used. When p = 0, the test statistic has an F distri
bution with (b-1) (t-1) and b (k-t) degrees of freedom in the
numerator and denominator, respectively. Therefore, the hy
pothesis is rejected in favor of the alternative hypothesis
H : p > 0 for large values of F .
a p
A brief discussion of the power of this test is
reserved for a later section, since we must first consider
the distributional properties of SSR when p > 0.
3.4 The Exact Distribution of SSR for p > 0
In the previous section, it was shown that when
p > 0, the distribution of the sum of squares for remainder
does not in general have a weighted chi square distribution.
This is because the matrix A2 of the quadratic form SSR does
not necessarily satisfy the equality A2VA2 = CA2, w^ere c
some constant and V is the covariance matrix of the observa
tions. In this section we shall seek to find an expression
involving independent chi square distributed random vari
ables for which the moments of the distribution of SSR can
be found.
The approach we shall use to find the distribution
of SSR involves rewriting the model in (3.1.1) in the form

31
yij = P + T + gj + (l-p)?Szijjl + p^Ujlj (3.4.1)
i 1 / 2 f t / ^ 1 / 2/ Id ^ dnd = 1 / 2 / j ,
where the x^ are the treatment effects, the Bj are the block
effects, p is the magnitude of the correlation between du
plicate responses to the same treatment in the same block,
and zj_j£ and uij are independent, identically distributed
normal random variables each with mean zero and variance a2.
Let us now define the random variable SSR|u^j to
be the usual sum of squares for interaction (which we have
chosen to call remainder in our additive model) given the
u- Since the conditional distribution of SSR given u-
1J J
can be found, the form of the unconditional distribution of
SSR is obtained by taking the expectation of the random
variable SSR|u^j with respect to u^j.
The distribution of the random variable SSR|u^j is
given by
SSR | u^ j ~ a2 (1 p) X2(b_i) (t-1) ( 2 (-p) r2) (3.4.2)
where x2 (1) denotes a random variable with a non-central
chi square distribution with v degrees of freedom and non
centrality parameter X. In the noncentrality parameter of
the distribution in (3.4.2), R2 is of the form
>2 _
= min l Z x* B*)2 ,
(3.4.3)
T*, B* i j
where x* and B are the parameters in the conditional distri
bution corresponding to y+x^ and y+Bj, respectively, in the
unconditional distribution. In order that the distribution

32
in (3.4.2) be expressed in a simpler form, it is convenient
to write R2 in terms of the design parameters t, b, k, and
r. To this end, let D~^ denote the diagonal matrix of cell
frequencies associated with a t x b table in a two-way cross
classification, that is,
n
11
n
12
n
lb
n
21
Then (3.4.3) may be written in the form
R2 = min (u-y)'D*1(u-y) ,
y :F y = 0
where
(3.4.4)
y' (yxl, ..., ylb, v21, ytb)'
u' (ui;l, ..., ulb, U21, utb) '
and F is a matrix of constraints for additivity in a t x b
cross classification. The form of the matrix F is given
shortly.
The quantity R2 equals the minimum value of the

33
quadratic form in (3.4.4). To find this minimum value, we
write
Q = (u-y)'D*1(u-y) + 2H'F'y
where II is a (b-1) (t-1) x l vector of Lagrange multipliers.
Differentiating Q with respect to y and setting the result
equal to zero, the minimum value in (3.4.4) is
R2
u'F(F'D*F)
u .
(3.4.5)
An expression for (3.4.5) involving the design
parameters t, b, k, and r for our problem requires the la
tent roots of the matrix F(F'D*F) ^F'. If these roots are
denoted by 0, then 0 are the solutions to the equation
I ?(r?*!,)~V ei(b-l)(t-l) I = 0 (3.4.6)
Using the following identity,
6(b-1) (t-1)
F' F'D*F
Qb+t-lj 0F.D*F p.F|
F'D*F
01
(b-1) (t-1)
- F(F'D
tF) 1fi
we find that b+t-1 roots of (3.4.6) are zero while the re
maining (b-1)(t-1) roots are positive. These latter
(b-1)(t-1) positive roots can be found by solving for 0' in
the equation
F'D*F
0 1 F F | = 0
(3.4.7)

34
and setting 9 = 1/9'. Since the 9' are non-zero, the frac
tion 1/9' is not undefined.
We should like to express equation (3.4.7) in a
simpler form to find the values of 9'. To this end, the ma
trix of constraints F is written as the direct product of
two other matrices. This direct product is
F = F(t-l) F(b-l) ,
where the two matrices are defined by
and
F(t-l)
14-i
-I
t-1
tx(t-1)
F(b-l) =
l-i
-I
b-1 J
bx(b-1)
Then
F'F
1
-b-1
f
where
L = I + J (3.4.8)
~a ~a
That is, L is an a x a matrix with 2's on the main diagonal
~a
and l's in all other positions. We now make use of the fol
lowing theorem and corollary to find the values of 9' satis
fying (3.4.7) .
Theorem. Let W be an m x m matrix with the distinct latent

35
roots w_. with respective multiplicities m., j = 1, 2. Let
R be an m x s matrix satisfying R'R = I Then the roots of
~ ~ ~ ~s
the matrix R'WR are the values of 0' satisfying the equation
R'WR 6'I | = 0. These values are
9' = w10" + w2(l-0") ,
where 0" are the solutions of | R'MR 0"I | =0 with
V v M = (W w2)
Proof; The proof follows directly by replacing M with
(W-w^Im)/(w^-w2) in the determinant | R'MR 0"I | = 0 and
simplifying.
Corollary. If W is defined as in the theorem, and F is an
m x s matrix of full rank s < m, then the solutions 0' of
the equation | F'WF 0'F'F | = 0 are
0 = w-^0 + w2 (1-0" ) ,
where 0" are the solutions of | F'MF 0"F'F | =0 with M
defined in the theorem.
Proof: There exists a matrix K such that K'F'FK = Ig. Let
r' = k'F' and apply the theorem.
In our problem, W is the matrix D*. Therefore
from the theorem, M is a diagonal matrix of ones and zeros.
Referring to the corollary to obtain the values of 0' satis
fying (3.4.7), we now need only to find the solutions 0" of

36
| F'MF 6"F'F |=0. (3.4.9)
Because the structure of the matrix F'F depends on the
matrices L^._^ and Lb_^ as defined in (3.4.8), a forward
Doolittle procedure is performed on La to find that the val
ues of 0" satisfying (3.4.9) are the same as the values of
0" satisfying
I e"i(b-l) (t-l) I = 0 > (3.4.10)
where F* = H(t) 0 H(b) with the matrix H(a) defined as the
first a-1 columns of the a x a Helmertz orthogonal matrix.
Since M = M* and M'M = M, then F;MF* = (MF*)'MF*, and the
positive values of 0" satisfying (3.4.10) are the positive
solutions 0" satisfying
| MF*F;M 0"Ibt |=0. (3.4.11)
Since we may write
f*f; = (Gt 0 Gb)/bt ,
where Ga is an a x a matrix with a-1 on the main diagonal
and -1 in the other positions, then the positive solutions
0" of (3.4.11) are functions of the positive solutions 0*
satisfying the equation
| M(Gt 0 Gb)M 0*Ibt |=0, (3.4.12)
where 0* = bt0".
At this stage, an expression for the random vari
able SSR is presently untenable for general t, b, k, and r.

37
In the remainder of this section, we shall derive an expres
sion for the random variable SSR when p > 0 for the special
case t = b and r = k = t+1. A similar expression for SSR
when p > 0 for the case b = 2t, k = t+1, and r = 2 (t+1) is
presented in Appendix 1.
For the special case considered in this section,
the matrix M(G 0 G, )M in (3.4.12) has the non-zero partition
~ ~t ~b ~
{(t-1)+ J /
for which the positive latent roots are t(t-2) and t(t-l)
with multiplicities t-1 and 1, respectively. Hence, since
these roots are simple multiples of the solutions 0" of
(3.4.9), the 0" are
0"
(t-2)/t with multiplicity t-1
(t-l)/t 1
0 (t-l)2-t
With the use of the corollary where w^ = % and w^ = 1/ the
values of 0' satisfying (3.4.7) are
0'
(t+2)/2t with multiplicity t-1
(t+1)/2t 1
1 (t-l)2-t
and the values of 0 satisfying (3.4.6) are
0 =
2t/(t+2) with multiplicity
2t/(t+1)
1 11 11
t-1
1
(t-l)2-t

38
In the distribution of the conditional random
variable SSR given u^ in (3.4.2), we may now express R2 as
R2 =
2t X2
2t 2
t+2 At-1 t+1
xi + x(t-D2-t
Furthermore, upon taking the expectation of SSR|u^_. with re
spect to u,, and using the notation SSR = E(SSR|u,.), the
lj 1 lj
sum of squares for remainder when p > 0 is distributed as
SSR/a:
al Xvx + a2 xv2 + a3 Xv3 '
(3.4.13)
where
and
al = 1 + p TT
a2 1 + P t+1 '
a3 1 ,
V1 = t-1 ,
v2 = 1 ,
v3 = (t-1)2 t
An approximate distribution to (3.4.13) will be obtained in
Section 3.5.
The conditional distribution of the usual sum of
squares for error given u^ is
SSE|uj ~ a2(1-p) X(k_t)(0) ,
and the random variable SSE|u. is independent of u_^
Hence, we have

39
SSDV ~ a2(1-p) x(k-t)(0) '
where SSDV denotes the expectation of the random variable
SSE|u^j with respect to j. Since the duplication var
iation sum of squares random variable is also independent of
the random variable SSR|u^j, then the random variable SSDV
is independent of the random variable SSR. By independent
random variables is meant, the distributions of the random
variables are independent. (The independence of the distri
butions of SSDV and SSR was established previously in Sec
tion 3.3 through the use of quadratic forms.)
To this point, it has been shown that for the
special case t = b and r = k = t+1, the random variable SSR
is distributed as a sum of weighted independent chi square
random variables when p > 0. We still do not have the exact
form of the density of the random variable SSR which is nec
essary in order to specify the distribution of the random
variable SSR/SSDV. The distribution of SSR/SSDV is also
necessary in order that we may calculate the power of the
test of the hypothesis HQ: p = 0 for non-zero values of p.
Since an exact form of the density of SSR would likely re
quire an excessive amount of work and since a simpler form
of an approximating distribution of SSR would suffice for
our problem in a majority of cases, an approximate distri
bution of the random variable SSR will now be considered.
A check on the accuracy of the approximate distribution when
compared to the exact distribution of SSR when p > 0 is pre
sented in Table A2 of Appendix 2.

40
3.5 An Approximate Distribution of SSR for p > 0
In this section we shall consider an approximation
of the distribution of the sum of squares for remainder when
p > 0 for the special case t = b and r = k = t+1. An ap
proximate distribution of SSR when p > 0 for the special
case b = 2t, k = t+1, and r = 2(t+1) is given in Appendix 2.
There are numerous approaches that could be used
to approximate the distribution of SSR. The approach used
in this section (and also used in Appendix 2) was introduced
by Box (1954). The rationale in selecting Box's ap
proximation lies not only in its relative ease of appli
cation but also in the fact that it was shown by Box that
the approximate distribution compared to the exact distri
bution of a quadratic form is fairly good except when small
differences in probability are to be examined. We now state
the theorem in his paper which we shall use.
Theorem. The quadratic form
is distributed approximately as where
g
(3.5.1)
and
h
(3.5.2)
In both of the expressions for the scale constant g and the

41
degrees of freedom h, the are scalars and the are the
degrees of freedom of the respective chi square random vari
ables that are summed to form Q, j = 1, 2, ..., p.
In our problem we seek to approximate the distri
bution of the random variable SSR, where
SSR/a2 ~ ax + a2 xv + a3 X*
v.
with a^ and vj, j = 1, 2, and 3, defined following (3.4.13).
From the theorem by Box previously stated then, if the aj
and Vj are substituted into (3.5.1) and (3.5.2) to find g
and h, respectively, we may say that SSR is approximately
distributed as a scaled chi square random variable with h
degrees of freedom. A tabulation of the values of g and h
corresponding to the integer values 3, 4, 5, 6, and 7 of t
and to some values of p in the interval between zero and one
is presented in Table 2, where h has been rounded to the
nearest integer and g has been rounded to four decimal
places.
The approximate distribution of SSR may be used,
when testing the null hypothesis Hq: p = 0 against the gen
eral alternative hypothesis H : p > 0, to compute the power
Cl
of the test under the alternative hypothesis for values of p
greater than zero but less than one. Since the distribution
of SSR is independent of the distribution of SSDV, then
under the alternative hypothesis we approximate the distri
bution of the statistic

TABLE 2
Values of g and h for the Approximate Distribution of SSR, I
p
t
V,
a.
a
g
h
1
2
3
1
2
3
0.1
3
2
1
1
1.025
1.05
1
1.0253
4
0.3
1.075
1.15
1.0776
4
0.5
1.125
1.25
1.1319
4
0.7
1.175
1.35
1.1880
4
0.9
1.225
1.45
1.2457
4
0.1
4
3
1
5
1.04
1.06
1.0205
9
0.3
1.12
1.18
1.0645
9
0.5
1.2
1.3
1.1121
9
0.7
1.28
1.42
1.1629
9
0.9
1.36
1.54
1.2166
9
0.1
5
4
1
11
1.05
1.0667
1.0173
16
0.3
1.15
1.2
1.0554
16
0.5
1.25
1.3333
1.0978
16
0.7
1.35
1.4667
1.1441
16
0.9
1.45
1.6
1.1940
15
0.1
6
5
1
19
1.0571
1.0714
1.0149
25
0.3
1.1714
1.2143
1.0485
25
0.5
1.2857
1.3571
1.0867
25
0.7
1.4
1.5
1.1291
24
0.9
1.5143
1.6429
1.1754
24
0.1
7
6
1
29
1.0625
1.075
1.0131
36
0.3
1.1875
1.225
1.0431
36
0.5
1.3125
1.375
1.0778
35
0.7
1.4375
1.525
1.1168
35
0.9
1.5625
1.675
1.1599
35

43
MSR
Fp MSDV
with a weighted F distribution with h and b(k-t) degrees of
freedom, where the weight is given by g/(l-p). That is,
1-P
g
' b (k-t)
(3.5.3)
approximately. The probability that Fp exceeds some value
Fq is approximately equal to the probability that the random
variable F^^-t) exceec^s (l-p)Fo/g*
A method for estimating the magnitude of the
correlation between duplicate responses observed with the
same treatment in the same block will be discussed in the
following section.
3.6 An Estimate of the Correlation
In Section 3.3 a procedure for testing the hy
pothesis Hq: p = 0 was outlined in detail. As mentioned at
the beginning of Section 3.3, the test of the hypothesis of
zero correlation is normally the first action to be taken
during the analysis of the experimental data. If the hy
pothesis is rejected, we should then want an estimate of p.
The estimate of p would be used when estimating the variance
th
of the intrablock estimate of the effect of the i treat
ment as shown in (3.2.5) and/or when estimating the variance
of the difference between the intrablock estimates of the
effects of two treatments as shown in (3.2.7). Still anoth
er use for the estimate of p would be when estimating the

44
efficiency of the extended complete block design compared to
the complete block design as shown in the paper by Cornell
(1974). The value of the relative efficiency of the two de
signs could be very useful when considering designs for sub
sequent experimentation, particularly in a sensory exper
iment where the same panelists are to be used in additional
experiments.
Referring to the formulae (3.3.6) and (3.3.7), we
see that the expectations of the mean squares for duplica
tion variation and for remainder variation are
E(MSDV) = (l-p)CT2 (3.6.1)
and
E(MSR) = (l+4>p) CT2 (3.6.2)
where is defined in Table 1. If a linear combination of
these mean squares and a ratio of two linear combinations of
them are considered, we can express the correlation p in the
form
p = {E (MSR) E (MSDV) }/{E (MSR) + c¡>E (MSDV) } (3.6.3)
As an estimate of p then, the expectations in (3.6.3) are
replaced by their respective mean squares resulting in the
formula
p = MSR ~ MSDV (3.6.4)
M MSR + <{>MSDV

45
Similarly from (3.6.1) and (3.6.2), an estimate of a2 may be
obtained as
^2 MSR + MSDV
1 + 4
Since the calculated value of MSR is always greater
A
than or equal to zero, then from (3.6.4) p > -l/cj>. Further
more, since the calculated value of MSDV is always greater
A
than or equal to zero, then p < 1. If these extremes are
considered as the endpoints of the range for the values of
A A
the estimate p, then -l/ < p < 1. However, since we are
interested only in the values of p in the interval between
A
zero and one, any negative value of p calculated is con
sidered meaningless and is set equal to zero in this case.
(Setting a negative estimate of a non-negative parameter e-
qual to zero is a procedure practiced when estimating vari
ance components in random and mixed models.)
A
The distribution of the random variable p depends
on the forms of the distributions of the random variables
SSR and SSDV. It was shown in Section 3.4 that the random
variable SSR is distributed as a weighted sum of independent
chi square random variables. Although an approximate distri
bution of SSR was given in Section 3.5, an approximation to
\
the distribution of p is at present untenable. Nevertheless,
A
the first two moments of the distribution of p could be ap
proximated using a Taylor series. That is, the formula in
A
(3.6.4) for p may be expressed in a Taylor series and the
A
mean and the variance of the distribution of p could be

46
approximated with a finite number of terms in the series by
taking the appropriate expectations.

CHAPTER 4
A PARTIALLY BALANCED
GROUP DIVISIBLE ECBD
In the extended complete block designs presented
thus far, the class of balanced incomplete block designs was
only considered in the extended portion of the b blocks.
That is, in making the extended complete blocks of size k,
we have considered in combination with the complete blocks
of size t only balanced incomplete blocks of size k-t. By
restricting attention to the use of balanced incomplete block
designs only in the extended portion, the extended complete
block designs retain the property of balance among the treat
ments. By balance is meant, the off-diagonal elements in
the matrix A (or NN') in (3.2.3) are all equal, resulting in
a single value of the variance for all pairwise treatment
comparisons. Hence, all pairwise treatment comparisons could
be made with the same precision.
When it is not necessary to have equal precision
for all pairwise treatment comparisons or when to achieve
balance the use of balanced incomplete block designs requires
a large number of replications of the treatments or possibly
too many blocks, a partially balanced incomplete block design
(PBIBD) might be used in the extended portion of an extended
complete block design. To illustrate this point, consider
an extended complete block design consisting of six treatments
47

48
in blocks of size nine. If a BIBD were used in the extended
portion of the.blocks, the balanced design would require ten
extended blocks supporting fifteen replicates of each of the
six treatments. On the other hand, if a PBIBD were used in
the extended portion, only six blocks supporting nine repli
cations of each treatment would be required.
In this and subsequent chapters then, the use of
PBIB designs in the extended portion of the extended complete
block design will be considered. Specifically, we shall
limit our attention to the use of PBIB designs with two asso
ciate classes. By relaxing the requirement of balance, in
most cases we do not sacrifice that much precision when con
sidering PBIB designs with two associate classes where in
stead two variances are required for making all pairwise
comparisons of the treatments. The two variances arise be
cause with each treatment a subset of the t-1 other treat
ments are first associates while the remaining other treat
ments are second associates. One variance is used for pair
wise comparisons among the treatments that are first asso
ciates while the second variance is used among treatments
that are second associates. The generalization to partially
balanced incomplete block designs with more than two asso
ciate classes should be straightforward.
For the balanced extended complete block designs,
the case where responses to the same treatment in the same
block were positively correlated was presented in detail.
The general theory developed in Section 3.4 on the exact

49
distribution of SSR when p > 0 with the additive model could
be used with partially balanced extended complete block de
signs. Hence, since the theory is general, the analysis of
partially balanced ECB designs with correlated observations
will not be presented. Our discussion will be limited to the
additive and non-additive models when all observations are
uncorrelated.
The group divisible association scheme for t = mn
treatments where m and n are integers is derived by parti
tioning the treatments into m groups of n treatments each
with those in the same group being first associates and those
in different groups being second associates. For example,
with six treatments (denoted by the numbers 1, 2, 3, 4, 5,
and 6) a group divisible association scheme for three groups
of two treatments each would be given by the 3x2 rectangu
lar array
1 2
3 4
5 6 .
In the array, treatments in the same row (1 and 2, 3 and 4,
5 and 6) are first associates. Treatments not in the same
row as a specified treatment are second associates of that
treatment. For example, the set of second associates of
treatment 1 consists of the treatments 3, 4, 5, and 6.
For a PBIBD with the group divisible association
scheme and incidence matrix N*, the matrix N*N*' may be

50
arranged in a particular pattern that will be described in
detail in Section 4.2. The particular pattern of the matrix
N*N* facilatates finding a solution of the normal equations
for the intrablock estimates of the treatment effects. The
pattern of N*N*' carries over to the matrix NN', where N is
the incidence matrix of the extended complete block design.
Extended complete block designs generated by the class of
PBIB designs with the group divisible association scheme
will be called "partially balanced group divisible extended
complete block designs."
4.1 Definitions and Notation
An extended complete block design generated by a
PBIBD is defined as a connected, two-way classification with
the following properties:
1. Each treatment is applied either Cq or c-^ times in
a block, c. >0, i = 0, 1.
i
2. In the incidence matrix of the design, replacement
of Cq by zero and c^ by unity results in the inci
dence matrix N* of a PBIBD (with two associate
classes).
It follows from this definition that the incidence
matrix N of such a design can be generated from the incidence
matrix of any PBIBD. That is, given a PBIBD with the inci
dence matrix N* and denoting by J a matrix of l's (the inci
dence matrix of a complete block design), the incidence
matrix of an extended complete block design (ECBD) generated

51
by a PBIBD is
N = cQJ + (C;l-c0)N* (4.1.1)
This equation is identical to the equation (2.1) for the
incidence matrix of a balanced ECBD except that N* is now
the incidence matrix of the generating PBIBD.
The parameters associated with an ECBD generated
by a PBIBD are as follows:
t = the number of treatments,
b = the number of blocks,
k = the block size,
r = the number of replications of each treatment in the
experiment,
= the number of distinct pairs of experimental units
which receive any fixed pair of i^ associates
while appearing in the same block, i = 1, 2,
th
n^ = the number of i associates of each treatment,
i = 1, 2, and
4-Vi
p., = the number of treatments that are both j asso-
th
ciates of treatment a and k associates of treat
. .th
ment 3 given that a and 3 are i associates.
The corresponding design parameters of the generating PBIBD
will be denoted by t, b, k*, r*, A*, n^, and Pl
owing to the definitions given above for the param
eters of the generating PBIBD as well as the ECBD generated
by a PBIBD, the following relationships are satisfied:

52
1.
r =
clr* + (b-r*)cQ = cQb + (crc0)r*
(4.1.2)
2.
k =
gxk* + (t-k*)cQ = cQt + (Gl-c0)k*
(4.1.3)
3.
r*t
= bk*
(4.1.4)
4.
rt
= bk
(4.1.5)
5.
Xi
= (cl"c0)2Xi + c0(2r-bcQ) i = 1, 2
(4.1.6)
6.
rk
- (cQ+c^)r + CQC^b = n-^X^ + n2X2 *
(4.1.7)
In
matrix notation, the non-additive model
is
y = + ?11 + + Y) + e (4.1.8)
where y is the overall mean effect, x is a t x 1 vector of
treatment effects, 8 is a b x 1 vector of block effects and y
is a bt x l vector of block x treatment interaction effects.
Letting lt denote a t x l vector of l's, It denote the t x t
th
identity matrix, and Yj_j£ represent the l response to the
th
i treatment in the j block, the vector y and the matrices
C, Xj, and X2 in (4.1.8) are of the forms
y =
ym
Ylln
Y211
11
Ytbn
tb
' ?1 =
it
it
'it
r Xr)
btxt
it
it J
btxb
bkxl
and

53
1
~n
11
C
bkxbt
4.2 Intrablock Estimation of
the Treatment Effects
Consider the model in (4.1.8) where the interaction
effects are all zero. This additive model is written as
y = C(ylb + xr + X28) + e (4.2.1)
Setting up the normal equations exactly as detailed in Sec
tion 3.2 results in the equality
G/bk
r a
y
kQ
=
/A
Ax
rB N'T
(rkl, N'N) §
~ D ~ ~ ~ J
where all symbols are defined in Section 3.2.
The solution of kQ = Ax in (4.2.2) depends upon
the form of the matrix
A
rkl NN'
t
(4.2.3)

54
which in turn depends upon the form of the matrix N* from
(4.1.1). For a PBIBD with the group divisible association
scheme, the matrix N*N* may be arranged in a particular
pattern as follows. For an association scheme consisting of
m groups each containing n treatments (so that t = mn), let
the treatments in the first group be labeled 1 through n; in
the second group, the treatments are labeled n+1 through 2n;
th
and so on, so that in the m group, the treatments are la
beled (m-l)ntl through mn. Now, with the corresponding
blocking plan and the labeled treatments listed in numerical
order in the incidence matrix N*, the matrix N*N*' becomes
N*N*' = (r*-A*) (Im In) + (Asj'-A*) (Im Jn) + X* (Jm Jn) ,
(4.2.4)
where denotes the direct (Kronecker) product of two matri
ces and A| and A| are the number of distinct pairs of exper
imental units over all blocks which receive any fixed pair
of first and second associates, respectively, in the same
block in the generating PBIBD.
Since the incidence matrix of the ECBD generated
by a PBIBD equals cgJ + (c]_-cq)N*, then
NN' = (c-l-Cq) 2N*N*' + cQ (2r-cQb) (Jm Jn) (4.2.5)
By substituting (4.2.4) into (4.2.5) and simplifying, we
obtain
NN* = [(c0+c1)r-c0c1b-X1] (Im In) + (XrX2) (Im Jn) +
(4.2.6)
X0(J J) .
2 ~m ~n

55
To facilitate finding a solution of kQ = At for x
in (4.2.2), an-expression for the matrix A may now be found
by substituting (4.2.6) for NN' into (4.2.3). By further
simplification using the identity (4.1.7) involving X-^ and
X2, the result is
kQ = [(nXi+njjkj) (Im 0 In) (X1-X2)(Im 0 Jn) -
x2h '
th
of which the i element is given by
kQi = (nX1+n2X2)xi (^i~^2^G^Ti^ X2T*
(4.2.7)
where G(x^) denotes the sum of all the estimated treatment
. th
effects of the treatments m the group containing the 1
treatment. In other words, G(x^) is the sum of the effect
th
of the 1 treatment plus the effects of all first associates
th
of the i treatment. Also in (4.2.7), x. denotes the sum
of all the estimated treatment effects. However, one of the
restrictions used on the treatment effects to obtain (4.2.2)
was to set x. equal to zero. Therefore, if G(kQ^) denotes
the sum of the kQ^ in (4.2.7) plus the kQ^1, i ^ i', corre-
. th
spondmg to all the first associates of the 1 treatment,
we obtain
G(kQ^) (nX j+n2 X 2) G (x ^) n (X^-X2) G (x^)
= tX2G(xi) (4.2.8)
By substituting G(Xj_) from (4.2.8) back into (4.2.7), the
th
intrablock estimate of the effect of the i treatment is

56
found by solving for x^ in the expression
X -X
(nX +n X )x = kQ. + G (kQ. ) .
1 2 2 i i i
(4.2.9)
The difference between the intrablock estimates of
the effects of the treatments i and i', i ^ i', can be writ
ten as
A A
T -X .
1 1
t
= MQ.-Q. ,) +
i x'
VX2
tX0
[g (kQ ) -G (kQ ) ] .
i l
Under the assumption that the errors are normally distributed
, 2
with mean zero and variance structure o I, the difference
e~bk
A A
x^-x^, has the properties
E(t.-t,
i i
T -T .
1 1
i
and
Var(x.-T.
i i
,)
2ka2
r i & i' are 1st associates
nXi+n2X2
2ka2
e
nXi+n2X2
(1 +
i & i' are 2nd
associates .
The intrablock analysis of variance table for the
partially balanced group divisible extended complete block
design is presented in Table 3. In the sum of squares ex
pressions in Table 3,
CM
_1
bk
l l
i j
l
a
i j Z
r

TABLE 3
Intrablock Analysis of Variance for Partially Balanced
Group Divisible Extended Complete Block Designs
Source
df
Treatments (adjusted) t-1 SSTA = (n~~Tn~~~r H^kQi)2 + t\ ^ (kQi)G(kQi)]
Sum of Squares
1' **22' i tX2
Blocks (unadjusted) b-1
SSBjj = | l B? CM
j
Remainder
(t-1) (b-1) SSR =11^- R?j "
1 D ID
SSTa SSB0 CM
Error
b(k-t) sse -III Yin 11 ht: RL
j H
i D ID
Total
bk-1
TSS
yij£
i j £
CM
* E (MSTa)
= ae + k(t-1) t'At E(MSR) = a* + y'Dy and E(MSE) =
- rr 2
a where
A = rklfc NN' and D = C'C
EMS*
E (MSTa)
E (MSR)
E (MSE)
cn
-J

58
As can be seen from Table 3, the ratio of
block, and
treatment
the mean square for remainder to the mean square for error
(previously called the mean square for duplication variation)
provides a statistic for a test on the validity of the addi
tive model. If the hypothesis is not rejected, a test of
the hypothesis of equal treatment effects could be performed
using the statistic the ratio of the mean square for treat
ments (adjusted) to the mean square resulting from pooling
the mean square for remainder with the mean square for error.
If the hypothesis is rejected, then we might wish to consider
a non-additive model. With the non-additive model, we could
concern ourselves with the estimation of the block x treat
ment interaction effects or concentrate on testing the hy
pothesis of equal treatment effects in the presence of inter
action effects.
The next section gives validity to both of the
above mentioned tests of hypotheses. As will be shown, the
sums of squares in Table 3 are each distributed as weighted
chi square random variables and each sum of squares is dis
tributed independently of the others.
4.3 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses
In order to validate the tests mentioned at the
end of Section 4.2, we must first obtain the distributions

59
of the sums of squares in Table 3. The approach that will
be used to derive the distributions of these sums of squares
is to express them as quadratic forms and use our knowledge
of the distributions of quadratic forms.
The sums of squares in Table 3 can be written in
the quadratic forms
SETA = X'dlX = X :,..,+n2y S'ibt k ;2525>?1
H5 X '
SSBu X'*2X = X' BE tbcx2xc J) y ,
SSR = y'A3y = y' (CD_1C' £ CX^C' A-^ y
SSE = y'A4y = y' (Ibk CD 1C') y ,
(4.3.1)
(4.3.2)
(4.3.3)
(4.3.4)
and
TSS = y'A5y = y' (Ibk ££ J) y (4.3.5)
where in (4.3.1), F = Im IR.
In the quadratic forms (4.3.1) through (4.3.5),
each of the matrices A^, A2, A^, A^, and A^ is real, sym
metric, and idempotent, and
4
I ip = *5
P=1 P
Also, the ranks of the five matrices, where r(Ap) denotes
the rank of the matrix A are
~ IT
riA^) = t-1 ,

60
r(A2) = b-1 ,
r(A3) = (b-1)(t-1) ,
r(A4) = b(k-t) ,
r(Ac) = bk-1 = l r(A ) .
P=i ~p
Hence, by applying Theorem 5 in Searle (1971) on the dis
tribution of quadratic forms, it is found that when
y
N( y,
aeibk
/
then
y'A Y ~ CTe xr(A ) ( y'^p!y2a ) (4.3.6)
P
for p = 1, 2, 3, 4 and the y'A y are mutually independent.
~ ~ IT ~
The distributional forms in (4.3.6) will be used in con
structing the aforementioned tests of hypotheses.
The test of the assumption concerning the validity
of the additive model corresponds to the test of the hy
pothesis of zero interaction effects when the non-additive
model is considered. To test the validity of the additive
model, the test statistic used is the ratio of the mean
square for remainder to the mean square for error. If the
additive model holds, then the test statistic possesses an
F distribution with the appropriate degrees of freedom, and
the hypothesis concerning the validity of the model is re
jected for large values of this ratio. If the additive
model assumption is valid, a test of the hypothesis of equal

61
treatment effects would be performed using the ratio of the
mean square for treatments (adjusted) to either the mean
square for error or the pooled mean square for remainder
plus error. Under the hypothesis of equal treatment effects,
this ratio possesses an F distribution. On the other hand,
if there is evidence to reject the assumption of the addi
tive model in favor of the non-additive model, then an ap
proximate test on the treatments could be performed if de
sired.
The intrablock analysis given in this section and
in the previous section is, of course, an analysis of a model
in which the treatment, block, and interaction effects are
considered as fixed effects. In comparative type experi
ments, usually we seek to draw inferences about the effects
of the specific treatments used in the experiment and hence
the assumption of fixed treatment effects presents little
argument. Now the block effects, on the other hand, may be
fixed or random. In this latter case the emphasis may be on
drawing inferences about the magnitude of the variance of
the population from which the sample of block effects was
assumed to be drawn. A model in which the treatment effects
are fixed while the block effects are random is called a
mixed model.
Since the partially balanced group divisible ex
tended complete block designs would frequently be used with
random block effects, we shall now present the analysis of
a mixed model for these designs.

62
4.4 Mixed Model Analysis
In the mixed model analysis, all symbols in the
model are defined exactly as in (4.1.8) with the exception
of 8 and y. In the mixed model, the parameters 8 and y are
assumed to be independently distributed normal random vari
ables with
8 ~ N( 0, oIb ) ,
I ~ N< btibt > '
and each distributed independently of the random errors e.
Under these assumptions, the expectation of the vector of
observations is
E (y) = c(ylbt + x-jt) = y
and the variance of the observations is
(4.4.1)
Var(y) = e[c(X28 + y) + e] [c (X28 + y) + e] '
= ?X2XC-a + CC'at + a^Ibk = V
(4.4.2)
The analysis of variance table for the mixed model
is presented in Table 4. The differences between the en
tries in Table 4 and Table 3 for the fixed effects model are
the replacement of the source of variation called blocks
(adjusted) for the source of variation called blocks (un
adjusted) and the expected mean square expressions. Of
course other differences exist in the use of the two tables,
namely in the interpretation of the tests of hypotheses.
The expected mean squares in Table 4 were obtained using the
identity

TABLE 4
Mixed Model Analysis of Variance for Partially Balanced
Group Divisible Extended Complete Block Designs
Source
df
Treatments (adjusted) t-1
Blocks (adjusted) b-1
Interaction (t-1)(b-1)
Error
b (k-t)
SS*
SST
A
SSB,
SSI
SSE
Expected Mean Squares**
1
e + t-1 * abt + k (t-1)
r 'Ax
e + (si- 7 + hT<>* 7F (r-k) s,}at2
b-1 V1 r 2' b b-11^ rk
ae + (b-i)Tt-l) {S1 k S2 ^*}abt
2J bt
Total
bk-1
TSS
* SSBa = SSTa + SSBy SSTy and the other SS are defined in Table 3 with SSI = SSR .
p
** A = rklt NN' s = l l ni- for p = 1, 2, and 3, X** = (cq+c-^Aj. CqCjT and
** = bk(n>^2A2) '1 + ) (kslS2-2slS3+S|) + (X1s22X**s1) } .

OJ

64
E(y'Apy) = tr(ApV) + u'Apu
involving the expectation of a quadratic form with
y ~ N( y, y )
and where tr(A V) denotes the trace of the matrix A V.
~ IT ~ ir ~
To test the hypothesis HQ: = 0, the statistic
takes the form of the ratio of MSI to MSE, where MSI equals
SSI/(b-l)(t-1) and MSE equals SSE/b(k-t) with SSI and SSE
defined in Table 4. The hypothesis is rejected for values
of this ratio larger than the appropriate tabular F value.
/s _
If the hypothesis is rejected, an estimate of could
be found using the analysis of variance approach. An es-
timate of the random blocks component of variance may
also be obtained using the analysis of variance approach.
When the hypothesis HQ: = 0 is rejected, we
may still wish to test the hypothesis that the treatment
effects are small relative to the magnitude of the inter
action variation. For this test the statistic is given by
s-,^- (1/k) s2~(j)5
(b-1) (t-lT~
MSTa -
t-1
MSI
s1~ (1/k) s2-(f)*
(b-1)(t-1)
t-1
MSE
where expressions for calculating the values of s^, s2, and
<})* are presented at the bottom of Table 4. The statistic FT
can be shown to possess an approximate F distribution with
f and b(k-t) degrees of freedom in the numerator and denom
inator, respectively, where the number f is computed by the

65
procedure given by Satterthwaite (1946) for approximating
the distribution of the estimate of a variance component.
If the hypothesis HQ: cr£t = 0 is not rejected, a
simpler test on the treatment effects may be performed. For
this case, the statistic is the ratio of the mean square for
treatments (adjusted) to the pooled mean square consisting
of the mean squares for interaction and error.

CHAPTER 5
A PARTIALLY BALANCED ECBD WITH
THE L2 ASSOCIATION SCHEME
The partially balanced group divisible extended
complete block designs presented in Chapter 4 comprise a
large class of partially balanced ECB designs. However,
because of its general applicability, still another class of
partially balanced ECB designs to be considered is the class
of partially balanced extended complete block designs with
the L2 (Latin Square) association scheme. The L2 asso
ciation scheme for t = n2 treatments is characterized from
the arrangement of the treatments in a square array. The
classification of the treatments to one another in the L2
association scheme is such that the treatments in the same
row or same column are first associates and the treatments
not in the same row or same column are second associates.
For example, with sixteen treatments (denoted by the numbers
1 through 16) an 1>2 association scheme would be determined
from the square array
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The first associates of treatment 1 are the treatments 2, 3,
66

67
4, 5, 9, and 13, while the second associates of treatment 1
are the treatments 6, 7, 8, 10, 11, 12, 14, 15, and 16.
In this chapter, we shall present only the intra
block analysis of the partially balanced extended complete
block designs with the L2 association scheme. The mixed
model analysis follows the procedure outlined in Section
4.4. In fact, the only difference in the final analysis of
the mixed model with the two association schemes is that with
the partially balanced ECB designs with the L2 association
scheme, the expected mean square expressions are slightly
more complicated than the corresponding expressions with the
partially balanced group divisible ECB designs.
In the intrablock analysis, we shall make use of
row sum and column sum operators which we denote by RS()
and CS () respectively. For the square array containing the
treatment effects x^ through Xg
the row sums RS(x^), RS(Xg), and RS(Xg) are given by
RSx^ = ti+t2+t3 = RS(x2) = RS(x3) ,
RS(x5) = t4+x5+t6 ,
and
RS(x 9)
t7+t8+t9

68
The column sums for the same treatment effects are given by
CS(t1) = T1+T4+T7 = cs(t4) = CS(x7) ,
cs = t2+t5+t8 ,
and
CS(t9) = t3+t6+t9 .
Note that for the i1" treatment effect,
RS (T) + CS(Ti) = s1(Ti) + 2xi ,
where S^(t^) denotes the sum of the effects of all treatments
that are first associates of the i*" treatment.
For a PBIBD with the association scheme, the ma
trix N*N*' may be arranged in a particular pattern which will
be described in Section 5.1. The particular pattern of the
matrix N*N*1 facilitates solving the normal equations to ob
tain the intrablock estimates of the treatment effects. As
expected, the pattern of N*N*' is reflected in the matrix
NN', where N is the incidence matrix of the ECB design.
5.1 Intrablock Analysis
The definitions and notation presented in Section
4.1 will be used again in this section with the exception
that N* now denotes the incidence matrix of the generating
PBIBD with the association scheme.
Let us consider an additive model consisting of
the overall mean parameter, a treatment parameter, a block

69
parameter, and a random error term. Since the interaction
effects are all zero, the model in (4.1.8) may be written as
y = C(ulbt + Xjj + X28) + e (5.1.1)
Using the same form of the normal equations as detailed in
Sections 3.2 and 4.1, we have
G/bk
r a -
y
A
kQ
=
At
A
rB N'T
(rklb N'N)B
where all symbols are defined in Section 3.2.
A
The solution of kQ = At in (5.1.2) depends upon
the form of the matrix NN', since A = rklt NN'. For a
PBIBD with the L^ association scheme, the matrix N*N* may
be arranged in a particular pattern as follows. In the
association scheme, let us label or number the treatments
from 1 to n2. Then, if the treatments are listed in numeri
cal order in the incidence matrix according to the particu
lar blocking plan used, the matrix N*N* is of the form
N*N*' = (r*-2A*+A*)(I 0 I ) + (A*-A*)(I 0 J )
- ~ 2 ~n ~n 1 2 ~n ~n
+ (A*-A*)(Jn 0 In) + A*(Jn 0 Jn) (5.1.3)
J. ^
where 0 denotes the direct product of two matrices and where
over all blocks in the generating PBIBD, A* and A* are the
number of distinct pairs of experimental units which receive
any fixed pair of first and second associates, respectively,
in the same block.

70
Since the matrix N can be expressed in the form
N = CqJ + (c-l-CqJN* ,
then
NN* = (C-Cq)2N*N*' + c0(2r-c0b)(Jn 0 Jn) (5.1.4)
Now if the form (5.1.3) of the matrix N*N* is substituted
into NN' in (5.1.4), the resulting expression for NN' after
simplifying is
NN = [(c1+c0)r-c0c1b+X2-2A1] (In 0 IR)
+ (Xl-X2)[(Jn In)+(In Jn)] +X2(Jn0Jn) .
(5.1.5)
This expression for NN' can now be substituted into
A = rklfc NN' ,
so that the intrablock estimates of the treatment effects
are obtained by solving the equation
kg = (n[2Aj+(n-2)X2] (Jn In> i.\1~X2) [(Jn Jn) + (Jn Jn>]
- X2 th
of which the i element is
kQi = n[2X1+(n-2)X2]xi (Xx-X2) [RS (t) +CS (t) ] (5.1.6)
since -A2t. = 0. In (5.1.6), RS(t^) and CS(t^) are the row
th
sum and column sum, respectively, of the estimated i treat
ment effect.

71
To obtain the expression for from (5.1.6), we
need the row and column sums of kQ^ in (5.1.6). These sums
are respectively
RS(kQ) = [nX1+n(n-l)X2]RS(Ti) (5.1.7)
and
CS(kQ^) = [nX-^+n (n-1) X2] CS (x^) .
(5.1.8)
Replacing RS(x^) and CS(x^) in (5.1.6) by their respective
equivalent expressions in (5.1.7) and (5.1.8), then kQ^ in
(5.1.6) may be rewritten as
*>i = n[2X1+(n-2)X2lT1-nX1+Mn-l)X2 [RS (kQi)+CS (kQl) ] ,
th
and hence the intrablock estimate of the effect of the i
treatment is given by the equation
n [2X^+(n-2)X2]Ti = +
A1 A2
nX^+n(n-1)X^
[s1(kQi)+2kQi]
(5.1.9)
where Sj(kQ^) is the sum of kQ^ in (5.1.6) plus the kQ^,,
th
i ^ i', corresponding to the first associates of the i
treatment.
The difference between the estimated effects of
treatments i and i' can be written as
Ti~Ti' =
X X
k (QiQi>) + nX +n(n-1)X
i.
[S1(kQi)-S1(kQi,)+2k(Qi-Qi,)] .
Under the assumption that the random errors in (5.1.1) are

72
normally distributed with mean zero and variance structure
A A
eIbk' the difference has the properties
A A
and
A A
Var(Ti-xi,)=
Kl> + X.+n (n-T)'rf i & i' are lSt associates
X1 X2
K[l + nx (n-)X 1' i & i' are 2nC^ associates,
where K = kcr* /n[2X-^ + (n-^)^] and n = /t*
The intrablock analysis of variance table for the
partially balanced extended complete block designs with the
association scheme is presented in Table 5. All symbols
in Table 5 are defined as they were defined in Table 3 with
the exception of S^(kQ^) which has been defined in this
section.
5.2 Distributions of the Sums of Squares
and Relevant Tests of Hypotheses
As in Section 4.2, before considering any relevant
tests of hypotheses, we shall obtain the distributions of the
sums of squares in Table 5. Resorting once again to the the
ory of the distributional properties of quadratic forms, the
sums of squares formulae in Table 5 are expressed as quad
ratic forms in the following matrix notation

TABLE 5
Intrablock Analysis of Variance for Partially
Balanced ECBD of the Association Scheme
Source
df
Sum of
Squares
EMS*
X.
-X,
Treatments (adjusted)
t-1
ssta =
2nX-
k
L+n (
n-2)X2
Kq? +
i
1
nX-^+n
2
(n-l)X2
[Qisi
(Q) +
2Q?]}
E (MST
Blocks (unadjusted)
b-1
SSBu
1
k
l
j
B2
D
- CM

Remainder (t-
1)(b-1)
SSR =
l
l
1
. R? -
SST -
SSB 7
- CM
E (MSR)
i
j
nij
ID
A
U
Error b
(k-t)
SSE =
I
i
1
j
I y
a
2 _
i j a
l l r
i j I]
R2 .
ID
E(MSE)
Total
bk-1
TSS =
I
i
l
j
C
2 _
i j £
CM
* E (MST ) = a2 + x1 At E(MSR) = a2 + y' Dy and E(MSE) = a2 where
A e k(t-l) ~ ~~ e ~ ~~ e
A = rkl NN' and D = C'C .
co

74
SST = y'A y = y'
A ~ ~1~ ~
C(I J. X X D) X,
2nA2_+n (n-2) X2 ~ ~bt k ~2~2~ ~1
A -A
IX + 1 2 h] X' (I 1 DX X' ) C' y (5.2.1)
(n-1)A o ~ ~1 ~bt k ~~2~2 ~
'~t nAi+n
SSB = y' A y = y' I(CX X'C I J) y ,
U ~ ~2~ ~ k ~~2~2~ b ~ ~
(5.2.2)
SSR = y'A y = y' (CD-1C' -ij-A -A)y, (5.2.3)
~ ~3~ ~ ~~ ~ bk ~ ~1 ~2
SSE = y'A y = y' (I CD_1C') y ,
~ ~4~ ~ ~bk ~~
(5.2.4)
and
TSS = y'A y = y' (I 1 J) y ,
~ ~5~ ~ ~bk bk ~
(5.2.5)
where in (5.2.1), the matrix H is given by
H = (I J ) (J I ) .
~n ~n ~n ~n
The matrices A., A., A-, A., and Ac are each real,
~1 ~2 ~3 ~4 ~5
symmetric, and idempotent, and
l A = A_ .
L. ~p ~5
p=l
Furthermore, the ranks of the five matrices, where r(A ) de-
~ hr
notes the rank of the matrix A are
~P
r(^l}
t-1 ,
b-1 ,
(b-1)(t-1) ,
b(k-t) ,
and
r(A ) = bk-1 = V r (A )
~ R _L, ~ D
p=l

75
Hence, by again referring to Theorem 5 in Searle (1971) on
the distribution of quadratic forms, we find that when
y ~ N( H' aeibk } '
then
yApy ~ e xr(A )( H'^pH/2ae ) (5.2.6)
for p = 1, 2, 3, and 4 and the y'A y are mutually independ-
~ ~ IT ~
ent. The distributional forms (5.2.6) can now be used to
construct statistics for the tests of hypotheses in the
usual manner.
The test of the assumption concerning the validity
of the additive model corresponds to the test of the hy
pothesis of zero interaction effects when the non-additive
model is considered. To test the validity of the additive
model, the test statistic used is the ratio of the mean
square for remainder to the mean square for error. If the
additive model assumption holds, then the test statistic
possesses an F distribution with the appropriate degrees of
freedom, and the hypothesis concerning the validity of the
model is rejected for large values of this ratio. If the
additive model assumption is valid, a test of the hypothesis
of equal treatment effects would be performed using the ratio
of the mean square for treatments (adjusted) to either the
mean square for error or the pooled mean square for remainder
plus error. Under the hypothesis of equal treatment effects,
this ratio possesses an F distribution. On the other hand,

76
if there is evidence to reject the assumption of the addi
tive model in favor of the non-additive model, then an ap
proximate test on the treatment effects could be performed
if desired.

CHAPTER 6
THE GENERAL PARTIALLY BALANCED
EXTENDED COMPLETE BLOCK DESIGN
In Chapter 4 the analysis of the fixed effects
model as well as the analysis of the mixed model for the
class of partially balanced group divisible extended com
plete block designs was presented in detail. In Chapter 5
the analysis of the fixed effects model was presented in de
tail and the analysis of the mixed model was mentioned for
the class of partially balanced extended complete block de
signs with the I12 association scheme. These two special
cases of the general partially balanced (GPB) extended com
plete block designs were presented in detail, not only be
cause of their general applicability, but also because the
constants (containing the parameters of the designs) were of
the same form for both special cases.
In this chapter, we shall present the analysis of
the GPB extended complete block designs. For this general
class of designs, it will be necessary to introduce new con
stants to aid in simplifying the forms of the necessary cal
culating formulae. The introduction of these new constants
stems from the desire to conform to the use of the standard
notation for the analysis of general partially balanced in
complete block designs. In particular, the new constants
are d., d and A which correspond to the constants c c ,
12s 12
77

78
and A as defined and used by Bose and Shimamoto (1952) in
the analysis of PBIB designs.
We now present the intrablock analysis of variance
for the GPB extended complete block designs.
6.1 Intrablock Analysis
Let us consider the model in (4.1.8) where again
the interaction effects are all zero. The additive model is
written as
y = C(ylbt + XjT + X23) + e (6.1.1)
where the symbols y, C, y, lbt, X^, t, X2, 3, and e are de
fined following (4.1.8). The normal equations are set up
exactly as presented in Sections 3.2 and 5.1, resulting in
G/bk
1
< 2-
\
kQ
--
A
At
A
rB N'T
(rklb -N'N)3
where G is the grand total of the observations, Q is the
vector of adjusted treatment totals, B and T are respec
tively vectors of the unadjusted totals of the blocks and
treatments, N is the incidence matrix of the design, r is
the number of replications of each treatment in the exper
iment, and k is the block size. As in the previous sections
where the intrablock analysis was discussed, the matrix A is
given by
A = rklt NN' ,

79
A A A
and y, x, and 3 are respectively the estimates of the param
eters y, x, and g in (6.1.1).
To find an expression for the intrablock estimate
th
of the effect of the i treatment, we note from the t x 1
a th
vector of equations kQ = Ax in (6.1.2) that the i element
can be written as
b t
kQi = rkTi J I nijnhjTh ,
(6.1.3)
th
where is the adjusted total for the i treatment and n^
and n^j are elements of the incidence matrix N of the de
sign. The quantities
1 l ijVTh
3 h J
can be expressed in terms of the parameters of the design as
follows:
l 1
j h
nijnhjTh
l nij*i
3
l xiTi
1
_ A
l Vi
i = h
s t
i f h, i & h are 1 associates
i ^ h, i & h are 2n<^ associates
(rk-n-^X j-n2X2^ Ti
XjS^(x^) ( i / h
^2^2 i = h
s t
i & h are 1 associates,
nd
i & h are 2 associates
where S-^ix^) is the sum of the estimated treatment effects
of all treatments (n-^ in number) that are first associates
th a
of the i treatment, and likewise, S2(x£) is the sum of the

80
estimated treatment effects of all treatments (n^ in number)
J_ T_
that are second associates of the iT'n treatment. By re
placing the quantities
£ l n. .n t
j h x3 h3 h
in equation (6.
,1.3) with their equivalent expressions in-
/\
A
volving S^(t_^) and S2(t^), we maY wr;i-te equation (6.1.3) in
the form
A A Z\
kQi = ^ni^l+n2^2^Ti ^lSl^Ti^ ^2S2^Ti^ (6.1.4)
Equation (6.1.4) is now summed over the first asso
ciates of the x treatment resulting in the expression
kS1(Qi) =
"hVi + Sl(i> (nlXl+n2A21lPl'i2p2)
+ S2(.)(-X1p^1-X2p2) (6.1.5)
Summing (6.1.4) over the second associates of the i1" treat
ment results in
kS2(Q.) =
X2n2il + S2(il> (Vl+n2X2'XlP12X2P22)
+ S1(.)(-X1p>2-X2P2) (6.1.6)
In (6.1.5) and
(6.1.6) px is the number of treatments that
3k
are both a jth
+ Vi
associate of treatment a and a associate
of treatment 6
given that treatments a and 8 are ix asso-
ciates. As in
Bose and Shimamoto (1952), we write equations
(6.1.5) and (6.
.1.6) in the forms
kS1(Qi) =
= -n1X1i + a11S1(?l) + a12S2(.) (6.1.7)
and

81
kS2(Qi) n2A2Ti
(6.1.8)
all nlXl + n2X2 Xlpl X2P12
a
12 Xlpl X2P2
a
21 "Xlp2 X2P22
and
X1P2 X2P22 '
In order to express the intrablock estimate x^ as
a function of Q^, S^(Q^), and S2 (Q) having arrived at the
equations (6.1.7) and (6.1.8), we interrupt the development
A A
briefly to see how the sums S^ix^) and S2(x^) can be re
placed in (6.1.7) and (6.1.8) with the quantities S^(Q^)
and S2(Q^). To this end, consider the linear combination
L = k2Q + d1kS1(Qi) + d2kS2(Qi)
(6.1.9)
1^1vvi
involving only the Qi and parameters of the design with d^
and d2 being constants consisting of linear functions of the
aij, i < j = 1, 2. If both (6.1.7) and (6.1.8) are substi
tuted into (6.1.9) for kS^Q-^) and kS2(Q^), respectively,
the resulting expression for is
= [k (njX^+n2X2)-djn^X-^-d2n2A2] x^ + (d^a^j+d2a22_~kA^) S^ (Q^)
(6.1.10)
+ (d1a12+d2a22-kX2)S2(Q^) .
The quantities d-^ and d2 in (6.1.10) are now chosen so that

82
upon equating the right-hand side of (6.1.9) to the right-
hand side of (6.1.10), the equation (6.1.10) becomes
k2Qi + d1kS1(Qi) + d2kS2(Qi) = k (n^+n^) (6.1.11)
A
That is, equation (6.1.11) expresses the estimate as a
function of the quantities S-^iQ^), and s2 (Q^) .
To obtain the values for d-^ and d2 so that equation
(6.1.11) is as shown, we require the identities
k^i = d^ (a-j^+n^A-^) + d2 (a2^+n2X2) (6.1.12)
and
kX2 d^ ^2 ^a22+n2^2^ (6.1.13)
Solving equations (6.1.12) and (6.1.13) simultaneously by
the use of determinants, we have d-^ = D-^/D and d2 = D2/D,
where D, D-^, and D2 are given by
D (a^^tn^X^) (a22+n2A2) (a^2"^~^2_^i^ (a2*^tn2X2) ^ (6.1.14)
~ kX^(a22tn2X2) kX2 (a2i+n2'*''2^ t (6.1.15)
and
D2 kA2(a^*^~^"^2_^}_^ """* kXi (a-^2+n^X^) (6.1.16)
Substituting for a^, a^2, a2^, and a22 in (6.1.14), (6.1.15),
and (6.1.16), simplifying, and writing D = k2A the fol-
b
lowing equations for d-^ and d2 are obtained
kAgd-j_ = A1 (n1X1+n2X2 + X2) + (X-^X2) (^ 2P2 ^ lp 2 ^
and

83
k^s^2 ^2 l"^^2^2^^ 1^ **" ^2^ ^2P12^lp12^ f (6.1.18)
where
k2A
s
^nl^l+n2^2+^l^ ^nl^l+n2^2+^2^
+ (X1-X2) [ (niAi+n2A2) (P3_2"P2^ + X2P12 ^lp12^ *
The expressions (6.1.17) and (6.1.18) for the values of d-^
and d2, respectively, are now substituted back into (6.1.11).
th
The intrablock estimate of the effect of the i
treatment is
Ti k(n1A1+n2A2) tk2Qi + ^i^Qd.) + d2S2(kQi)] (6.1.19)
which has the alternate forms
Ti k(n1A1+n2X2) t(k dl)kQi + {dl d2)s1(kQi)] (6.1.20)
and
- = 1
Ti k(n^X^+n2X2)
[Oc-a2)kQ. + (a2-a1)s2(kQ.)]
(6.1.21)
The use of one of the alternate forms would probably be more
convenient since only one sum, either S-^(kQ^) or S2(kQ^),
for each treatment need be calculated.
/\ /v
The difference of the estimated effects of
treatments i and i', i ^ i', using the alternate form in
(6.1.20) above is
Ti"Ti* =
k (n.X^+nX.) [(k"di)k(Qi Q,)
11 2 2
+ (d1-d2)k{S1(Q.)-Si(Q.I)}]
Under the assumption that the random errors in (6.1.1) are

84
independent normally distributed random variables with mean
r.
zero and variance a^r the difference has the properties
and
A A
Var(xi-xi,)
f2(k"dl)ae st
-:rr; i & i' are 1 associates
nl^l+n2^2
2(k-a2)a a
-;rrr 1 & 1 are 2 associates .
[nlh+n2X2
The analysis of variance table for the general
partially balanced extended complete block design is exactly
of the same form as Table 5 except that the calculating
formula for the sum of squares for treatments (adjusted) is
now given by
SSTa
1
k2(n]_^]_+n2^2)
I
[ (k-d1) (kQi) 2+ (d1-d2) (kQi) S1 (kQi) ] .
Also, the tests of hypotheses usually performed are con
ducted in the manner described in Section 5.2.
The recovery of the intrablock information and a
combined estimate of the intrablock and interblock treatment
effects can also be obtained with the straightforward appli
cation of the maximum likelihood method of Rao (1947).
The utility of the general partially balanced ex
tended complete block designs is at present limited. This
limitation is imposed by the complexity of the formulae for

85
the estimates of the treatment effects, in that the con
stants and d^ must be calculated for any design used. A
similar difficulty is encountered in the analysis of PBIB
designs, unless one has reference to the extensive listing
of PBIB designs (with two associate classes) and their asso
ciated constants necessary for an analysis as given by
Clatworthy (1973).

CHAPTER 7
CONCLUDING REMARKS AND A
SENSORY TESTING EXAMPLE
Throughout the development and presentation of
this work, it has been necessary to make certain assumptions
concerning the model as well as the type of correlation
present in the data in order to formulate our methods of
analysis. In Chapter 3 for example, in the development con
cerning the possible presence of the correlation p between
duplicate responses to the same treatment in the same block,
the additive model only was assumed. Without making the
assumption that the interaction effects are all zero, the
estimate of the magnitude of the correlation would be con
founded with the estimates of the interaction effects. In
this case, an estimate of the correlation free of the inter
action effects would not have been possible by the method
we used. Although the assumption of additivity for many
realistic situations is somewhat restrictive, the additivity
assumption was made as a matter of necessity and to be in
line with the assumption employed in the analysis of ran
domized complete block designs (of which our designs are
just extensions).
The assumption that the correlation between du
plicate responses to the same treatment in the same block is
constant and equal for all treatments and blocks may also
86

87
appear to be somewhat restrictive. It would seem more appro
priate perhaps to assume the correlation is not constant but
rather varies over the treatments and blocks. That is, for
many practical applications it may be more realistic to con
sider the correlations p^j, i = 1, 2, ..., t and j =1, 2,
..., b. However, this non-equality of the correlations
would give rise to the difficulty of having to estimate a
larger number of parameters than the number of observations
present in the experiment. Also, all the correlations p^j
could not be estimated in a given experiment since not all
block-treatment combinations would have duplicate responses.
A simplification of the problem of non-equality of
the correlations would be to consider the correlations pj,
j = 1, 2, ..., b, that is, a different correlation is asso
ciated with each block. Such a case might arise when pan
elists of varying degrees of proficiency are used in sensory
testing experiments. The estimation of the Pj and the sub
sequent test on the treatment effects is being considered
for future work.
The methods presented for testing the hypothesis
of zero correlation and for estimating p may appear to be
somewhat intuitive. First attempts in finding a likelihood
ratio test of the hypothsis of zero correlation and a maxi
mum likelihood estimator of p resulted in complex expres
sions which did not seem to simplify. These likelihood pro
cedures could be investigated in future work. At that time,
it may be of interest to compare the likelihood results with
the results contained in this work.

88
The possible lack of utility of the general par
tially balanced extended complete block designs developed in
Chapter 6 was mentioned at the end of that chapter. Inves
tigations into the possibility of expressing our solutions
in terms of the parameters and constants already tabulated
for PBIB designs by Clatworthy (1973) could be considered.
Hopefully, this would enhance the utility of the GPB ex
tended complete block designs with respect to the calcula
tions involved in the analysis.
As mentioned at the end of Section 3.2, we shall
now suggest a test of the hypothesis of equal treatment ef
fects in the presence of a non-zero correlation. The test
procedure is just a suggestion since the properties of the
procedure have not been studied in detail at this time and
remain for future consideration.
The quadratic forms for the sums of squares for
treatments (adjusted) in Table 1 of Section 3.2 and for re
mainder in (3.3.2) are not independently distributed in gen
eral. However, the quadratic forms for SST^ and the sum of
squares for duplication variation in (3.3.1) are independ
ently distributed. In fact, we have that
SSTa ~ a2{1 + [(k+t)X 2rk]} x^-i
when the hypothesis of equal treatment effects is true, that
SSDV ~ o2(1P) X(k_t) ,
and that these random variables are distributed independently

of one another. Thus, to test the hypothesis under con
sideration, we may use the test statistic
89
MST /{I + |£ [ (k+t)A 2rk]}
pi A AK
t MSDV/(1 p)
which possesses a central F distribution with t-1 and b(k-t)
degrees of freedom in the numerator and denominator, respec
tively. When the hypothesis is not true, the test statistic
F has a non-central F distribution which depends upon the
true value of p as well as the unknown value of the ratio of
£ x2 to ct2. Thus, the power of the test could be calculated
for various values of p and the ratio J x2/a2.
The value of the test statistic F depends upon
the true value of p which is usually unknown. The suggested
A
procedure, therefore, is to replace p with p resulting in
the approximate test statistic F* given as
MST /{I + ||. [(k+t) A 2rk] }
p* A AK
t MSDV/(1-p)
The distribution of the approximate test statistic
F* depends upon the unknown distribution of p. At this
point, complications arise in arriving at the exact or an
approximate distribution of F* since p is calculated using
the value of SSR which, as a random variable, is not inde
pendent of the random variable SST Hence, until more
A
investigation may be made into the distribution of the es
timator p, the distribution of the test statistic F* might
be approximated by an F distribution with t-1 and b(k-t)

90
degrees of freedom (the distribution of F ). The closeness
of this approximation to the exact distribution is being
considered and will hopefully be reported in later work.
The following is a numerical example of a taste
testing experiment. The objectives of the experiment were
twofold. First, it was of interest to compare the degree of
preference for the treatments by the specific panelists used.
Second, it was suspected that correlation would be present
in the data and hence a test for its presence was to be per
formed.
Each of the trained panelists (denoted by the num
bers 1 through 10) was asked to evaluate five different
treatments (denoted by the letters A, B, C, D, and E) by as
signing a numerical value of 1 through 9 according to his or
her degree of preference for the treatments. The lower end
of this hedonic scale reflects an extreme non-preference
while the upper end reflects an extreme preference for the
treatments. Since there was an interest in measuring the
consistency of the panelists used and since each panelist
could evaluate six food samples effectively at one sitting,
each panelist was asked to evaluate each of the treatments
plus a replicate of one of the treatments. The data with
some calculations is

91
Treatments
where the kQ. are
i
calculated using the formula
which follows
formula in (3
tions are
kQ. = kT. l n..B
13 3
(3.2.3) and the are calculated using the
2.4). For treatment A, the necessary calcula-
kQ = (6) (89) {(2) (38+19) +(1) (34+27+23+36+31+42+30+22)}
A
= 175
and
£ 175 o c
ta JT4TT5T ~ *b *
The sums of squares for treatments (adjusted) and
blocks (unadjusted), the total sum of squares, and the sum
of squares for residual are calculated using the formulae in
Table 1. The results of these calculations are

92
SSTa = 127.919 ,
SSB0 = 83.933 ,
TSS = 225.933 ,
and
SSR = 44.081 .
e
The sum of squares residual is partitioned into the sums of
squares for duplication variation and remainder by using the
formula prior to equation (3.3.1) and by finding the value
of the difference SSR SSDV. These results are
e
SSDV = 3.5 and SSR = 40.581 .
To test the hypothesis HQ: p = 0 at the 0.05 level
of significance, the ratio of the mean square for remainder
to the mean square for duplication variation is compared
3 6
with the tabular value F-^q q = 2.68. The result of this
ratio, denoted by Fp in the discussion prior to equation
(3.5.3), is Fp = 3.22. Hence, the data presents sufficient
evidence at the 0.05 level of significance to indicate that
p is greater than zero.
An estimate of p is obtained using formula (3.6.4),
which for this data is
- 40.581 3.5
p 40.581 + (0.159)(3.5)
0.901 ,
where the value 0.159 of A ,
given in Table 1. Since p is very close to 1, the conclusion

93
is that the panelists are consistent in their evaluations of
a treatment and its duplicate. Future experimentation using
these same panelists could be performed using any appro
priate experimental design without specific emphasis placed
on duplicating any of the treatment responses for each pan
elist.
Using the procedure suggested in this chapter for
testing the hypothesis of equal treatment effects, the re
sult is
F* = -099 127.919/4 = 7>45
t 1.214 3.5/10
Since the tabular value for is 3.48, the hypothesis
10/0 *05
is rejected and the conclusion is that the treatment effects
are not equal. The use of a multiple comparison procedure
for comparing the treatments could now be used where the
formula for the variance of a specific effect or the dif
ference between two treatment effects is given in Section
3.2 (in equations (3.2.5) and (3.2.7), respectively).

APPENDIX 1
THE EXACT DISTRIBUTION OF SSR
FOR b=2t AND k=t+l WHEN p>0
In Section 3.4, the exact distribution of the sum
of squares for remainder was developed when p > 0 for the
special case t = b and r = k = t+1. In this appendix, we
shall extend the development of the exact distribution of
SSR when p > 0 to another special case, namely the case
b = 2t, k = t+1, and r = 2(t+1). In other words, we shall
now consider the situation where the number of blocks is
twice the number of treatments but the block size remains
equal to t+1.
Following the procedure outlined in Section 3.4,
we note that in this situation the matrix M(Gt 0 G^JM in
(3.4.12) contains the non-zero partition W, where on the
main diagonal of the matrix W there are t matrices of the
form
(b-1)(t-1) 1-t
1-t (b-1)(t-1)
while the number 1 is present in all other positions. If we
can find the latent roots of W, then the latent roots of the
matrix F*MF* in (3.4.10) can easily be obtained, since the
roots of FMF* are simple multiples of the roots of W. With
the aid of the corollary in Section 3.4, the roots of the
94

95
matrix F(F'D^F) ^F' in (3.4.6) can be obtained, from which
an expression for the minimum value R2 in (3.4.5) and (3.4.3)
can be specified. Once R2 is expressed in terms of the de
sign parameters, the distribution of the conditional random
variable SSRlu.. is specified.
1 iD
To obtain the latent roots d of the matrix W, we
seek the solutions of the equation
|W dl | = 0 (A.1.1)
This determinant can be expressed in a simpler form by per
forming elementary row and column operations on the matrix
W-dl^r resulting in the expression
[b(t-1)-d]t+1[(b-1) (t-1)- (t+1)-d]t_1 = 0 (A.1.2)
for which the roots d are
b(t-l) with multiplicity t+1
d =
bt-2t-b
t-1
Since b = 2t, the roots d may be written in terms of the
single design parameter t in the form
2t (t-1)
d =
2t(t-2)
with multiplicity t+1
t-1 .
Thus, the roots 0" of the matrix F^MF* in (3.4.10) are

96
," ^
(t-l)/t with multiplicity t+1
(t-2)/t t-1
0 2t2-5t+l
while the solutions 0' satisfying (3.4.7) are
(t+l)/2t with multiplicity t+1
0' = (t+2)/2t t-1
1 2t2-5t+l
Finally, the roots of the matrix F(F'D*F) ^F' in (3.4.6) are
0 =
2t/(t+l) with multiplicity t+1
2t/(t+2) t-1
1 2t2-5t+l
Using the above values of 0, an expression for R2
in the distribution of the conditional random variable
SSR|u^j in (3.4.2) may be written as
R2
2t 2 2t 2
t+1 xt+l + t+2 xt-l +
X
2
2t2-5t+l
That is, R2 is again a linear combination of weighted
independent chi square random variables. Upon taking the
expectation of the conditional random variable SSR|uj_j with
respect to u^j and writing SSR = E(SSR|u^j), we find that
the random variable SSR is distributed in the following
manner:
SSR/ct2 ~ a-L Xv + a2 xj + a3 xj r
12 3
(A.1.3)
where

97
and
ai = 1 + p fir '
v1 = t+1 ,
v2 = t-1 ,
= 2tz-5t+l .
An approximate distribution for the random variable SSR in
(A.1.3) is given in Appendix 2.

APPENDIX 2
AN APPROXIMATE DISTRIBUTION OF SSR
FOR b=2t AND k=t+l WHEN p>0
In Appendix 1, it was shown that the random vari
able SSR is distributed in the following way:
SSR/a2 ~ ax Xv + a2 X* + a3 X2 (A.2.1)
J- ^ *J
where a^ and v j j = 1, 2, and 3, are defined following
(A.1.3). In this appendix, we shall consider an approximate
distribution of (A.2.1) for the given special case b = 2t,
k = t+1, and r = 2 (t+1). The form of the approximate dis
tribution follows from the theorem stated in Section 3.5 on
the approximate distribution of a quadratic form. That is,
SSR/a2 ~ g X (A.2.2)
approximately, where the values of g and h are computed
using (3.5.1) and (3.5.2), respectively.
For the integer values 3, 4, 5, 6, and 7 of t and
for some values of p between 0 and 1, values of g and h are
presented in Table A1. The value of the scale constant g
was computed to four decimal places and the approximate de
grees of freedom h were expressed to the nearest integer.
To justify the use of the approximate distribution
in (A.2.2) for testing the hypothesis Hq: p = 0 against the
alternative hypothesis Ha: p > 0, the approximate 0.05 level
98

TABLE Al
Values of g and h for the Approximate Distribution of SSR, II
p
t
V,
v
a
a^
a
g
h
1
2
_3
1
2
3
ZJ
0.1
3
4
2
4
1.05
1.025
1
1.0255
10
0.3
1.15
1.075
1.0792
10
0.5
1.25
1.125
1.1361
10
0.7
1.35
1.175
1.1958
10
0.9
1.45
1.225
1.2581
10
0.1
4
5
3
13
1.06
1.04
1.0207
21
0.3
1.18
1.12
1.0658
21
0.5
1.3
1.2
1.1156
21
0.7
1.42
1.28
1.1695
20
0.9
1.54
1.36
1.2271
20
0.1
5
6
4
26
1.0667
1.05
1.0174
36
0.3
1.2
1.15
1.0563
36
0.5
1.3333
1.25
1.1004
35
0.7
1.4667
1.35
1.1491
35
0.9
1.6
1.45
1.2022
34
0.1
6
7
5
43
1.0714
1.0571
1.0150
55
0.3
1.2143
1.1714
1.0493
55
0.5
1.3571
1.2857
1.0887
54
0.7
1.5
1.4
1.1330
53
0.9
1.6429
1.5143
1.1818
53
0.1
7
8
6
64
1.075
1.0625
1.0132
78
0.3
1.225
1.1875
1.0438
78
0.5
1.375
1.3125
1.0795
77
0.7
1.525
1.4375
1.1200
76
0.9
1.675
1.5625
1.1650
74

100
of significance for the distribution in (A.2.2) is compared
with an exact probability level as well as a probability
level obtained using a second approximate distribution. We
shall feel confident in using the approximate distribution
in (A.2.2), if the exact probability levels are close to
0.05 for reasonable values of t and p.
To compute the exact probability levels corre
sponding to the approximate probability 0.05, the method
suggested by Box (1954) can be used. However, the calcula
tions of the exact probability values are limited to the
case t = 3 for all values of p because of the large number
of significant digits necessary to ensure accuracy to the
fourth decimal place. This limitation prompted us to con
sider a second approximation, which is more exact than the
distribution specified in (A.2.2), to check the probability
levels of the distribution in (A.2.2). The second approxi
mation is now described in detail.
Corresponding to the form of the distribution of
the random variable SSR/p2 given in (A.2.1), the moment gen
erating function m(0) of SSR/ct2 may be written as
-v-,/2 -V-/2 -Vo/2
m (0) = (l-2a10) (l-2a20) (l-2a30) (A.2.3)
Taking the natural logarithm of m(0) results in the sum
3 vi
In m(6) = £ ln(l-2a^0) (A.2.4)
i=l
The quantity ln(l-2a^0) can be expressed in terms of a new
constant a as

101
ln(l-2ai6)
In (l-2a0)
+
In
l-2ai9
l-2a0
(A.2.5)
The reason for expressing ln(l-2a^0) in this form will become
evident as the development of the second approximation con
tinues .
Using some simple algebraic manipulations and the
expansion of ln(l+x), we may write the second term on the
right-hand side of (A.2.5) in the form
In
l-2ai0
l-2a0
In(l-2a0)
+
1
P
l-2a0

(A.2.6)
where = (a/a^) 1. Substituting (A. 2.6) into (A. 2.5)
and using the resulting expression in (A.2.4) gives
v
v.
In m(0) = 2 ln(l-2a0) -
3
X 5
1 = 1
1
I |
P=1 p
Oi-i
l-2a0
(-1)
p-1
(A.2.7)
where v = + \>2 + Vg.
A value a of a is now chosen so that when we set
o
p = 1, the second expression on the right-hand side of
(A.2.7) is equal to zero. The value aQ is given by
a = v v
o
(A.2.8)
which satisfies the equation
3
2 (l-2a0) = 0

102
With this value of aQ substituted into (A.2.7), the resulting
expression for m(0) in (A.2.3) is given approximately by
m(e)
= (l~2a_6) V/2
exp
oo 3
I I
p=2 i=l
2p
ai
P
(~1)P
l-2a 0
o
j
(A.2.9)
If we further define to be
~ P
3 v-a^ _
J -%r <-p
and define to be
\p = l-2ao0 ,
then m(0) in (A.2.9) may be written in the forro
m (0) =
(l-2aQ0) V//2 exp
I
p=2
(A.2.10)
Since we require the equality m(0) = 1, the approximation in
(A.2.10) is altered to satisfy m(0) = 1 resulting in
m(0)
(l-2aQ0) V^2 exp
p=2
TTp (l/j P-l)
(A.2.11)
An approximation to m(0) in (A.2.11) is made by
using the first three terms of the infinite sum on p and by
approximating e by the formula
eX = 1 + x + (x2/2) .
The approximation becomes

103
m(0) (l-2ao0) [ (1-S+sS2) + tt^I-S)^ ^ + ir^d-S)^ ^
+ [tt4 (1-S)+35tt2] \p 4) ,
where S = 7^ + it 3 + Hence, the second approximation of
the required probability is given by
Pr[ SSR/a2 > S ] = (I-S+5S2) Pr[ aQ x* > <$ ]
+ tt2 (1-S) Pr[ aQ x*+4 > <$ ]
+ it3 (1-S) Pr [ aQ x+6 > 6 ]
+ [tt4 (l-S)+is7r2] Pr [ aQ xJ+8 > 6 ] .
(A.2.12)
The probabilities presented in Table A2 are the
values resulting from the use of
Pr[ SSR/ct2 > gxQ ] ,
where xQ is the tabular value of a chi square random variable
with h degrees of freedom such that the probability of ex
ceeding xQ is equal to 0.05. That is, the value of gxQ was
determined by the approximate distribution in (A.2.2) and
this value of gxQ was then used to determine the exact prob
ability values using the method given by Box (1954) as well
as the second approximate probability values computed using
(A.2.12) .
From the entries in Table A2, we note that the
probabilities for the approximation in (A.2.2) are nearly
equal to the probabilities corresponding to the exact

TABLE A2
Comparison of the Exact and Two
Approximate Distributions of SSR
t
£
Exact
Approx I
Approx II
3
0.1
0.0499
0.0500
0.0499
0.3
0.0492
0.0492
0.5
0.0480
0.0480
0.7
0.0465
0.0465
0.9
0.0448
0.0448
4
0.1

0.0494
0.3

0.0480
0.5

0.0455
0.7

0.0578
0.9

0.0532
5
0.1

0.0497
0.3

0.0476
0.5

0.0554
0.7

0.0498
0.9

0.0546
6
0.1

0.0497
0.3

0.0470
0.5

0.0508
0.7

0.0533
0.9

0.0434
7
0.1

0.0497
0.3

0.0465
0.5

0.0479
0.7

0.0463
0.9

0.0492

105
distribution in (A.2.1) as approximated by the more exact
second method. Therefore, the approximate distribution in
(A.2.2) can be used with confidence, at least for the values
of t included in the table. These values of t cover most of
the cases which might be encountered in the application of
our problem.

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1062 .

BIOGRAPHICAL SKETCH
Jack Franklyn Schreckengost was born on January 6,
1944, in Summerville, Pennsylvania. After graduation from
the Brookville Area High School in 1961, he attended nearby
Clarion State College. During his enrollment there, he ma
jored in chemistry and mathematics receiving the degree of
Bachelor of Science in Secondary Education in 1965. While
he taught mathematics in the Penn Hills School District in
>
a suburb of Pittsburgh, he studied part time at Bucknell Uni
versity, Lewisburg, Pennsylvania. Having been awarded the
degree of Master of Science in Mathematics (Teaching) in 1968,
he assumed a position at Bucknell University teaching fresh
man calculus and elementary statistics. In 1970, he enrolled
in the Graduate School of the University of Florida and re
ceived the degree of Master of Statistics in 1972. While
attending the Graduate School, he worked as a teaching assist
ant and as a research assistant in the Department of Statis
tics and, until the present time, has pursued his work toward
the degree of Doctor of Philosophy.
He is married to the former Donna Rae Leach of
Arnold, Pennsylvania, and is the father of two children. He
is a member of Theta Chi and Pi Mu Epsilon fraternities and
the American Statistical Association.
109

I certify that I have read this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
John A. Cornell, Chairman
Associate Professor of Statistics
I certify that I have read this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Frank G. Martin
Associate Professor of Statistics
I certify that I have read this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
John G..Saw
Professor of Statistics

I certify that I have read this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and guality, as
a dissertation for the degree of Doctor of Philosophy.
(v'.Aj .d0
P .V. Rao
Professor of Statistics
I certify that I have read this study and that in
my opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
/U2 J
Frederick W. Knapp
Associate Professor ;of Food Science
This dissertation was submitted to the Graduate
Faculty of the Department of Statistics in the College of
Arts and Sciences and to the Graduate Council, and was ac
cepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
August, 1974
Dean, Graduate School