*This research was supported in part by the National Science Foundation
under Grant GP 42325 •.
'.
\
......
.'
'r
DESIGN AND ANALYSIS OF SERIAL EXPERIMENTS*
Lynn Weidman
Department of Statistics
University of North Carolina at Chapel Hill
"
Institute of Statistias Mimeo Series No. 997
April, 1975
\
LYNN WEIDMAN. Design and Analysis of Serial Experiments (under the
direction of I.M. CHAKRAVARTI.)
Several cases of the following situation- are considered.
Each exp-
erimental unit receives a prescribed series of treatment applications,
and a particular characteristic of interest is measured in association
with each application.
The experimenter is not only interested in the
effect a treatment has on the observation associated with its application,
called direct effect, but also in the effects it has on succeeding observations , called residual effects, and its combined effect with following
applications on their associa.ted observations, called interaction effects.
Comparing the treatment effects to select the most effective overall
treatment is the purpose of these experiments.
The design problem is to select the series in such a manner that
variances for all the effects in a particular class of effect, e.g.,
direct effects, are as equal as possible and covariances between any two
members of different classes are the same.
The following cases are considered:
i)
effects.
Each application has a direct. two residual, and two interaction
Every combination of three consecutive treatments, where no
treatment is repeated, occurs equally often.
ii)
No restrictions on treatment order and any specified number, say
n-l, of residual or interaction effects may be associated with all applications.
Every combination of n consecutive treatments occurs equally
often, and designs are called serial arrays.
when interactions are not present.
Simplified analysis results
Method of analysis using orthogonal
polynomials is available when the treatments are different levels of
a single factor.
iii)
Several series of dependent random variables are given, where
the observations for each series follow a set of conditional density
functions.
E~perimenter
wants to estimate and compare parameters of the
densities for the different series.
If the conditional densities satis-
fy certain conditions the following usual maximum likelihood results
follow:
(a)
parameter estimates are consistent and asymptotically
normal; (b) likelihood ratio tests are asymptotically
X2•
Designs
possibly useful for data collection in this case are constructed.
iv)
Consider 2w factorials, w = 2,3,4,5,6, where the factor combin-
ations are applied in series in order to estimate the direct and one
residual effect for the two levels of each factor.
designs are derived for the given values of
units each receive 4 different combinations.
.~,
Serial factorial
where
.~
experimental
Construction consists of
defining fractions and their order of application for each unit.
In cases (i), (ii), and (iv) the response model is linear and
analysis is by least squares.
TABLE OF CONTENTS
Acknow1edgements---------------------------------------------iv
Notation----------------------------------------------------- v
I.
Introduction and Summary------------------------------------- 1
II.
Triple Designs and Their Analysis
2.1 Properties and Construction----------------------------- 5
2.2 Response Model and Estimation of Parameters------------- 9
2.3 Variances and Covariances of the Estimates--------------17
2.4 Tests of Hypotheses-------------------------------------23
III.
Serial Arrays and Their Analysis
3.1 Description and Construction----------------------------26
3.2 General Ana1ysis- ------- ----- --~..----- ---- -------- -------30
3.3 Analysis and Hypothesis Testing When Interactions
are Excluded----------------------··-~------------------38
3.4 Blocking of Units---------------------------------------41
3.5 Levels of a Single Factor-~-----------------------------43
IV.
Maximum Likelihood for Series of Dependent Random Variables
4.1 The Problem of Comparing Dependent Observations--------50
4.2 Preliminary Results-------------------------------------51
4.3 Notation and Asswnptions--------------------------------53
4.4 Properties of Maximum Likelihood Estimates and
Likelihood Ratio Tests---------------------------------S7
4.5 Sillfu~ary-------------------------------------------------66
V.
Construction of Some 2- and 3-Repeating Designs
5.1 n-Repeating Designs- -- -------..;------------- -----------·--68
5.2 Construction of 2-Repeating Designs---------------------71
5.3 Construction of 3-Repeatir.g D~sigllS---------------------78
VI.
Serial Factorial Designs
6.1 The Serial Factorial Model and Notation-----------------82
6.2 23 Designs----------------------------------------------87
ii
iii
6.3 24 Designs---------------------------------------------89
6.4 2S Designs---------------------------------------------91
6.5 26 Designs---------------------------------------------96
6.6 General Analysis of 2w Designs-------------------------98
6.7 Simplified Analysis for w Even----------------------lOS
Bib1iography-----------------------------------------------------109
ACKNOWLEDGEMENTS
To the members of the faculty of the Department of Statistics I
wish to express my gratitude for their contributions to my graduate training.
I especially acknowledge the suggestions and encouragement given to
me by my advisor, Professor I.M. Chakravarti, during the course of this
investigation.
I wish to thank Professor N.L. Johnson for his careful reading of
the manuscript and the other members of my doctoral committee, Professors
D.G. Kelly, K.J.C. Smith, and T. Brylawski, for their review of my work.
Additional thanks go to Dr. H.D. Tolley for suggesting the topic of
Chapter 4.
I am indebted to the National Aeronautics and Space Administration
and the Department of Statistics for their financial support.
For her excellent typing of the manuscript I thank Ms. June Maxwell:
I would like to extend special appreciation to my parents who continually encouraged my work.
Finally, I thank my fellow graduate students and other friends on
this campus who have helped to make my non-statistical hours pleasant
and entertaining.
iv
r~OTATION
is a column vector, and !'
a
A = (a .. )
is a matrix with
j-th column.
A'
is the transpose of A.
1)
a..
is
~he
corresponding row vector.
as the element in the i-th row and
1)
is any generalized inverse of A and is defined by AA-A
is the Kronecker product of the matrices A and
(i,j)-th partition of A~B is defined as a .. B.
A~B
= A.
B, where the
1J
In
is the identity matrix of order n.
J
n
is the (nxn) matrix with all entries one.
m,n
is the (mxn) matrix with all entries zero.
0
0
is the (nxl) vector of zeros.
1
-n
is the (nxl) vector of ones.
--n
For x
= (x1, .. ·,xm)'
and X.
= (Yl'",Sn)'
random vectors,
E(x)
is the (mxl) vector with i-th element
Cov(!.,y)
is the (mxn) matrix with Cov(x. ,y.) as the (i,j) -th element;
Var(~
is the (mxm) matrix
x ,..,
N(]l,a 2 )
x ,.., Nr(]l,E)
-
-
x ,..,
x""t
x2
r
r
1
E(xi) ;
J
Cov(~,~ .
means the random variable x has the univariate normal
distribution with mean ]l and variance a2 •
means that the random vector x has the r-variate multinormal distribution with mean vector ~ and variance matrix
means that the random variable x has the central chi-square
distributionwith r degrees of freedom.
means that the random variable x has the central Student's
t distribution with r degrees of freedom.
v
E.
vi
x
F
IV
r,s
L
means that the random vector !m converges in law to the
random vector x.
x + x
.....
m
x
-'JIl
means that the random variable x has the F distribution
with r and s degrees of freedom.
L
+ G
means that the random vector ~m converges in law to the
random vector x which has distribution function G.
p lim x =c means that
JJ1+CO
m
lim x a+s· c
m
11\"+00
p lim X =C
J1t+oom
o
xm converges in probability to the constant
means that
~m
c.
converges almost surely to the constant
Xm
means that each element of the matrix
converges in
probability to the corresponding element of C.
denotes the conclusion of a proof or example.
c.
CHAPTER I
INTRODUCTION AND SUMMARY
Comparison of the effectiveness of several treatments under certain
assumptions commonly refered to as the general linear model has supplied
much of the impetus for developing the subject area of experimental design.
Among the situations which have been considered is that in which each
experimental unit receives a prescribed series of treatment applications,
and a particular characteristic of interest is measured in association
with each application.
In this case the experimenter is not only inter-
ested in the effect a treatment has on the observation associated with
its application, but also in the effects it has on succeeding observations.
Definition 1.1:
The effect a treatment has on the observation associated
with its application is called its direot effeot.
For the first obser-
vation following that one its influence is the first residuaZ effeot,
for the next observation it is the s.90ond residuaZ effeot, etc.
ment
i+j
h applied at time
i
combines with treatment
If treat-
k applied at time
to influence the observation associated with the application at
time i+j, this is called the j-th interaction effect of treatments
h
and k.
Designs used in this situation to estimate direct, residual, and
interaction effects are commonly called change-over designs.
mary of designs of this type see [20] and [22].
For a sum-
2
Some constructions for change-over designs which will be important
here were given by Williams [28],[29].
He assumed that each unit was to
receive each treatment exactly once and that the designs were to be balanced either for a single residual effect or for pairs of residual
effects.
Balance means that all estimates in the same cl a s s ( e.g.,
first residuals, direct effects) have equal variances and that covariances
between the members of two classes are all equal.
We use the following
definition in describing these designs.
Definition 1.2: In a design with a series of treatments applied to each
unit, each n consecutive applications in which no treatment can occur
more than once is called an n-pZet.
any n
If a treatment can be repeated in
consecutive applications they are called n-tupZes.
If v treatn
ments are used, there are v(v-l) ... (v-n+l) possible n-plets and v
possible n-tuples.
The designs for a single residual effect given in [28] consist of
either one or two latin squares, where the number of times each 2-plet
occurs is equal to the
n~~bBr
of latin squares.
pairs of residual effects as given in [28]
latin squares.
Designs balanced for
and [29] consist of a set of
In such a set each 3-plet occurs once a11d each 2-plet
occurs at the beginning of some row and at,the end of some other row
exactly once.
Each row of a latin square is the series of treatments for
a single experimental unit in these designs.
Finney and Outhwaite [12],
[13] considered applying a. long series of treatments to a single unit in
order to estimate the same effects.
Further designs along this line were
given by 5mnpford (26].
A combination of these two approaches was used by Federer and
Atkinson [llJ
under the title tied-double-change-over designs.
For these
3
designs each unit is assigned several series of treatments which are
applied successively and are balanced for the estimation of a single
residual effect.
Chapter 2 presents designs and analysis for what we
call triple designs, which assume the presence of first and second residual effects.
The method of construction involves combining in a
specified manner the latin squares used as designs balanced for pairs
of residual effects by Williams.
In general, let n represent the maximum number of observations
on which any given treatment application has an effect.
We now drop
the restriction that no treatment may occur more than once in any n
consecutive applications) and in Chapter 3 we introduce serial arrays
for estimating in this case all treatment effects in a balanced manner.
Thus we will consider n-tuples rather than n-plets as we did in Chapter
2.
The primary property of serial arrays is that each n-tuple occurs
equally often.
This along with the other properties of serial arrays
allows us to derive general formulas for estimating all the effects being considered.
Serial arrays are also applicable to the case of exam-
ining v different levels of a single treatment, under assumptions that
the treatment effects are pol}rnomials.
This case is discussed in
Section 3.5.
When each unit in an experiment repeatedly receives a single treatment and the response pattern is assumed to fit a particular linear
model, we have what is commcnly called a growth curve.
An early exam-
ination of this case was given by Wishart [31],[32], and extensive work
has been done since that time.
[25].)
(See, e.g., [8],[10],[16],[17],[23],[24],
In Chapters 4 and 5 we allow a more complicated response pattern
in which each response is dependent on previous responses in addition to
4:
the direct, residual, and interaction effects of the treatments.
Chapter
4 considers the problem of estimating and comparing parameters included
in several independent series of dependent random variables.
The case
of primary interest to us is when each series of dependent variables is
the set of observations for a single unit, and the series for each unit
is independent of all others.
For each series the conditional probab-
ility density functions.must satisfy the conditions specified by Cl
through CS.
This work is an extension of results given by Bhat [4] and
Billingsley [6].
Chapter 5 presents some designs which may be useful in
establishing the verification of the conditions on the conditional density
functions.
In the final chapter we consider using series of treatment applications in the factorial case.
Factorial designs consist of treatments
which are coniliinations of several factors, and in fractional factorials
a particular subset of the total set of possible combinations is used,
each combination being applied to a different unit.
Main effects of and
interactions between factors are of interest in the designs.
If we apply
several of the ccmbinations in a fraction, in a given manner, to each
unit available, we will perhaps be able to estimate the direct and
residual effects of the individual factors.
Such information would be use-
ful in deciding which factor combination should be used over some future
extended period of application.
This general situation was discussed by
Patterson [21] under the title of serial factorial designs.
For the
special case of 2w factorials, w'= 2,3,4,5,6, some designs on fewest
possible units, and their analysis, are derived in Chapter 6.
CHAPTER II
TRIPLE DESIGNS AND THEIR ANALYSIS
2.1
Properties and Construction
In 1955 Federer and Atkinson [11] presented what they called tied-
double-change-over designs which are used to compare effects of v
different treatments.
Definition 2.1:
Let
v and
s
be integers and let
I
be the number of
A tied-doubZe-ahange-ovep design has
experimental units being treated.
the following properties:
(i)
(ii)
(iii)
(iv)
each experimental unit receives a series of sv treatment
applications;
no treatment occurs in two consecutive applications;
all 2-plets occur equally often in the whole design; and
at each time of application every treatment is applied to the
same number of units, i.e., I/v is an integer.
Since' there are v(v-l) 2-plets,
such a design is
the total number of applications in
dv(v-l) + I, where d is some integer.
Each obser-
vation is affected by one direct and one residual treatment effect and,
possibly, by a single interaction effect.
in each group of effects are equal.
Variances for all estimates
Also, all covariances between
estimates from two given groups are equal.
In this chapter we extend these assumptions to include second residual effects and interactions for each 3-plet of consecutive treatments.
Analogous to the above case we require that all of the v(v-l) (v-2)
6
3-plets occur equally often in consecutive applications.
be assigned a series of the v
times.
treatments which is to be repeated s
The last two treatments of this series are to be
the unit in the first two application periods.
until after the third
9i
Each unit will
application~
No
~pplied
ob~ervations
to
are made
so these first two may be called
conditioning" and are used only in estimating residual and interaction
effects.
Before proceeding we introduce some terminology.
Definition 2.2:
Let
V=
{Tl~T2~
pectangZe is a matrix with r
... ~Tv} be a set of v symbols.
rows and c
columns whose entries are
An rxr rectangle is a squape.
members of V.
A
If each member of V occurs
exactly once in each row and column of a square it is called a latin squape.
The designs examined in this chapter are constructed from Williams'
[29]
for
designs balanced for pairs of residual effects.
v
Williams' designs
treatments consist of sets of v-I latin squares of size v, as
described in Chapter 1.
Each 3-plet occurs exactly once and each 2-plet
occurs once at the beginning of some row and once at the end of some
other row in those designs.
Definition 2.3:
s
are integers.
Suppose we want to compare v
rand
A triple design has the following properties:
(i') there are
(ii') sv
treatments, and
r(v-l)(v-2) experimental units;
is the nwnber of observations made on each unit;
(iii') observations are made on all the units in the same time periods;
(iv') each treatment is applied rs(v-I) (v-2) times;
(v') all 3-plets occur rs times in the whole design;
(vi') the v-th treatment occurs r(v-l) times at application period
j, j=nt-l or nt, n=1.2 •...• s. and 0 times at periods nt-l and
nt; and
7
(vii') all other treatments occur r(v-3) times at period j,
j~nt-l or nt, and r(v-2) times at periods nt-l and nt.
We give a procedure for constructing triple designs for any integral
(r,s,t), where the set of treatments is
V = {O,l, ... ,v-l}.
(1)
Take one of Williams' designs described above for v-I treatments, where the first row of each latin square ends with a
different number. Write down these squares one below the
other.
(2)
Place a v-I at the beginning of each of the (v-l)(v-2) rows.
We now have v-2 (v-l)xv rectangles.
(3)
In the i-th rectangle, i=2,3, •.. ,v-2, move column j,
j=1,2, •.. ,v-i+l, to column (j+i-l) and the last i-I columns
to the first i-I, retaining their order. This cyclically
rearranges the columns so that treatment v~l now occurs in
column i in rectangle i, i=l,2,3, ... ,v-2, and never occurs in
columns v-I and v.
(4)
Repeat each row after itself 5-1 additional times and place
the two treatments at the· end of each row also at its begin-"
ning. These are the conditioning treatments. Repeat the
entire resulting rectangle r-l additional times.
(5)
Let each row of this r(v-l) (v-2)x(sv+2) rectangle represent
the treatment application series for a single unit and each
column represent an application period.
All of the properties ofa triple design, except perhaps (v'), are
clearly included in these designs.
Lemma 2.1:
We show that (Vi) is also satisfied.
All 3-plets occur exactly rs times for the rectangles gener-
ated as above.
Proof:
We need consider only the upper left quadrant of size (v-l)(v-2)
x(2+v) from our rectangle.
If each 3-plet is included once then the whole
rectangle will include each rs times because of the method of repeating
8
rows and rectangles in its construction.
From
~he
properties of the original design for v-I treatments
each 3-plet without v-I occurs exactly once.
We show that every 3-
plet including v-I occurs exactly once.
(a)
Since every 2-plet without V-I occurs once at the end of a
row in the original design, every 3-plet of the form
a b v-I occurs exactly once in the new design. Similarly,
every 3-plet of the form v-I a b occurs once.
(b)
Each initial row of a square in the original design begins
with 0 and ends with one of the other treatments 1,2, .•. ,v-2,
such that each different possible last element occurs once.
Our construction method has placed v-I between the first and
last elements of the first row of each square so that all
3-plets of the form 0 v-I a , a=1;2, ... ,v-2, occur exactly
once. A similar result holds for all the second rows, third
rows, etc. In this way every 3-plet with v-I as the middle
element occurs exactly once.
0
Thus every 3-plet occurs exactly once in the quadrant of interest.
Example 2.1:
r=l, s=2, v=4.
We demonstrate the construction of a triple design for
The original design balanced for pairs of residual effects
consists of two latin squares of size v-I = 3.
0
1
2
0
2
1
1
2
0
I
0
2
2
0
1
2
1
0
The derived triple design is then
(2.1.1)
9
1
2
3
0
I
2
2
0
3
I
2
0
0
I
3
2
0
1
0
2
0
2
1
0
I\ 3
2 3
3
3
3
I
0
2
I
0
I
2
1
3
0
I
2
I
2
0
2
0
I
3
0
2
2
3
0
0
3
I
2
(b)
(a)
I
(2.1. 2)
(c)
(a) represents the conditioning periods, (b) is the design derived
where
directly from the squares (2.l.lL and (c) is a repetition of the rows
of (b) •
Note that in (b) the treatment v-I
=3
occurs in the first column
for the first 3 units and in the second column for the last 3.
2.2 ResEonse Model and
E~timation
of
0
~arameters
As mentioned above, these triple designs enable us to estimate all
the direct, first residual, and second residual effects for all
v treat-
ments, as well as interactions between treatments occurring within any 3
consecutive periods.
Because of the special position given to treatment
v-I in the construction of these designs, the variances for and covariances between classes of estimates will not be the same for all treatments.
However, they will be the
sa~e
for all treatments except v-I.
This will allow us to derive simplified equations for estimating the
treatment effects of all the other treatments.
Also, the experimenter
,
should decide how best to handle this special relationship of treatment
v-I before using a triple design.
One appropriate method is to use a
control or standard treatment as v-I and randomly assign the others to
the remaining numbers.
For the model presented here we consider any interactions to be
zero, so that our interest is centered on deriving estimates of the
10
direct and residual treatment effects.
Assuming a linear response
model we have
v-I
Y" h = N, .h (II + y, +
1J
1J
1
t3.J + 0h + p=O
L N,l'(.J- 1) PPP
+
v-I
)
LN'('2)n+€"h'
q=O 1 J- q q ~J
(2.2.1)
i=1,2, ..• ,r(v-l)(v-2), j=I,2, ... ,sv. h=O,l, ..• ,v-l. where
Y"h
1J
is the ooservatioh made following the application of treatment h
•
to unit i at time j •
= {I
if treatment
h was applied to unit
i
at time
j
o otherwise
II
is the overall mean,
Yi
is the effect of unit
13.
is the effect of time period j,
J
i,
0h,Ph,n are respectively the direct, first residual, and second resih
dual effects of treatment h, and
ijh is a random error term for observation Yijh .
Note that for each i,j only one N, 'h is non-zero.
E
1J
of observations we have is
r(v-l) (v-2)xv.
Thus the number
We consider only these re-
sponses in our analysis, ignoring all Yijh which are identically zero.
We make the additional assumptions that the
and identically distributed as
r(v-1l(v-2)
1
i=l.
sv
y.
1
= L 13, =
j=l J
ijh are independent
E
N(O,cr 2 ) and
v·l
v-I
r
0h = r P
h=O
p=O
P
=
v-I
rn
q=O q
= O.
The number of independent effects to be estimated is then
r(v-I)(v-2) + v(s+3) - 4.
(2.2.2)
11
Due to the distributional assumptions on the
E"
1J h
we may esti-
mate the effects by the method of maximum likelihood by calculating
A
the estimates
Yijh which minimize the sum of squares
r(v-1l(v-2)
1
G=
i=l
SV
V-I.
I I
j=l h=O
under the restrictions (2.2.2).
2
A
(2.2.3)
y. 'h)
()T"h
1J
1)
We do this in the usual manner by taking
partial derivatives with respect to the effects and setting them equal
to O.
(a)
-
1
aG
I. I. hIN.1J'hY"h
- rsv(v-l) (v-2)J.! - Sv
1J
J
~-
aJ.!
1
ry. - r(v-l) (v-2)Ia.
• 1
1
•
J
- rs(v-I)(V-2)I (oh+P.h+nh)
h
= O.
J
(2.2.4a)
(The summations range over all possible values of the given subscript
unless otherwise indicated.)
Using (2.2.2) this reduces to
rsv(v-l)(v-2)J.! = Yooo
(2.2.4b)
>
where Yooo is the sum of all observations.
(b)
1
dG
-':l~:
oy.
rj hIN. 'hY"h - sVj.! - svy.
1J
1
1)
1
- Is.- s I(oh+Ph+nh)
j J
h
=0
(2.2.5a)
which reduces to
sV(J.!+Y.)
1
where y.1 ••
= y.1 ••
(2.2.5b)
,
is the sum of observations for unit
Before taking derivatives with respect to the
i.
f3 j , we note that the
effects for treatment v-I CO not occur in all periods.
perties of these triple designs.)
(i) 0V_l
occur with
a.,
J
In
fact~
(Recall the pro-
if n=I,2, ... ,s, then
Sj' j=nv-I and nY, (ii) Pv- I' does not
S., j=nv, nV+I, and 1, and (iii) n I does not occur with
does not occur with
J
j=nv+I, nv+2, I, and 2.
v-
12
(
c
)
-
t
~
aG •
as..
J.
= nY-I,
If j
n=l,2, ... ,s, then we have
L L N. 'hY' 'h - s(v-1)(v-2)~ - iLY,1 - r(v-2) hrv-1
L 0h
i h 1J 1J
- r(v-1)(p 1+nv 1) - r(v-3)
v-
-
- s(v-1)(v-2)S.
J
=0
L (p +n )
prv-1 p P
(2.2.6a)
.
Using (2.2.2) this becomes
n 1) = Y'$J' ° ,
s(v-1)(v-2) (~+S.)
1+v
J - r(v-2)o v- 1 + 2r(pv -
whereYojO
is the sum of observations for period j.
(2.2.6b)
Similarly we
have 4 other equations for different values of j.
j
S(V-1)(V-2)(~+Sj)
S(V-l)(v-2)(~+Sj)
=Y~j~
+ 2r(ov_l+ Pv_l) - r(v-2)nv _l
j = nv + 2
= rojo'
- Jz
,
(2.2.6d)
(2.2.6e)
2r(ov- I+ Pv- l+nv- 1) =y.
,
0J~
all other j .
(d)
(2.2.6c)
+ 2r ov _1 - r(v-2) (Pv-l+nv-l)
j = nv + 1
s(v-l)(v-2)(~+S,) +
J
= nv
(2.2.6f)
aG
aT :
h
For h
r v-I
our equation is
L L ~. 'hY' 'h - rs(v-l)(v-2)(~+Oh) - sLY, - r(v-3) I
s.
i j . 1J 1J
i 1
j rnv-l, J
nv
- r(v-2)
I s. - rs prh
L N,i)'hN1.(.J- 1) PPP - rs qrh
L N.1J'hN1,(,J- 2) qnq = 0
j=nv-1,J
nv
(2.2.7a}1
13
Using (2.2.2) and the property that any two treatment pairs occur in
designated positions in the same 3-tuple rs(v-2) times, we can rewrite this equation as
rs(v-l)(v-2)(~+o ) + r
n
r
s. - rs(v-2)(Ph+nh)
j=nv-l,J
nv
= Yooh '
(2.2.7b)
where Yooh is the sum of all observations where the direct effect of
treatment h occurs.
Ror the direct effect of treatment v-I we have a slightly different
equation:
r r N.1J·v-lY1JV
·· -1 - rs(v-l)(v-2)(~+oV_l)
i j
rY
i i
L Sj
j;nv-l,
nv
- rs
N•• IN.(. 1) P - rs
N•• IN.(. 2) nq=O.
p=v-l 1JV- 1 J- P P
q;v-1 1JV- 1 J- q
(2.2.7c)
r
- s
- r(v-l)
r
Again we can reduce this equation to a simpler form.
rs(v-l)(v-2)(~+o
r
v- 1) - r(v-I) j=nv-l,S.J - rs(v-2)(pv- l+nv- 1)
nv
-y
° ov-1
(2.2.7d)
The equations for residual effects are derived in the same manner.
(e)
aG
- ~r:
Pp
rs(v-l)(V-2)(Pp+~) + r
r
j=nv,
nv+l,l
Sj - S(V-2)(op+np ) = Yo(l)p ,
p ; v-I,
(2.2.8a)
r
rs(v-l)(v-2)(p v- 1+~) - r(v-I) .
8. - s(v-2)(0v- l+nv- 1) = Yo(l)v-l'
J=nv, J
nV+l,1
p
where
= v-I
,
(2.2.8b)
Yo(l)p is the sum of observations where the first residual effect
of treatment p occurs.
(f)
aG
-!.a-an
q
rs(v-l)(V-2)(nq+~) + r
L B. -
j=nv+l, J
nv+2,1,2
s(v-2) (Oq+Pq ) = Yo (2)q ,
(2.2.9a)
q .; v-I ,
rs(v-l)(v-2)(n v_l +V) - rev-I)
r
e.
j=nv+l, J
nV+2,1,2
= YO(2)V-I'
-
s(v-2) (ov-l+Pv-l)
(2.2.9b)
q =v-l ,
where Y.(2)q is the sum of observations where the second residual treatment of q occurs.
The system of normal equations for model (2.2.1) is given by
(2.2.4b), (2.2.5b). (2.2.6b-f), (2.2.7b), (2.2.7d), (2.2.8a-b), and
e2.2.9a-b).
Direct solution of (2.2.4b), (2.2.5b), and (2.2.6b-f)
gives the following set of equations, where the effects of all treatments except v-I have been eliminated.
....
~
=
(2.2.10a)
Yooo
?i = Yioo - Yoco
....
e.J
e·J
e·J
e·J
e.J
....
A
A
A
=
(YOj
0
= (Yo j
0
= (y j
0
0
= (raj 0
=
(Y_ j
0
2
v-2 0
j=nt-I
sv(pv-I+n
+
sv v-I
v - I )'
- Y"
2
v-2
, j=nt
- Y"ooJ + -(0
sv v- I+Pv- 1) - sv nv- I
2
v-2
j=nt+l,l
-1
- -sv 0v-I + sv(pv- 1+nv - l )
2
v-2n
, j=nt+2,2
- ~aoJ - -(0
sv v-I
sv v-I +Pv-I ) + 2
) all other j
- rooo ) - --(o+p
sv v-I v-I +nv-I'
(2.2.10b)
(2.2.10c)
0 0)
HO
)
A Y indicates the mean of the given y and a
.
A
(2.2.10d)
(2.2.IOe)
(2.2.10f)
(2.2.IOg)
over a parameter in-
dicates that it is the maximum likelihood estimate.
Substitution of these results into equations (2.2.7d), (2.2.8b) and
(2.2.9b) gives a set of 3 equations for estimating the effects of
treatment v-I.
15
sv-2 0
(v-4)(v-l) - sv(v-2)
4(v-l) - sv(v-2)
Pv - I + -sv(V:::i)(v-l) nv - I
sv v-I + _·sv(v':2) (v-I)
=
(Yoov - 1 - Y
+ S(V:2)
ooo )
(v-llJv-4L.- sv(v-2) 0
sv(v-2) (v-I)
v-I
-
= (Yo (l)v-I-
+
L (Yojej=nv-l,
nv
sv-2 P
sv v-I
-
OGo )
+
s(v-2)
(2.2.11a)
(v-I)(v-4) - sv(v-2)
. nv _I
+ _. sv(v-~-2) (v-I)
~
I
Y
Yoo )
(2.2.11b)
l
j=nv~
nV+l,l
+ (v~l)Jlv~jl
4(v-l) - sv(v-2) 0
sv(v-2) (v-I)
v-I
- sv(v-2) ..
sv-2
sv(v-2) (v-I)
Pv-1 + SV- nv- I
Denote the 3 coefficients on the left hand side
(LHS)
of (2.2.lla) by
cvl ' cv2 and cv3 ' respectively, and denote the RHS of the 3 equations
by YOv'Y lv ' and YZv ,respectively. Then the estimated effects for
treatment v-I are given in matrix form by
(2.2.I2a)
(2.2.120)
A
Once we have obtained
(2.2.l0a-g) to give us the
~v-l
we can substitute its entries into
A
Sj's.
More importantly, we can now sub-
stitute into (2.2.7b), (2.2.8a), and (2.2.9a) to derive the estimates
of the remaining treatment effects.
These equations reduce to the
16
following 3.
o
1
h - v-I
(
Ph+nh
-
) +
;S
2
sv(v-l)
-
v-I
1
(0)
q+Pq
+
nq
4
- sv(v-l)(v-2)
4
"
sv(v-I)(v-2) Pv - l - sv(v-I) (v-2) nv - l
1
= (Y ooh - Yoo .) - s(v-2) (v-I)
- v-I
+ _Jv-4);"
L
(Yojo- YooJ ' h ~ v-I
j=nv-l,
nv
8
v-I
(2.2.13a)
+ _(v-4)
"
2;..
sv(v-l)(v-2) Pv - l + sv(v-l)nv- l
A
A
;..
In (2.2.13a) denote the coefficients of 0V-l'Pv - 1 ' and nv - l by chI'
ch2' and ch.3' respectively.
(2.2.13a-c), respectively.
A
eh
A
P.n
A
nh
1
1
= - v-I
1
- v-i
- 1
- v-I
Let
YOh ' YIh' and y 2h denote the RHS of
Then the estimates are given by
1
- v-I
1
1 .
- v-I
1
1
- v-I
-1
A
'hI cl\2 ch3
YOh
Ylh
0
v-I
A
-
ch2 ChI ch?
Pv - l
;..
Y2h
ch3 ch2 ChI
nv- l
(2.2.14a)
or, with the obvious notational changes,
where
1
v-2
1
17
_-_
--_._._-
2.3 -------_
Variances and. Covariances
of- ....the Estimates
....
--_._--_.~.~
For the purpose of making comparisons among the effects estimated
A
A
A
Cov(~I~V_I)1
A
and
A
Cov(~'~k),k ~
~
hand h1k
A
Var(~_I)1 Var~)1
in Section 2 we need to calculate the matrices
v-I.
We will calculate
each of the elements of these matrices individually.
First note that if
element of
Xh or Iv
E(yx) = Ex
I
yx is any of the means that appears as an
and
Ex
is the corresponding mean of
€ijh lSI
then yx ,where E denotes expectation~. This is a consequence of the assumptions that all of the effects in (2.2.11) are
fixed and E(€ .. h) = 0 I all (i I j ,h) .
Now let
1)
a, ' denote the number
x
of sununands in €x and aw the same for E w' Then E(€x€y) = (1~.
(number of summands common, to €x and €w)";' axaw' We give, an,
example of a covariance calculation using this method.
Cov(yOv'Y lh )
=
x
E{[Yov-E(Yov)][Ylh-E(Ylh)]}
= E{[ (Eoov -
E000 )
+
1
s(v-2)
1
~
()]
€ojo- €ooo
j=nv-l,
nv
L
.r
s(v-l) (v-2) )=l,nv,(EOjO - EooJ]} .'
nv+l
(2.3.1)
= 1.
(For simplicity of notation in this example we will take
(12
This does not affect the calculations in any way and
is placed in
(12
(2.3.6).)
The number of summands in
~o 0v and Eo (l)h is sr(v-2) (v-I)
{ €.jo is r(v-l) (v-2)
€o~o
is srv(v-l)(v-2)
18
Every Eo1J0h is a summand for
E'
•
0 0
Now we calculate the expectations term by term.
_
E(E oov - Eooo)(Eo(l)h- Eooo ) - [
r(v-2)
)2 sr(v-l) (v-2) .
2sr(v-I) (v-2)
2
[sr(v-l)(v~2)] v
+
§rv(v-l) (v-2)
2
[srv(v-l) (v-2)]
_
2
(v-2)
rv_
-_ . . - rsv(v-l)(v-2)
2
(2.3.2)
[srv(v-l) (v-2)]
-
-
~
[1
E(E oov '- Eooo ) -I(v-~T)(v~2)
0_1..
-
J-l,nv,
nv+l
depends on the different values of j. For j
have no summands in common, and for j
common.
-
Thus if j
-
E(Eoov - Eooo
-]
(E ojo - EooJ
= nY,
= nV+l,l,
(2.3.3a)
EooV and
they have r(v-2) in
= nv,
) =0
)(-Eojo - Eooo
_
sr(v~1)(v-2)
2
[sr(v-l)(v~2)] v
_
r(v-l) (v-2)
2
[r(v-l)(v-2)] sv
+ srv(v-l) (v-2)
[srv(v-l)(V~2)J2
srv(v-l) (v-2)
2'
[srv(v-l) (v-2)]
= - ----
and if j
= 1 or
(2.3.3b)
nV+l,
srv (v-I) (v-2)
[srv(v-I) (v-2)] 2 '
where the last term follows from (2.3.3b).
This expected value sim-
plifies to
~y2(v-l)~rsv(v-l)(v-2)
[srv(v-I) (v_2)]2
(2.3.3c)
19
There are s of each of the terms (2.3.3b) and (2.3.3c) in (2.3.3a),
so that this last becomes
2
·s
{-srV(V-l)(V-2) + rsv (v-l) - srv{v-2)(v-l) }
- s(v-I) (v-2)
[srv(v-I) (v-2)J 2
_ ~s~lv-2) - rsv 2
2
(v-2)[rsv(v-I) (v-2)]
(2.3.3d)
.
Next we consider
E(Eo(l)h- E oo J[S(V=2)
If j :: nv-l,
if j
= nv,
and
E.
aJ
they have r(v-2).
2
j=nv-l
nv
a
(Eojo - EooJJ •
(2.3.4a)
have r(v-3) summands in common, and
Thus if j
= nv-l,
(2.3.4b)
and if j :: nv ,
E(Eo(l)h - E
000
) (eoja -
e )::
000
s
2
[r(v-I) (v-2)] s
::
There are
r (v-2~
rsv(v-2)
2
[rsv(v-l) (v-2)]
rsv(v-l) (v-2)
[rsv(v-2) (V_I)]2
•
(2.3.4c)
terms each of the form (2.3.4b) and (2.3.4c) in (2.3.4a),
so it becomes
2
--:--'s-:-::- { rsv (v-3L::-!sy(v-I) (v-2) + rsv(v-2) }
s(v-2)
[rsv(v-I) (v_2)]2
::
Next look at
2
~ rsv(v:~+
2
rsv (v-3)
(v-2) [rsv(v-I) (v-2)i
(2.3.4d)
20
When looking at the products of the given sums we see that there are
s
terms with E(E:- 0jo
E(Eojo -
€
-
B(E: 0J'o -
000
-E
-
) (e0ko - €
000
)
-
E ooo )
2
and 4s 2 -s terms with
J' r~ k .
)
000'
2 =r c.v..: 1) (v-2)
2
[r(v-l) (v-2)J
2r(v-l) (v-2)
[r(v-l) (v-2)]2sv
+
rsv(v-l) (v-2)
[rsv(v-l) (v-2)J 2
2 2
= rs~Jly-l)(v-2)
- rsv(v-l) (v-2)
[rsv(v-l) (v_2)]2
E(E" , - e
0Jo
)(e
°k o
000
- E
)
000
=0
_ rsv(v-l) (v-2)
[
rsv(v-l) (v-2)
(2.3.5b)
J2
•
(2.3.5c)
Putting these last two results in (2.3.5a) we have
_
2}
22
s[rs v (v-l)(v-~)-rs~(v-l)(v-Z)]-(4s -s)rsv(v-l)(v-Z)
2
2
2
s (v-l)(v-2)
[rsv(v-l)
(v-2)]
1
{
=
- rsv Z + 4rsv
Z
[rsv(v-2)(v-l)] (v-Z)
(2.3.5d)
-~---
Adding (Z.3.2), (Z.3.3d), (Z.3.4d), and (2.3.Sd) we get
Cov(YOV'Ylh) =
1
-Zo:--[rsv(v-Z)(v-l)J (v-Z)
x
{rv Z(v-Z) Z - rsv[ (v-l)(v-Z) Z + v(v-4)
- 2(v-Z) + (v-l)(v-Z)
=
rv Z(V_2)Z _~y(v3_3v2_Z)
[rsv(v-2)(V-l)J4(V~2)
+
(v-4)J}
(]2
(2.3.6)
o
21
Denote the value (2.3.6) by wvh2 ' We list the rest of the
needed covariances. Let d = [rsv(v-l) (v_2)]2. For Cov(lv~Xh) .
the rema.ining entries are as follows:
=W
vhl
(2.3.7a)
(12.
(2.3.7b)
(2.3.7c),
Using these results we have the matrix
Cov(lv~.q/ =
= w"
(12
• 1111
Hvhl
\'lvh2
wvh4
lilVh3
wYnl
wvh2
wvh4
wvh3
wvhl
(12
.
(2.3.8)
For Var(y ) we use the following entries
=-:v
= rsv2(v~})J~~2)
(V-3)
(12
,vOV
. . (YOv,Y ) = _ ~.Y~Jv:.!lJV-3)
d(v-2)
(12
Var(yOV )
d(v-2)
1v
Cov(YoV 'Y2v) = -
rsv 2 Cv-l) (v-4)
d(V-2)
=
2
(2.3.9a)
= Wv2 (12
(2.3.9b)
Wv3(12
(2.3.9c)
(12 =
Wvl (1
22
Then
\if
t'7
VI
"j
~'l
W
N
vI
=
Var(I.y)
H
t'!
v2
v2
v3
(12
' v2
vI
"v2
v3
vI
.
= WV (T2
(2.3.10)
For Var(lh)' h f v, we use the following entries.
Var (YOh)
= !.?y_(".,":.2)h.l".-I)
d(v-l) (v-2)
3
(v-2) -2J
=W
02
hI
02
(2.3.11a)
(2.3.l1b)
Cov (YOh ' YIh )
(2.3.He)
Then
11!
Var(lrt)
=
Cov(~~,~),
,."
"h2
"hI
VI
w
(
ov YOh'Y1k
h3
'tl
"h2
"h3
fJI
'h2
(12
~11
"hI
h2
.
(2.3.12)
h f k, we use the following entries.
C (
) = .,
oVYOh'YOk
.
C
liT
T;f
= \f h0 2
For
hI
2
2
£Sy[(v-:!2._(:y- 2) +4J
d(v-l) (v-2)
2
(12
232
=
W
hkl
) = !X..J~.cy",:,l)~_-:.~.L..:.~(y +6v +12v-8)
d(v-l) (v-2)
(12
J
(2.3.13a)
02
=W
hk2
(12
(2.3.13b)
COV(Y Oh 'Y2k)
(2.3.13e)
23
Then
W
hkl
Cov(Xn'~)
=
whk2
~~k3
1;i
hk2
,whk1
2
whk2 a
lJhk3
whk2
whkl
=Whka2 .
(2.3.14)
Wvh ' Nv ' ~11' andWhk with the estimates
given by (2.2.l2b) and (2.2.14b), we can calculate the covariances
Using the matrices
between effects.
(2.3.15a)
(2.3.15b)
Var(~)
~
~)
Cov (~-l'~
= [C-W
v vh
-
Z C Jl,-l~2
vh
v
= Zvh~2
v
(2.3.15c)
2.4 Tests o(.Hypotheses
Once the estimates of all the treatment effects have been made
as in Section 2 and all the variances and covariances have been calculated as in Section 3, we will be able to test for the superiority of
certain treatments over the others.
If the treatment we select as best
will be used in the future for repeated application to a single unit,
we are not interested specifically in the individual effects, but in
the sum of the direct and all residual effects for each treatment.
Thus we want to find the treatment with the largest value of
(Oh+Ph+nh)' if it is significantly larger than those of all other treatments.
If there is a group of treatments with their sums of effects
significantly larger than all other treatments but not significantly
24
e~perimenter must
differing among themselves, then the
use some other
means of choosing a treatment from this group for future use.
Criteria
for this selection might include cost" availability, ease of application, etc.
The number of
degrees~of
freedom
available for error estimation
in model (2.2.1) with restrictions (2.2.2) is
• e = rsv(v-l) (v-2)
- r(v-l) (v-2) - v(s+3)+4.
(2.4.1)
"
Also, the maximum likelihood estimates Yijh,calculated
by minimizing
the sum of squares
G of (2.2.3) are
v-I
v-I
A
A
A
A
A
r
A
Y"h
= N. 'h Cll+y ,+(3.+0h + t.. N. C' l)h P + L N.(. 2)
1J
1J
1
J
p=o 1 J P q=O 1 J- q q
n) .
J
(2.4.2)
Using these two results we get the usual maximum likelihood estimate
of
given by
(J2
sv v-I
l l
j=l h=O
"2
(Y ijh - YiJ'h)
(2.4.3)
:r
We are now ready to test the hypotheses of interest.
In order to
compare SQms of treatment effects for two different treatments we use
the usual analysis of variance statistic, the exact form of the statistic depending on whether or not one of the treatments being compared
is the (v-I)-th.
h
~
k.
h~pothesis
be H: (oh+Ph+nh)
= (ok+Pk+nk)
,
Then if H is true, tl1e statistic
A
t
Let the
hk
=
A
A
A
A
A
(Oh+Ph+nh) - (Ok+Pk+nk)
----- ...:.:....-.::.-.::.::.-.._------
Estimated{varC8h+Ph+~) + VarC8k+Pk+nk)
- 2 cov[
(5h +Ph+nhL
C8k+Pk+nk)]}
(2.4.4)
25
has the
ne distribution.
t·
To test HO: (oh+Ph+nh)
use of the matrices
= (ok+Pk+nk)'
Zv' Zh' Zvh' and Zhk
h
~
k, h, k , v-I, we make
in order to derive the de-
nominator of (2.4.4).
The estimated variances and covariances are
obtained by replacing
0
2
~2
~
~
test,
with cr .
If we want to make a size
0 <
~
< 1, for HO' then we
reject tlle hypothesis of interest if I~hk'
is greater than t n .l-a/2'
e
Other tests for relations between
where P(tn > ~n 1-~/2) = ~/2.
e
e'
treatment effects are carried out according to the usual analysis of
variance methods, making use of the results of Sections 2 and 3.
CHAPTER III
SERIAL ARRAYS AND THEIR ANALYSIS
3.1 Description anE__Cons~!uction
For the triple designs of the previous chapter we assumed that no
treatment was avplied more than once to a given unit in any 3 consecutive periods, where 2 was the number of residual effects being considered for each application.
Here we consider n-l, n=2,3, ••. , residuals
and allow a treatment to be repeated up to n times in any n consecutive periods.
Because of this we will also be able to estimate inter-
actions between the sarne treatment applied at different periods.
Thus
these designs are an extension of those commonly referred to as changeover designs. (See, e.g., [11],(20),[22],[28],[29]).
It is also a gen-
eralization of Sampford's [26J type I designs as those use only one
experimental unit.
perty
~f
The method we use requires us to sacrifice the pro-
the designs of the previous chapter that each treatment is
applied equally often to every unit.
Note also that the estimates of
interaction effects will not be balanced with respect to the effects of
application periods.
Section 5 considers a special application of the
designs given in this section.
Before giving the method of design construction we introduce some
terminology.
Let
V = {O,l, ... ,v-l} (modulo v) be a set of symbols as
used in definition 2.2.
Definition 3.1:
An ordered collection
D = (d l ,d 2 , .•. ,dw) of w
symbols from V is called a differenae set of size w (in short,
27
differenae set) if it is used to generate a vx (w+l) rectangle R = (r ij )
with the following entries.
r ll
=0
(3.1.Ia)
= 2 dk
(3.1.1b)
j-l
(modulo v) j =2,3, ••. , w+1
k=l
r .. = r lj + (i-I) (modulo v) i=2,3, .•• ,v
IJ
j =I , 2, . . • ,w+1
r 1j
.
(3.1.lc)
In words, the j-th element of the first row of R is the sum (mod v)
of the first j-l elements of D: and the i-th row is derived from the
first by adding i-I (mod v) to each of its elements.
Example 3.1:
Let V = {O,1,2} and D = (2,0,1).
The rectangle generated
by D is
·0
2
2
0
1
0
0
1
2
1
1
2
(3.1.2)
We shall use this same example to exhibit later results.
Definition 3.2:
V.
(Bose [7]).
Consider an rXc rectangle
0
0 from the set
If each rxn sub-rectangle of 0 contains all possible,n-tuples
with the same frequency.; A, . the rectangle is called an orthogonal
array of strength v) size r, c aonstraints, and v
may be denoted by (r,c,v,n)
r
n.
=,
AV , SInce
Definition 3.3:
and A is the
inde~
7..~ve Zs.
The array
of the array.
Clearly
t here are vn different n-tuples.
Consider an rxc rectangle
~
with entries from V.
S is called a seriaZ array (SA) of strength n,. r
v Zeve.z.s, and indii::c A if
(i)
!'OfPS,
c aoZwnns,
all of the v symbols occur r/v times
in every column and (ii) all n-tuples of the v symbols occur A times.
Such an SA is denoted by (r,c,v,n,A).
28
Note fron
definitio~s
3.2 and
~.3
that an (r,c,v,n)
orthogonal array
with index A is also a (r,c,v.n,A(c-n+I)) SA, so that the class of
orthogonal arrays is a subclass of the class of serial arrays.
There are vn different n-tuples, (c-n+l) of which occur in each
ra~
of a (r,c,v,n,A) SA. Thus the parameters must satisfy the relation
r(c-n+l)
One way of choosing c and r
= AVn
is to let
(3.1.3)
r
= vn-l
and c
= Av+n-l.
Only SA's of this type will be investigated here, and we shall give a
recursive method of construction.
~nen
generating a rectangle from a difference set each (n-l)-tuple
of the set gives rise to v n-tuples, where the differences between consecutive members of the n-tuples are simply the elements of the related
(n-l)-tuple of the difference set.
each row of the rectangle.
There is one of these n-tuples in
If every (n-l)-tuple occurs exactly once in
a difference set, then every n-tuple will occur exactly once in the generated rectangle because each different (n-l)-tuple of the difference
set gives rise to a unique set of v n-tuples in the rectangle. In other
words, the rectangle will have v ' vn-l = vn different, or all possible,
n-tuples. Similarly, if we can find vn-2 difference sets of size c-l
such that every (n-l)-tuple occurs
A times, then by generating a vxc
rectangle from each difference set we will have each n-tuple occurring
a total of A times in these rectangles. The total number of rows in
these rectangles is v· vn-2 = vn-l
Now take these vn-l rows and
place them one below the other to form a vn - l x c rectangle. This rectangle is a (vn-l ,c,v,n,A) SA and from (3.1.3)
the following.
c
= Av+n-l.
We have proved
29
Lemma 3.1:
v n-2
I f we can find
difference sets of size Av+n-2
such that every (n-l)-tuple occurs
(vn-l ,Av+n-l,v,n,A) serial array.
A times, then we can construct a
Notice that if we consider the vn - 2 difference sets of the ab6ve
,
lemma as the rows of a rectangle then they form a (vn-2 ,Av+n-2,v,n-l,A)
SA.
By continuing tIlis reverse process of finding difference sets to use
in generating SA's, it is seen finally that all we need to construct is
a difference set of size Av.
Then by successive generation of rectangles,
using the rows of each SA as difference sets, a (vn-1,Av+n-l,V,n,A) SA
results after n-l generations.
Lemma 3.2:
Let V
lVe have proved the following.
= {O,l, ... ,v-l}.
AV such that each member
o~
Write down a difference set of size
v occurs
A times.
From it we can gener-
ate a (vn-l,Av+n-l,v,n,A)serial aTray, n any integer ~ 2.
There are (AV)!/(A!)V
such difference sets.
Once we have an SA with specified parameters we can consider the
rows as experimental units, symbols as treatments, and columns as orderof treatment application in a design for comparing effects of different
treatments.
With these designs we can estimate direct and residual
treatment effects balanced with respect to period effects.
estimate interaction effects, but they
to period effects.
~ill
We can also
not be balanced with respect
The designs given by Williams [28] as balanced for
one residual effect with v even are (v,v,v,l,l) SASs
and for v odd
are (2v,v,v,1,2) SA'S, when the second latin square is written below the
first.
30
E~ample
3.la:
Rectangle (3.1.2) is a (3,4,3,2,1) SA.
We generate a
2
I
(3 ,4+1,3,2+1,1) = (9,5,3,3,1) SA by using the rows of that SA as
diffe=ence sets.
0
2
2
0
0
0
1
2
1
1
2
1
0
0
2
1
1
0
1
1
1
2
0
2
0
1
1
1
0
2
2
1
2
2
0
1
0
1
2
2
2
1
0
0
2
2
0
0
,1
0
0
1
2
1
2
0
2
(3.1.4)
Now place the second rectangle under the first and the third under the
0
second to get the desired SA.
3.2 General Analysis
Before using an SA to compare effects of different treatments, the
experimenter should have a good idea of the maximum number of residual
or interaction effects to be included in his model.
If this number is
n-1, an SA of strength n is needed. For each complete replication of
such a design vn - l experimental units are required, each receiving a
series of Av+n-l treatments.
tions of an SA,
Assume that we are to use m rep1icam integral, and that we have available mvn-l homogen-
eous experimental units.
Then we need not allow for unit effects in
our response model and it can be written as
Yi'h = N. 'h(~ +
J
1J
n-l v-I
S.J
+
L L N. ('-k) 0kq
k=O q=O 1 J q
n-l v-I
+
r r NiC'-k)
~k h+ ci ')
J
q q
J
k=l q=O
.
(3.2.1)
31
Yijh
is the response for unit i at time j receiving treatment h,
II
is the overall mean response,
a.J
is the effect of time period
q is applied to unit
if treatment
time (j-k)
1'1.1 (.J- k) q =
j,
i
at
otherwise ,
0kq
is the k-th residual effect of treatment q and the O-th
residual is the direct effect,
~kqh
is the interaction effect when h is the currently applied
treatment and q was applied k periods previously, and the
ara independent, identically distributed N(O,a 2 ).
e..
1J
Note that as in model (2.2.1) a large number of the y. 'h's are identi1J
n
cally zero. Here there are mv observations of interest.
In order to sinrr H:':;", the an~.YS1s, we p 1ace the
n
l '
following~restrict-
ions on the parameters.
v-I
v-I
v-I
a. = 0kq = h=O ~kqh =q=O
L ~kqh = 0 ,
J=l J q=O
~v
.r
r
r
(3.2.2)
where the subscripts not included in the summation are held fixed.
number of independent parameters, excluding
1 + (Av-l) + n(v-l) + 2(n-l)(v-1)
2
2
0 ,
The
in (3.2.1) is then
2
= v (n-i) + v(A+2-n) - 1. (3.2.Z)
Since we are considering the first n-l applications in the design as
conditioning and make AV observations on each unit, the total number
n
of observations is mAVn • The design we use must have mAv
large
2
enough to allow for the estimation of the v (n-l) + v(A+2-n)-1
independent parameters and still have sufficient degrees of freedom
available for the estimation of
02.
This can be accomplished by choos-
32
ing m or A large enough, and which of these we choose to increase
depends upon the number of treatments being used and the number of
homogenous units and time periods available.
In the following analysis
we assume that a sufficient number of observations are available for
estimation of all parameters.
For the model (3.2.1) we will calculate the maximum likelihood
estimates of the parameters.
This will be accomplished by minimizing
the sum of squares
n
mv - l AV V-l[
G=
i=l j=l h=O
I I Iy· °h 1)
+
'h(~
1)
N,
n-1 v-I
+
S.
)
+
I I N·(·_k)·q8kq
k=O q=O
1]
J2
n-l v-I
~ ~ N - )q1';kqh)'
k-l q-O iO k
(2.3.4)
We will derive the normal equations by taking the partial derivatives
of G with respect to the effects. We use r = mvn - 1 •
(a) -
!z aCL.
as.J
I I No, hY .. h i h
1]
1)
r~ - rS J
0
n-l
-
Ii hI k=OI IN.,
h N, (' -k) Ok
q q
,
Q
1]
1)
n-!
- Ii k=l
I Iq Ih N,1)'hN'('_k)
1';k h = 0
1 J
q q
•
(3.2.5a)
(The summations range over all possible values of the given sUbscript
unless otherwise indicated.)
k=k' we have the sum
Looking at the fourth term, for fixed
Ii hI Iq N,1]'hN'('_k~)
0kiq'
1)
~
residuals occurring in period
j.
But in an SA every treatment occurs
equally often in every column, including the
becomes
~ I Ok!
v q
q
= 0 ijy
which is the sum of k'-th
(j-k~J·-th.
Thus this sum
(3.2.2), and the fourth term vanishes.
We
33
write (3.2.5a) as
(3.2.5b)
(b)
aG
- ~ ~--':
aOk 'q'
?/?J
hI NijhNi(j-k')q'Yijh -
1
r~v~ - ~ J? Sj
- I
h
?
1
- li lj
n-l
~ I I
J k=O q
NijhNi(j-k)qNi(j-k')q'~kq
.
n-l
I qI Ih N,1J'hN'('-k)
N'('_k') q "kqh=·O.
k=l
1 J
P 1 J
(3.2.6a)
The simplified form of the second and third terms arises from properties
of SA's and by (3.2.2) the latter term is O.
By rearranging the order of
sumnation in the fourth term it becomes
n-l
+
I
k~k'
qI iI Ij hl
N,
'1
N,
1)11 1
C'J- k) qN.1 ('J- k') q I 0kh '
(3.Z.6b)
=0
since every k'-th residual occurs equally often.
For fixed k, the k-th
residual effect of each treatment occurs equally often with the k'-th
residual of treatment q' because every n-tuple occurs exactly mA times
in our design.
vanishes.
Combining this with (3.2,2) the second term of (3.2.6b)
Similarly, the fifth term of C3.2.6a) is 0 and we have
(3.2.6~)
(c)
~ ~
rAv
t ~
.
C' k') q ,y,1)'h - 2-l..l
- 1f )f NiJ'h,Ni(J'-k')q,BJ,
il. jl. N'jh,N,
1
1 JV
34
n-l
- iI Ij k=O
I qIN·1 CJ·_ k ) qN.1J'h,N'('_kt)
'Okq
1 J
q
- Ii
n-l
rj k=lr qLLh N, 'hN. ('-k)
N. ·h,N. ('_k1)qt?;k h = 0 •
J
q
q
1J
1
1J
(3.2.7a)
1 J
First look at the fourth term. Wllen k = k',
N.1
C'J- k') qN.1 C'J- k 1 ) q ,
~ ~
l l
and the sum is
i j
N. 'hi ok'
1J
=
q
I
{Io
if q = q'
otherwise
rAv
= --2-V
0k'q' , since each treatment occurs
equally often as the kIth residual with the direct effect of hi.
k
When
1 k', the sum is 0 because each n-tuple occurs equally often, as
described previously.
h 1 hI
In the fifth term we have N.1J·hN1J
. 'h'
and N.1 (.J- k') qN.1 (.J- k') q \ = 0 if q 'F q'.
=0
if
Thus we can split it
into two further terms:
(3.2.7b)
and
n-l
ri jL k=lL
(3.2.7c)
k1k l
where (3.2.7b) follows as above and (3.2.7c) as for the fifth term of
(3.2.6a).
Thus we have
where y(k')q'h'
is the sum of all observations where the direct effect
of h' and the k 1 -th residual of q' are present.
35
(d)
ri rj hIN"hY"
1J 1J h
n-I
rAVV - r
rj s.J - ri rj hr k=Or rq N.1J·hN1. (.J- k) qokq
n-l
- ri Ij k=lI rq hL N. ·hN. (.J-k) q7;kqh = 0
1J
(3.2~8a)
1
Using (3.2.2) and the fact that each n-tuple occurs equally often this
reduces to
rAv].l
=Y
000
(3.2.8b)
•
(3.2.5b), (3.2.6c), (3.2.7d), and (3.2.8b) form the set of normal
equations.
The
solutions of these equations depend on the particular
SA used for the design because of the sUDmlations appearing in (3.2.5b)
and (2.2.7d).
We will give a solution in matrix form for which we use
the following vectors.
.@.'
= (Sl,S2""'SXv)
= (° 00
IV
,°
(3.2.9a)
01 "" 'OO(V_l),olO""'o (n-1)0"" 'O(n_l) (V-I)) (3.2.9b)
= (7;100'7;110.···' 7;l(v-l) 0,7;101' .. O$/; (n-l) 0 (v-I) ,
'.! •
/;(n-l) (v-I) (V-I))
(S.2.9c)
(3.2.9d)
Y-a
=
(3.2.ge)
(Yo'loSo2",,,,S"Av J
r6 = (Yo(k)q)
(3.2.9f)
,
,
where the SUbscripts are ordered as those in 0'
r;;
= (y (k)qh)
(3.2.9g)
•
where the subscripts are ordered as in
I'
(3.2.9h)
36
Let
A be the matrix of coefficients for the parameters of the normal
equations arranged so that
Ie = AS
(3.2.10a)
Then the usual least squares estimate of S is given by
A
~
where A-
_
= A Ie '
(3.2.l0b)
is any generalized inverse of A.
Example 3.1b:
vious example.
Consider the analysis of the (9,5,3,3,1) SA of the preFrom (3.1.3) we note two things: (i) every 2-tup1e
occurs in each pair of adjacent columns and (ii) every 2-tuple occurs in
columns separated by one other column.
As a result the sums in equations
(3.2.Sb) and (3.2.7d) are both 0, where we also use restrictions (3.2.2).
The vectors (3.2.ga) - (3.2.9h)
are
8'
= (81 ,8 2,B3)
0'
=
~'
= (~100'~llO'~l20'~lOl'
8'
= (ll,~' ,..9.' ,~I)
re
= (Yulo,Yo2"'Yo3J
(3.2.11e)
~
= (Yo(0)O'Y~(O)1'YO(O)2'YO(1)0"'.,Yo(2)2)
(3.2.11f)
~
= (Y(l)OO'y(1)lO·y(1)20· Y(1)Ol·····y(2)02· Y(2)12'Y(2)22)
(3.2.l1a)
,
(°00,°01,°02,°10,°11,°12,°20.°21,°22) ,
····~202'~212'~222)
,
(3.2.l1b)
(3.2.lIc)
(3.2.11d)
(3.2.l1g)
and
(3.2.lIh)
The numbers of elements in (S.2.Ha) .:. (3.2.Hd) are, respectively,
3,9,18, and 30, and the same is true for (3.2.11e) - (3.2.llh).
37
A has the following form'
18
139
~
1
3
27 I
I
\. F
o,ti
01
-
-3
----."
~,...-"'-&
,
I 0'
I -18
9
-------------------_._-gls
I
I
I
I
9I S
I
I °3X18
0~9
- -- -1- - - - -1- - - - - - - - 9lg
I
I
09x3
I
I
+- - - - - I . 09X18
I
9I g
----------------------311 8
J.
18
I
I
I
I
1
I
I
I
1 3I
31
.3
I
1
I
I
3I S I °9X3 I
S
I
I
I
°18x3 I l8x3 1
'0
I
I
I
I
I
I
I SI 3
I
1
I
I °9x31 31 3
I
I
I
I
T 31 3
j
I
I
I
SI18
(3.2.12)
The numbers at the side and top indicate the numbers of rows and
colunms; respectively; in the given partitions.
0
The variance matrix of the estimated parameters is
Although the generalized inverse A- is not unique unless A is nonsingular; any estimable function
variance given by ~'A-Qa2.
~'~
of the parameters has a unique
The maximum likelihood estimate
a of
2
a2
is calculated by replacing the parameters of (3.2.4) by their maximum
likelihood estimates and dividing the sum G by
n·e
= mAvn
- v2 (n-l) - v(A+2-n) + 1 .
(3.2.14)
38
Tests of hypotheses are carried out as in the previous chapter and are
discussed for a special case in the next section.
3.3 Analysis and
Hypot~~i~-.!e~!~!11Lt~~~_ In_t~ract~ons
are Excluded
In many cases the experimenter will test for the absence of interaction effects in an attempt to simplify the form of his response model.
For this test he will need to estimate the parameters of the model
Y"h = N. 'h(ll
1.)
1.J
n-l V-I
e.J
+
I
+
I
k=O q=O
(3.3.1)
Ni (· k) Ok + E •• ) ,
J- q q
1.J
where the symbols and subscripts have the same meaning as in (3.2.1)
and the relevant restrictions from (3.2.2) hold.
The normal equations
are (3.2.5b), (3.2.6c), and (3.2.8b), where the summation of interactions in (3.2.5b) is no longer present.
(3.3.2a)
rAvll = Yo ~
0
r (ll +
f3.)
J = y J.
0
(3.3.2b)
0
okq) = Y
rA (ll +
Q
(3.3.2c)
(1.)
" q
TIlese equations have the solutions
(3.3.3a)
A
f3 j = Y" j
0
-
Yeo
(3.3.3b)
0
° -Y
and
A
kq-
where
(3.3.3c)
o(k)q
Y indicates the mean of the sum of observations y.
From
these equations we can determine the variances of and covariances between estimates in a manner similar to that used in Section 2.3.
We make use of the fact that there are r
~
v
= r'
A
'
summand
Sln
Y.(k)q·
summands in YOj. and
39
(3.3.4a)
var(8
kq
) = ( ~_ ~ + +)cr2
rA rAY rAY
= v-I
cr 2
rAY
(3.3.4b)
(3.3.4c)
since by construction the estimates are orthogonal,
(3.3.4d)
- rtv cr 2 if k=k
ql'q t
i,
(mAvn - 2 __2_ + -L)cr 2
r2A2
rAY rAY
n-l
mAy
-rA cr 2
= _0_0,2,2
r
A
if k¢k I
(3.3.4e)
v
The number of independent parameters, excluding CT~ in (3.3.1)
is 1 + (Av-l) + nev-l) = v(n+A) - n , which leaves
new
= mAvn
degrees of freedom for estimating cr 2 •
=-- r
w new i=l
"2
cr
I
r
rr
j=lh=O
AVV-I[
(3.3.5)
- v(n+A) + n )
A
A
This estimate is
r
n-l v-I
I
e·_
...
Y" h - N·· h (lJ+l3. +
N·
) <\)
1J
1)
)
k=O q=O 1 J k q q
J2
.
e3.3.6)
It is then clear that the usual F-test, which is a function of the likelihood ratio test, for the hypothesis
HO:~kqh
= 0 , all (k,q,h),
is based on the statistic
F
=
If HO is true,
1'.2
(3.3.7)
CT
F has the Fn -n n distribution.
ewe' e
Thus for a size
40
a test,
0 < a < 1, we reject HO if F > F , where P(F
>F ~ = a.
a
new-ne,n e a,
Now to compare treatment effects when interactions are taken to be
zero, either after performance of the above test or by assumption, we
can apply directly the method of Section 2.4.
The test statistic for
q t: q', is
"
"
= <\q - (\q '
(3.3.8)
2aw2 /rA
is the hypothesis
distribution.
new
that two different treatments have equal k-th residual effects,
which under HI
has the
t
k=O,I, ..• ,n-l.
Again we may want to select a Hbest" treatment, Le.,
n-l
one which has the greatest overall effect as measured by
ok • For
k=O q
comparison of these sums of effects for treatments q and q' we use the
r
statistic analogous to (2.4.4),
n-l
"
- °kq')
k=O q
t
=
----n--""'=l;----qq'
Estimated variance of
(5 -5 ,)
k=O kq kq
L (5k
(3.3.9)
r
which under HI
has the t n distribution.
ew
We now calculate the required
variance.
n-l
n(Av-l) 2
(l)(mAVn-I-rA) ~2
yare 5k ) = rAv
cr + n n2 2 '\
k=O q
r A v'
r
(3.3.l0a)
\,I
n-l
= -~ cr2 + n(n-I) (mAv
-rAJ cr2
rAY
2~2
r A v
n-l
yare
L ~kq
k=O
n-l
-
r ~kq')
k=O
[n-l
n-l
n-l
= 2 var 5k - cov( ~k '
5 ,)
k=O q
k=O q k=O kq
r
r
r
(3.3.10b)
J
(3.3.10c)
41
Using this result (3.3.9) becomes
n-l
k~O(~kq
(3.3.11)
3.4 Blocking of Units
For the type of serial arrays we are considering in this chapter,
.
t a 1 unl't s are requlre.
. d
a total of mvn-1 experlmen
As m,v, or n
increases in size the difficulty of finding such a large number of homogeneous units also grows, and we would like to have some method for
mitigating this problem,
A clue to such a method is given by the re-
cursive nature of the SA construction which we have given.
each column includes all
v treatments in the first
Recall that
v rows, second
v rows, and each succeeding v rows. Let u be any integer such
that mvn-l luv = s is an integer. Now we need only find s groups of
uv homogeneous units each and consider each successive group of uv rows
in our SA as a block.
Assume that the differences in responses between
blocks may be accounted for by independent random variables with mean
o and
common variance.
That is, for the case of no interactions,
(3.3.1) becomes
(3.4.1)
where
1
M.19 = { 0
if unit i is in block g
otherwise ,
ag ,.,. (~(O,a~)
is the random effect of block g
cov(ag,ag ,) = 0
cov(E ij .• a g)
=o ,
g¢g' , and
all (i,j,g)
42
This model is analogous to the one used for inter- and intrablock analysis in randomized block designs.
If we carry out that
analysis here we will see that no information about treatment effects
is gained by using the totals of observations for the blocks, due to
the orthogonality of block totals and these effects.
Recalling from
the previous paragraph that each treatment occurs exactly u times in
each column of a block in our SA, the total of all observations in block
g, denoted by yg
is
yg
and
EYg
= UAV~
= UAV~
+
r
i,jE:block g
(3.4.2a)
E.. + uAvag
1J
(3.4.2b)
,
E(Y g - uAvD)2
= UAV(cr2+UAVcr~)
,
(3.4.2c)
A
where
~
is the maximum likelihood estimate of
~
derived previously.
Thus, in the case where we assume no interactions, the expected values
of the block totals give us no additional information about the treatment effects.
Only the variance structure of the model is changed,
and the estimates of the variances are derived from (3.3.6) and (3.4.2c).
In the case where interactions are included in the model, these
results no longer hold.
Because the interactions do not occur in any
particular pattern in an SA, in general, the block totals will now
include the sum of all interaction effects occurring in the given block.
(3.4.2b) is now
r
AV . n-l v-I v-I
r r r r
r
Ey = UAV~ +
N. ·hN. (. k) . 1';, ••
g
i€block g j=l k=O q=O h=O 1J 1 J- q xqh
(3.4.3)
We will not further consider the analysis of this case.
3.5
Levels of a
Singl~-!~ctor
In this section we consider the adaptation of serial arrays to a
different experimental situation.
Instead of v different treatments,
suppose we have v equally spaced levels of a single factor whose
direct and residual effects can be represented as polynomials.
x"1) q denote level
q being applied at time
Let
to unit i. where the
j
lowest level is denoted by 0, the next by 1, etc., the highest being
denoted by v-I.
The application order of these levels is determined by
the SA being used for the design, the rows representing units and the
columns order of application.
If we again assume that there are no
interaction effects, we write the response model as
n-l v-I
y.. = 1.I + f3. +
1J
r r
N. (. k) Pk (x. (. k) ) + e:.. ,
J k=O q=O 1 J-
1 J-
q
1J
q
(3.5.la)
where Pk represents the polynomial for the k-th residual effect, and
are independent and identically distributed N(O,a 2 ).
1J
restriction
AV
the
E ••
r a.
j=l J
The
(3.5.lb)
= 0
is still assumed to hold.
Orthogonal polynomials will be used to simplify estimation of the
unknown
coefficien~in
the Pk'
Let
~dij
denote the d-th degree
orthogonal polynomial for the level applied to unit
i
at time
j.
Then
+ •••
k
adk~dki(j-k) .
(3.5.2)
where dk is the degree of the polynomial for the k-th residual effect.
44
The number of parameters, excluding 0 2 , to be estimated is
n-l
n
AV + l dk ,.,'
In the SA we choose for our design we must have mAv
. k=O
sufficiently large to allow for estimation of all these parameters.
For model (3.5.1) we assumed that there were no interaction effects.
Actually, because of (3.5.2). if we assume a certain additive form for
interaction effects, they will fit into the given model.
Let the k-th
interaction effect be of the form
(3.5.3)
a sum of the respective direct and residual orthogonal polynomials, dk
determined by k. Then by rearrangement of terms these effects will fit
into the form (3.5.2), b~ contributing to both a~ and a~, all k
and d.
Now we derive the normal equations from the sum of squares
r
G =
1
AV
L L
i=l j=l
(y .. -lJ
1J
-
n-l
k
2
,
(3.5.4)
We introduce the further notation
is the d-th degree orthogonal polynomial for level h.
~dh
aG
Ca)
k
e.J - k=O
L d=l
L ad~di('-k))
J
where we have made use of (3.5.2).
that
d
l
- !:aas.
J
L
y..
i 1J
- rlJ - r8. J
L
L L adk~d"1 (.J- k)
i k d
= 0 •
(3.5.Sa)
The summations are over the entire range of the indices unless otherwise
noted.
For fixed k and d the fourth term may be written
(3.5.5b)
since every level occurs !
v
times in each column of an SA.
By construction of sets of orthogonal polynomials this sum is 0 for
all k and d, so we have
r(B j +
~)
(Note that the final subscript on
= Yoj
(3.S.Sc)
~di(j-k)
may be 0 or negative.
Such values indicate application during the conditioning periods before
observations were taken.
A value of -(n·2) represents the first con-
ditioning period and 0 the last.)
aG I
(b)
- ~
k':
aad ,
f 1[~d'i(j-k')(Yij-~-Bj - t ~ a~~di(j_k»J = 0 •
For fixed
j, the terns containing
=0
For fixed
d and k#k'
and
~
Bj
(3.S.6a)
have the form
(3.5.6b)
.
the quadruple sum is
(3.S.6c)
which holds because every n-tuple occurring equally often implies that
every pair of levels occur together as k'-th and k-th residuals, respectively, equally often.
Fixing h in the RHS sum we have
L ~d'h~dq = 0
q
,
(3.S.6d)
again by construction of orthogonal polynomials.
Thus for k#k' all sum-
mands in the last term of (3.5.6a) reduce to O.
If k=k', the sum becomes
(3.S.6e)
since only one level is applied at any particular time (j-k') on unit
46
i , and each level is applied equally often as the k'-th residual
in the design.
Thus (3.5.6a) may be written
(3.5.6f)
(c)
ri jL y..
1J
-
rA~
- r
L a·J - ri jL kr dL a~~d'('_k)
1 J
j
= 0 .
(3.5.7a)
For fixed k=k' and d=d' the quadruple sum is
a~: 1~ J~ ~d'i(j-k') =
~d'h = 0
rA hL
.
(3.5.7b)
Using this and (3.5.1b) we have
rAv'll
=Y
(3.5.7c)
00
The normal equations are given by (3.5.5c), (3.5.6f) and (3.5.7c)
and their solution is
,..,
(3.5.8a)
'll = Yoo
- Yo
0
(3.5.8b)
'
AV
v-I 2
L ~d' (·-k)Y'1J./rAh=O
r ~dh
j=l 1 J
(3.5.8c)
•
\
In order to show that these estimates are uncorrelated and to find the
variance matrix for them we write the normal equations in matrix form.
We use the following two vectors.
(3.5.9a)
(3.5.9b)
The design matrix Bi
is of the form
47
I l AV I ...
1- - - --1-- --H' -I lAV I -2
1- - :- -/"- "--.-'-- - ~ _:I lAV I---.,.......
-. r
~
lAY
4v
B' =
1
-AV
~l
}
~v
AV
(3.S.9c)
-'-
} AV
---.,.......
n-1
AV
1
r dk
k=O
where the partitions are as indicated, its size is rAY x
n-l
(l+Av+
-
~i
I d ), and
k=O k
F;lil
F;2il
F;dOil 'F;liO
F;li2
F;2i2
~doi2 F;lil
.
=
F;d1iO ···F;dn-1' i , (2-n)
F;dn- 1,i,(3-n)
~dn- 1,i,(Av-(n-l))
~li,AV ~2i,AV ••.
(3.5.9d)
is the set of orthogonal polynomials for unit
i.
Using this notation
the model (3.5.1) can be written
l.
where
E
(3.S.Sc),
= B'1 + §.
is the rAY-Vector of errors
(~.5.6f),
(3.5.10)
,
ij .
and (3.S.7c) are also
E
Then the normal equations
BB't = Bl. ,
(3.5.11)
whose solutions (3.S.8a), (3.5.8b), £nd (3.5.8c) are
A
52.
= (BB')
_
By •
(3.5.12)
48
But
AV 2
. 1~l"1J
r
J=
o
. Av 2
r ~d
j=l
..
01J
Av-l
r ~2l"1J
j=O
Av-l
L ~~
j=O
o
ij
1.
AV-fn- 1) 2
j=2-n
~d
..
n_1 1J
(3.5.13)
So letting
v-I
rA
L ~ih
h=O
(3.5.14)
.
v-I
rA
r ~~kh
h=O
we have
BB'
o
=
(3.5.15)
o
D
n-1
Because of the block diagonal form of BB', its generalized inverse is
obtained by replacing each of its non-zero matrices on the diagonal
with its own generalized inverse.
Each Ok' k=O,l, ... ,n-l, is non-
49
singular and has a true inverse obtained by replacing each of its
diagonal elements with its inverse.
In the upper left of BS' we
have
(3.5.16a)
which has as a generalized inverse
(3.5.16b)
" = (BB,)-02
Thus, as discussed on page 37, var(¢)
o
=
I
____ l
I
QAv+!:d
k k I
-0' Av+!:dk
k
r
-1
_
(3.5.17)
IAV
o
I
I
I
o
. -1
D
n-1
Thus the period effect and coefficient estimates are orthogonal, and
tests of hypothesis can be made using (3.5.8a), (3.5.8b), (3.5.8c) and
(3.5.17) .
Again we note that we have v factor levels,
n effects for each
level, and a polynomial of degree dk for the k-th residual effect,
k=O,l, ... ,n-1. We must select a (vn-l ,Av+n-1,v,n,A) SA and repeat it
m times for our design, where m and A are chosen so that the number
of observations -mAvn is sufficiently larger than the number of fixed
n-1
parameters, AV + I dk ' to allow for estimation of all of them and 0 2 •
k=O
CHAPTER IV
MAXIMUM LIKELIHOOD FOR SERIES OF
DEPENDENT RANDOM VARIABLES
4.1 The Problem of Comparing Dependent Observations
In Chapters 2 and 3 we considered designs that were appropriate for
estimating direct, residual, and interaction effects of treatments under
certain assumptions about the covariances between observations on a single
experimental unit.
For the cases discussed we assumed that all observa-
tions were independent.
Now we want to consider the case where the obser-
vations on a unit are dependent, and the conditional distribution of an
observation, given all previous
ob~0~ations,
unknown population parameters.
Observation~
is determined in terms of
on different units are assumed
to be independent.
There are two specific experimental situations £Qr which this
analysis would seem to be appropriate.
~~e
of
In the first situation each experi-
mental unit receives a single treatment which is applied repeatedly at
equally spaced time points.
The ntmber of units must exceed the number of
treatments under consideration, so that each treatment is applied to at
least one unit.
Interactions between treatments cannot be estimated in
this case, but direct and residual effects can.
The experimental units
are divided into groups in the second situation.
Each unit in a group
receives the same series of treatment applications, and each group is
assigned a particular series.
The series are like those of the previous
chapters, the exact treatment order depending on the number of residuals
51
and interactions under consideration and the specific design selected.
In the next chapter we will consider some designs to be used in this
situation.
A particular case is that every unit receives a different
series of treatments.
If we are dealing with series of independent observations, we can
use the method of maximum likelihood to estimate all unknown parameters,
as we have done previously.
For comparing parameters in that case the
likelihood ratio test has certain desirable statistical properties.
In
Section 4 certain results are proved which generalize the work of Bi1lingsley [6] and Bhat [4] and allow us to apply these likelihood methods to
the case of dependent observations.
give the results of interest.
Theorems 4.4.1, 4.4.2, and 4.4.3
Known statistical results which are used in
the proofs of these theorems are given in Section 2, and in Section 3 the
assumptions concerning the conditional distributions are presented.
4.2 Preliminary Results
Theorem 4.2.1:
(Loeve [18; p. 387]).
Let {x.} be a sequence of dependent random variables.
J.
m
~ 00 ,
lJ.·m I..~ var(xi) < co as b .....,m-+co i=l b~
J.
J.
then
a.s.
0•
-io'
(E(Xklxk_P'" ,xl)
If
(4.2.1)
(4.2.2)
denotes the conditional expectation of xk given
xk_l,xk_2"",xl')
The next theorem follows from Theorem 9.1
and its extension to prove
Theorem 1.2 as given in Billingsley [6; chapter 9].
52
Theorem 4.2.2:
(Billingsley).
Let
{x.}
be a sequence of r-dimensional
-1.
random vectors with moments of order 2+0, for some
0>0, and let
fO,f l , ..•
be a non-decreasing sequence of Borel fields such that E{x~lfm_l} = 0
almost surely, m=1,2, ••• , where x~ is the j-th component of
j=I,2, .•• ,v.
!m '
Suppose that
(4.2.3)
and
(4.2.4)
for all
i,j=1,2, ••• ,r, where the matrix B = (S .. ) is positive definite.
1.)
Then
(4.2.5)
Theorem 4.2.3:
vector in Er
(Billingsley [6; p. 62]).
Suppose
!m
is a random
and l.l is a probability measure in Er
Also suppose that
is a second random vector in Er
Im
L
!m-+P'
such that
satisfying either
(4.2.6)
or
Ix
-y I s €'Iv
I ' p JIl"+OO
lim E'm = 0 •
-m .<;n
m """1ll
In either case p lim(x
Jlt"+oo
-m
-Y )
4.Jll
= 0, so that
y
4.Jll
L
-+ l.l.
(4.2.7)
(1 1 denotes the
0
usual Euclidean length in ET .)
We will use a result on sums of quadratic forms (see Graybill [14;
p. 88]) to give a small extension to a theorem of Billingsley [6; p. 64].
Theorem 4. 2.4:
!
Let .r"" Nr (.Q., V) • Suppose that r.'A.r =
r.'A.l. and
. 1
1.
1.=
2
y'Ar.,.r'Aly, •.• ,r.'Ap_l.r have X distributions with n,nl, ••• ,np _l
degrees of freedom respectively.
If .r'Apl. is non-negative, it is
53
=
p-l
~ , n
n - L n. ,
p
p,.
1=1 1
and the
~'A.~,
].
i=1,2, .•. ,p, are independent.
Theorem 4.2.5: (Billingsley). Let lsr l sr 2s .•• srk . For i=1,2, ••• ,k-l
r.
r. 1
let Hi: E 1 ~ E 1+ be a linear mapping of rank rio Define Mk = I
and Mi = Hk_l •.•Hi ' i=1,2, ••. ,k-l. Suppose !m and x are random
vectors in Erk, x ~ x, and x - N (O,V). Then MJ.!VM1. is non-singu;.;;II
r.k lar and
(4.2.8)
z.(m) = x'M.(M!VM.) -1 M!x ~L X;2
1
,-m],.
J. 1
1-m
i
and the k-l random variables
zi+l(m)- zi(m), i=1,2, •.. ,k-l, are
asymptotically independent.
Corollary 4.2.1:
In Theorem 4.2.5
zl(m) is asymptotically independent
of zi+l(m) - zi(m) , i=1,2, •.• ,k-l.
Proof:
Since
z.(m) ~
1
Consider z.(m)
= (z.(m)-z.
1
1
1- l(m)) + z.1- l(m), i=2,3, ••. ,k.
r. 1
r.
E 1- c E ]. , zi(m)-zi_l(m) is non-negative. From Theorem 4.2.5,
y2
'~.
and
z'_l(m) ~
1
1
y2
'~.
1-
1
•
Denote the limiting forms of the
random variables by !.'Aix and .!'Ai _l .!, respectively. Applying
2
Theorem 4.2.4 with p=2, we get z1.(m) - z1'_I(m) ~L yand asympt''r. -r. 1
1
1otically independent of z. l(m).
k
Now let
zk(m)
1-
= . 2Cz.(m)
2
- z'_l(m)) + zl(m). By the above discus1
1= ~
sion and Theorem 4.2.4 we have that zl(m) and z.1 (m)-z.1- I(m),
i=2,3 •••. ,k, are asymptotically independent.
4.3 Notation and Assumptions
im.
..
.
111
1
Let x
= (xl'X2""'~.) , i=1,2, ••• ,s, be s
o
independent se-
1
quences of possibly dependent random variables, each having its own
joint probability density function (p.d.f.) with respect to a a-finite
measure ~
m.
1,
given by
(4.3.1)
i
~
depending on a set of real parameters
.
tiona1 p.d. f . 0 f xki given xik-l 1S
r
e0eE,
for some r.
The condi-
(4.3.2)
so that
m.
= n1 f1.'k1
(4.3.3)
k=l a
iO
which we assume to be continuous in ai . Let a
be the true value of
.
xik-l .
ei and E~ denote the conditional expectation of x~ g1ven
For clarity in presenting the assumptions below we will drop for the
ik = f k , ai = e, etc.
moment the index i in the above notation so that f.
1
e
Partial derivatives with respect to components of
subscripts.
E.g.,
~
u
k
af
= ae-u '
k
f uv
a fk
2
= aau aev
--
e
will be denoted by
-
k
a3 f k
and f uvw = aa ae ae
u v w
The range of x given xk-l does not depend on~. Difk
ferentiation with respect to ~ through the third order may be carried
Condition 1:
out under the integral sign, integration being with respect to xk •
These derivatives are also continuous in e.
This condition ensures that in fk
= gk
is well defined and that
its first three derivatives exist and are continuous with respect to
Since
e.
J fkd~ = 1,
x
where x is the range of xk ' we have
k
guk f kd~' = f f kd~ = f~ d~ = O.
x
x
x
f
fj
f
(4.3.4)
55
Thus
(4.3.5)
and
k
- Ekguv
k k
= Ekgug
v
= O'~V(e).
(4.3.6)
To ensure independence of the parameters we assume that the matrix
~~
= (O'~v(e))
is nonsingular.
Relation (4.3.5) also implies that the
u
= 1,2, ... ,r
(4.3.7)
are martingale sequences.
Condition 2:
p lim -I m
L Igk I
~
<
m k=l uvw
(4.3.8)
00
for some neighborhood of eO and all u,v,w.
Condition 3:
m .
•
I ~ o;ok a. s.
11m
-m [, L.. 0 -+
m~
where
r
0
e
k=l
e
0;0
L..
0=
e
(
0'
uv
(eO)) ,
is a positive definite matrix of constants.
1 (Ct) this condition holds trivially for
(4.3.9)
Under condition
gk identically distributed,
k=l ,2, •.• ,m.
Condition 4:
lim
JI1"+OO
for some 0 > O.
1+~/2
m
m
I Ekl~12+0
(4.3.10)
If the fk possess some additional properties this con-
dition may be relaxed or omitted.
[27].)
a..;,s. 0
k=l
(See e.g., Billingsley [5], Silvey
56
Condition 5:
1 m
k
lim ID2 L var(guv) <
~
k=l
for all
(4.3.11)
00
u,v.
im.
Suppose that each of the sequences x
fi~1
respective conditional p.d.f.'s
1
, i=l,2, •.• ,s, and their
= fik
satisfy CI-C5. Suppose
s
further that the number of observations . m =. L1m.1 0+ 00 in such a way
e
1=
that
m.
lim...!. = \
, i=l,2, .•• ,s, A.], -:) O.
~m
= ers
(4.3.12)
Ers
be the total parameter space and q ~ rs be
be the number of distinct parameters occurring in 8 1xe 2x... xe s € ers .
Let
exex... xe
€
Using C1, C3, C4, and the I-dimensional version of Theorem 4.2.2,
_
~i
converges in distribution to a normal
we have that 1
L. g
rm-::-1 k=l u e1. =8 1'0
ikl
distribution with mean O.
Thus
m.1
ikl
P lim -!-- 1: gu
m.o+oo mi k=l
1
i
e =8
iO
(4.3.13a)
=o .
Using CI , C3, C5, and Theorem 4.2.1 we have
m• .
t
lim.l..
gik/. . a....s·_cri (e iO ) ,
m.O+OO mi k=l uv e1 =e 10
uv
].
i=1,2, •.• ,s, and u,v=1,2, ..• ,r.
- -
If some of the parameters are common
to more than one iiO, i=1,2, ..• ,s, then q < rs.
~
= (8 1,e 2, .•• ,8 q)
want to estimate.
eu ¢ ~i
In any case, let
be the vector of the q distinct parameters which we
Now considering the derivatives of gik with respect
to the components of
to
(4.3.13b)
~,
if a partial derivative is taken with respect
, then its value is O.
Thus (4.3.l3a) and (4.3.13b) hold for
57
u.v=1.2 ••••• q. differentiation being with respect to the components of
i (ao) be the resulting limit in (4.3.13b). Let
i . and let auv
s
.
~i= (ai (ao)) a (aO) = ~ A.a i (a O) and LO = I A.E 10 • Since by C3 each
uv
• uv
.f. 1 uv
•
i=l 1
1=1
°
t:
10 is positive definite. so is to' Combining these results with the
manner in which each m. increases as m increases. we have
1
m.1
'k.
1
=°
(4.3.14a)
1 t t l a.s.
11m - l.
L g
+
JIt+CO m i=l k. =1 uv
(4.3.l4b)
1
5
I Ig
m i=l k.=l u
P lim JIt+CO
and
.
1
1
mi
5
ike
1
u.v=1.2 ••••• q.
These limits will be used in the proofs of results given
in the next section.
4.4 Properties of Maximum Likelihood Estimates and Likelihood Ratio
Tests.
im
For each of the s independent sequences x i • 1=1.2 J • • • • S J
im.
the likelihood function is the joint p.d.f. of x 1 as given by equation (4.3.3).
Since these sequences are independent the joint likeli-
hood function is the product of the individual likelihood functions
s mi
II f
k.
=1
i=l
II
ik
i
In order to calculate the maximum
likel~hood
estimate
1
"a of a it will be convenient to work with the log likelihood function
given by
s
= I
mi
I
i=l k.=l
1
ike
g 1
J
m=
r
5
(4.4.1)
m. ,
. 1 1
1=
which exists under Cl.
Theorem 4.4.1:
of
aO.
There is a consistent maximum likelihood estimator
Moreover,
"
~
is a local maximum of Lm(S)
a
with probability
58
tending to one, and it is essentially unique.
Proof:
By C1 and the mean value theorem
a
~
-1
m
Lm(S) = m-
+
u=1,2, ••• ,q,
for any
~
1
I
L
g 1(6)
m.
°
1 s
J.].. °
V~l (av-ev)m- i~l k~=lguVJ.(e )
'k
~
°I.r r
1
1
2 s mi
ik.·/
(4.4.2)
m- le-~
~iG J.(a') ,
1=1 k.=l
J.
contained in the smallest of the neighborhoods
of ~o, say N, defined by C2,
length in Eq, each
°
'k
1.
j=l k.=l u
J.
u
+
m.
s
1
i'
some point in N, le-iol denotes
I~il s q2/2 , and
I
Gik.1 = sup gik i ca )
ee:N UVW-
I
•
Since ~O is positive definite it follows that there is a
that
U~l V~laUVCeo)tutv ~ a
Suppose
€
>
° is given.
for any vector
Choose
a = aCE) >
a > Osuch
te:Eq such that
= 1.
It I
° in such a way that
C4.4.3)
Then choose mO(€) large enough so that if m ~ mOCE), then with
probability exceeding 1-E ,
m. 'k
Im-l.r
iceo)1 <
1=1 k.=1
J.
r]. g:
°
m.]. ].'k .
s
1
s mG ]. < M+l
i=l k.=l
].
r r
a
u=1,2, ••. ,q
(4.4.4a)
u=1,2, ..• ,q
C4.4.4b) .
59
<
where M is some finite constant.
(4.3.l4a), and (4.3.l4b).
1~_~OI s
<5 ,
<5
u=l,2, .•• ,q,
(4.4.4c)
These relations follow from C2,
If these relations are satisfied and
then by (4.4.2)
o2
I!-~
+
I
1 s
m.~
'k
1.
m-.r
r G
1=1 k.=1
l~i
1
S
3l<52 (M+l)
Hence,if
•
1~-aOI = 0 and using (4.4.3),
m.
'k
r r
i=1 k.=1 u
9~
L m- l s
u=l
1J..
g
1(8)
°I + 30'
al~-~
(8 _aD)
u u
~
S-
]
3 q3 (M+l)
< 0.
By Lemma 2 of Aitchison and Silvey [1] there is a value e of e
. 1 s m"k
1 1. A
such that mg 1 (a) =
u=l,2, ... ,q, and such that
i=l k.=l u
1
I~-eol S <5 < E. Since E was arbitrary e is a consistent solution of
r r
A
(4.4.2) •
°,
A
60
e
We now show that
is a local maximum and essentially unique.
By the mean value theorem, if
_~£N
m.1 1ok0
r
1
f7
,
0
C8 ) +
k,=louV
1
for all
u,v, lail
q.
$
m.1 1
'k
lOS
,
m- I.~.-.~. I): a I G 1
1=1 i k.=l
(4.4.5)
1
Choose
so small that
0
{~:I!-eOI
$
o}
£ N
uv is a qxq symmetric matrix with IT uv'-auv ceO) I <
0[1 + q(M+1)] for all u,V then Tuv is positive definite. Now equaand that
T
p
tions C4.4.4) hold with probability going to one.
2
_la Lm(S)
0
m
+ 0uv(e)
u v
I
ae-a-e-
for any .~ such that
Ia_eO I < O.
- --- a2 Lm(6)
probability going to one,
such that
1~-eOI <
o.
I
S
0
+
Then by (4.4.5)
qo(M+l)
Then by our choice of <5, with
.
.
..----definite for every
a8 a6 lS ne!Yatlve
(>
u
a
v
It follows that with probability tending to one
there is a unique solution of the maximum likelihood equations and any
o
such solution is a local maximum of Lm(S).
The next theorem gives the asymptotic distribution of the likelihood ratio test for the hYFothesis that
points in
~.
e=~O, where
eO
is a set of
It also shows that the maximum likelihood estimates
have asymptotically a normal distribution.
1
m,1
s
= m-~ I
'k
1.
I
First we define
0
g l(e)
i=l k.=l u
(4.4.6)
A
(4.4.7)
°
= m~Ceu-eu)
1
and
Let
e (m)
u
~Cm)
and
respectively,
~Cm)
1
.
be the vectors with components
yu(m) and euCm)
Also let max L be the maximum of Lm(S)
L~ be the maximum on
e
eO.
m
over
e
and
61
Theorem 4.4.2:
o
Let
A
e
be the true set of parameter values and
~
be the ma.ximum likelihood estimates which exist by Theorem 4.4.1. Then
p lim
~Cm)
=
m~
.rCm)
= to-1y..cm)
P lim ~.(m)
EO~Cm)
m~
L
q CQ, r O)
~ N
and
o
2[max L -L ]
e
m m
T.
~
~_Cm) ~ Nq(Q,E~l)
(4.4.8)
2
q
(4.4.9)
X .
Proof:
Using Cl, C3, C4, Theorem 4.2.2, and equations (4.3.14), we
eLm(S)
(O,r O)' On setting
have that y(m) ~ Nqasu = 0 and using (4.4.2),
we have for eeN
m. 'k
SAO 2
S
[: 1 S
J.
1.
0 ]
/ /e(m)
I
-yu(m) = l ev(m) m-.r
guv 1(8 ) + a./e-e
i=l 1 - v=l
1=1 k.=l
r
r
1
where
la.1 I
s q2/ 2 and --e'eN.
Now p limla-eo,
nr+oo - -
= °, so
by equations
(4.4.4b-c) for m ~ mo(€) ,
IYu(m) where p lim € (m)
~
u
q€ml~(m) I, where
V~leV(m)cruv(so) Is €u(m)l~(m)/
= O.
E
= max
u
p lim qEm = O.
Also
Let
m
€ (m).
u
IEO~(m)
Then ~(m) - rne(m)I s
v-
I
= cml~(m) I ' where
m~
m is a finite constant for each m ~ mO(€)' Then
Em
l.r(m) - E~(m) / s q
l EO~(m) I
C
c
m
€m
where p lim q -- = O.
JJt+<lO
p lim
m-+oo
~(m)
cm
= EO-1~(m)
By Theorem 4.2.3 p lim
EO~(m)
=
~(m)
and thus
~
and
L-1
Nq(Q,E o )'
~(m) ~
Again by the same theorem
62
we now have p lim rCm) =
Lo~.cm).
lJl"+IlO
The asymptotic distribution of 2[max,L _LO] remains to be found.
e
For
~€N
ike
g l
m m
we again use the mean value theorem to get
ike
ce ) = g
l
+ ~
+
~
i
0
ca )
+
0 ike
0
i C8u -8)g
l ca )
u u
u= 1
g o o ike 0
i ca -8 )(8 -8)g l ce )
~
l
u=l v=1
u u
~
0
A
A
v v
0 s
0 3 5
m.
1
UV
mi
+16-8
ILL a.G
--.
1
1=1 k.=l
1
ike
L L
1
1
By C4.4.4b)
~
Then
+
0 3 ike
a.18-a I
1 -- -
ike 0
g 1 ce )
v v i=l k.=l uv
i C6u -au )C8 -6 )
u=l v=l
A
~
0 in probability.
G 1
(4.4.10)
63
using (4.4.4c). Applying the first two limits of (4.4.8) to this resuIt we have
0
A
-1
P lim 2[L (6)-Lm] = ~'(m)Lo
J11:+OO
m
rem) .
)
Finally, using the distributional convergence of
and Theorem
~(m)
o
4.2.5, (4.4.9) follows.
Now we come to the main problem of interest in this
the
chapter~
testing of equality of parameters between different sequences of dependent random variables, i.e., that the true parameter values lie in some
im.
subset ~ of e. Suppose that {x 1}
i=l,2, ... ,s, satisfy Cl-CS.
J
Let
e
d: W~
be a mapping of an open subset
Denote any point
the components of
~
in
~
d(~).
by
(~l'~2""'~c)
Also suppose that
~EEc, c~q, into
and by
dj(~)'
e.
j=l,2, .•• ,q,
d satisfies the following
condition.
Condition 6:
The qxc matrix
D(~)
with entries
ad. (~)
(D(~))ju = -~
(4.4.11)
j=l,2, ••• ,q, u=1,2, .•. ,c
u
has rank
c throughout
~
and the
d.
J
have continuous third partial
derivatives.
Example 4.1:
Let r=3, 5=3, q=rs=9
parameter sets for the 3 sequences.
and 6 = (6i,e~,e~)
i
Let
e = (~1,~2,e3)
suppose that we want to test the hypothesis
Then
~EE7 and we may write
and
/
HI:
be the
E Eq and
e; = 63, i=l,2,3.
64
The matrix 0('1»
is given by
=
O(~)
1 0 0 0
0 1 0 0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1 0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
which is of rank 7.
0
ii
Example 4.2: If we have a subset ei = (8 i
,8 , ... ,S )
""1?
PI Pz
Pp
i
from each e and we want to test th~ hypothesis
HZ: Si
-p
= (6PI ,6. Pz , ... ,6Pp ),
of size P
i=1,2, ..• ,5, then the representation given
above may be extended directly to this case, where q = rs and
,
1
2
s 5+1
c = rs - (s-l)p. We wr1te 1 = (~ , a , •.. ~ ,a ) , where
ii
a s+l = (6 ,6 , .. .,6 ) and a i
= (6
,e
, .... 6i L i=l,2,. .. ,s.
PI Pz
Pp
Pp+l Pp.+2
Pr
Then the components of the mapping are given by
dj ('1» =
e
j
ei
j = Pk+(i-l)r
Pk
Pk
= Pk+(i-l)r
i=1,2, .•. ,s, k=l,Z, .•• ,p
i=1,2, ... ,5, k=p+1,p+2, .•. ,r.
0
If we let
¢i be the set of components of !
which relate to the
i-th sequence, i=1,2 •...• s, then we can let di(!i) represent the
,
,
, c,
mapping d : '1>1 + e1 , which is a restriction of d. If '1>l eE 1, then
i
the matrix O,('1>i) defined similarly to O(¢) has rank c. through1.
'k"
'k"
l'k
.
1
1
1'k ,
1 •
out ¢1. Let f 1(¢1) and g 1(¢1) represent f 1 and g 1
respectively under the assumption that
~
is the true parameter space.
65
Theorem 4.4.3:
ik.
Suppose that d: ~
satisfies C1-CS, i=1,2""$s.
0 satisfies C6.
+
Moreover~ if ~O
Then f
= dC!O)
.
1(~1)
is the true
value of the parameter set, then
C4.4.12a)
and
2[max L - max L ]
o m ~ m
L
+
2
C4.4.l2b)
X
q-c
and these statistics are asymptotically independent.
Proof:
By C6 the matrices
diC~i)
indicates that ~i is the
k.
set of parameters we are considering and the L 1 are defined by (4.3.6).
are nonsingular, i=1,2, ••. ,s, where
is a linear
By the chain rule for differentiation
1
function of the m:~
1
m.1
'k.
1
r
g lce l ) , u=1,2, ... ,r.
k =1 u
.
'k.
1
i
isfies CI-CS, so does each f
1
ik.
Since each f.11
e
sat-
.
C!1)
and Theorem 4.4.1 holds.
Then
there is a consistent solution of the maximum likelihood equations
u=1,2, ... ,c.
Suppose that
and ~O
~
is the
true set of values, Le. ,
LmCdC!O)) =
Let
LmC~O)
~(m)
=
L~
and
Lmci) = max Lm.
~
be the random vector with components
m.1
'k,
1 s
a 1
W
U
(m)
=
I I --m1 i=l k.=l
a~O
1
u
-1
g
1
C4.4.l3)
66
By the chain rule
~(m)
1
= D'(~ °)r(m).
m.
°
r
O'uv (~))
1=1 k.=l
s
'k
~ i
1
em)
~
is the variance matrix of rem)
and the variance matrix of
~(m)
is
D,(~O)E~D(~O). From the proof of the previous theorem we know that
= p lim r'
(m)D(~O) [0' (~O)E~(~O)] -10", (~?)y(m)
J1t+OO
=p
lim r'(m)O(~O)[D'(~O)EoO(~O)]-IOI(~O)r(m).
1Jl-+oo
Now let
z2(m)
= 2[max
o
= 2[max
o
Lm-LO]
m
L -max L ].
m ~
m
and
zl(m)
= 2[max
cp
But y(m) +L
L -LO] ,
N(~,EO)
m m
50
that
and by Theorem
4.2.5 and its corollary the results of this theorem follow.
0
For examples 4.1 and 4.2, respectively, we have that
2[max L -max L 1 L X2 and 2 [max L -max L ] L X2
o mH ~ + 2
0 m H2 m + (s-l)p'
1
Exa~le
4.3:
4.5 Summary
Our objective in this chapter has been to find a method for estimating and comparing treatment effects when our observations consist of a
number of observations of independent sequences of dependent random
variables.
If the conditional probability density function of each
observation in a series exists and satisfies conditions I through 5,
then by using the maximum likelihood method we have the following
results:
67
(1)
by Theorem 4.4.1 the estimates of the parameters of the conditional
densities are consistent;
(2)
by Theorem 4.4.2 the estimates have asymptotically a joint multi-
normal
(3)
distributio~;
and
by Theorem 4.4.3 the likelihood ratio tests of hypotheses of the
form "H: ~ce
is the true set of parameters ri have asymptotically a X2
distribution if the mapping d:
6.
~ +
0, defined by H, satisfies condition
These results are exactly analogous to those for the usual case of
sequences of independent random variables.
It should be noted that some of the conditions 1-5 are weaker than
other assumptions used to prove asymptotic normal distribution in the
dependent case.
See, e.g., Diananda [9J, Hoeffding and Robbins [15],
Silvey [27], and Billingsley [5],[6].
If the conditional densities
under consideration satisfy one of these other assumptions, then some
of Cl-C5 will be unnecessary for the results to hold.
CHAPTER V
CONSTRUCTION OF SOME 2- AND 3- REPEATING DESIGNS
5.1
n-Repeating Designs
Here we consider some designs which may be useful in collecting ob-
servations for which the analysis of the previous chapter is applicable.
Consider the situation in which each experimental unit receives a specified
series of treatments, and we wish to estimate direct, residual, and interaction effects for dependent sets of observations.
If we suppose that we
know the form of the conditional densities and that they satisfy the conditions of Chapter 4, then we can use the method of maximum likelihood
to estimate the treatment effects if they are present in the conditional
densities.
An obvious requirement of any design to be used for this pur-
pose is that as the total number of observations on a unit increases so
does the number of appearances of each treatment parameter of interest
occurring on that unit.
If some parameter, say the u-th, appears only a
finite number of times with unit
i,then t~ will have a zero in position
(i,i) and will not be positive definite.
This is a violation of C3.
Another desirable property of such designs is for each direct and
residual effect to occur equally often, each first interaction to occur
equally often, each second interaction to occur equally often, etc.
Also
we would like any member of a class of effects to occur equally often
with each member of another class.
In the particular case of asymptotic
estimation, the estimates of elements in the variance matrix for the
estimated parameters are given by the negative of the summations in the
LHS of (4.3.14b).
If the above properties are satisfied by the designs,
then an equal number of terms will contribute to each estimated variance
for a particular class of effects, and similar results hold for the
estimated covariances.
These conditions are satisfied by letting each
n-tuple or n-plet, whichever is being considered, occur equally often in
the design.
If we allow a treatment to occur more than once in any n
consecutive applications,
carefully chosen.
the serial arrays of Chapter 3 can be used if
If we allow a treatment to occur at most once in any
n consecutive applications then we could use tied-double-change-over
designs, in selected cases, for n=2 and triple designs for n=3.
For these
latter two cases we will give generally applicable methods of constructing
designs based on the work of Williams [28],[29].
In fact, the designs for
n=3 are constructed similarly to the method used for triple designs.
How-
ever, triple designs required at least (v-I) (v-2) units, and here we will
often be able to use fewer units.
E.g., when v-2 is divisible by 3 only
(v-I) (v-2)/3 units are required.
Let n-l
be the maximum number of periods for which residual or
interaction effects are assumed to be present, and let v be the number
of treatments being compared.
Definition 5.1:
Let n,r,s, and v be integers.
An n-repeating design
(n-R design) is defined by a set of s basic treatment series, each of
length rv.
If we consider the last n-l treatments of each basic series
to be applied initially as conditioning preceding the basic series, rv
n-plets are included in each series of applications.
these series have the following three properties:
For an n-R design
70
(i)
Each successive set of n treatments in a basic series
includes each treatment at most once.
(ii)
The rv n-plets occuring in each basic series are all disdinct.
(iii)
In the srv n-plets occurring in the basic series, all the
different n-plets occur equally often.
In a given experiment p units are assigned each basic series and
the total number of units is
sp.
The basic series are continually
repeated until the experiment ends.
If the basic series is repeated
m·
1
the same number of times for each unit then A. = lim ~ = -- ,
1
m-+co m sp
i=1,2, ••• ,sp. Thus if the likelihood method of the preceding chapter is
used to estimate the parameters of a conditional density, the information
from each unit contributes equally.
Because of the asymptotic nature of the results of Chapter 4, we
would like each n-plet to occur as often as possible.
For a fixed amount
of time available for an experiment this can be accomplished by making r
small.
Condition 3 of Chapter 4 requires the nonsingularity of the
limiting variance matrix for each experimental unit, so that as the
number of parameters increase, r may have to also.
For these reasons
we will consider construction of n-R designs for n=2, r=2, and for n=3,
Before proceeding with the constructions we give a simple example
of a 2-R design in order to clarify the definition.
71
Example
If v=3 we have a special case because only one basic
5~:
series is required for a 2-R design.
If V
= {O,l,21
is the set of
treatments, then a single basic series defining a 2-R design is
o1
2 0 2 1.
plets is
Using
I
1 I
O~l
1 for conditioning, the series which gives our 22 0 2 1 , which includes all 6 different 2-plets.
The values of the parameters for this design are
n=2, r=2, s=l, and v=3.
5.2 Construction of 2-ReEeating Designs
Assume that each treatment has an effect for one period after its
application so that we are interested in estimating at most one residual
and one interaction effect for each application.
Then n=2 and we will
give methods of constructing 2-Rdesigns with r=2.
Different construct-
ions will be/given for v odd and v even, but they both use the designs
balanced for a single preceding
effec~
with an odd number of treatments,
as given by Williams [28] .
. First consider the case where v is odd.
Theorem 5.2.1:
For any odd integer v = 2p+l, P
= 1,2, ... ,
there exists
a 2-repeating design.
Proof: We first give a method for constructing such designs and then show
that the properties of 2-R designs are satisfied.
From Williams [28]
there exist a pair of Latin squares of size v such that every 2-plet
occurs twice.
The difference set used to generate square I includes each
odd member of the set
V' = {1,2, •.. ,v-l}
exactly twice, and the differ-
ence set used to generate square 2 includes each even member of V'
exactly twice.
o
1
For square 1 the first row is
v-I
2
v-2
3
v-I
-2-
v+l
-2-
72
and for square 2 the first row is
o
v-I
1
v-2
2
v+1
-2-
v-3
Letting row (i,h) denote row h of square
v-I
-2-
(S.2.1b)
i, i=1,2, and h=1,2, .•• ,v,
we may write the j-th element of row (1,1) as
l-j
2
.1
2
if j
if
j
is odd
,
(S.2.2a)
is even ,
(S.2.2b)
and the j-th element of row (2,1) as
v+j
2 + P if j
is odd
_.i
is even
2
if j
,
(S. 2. Sa)
.
(S.2.3b)
(All arithmetic operations are to be performed modulo v.)
By the method
of generating the Latin squares we may also write the j-th element of
row (l,v) as
I-j +
..........
:v-I
2
.i + v-I
2
if
j
if j
is odd ,
(S.2.4a)
is even •
(S.2.4b)
and the j-th element of row (2, v) as
v+'
~+ p + v-I if j is odd,
-! + v-I
if j
(S.2.Sa)
(S.2.Sb)
is even.
rrom these two squares we construct a 2-R design with v-I = 2p basic
series.
The first
p basic series consists of row (l,i) followed by
row (2,p+1 - i), i=1,2, •.• ,p, and the second p consists of row (l,i)
followed by row (2,p-i), i=p+I,p+2, ..• ,2p.
We now show that every 2-
plet occurs exactly twice in the basic series.
By the method of selecting the 2 Latin squares, each 2-plet occur:s
in them twice.
In forming our basic series each of the 2-plets in row
73
v of the two squares is omitted.
To each of the basic series we have
added two 2-plets, one where the row from the second square follows the
row from the first and one where the first element of the row
from square 1 is repeated.
These added 2-plets must be the same as the
2-plets omitted from the final rows if we are to have a 2-R design.
For row (l,v) the v-th element is p, the elements in all other oddnumbered columns are >p, and the elements in the even-numbered columns
are <po
This may be seen by substituting the numbers
in (S.2.4a-b).
j
(i)p
(ii)
(iii)
O,l, ..• ,v-l
for
Thus row (l,v) has the following 2-plets:
is not followed by any other number;
i < P
is followed by v-i-2; and
i > P is followed by v-i-l.
We show, for example, that (ii) holds.
even-numbered column, say j, so
l-(~+l) + v-I
= v-I
- ~.
But
i
!
From (S.2.4b), i occurs in an
=! + v-I. Then it is followed by
= i+l - v, so v-I - ! = 2v - i-2 =
v-i-2 (mod v), and (ii) holds.
For row (2,v) the odd and even positioning of the elements is not so
apparent.
It is easy to see that v-I
= 2p
occurs in position 1.
Begin-
ning with the pair in positions 2 and 3, every 2-plet with initial element
in an even-numbered position has been reversed from the corresponding
pair in row (l,v).
(a)
P
(a)
for
j
To prove this we must show that
even, - i + v-I = 1-(j+1) + v-I and
(b)
for
j
odd,
is obvious.
= ~ = V;l ,
2
2
50
v+j
2
+
p+v-l
j-I
= -2+ v-I
.
.
For (b) we must show that v;J
. I
+
p =~.
that the LHS of this last equation is
But
2V;j-l
By making use of these results it is easily shown that the following
2-plets occur in row (2,v):
(iv)
(v)
(vi)
p-l is not followed by any other number ;
i, 0 SiS P - 2 and i
= 2p
i, pSi S 2p - 1, is followed by v - i - 2.
We show, for example, that (v) holds.
and i
i
= 2p
= V;J.
, is followed by v - i - 3; and
From (S.2.Sa)
i, 0 siS P - 2
, occurs in an odd-numbered column, say j (j<v), so
'-1
=~
+ v-I, using
But j = 2i - 2v+3 , so
+ P + v-I
-(~+l) + v-I.
(b).
It is followed by
-(j;l) + v-I
= 2v
- i - 3
=
v - i - 3, and (v) holds.
Now we show that the new 2-plets formed from the v-th element of the
square 1 row and the 1st element of the square 2 row in each basic sequence
are exactly those 2-plets that occur in row (l,v).
For series i,
1 S· i s p, row (l,i) is followed by row (2,p-i+l). Element v of row
.
I-v
.
(1 ,i ) 1S --2- + 1 - 1 = p+i which is in the range p+l to 2p. Element 1
of row (2,p-i+l) is V;l + P + (p-i+l)-l
= p-i.
1 s i s p, is followed by v - (p+i) - 1
= (2p+l)
from (iii).
For series
In row (l,v), p+i,
- (p+i) - 1
= p-i
But this is the same as the 1st element of row (2,p-i+l).
i, p+l sis 2p , row (l,i) is followed by row (2,p-i).
Element v of row (l,i) is p+i which is in the range 0 to p-l.
1 of row (2,p-i) is V;l + P + (p-i) - 1
o sis
,
p-l, is followed by v - (p+i)-2
= p-i-l.
= p-i-l,
is the same as the first element of row (2,p-i).
Element
In row (l,v), p+i,
from (ii).
But this
Thus the result has
been shown.
In an analogous manner it can be shown that the new 2-plets formed from
the v-th element of the square 2 row and the 1st element of the square 1
row in each basic series are exactly those p 2-plets occurring in row
(2,v).
15
As a result, in our 2p basic series each 2-plet occurs exactly 2
times and we have in fact constructed a 2-repeating design.
Example 5.2:
Let p=4 and v=9.
The two Latin squares used to construct
the 2-R design have the following initial rows.
o
o
2
7
3
(See (5.2.la-b)).
I
8
645
(S.2.6a)
8
1726354
(S.2.6b)
The series are shown in Figure 5.1.
Theorem 5.2.2:
0
0
For any even integer v there exists a 2-repeating
design.
Proof:
Again we consider the pair of Latin squares with initial rows
(S.2.la-b), this time of size v-I.
The elements of these initial rows
are calculated from (S.2.2a-b) and (5.2.Sa-b).
In order to construct a
2-R design for v even, we use the following procedure.
(i)
(ii)
(iii)
Write down the pair of squares of size v-I.
Place a v-I at the end of each row in the two
two (v-l)xv rectangles.
sqUQre~,
si";ng
The i-th basic series is defined by row i of the first rectangle
followed by row .i-l of the second, i=2,3, ••• ,v-l. For the
other series row I is followed by row v-I.
Recall that by the method of constructing the two Latin squares each
2-plet of the numbers O,I, ... ,v-2 occurs in them exactly twice.
Since
each of these numbers also occurs as the initial element of one row in
each square and as the final element in some other row, it is apparent
that each 2-plet including v-I occurs exactly twice in the design.
remains to be shown that no 2-plet occurs twice in any basic series.
We consider two separate cases.
It
76
Figure 5.1
first p=4
rows
4
S¥t
0
1
8
2
7
3
1
2
0
3
8
4 7 5 6
2
3
1
4 0 5 8 6
3
4 2 5 1 6
6
7
8
1
7
2
6
3
5
4
0
2
8
3
7
4 6
5
2 1 3 0 4
8
5
7
6
4 1 5 0 6
8
7
:'71
~
'I
0
7
8
3
2
----------------------------4 5 3 6
second p=4
rows
2
7
1
8
2
0
lXS
4 3 5 2 6 1
0
5
6
4 7 3 8
6
7
5
8
4 0 3 1 2
7
8
6
0
5
~6
4 6 3
7
7
0
8
8
1
0
2
1
4 1 3
2
2
5
7
4 8 3 0
6
8
5
'I
last row
onunitted in
design
{
1
4 2 3
7
0
----------------------------8
0
7
1
5
2
5
SQUARE 1
3
4
8
7
0
6
1
5
2
SQUARE 2
Arrows indicate rows of the second square which follow rows of the
first in forming the basic sequences.
4 3
77
(a)
The differences between adjacent elements in square I are odd
and in square 2 are even.
Thus no 2-plet which does not include v-I
is duplicated in a basic series.
(b)
For the 2-plets including v-I to occur singly we must have the
initial elements of the two rows used for a basic sequence different,
and the same for the final elements.
since row i
The initial elements are different
of each square has initial element i-I.
Now suppose that
the (v-l)th elements of rows (l,i) and (2,i-I) are identical.
Then we
must have (mod v-I = 2p+l)
1- (~-l) + (i-I) = (V-I); (v-I) + V;2
using (5.2.2a) and (5.2.3a).
+ (i-2)
(5.2.7a)
This reduces to
p + i
=P +
(5.2.7b)
i - 2 ,
)
which is impossible unless v-I
= 2.
But v-I is odd, and the 2-plets
o
including v-I in each basic series are unique.
By the methods of Theorems 5.2.1 and 5.2.2 we have constructed 2-R
designs for all v, where the values of the remaining parameters are
r=2 and s
=
Example 5.3:
even.
v-I.
We demonstrate the construction of a 2-R design for v
Let v=4, r=2, and s
3
3
3
(a)
.
.
0
I
2
= v-I = 3.
1
2
2
0
0
1
3
.
2
1
0
2
1
0
3
Square 2
(b)
y"'70
. 3~l
Square 1
3 . 2
(b)
3
.
3
..
Rectangle 1
Rectangle 2
78
(a)
Conditioning application
(b)
Final elements v-I = 3 added to end of each row.
The arrows indicate which row of rectangle 2 is·to follow a given row of
re~tangle
0
I to form the basic series.
5.3 Construction of 3-Repeating Designs
Now assume that each treatment has a residual and/or interaction
effect that is present two periods after application, in addition to its
direct effect.
For a design to be balanced for every pair of residual
effects and interaction effects it must have every 3-plet occurring
equally often.
In order to construct a class of 3-repeating designs
we will again make use of the designs balanced for pairs of residual
effects as given by Williams [28],[29].
If v treatments are being con-
sidered these designs consist of v-I Latin squares of size v.
The i-th
row, i=I,2, ••. ,v, of each square has initial element i-I, and we will use
only those designs for which the final element of the first row of each
square is different.
Then the final element of the i-th row of each
square is different.
Recall from Chapter 1 that every 3-plet occurs in
these designs exactly once and that every 2-plet occurs at the beginning
of some row and the end of some other row exactly once.
Whether the
designs used are orthogonal sets of Latin squares or constructed from
difference sets does not matter.
First consider the case where v-2 is divisible by 3.
We shall give
a method for constructing 3-R designs for this case which can be extended
to cover the cases when v-2 is not divisible
py
3.
79
Theorem 5.3.1:
A 3-repeating design with r=3 exists for v-2 divisible
by 3, where v is the number of treatments being considered.
To define the basic series, first take a design for v-I treatments
Proof:
balanced for pairs of residual effects, as described above.
Add a final
column with all entries v-I to each square, and write down side-by-side
the resulting v-2 rectangles of size (v-I) xv.
In each row of this
(v-l)xv(v-2) array let the first 3v entries, second 3v entries, etc.,
constitute the basic series.
The last 2 elements of each basic series
are placed at the beginning of that series as conditioning in order to
determine the rv 3-plets which occur.
It is obvious from the discussion
preceding this theorem that every 3-plet not containing v-I occurs exactly
once in some basic series.
We show that every 3-plet containing v-I
occurs exactly once.
(a)
Since every 2-plet occurs at the beginning of some row in the
original squares, every 3-plet with v-I as the initial element occurs exactly once.
Similarly, every 3-plet with v-I as the third element occurs
exactly once.
(b)
The first row of each of the original squares begins with 0
and ends with one of the elements 1,2, .•. ,v-2, where the last element for
each square is different.
of (v-I)'s
Because of the addition of the final column
to each square, each 3-plet of the form 'j
(v-I)
0,
j=l,2, ••• ,v-2, occurs once in the basic series constructed from the first
rows of the rectangles.
plets
j
(v-I)
from that row.
i,
i~j
Similarly, for row i, i=1,2, •.• ,v-l, the 3, each occur once in the basic series constructed
Thus every 3-plet with v-I as the second element occurs
exactly once in the design. Since each 3-plet occurs exactly once in some
basic series, no 3-plet is lepeated in a given series.
0
80
Example 5.4:
Let v=5 and v-2 = 3 be divisible by 3.
Each row of
the following rectangle represents a basic series and its conditioning
treatments.
2
4
0
1
2
3
4
1
0
3
0
4
2
3
0
I
4
3
2
1
3 I
I
2 I
I
1 I
I
0 I
•
Square 1
(a)
4
0
2
3
1
4
0
3
1
4
1
3
2
0
4
1
2
0
4
2
0
1
3
4
2
1
3
4
3
1
0
2
4
(b)
3
0
2
(b)
.
,
Square 3
I' 'I.
" '"
Rectangle 2
Ca)
Conditioning treatments
(b)
Columns of v-I = 4.
I
I 4
I
1 I 4
0
(b)
Square 2
Rectangle 1
2 I 4
I
3 I 4
Rectangle 3
Note that the corresponding rows of each square have the same initial
element.
Because of this and the fact that the 3 Latin squares are
mutually orthogonal, no two last elements of the same row in different
o
squares can be identical.
If v-2 is not divisible by 3 we can use the same type of procedure
to construct 3-R designs, but we must increase either r or s.
This is
because the number of elements in any row of the derived CV-l)xv(v-2)
rectangle is not divisible by 3v.
when we
co~e
We cannot just go to the next row
to the end of a given row in constructing the basic series.
In doing so we might repeat a 3-plet with v-I in the second position and
leave out some other 3-plet.
There are 2 different procedures which over-
come this problem.
(i)
If v-2 is divisible by the integer
q~3,
use the first qv
entries, the second qv entries', etc., of each row of the
derived (v-l)xv(v-2) rectangle as the basic series.
Then
81
r=q and s
(ii)
= (v-2)(v-l)
q
•
If s = (v-l)(v-2) experimental units are available and r=3 is
to be used, write down the derived (v-l)x(v-2) rectangle 3 times
to form an extended (v-l)x3v(v-3) rectangle.
Now use each suc-
cessive set of 3v elements in a row as a basic series.
That (i) and (ii) in fact give 3-repeating designs is clear from the
proof of Theorem 5.3.1.
CHAPTER VI
SERIAL FACTORIAL DESIGNS
6.1 The Serial Factorial Model and Notation
Instead of v distinct treatments, we now consider the case of
having' W different factors, each of which may occur at several different quantitative levels.
A treatment combination then consists of
the simultaneous application of one level of each factor.
When one unit
is assigned to each possible combination we have a factorial experiment,
and when only a certain subset of the combinations is used we have a
fractional factorial.
These designs are used to estimate the main effects
of each factor level and the interaction effects between levels of different factors.
In some situations an experimenter must decide on a single treatment
combination to be used repeatedly in the future.
Then the experimenter
might be more interested in the direct and residual effects of each factor
level in a series of treatments rather than in interactions between factors in a single application.
For this case we would like to be able to
arrange series of applications for several experimental units that would
\
allow us to estimate all the direct and residual effects with equal variances within,and covariances between,classes of estimates (e.g., direct
effects, first residuals), and at the same time use as few combinations
and units as is practicable.
A natural approach to this problem is to use fractional factorials
and apply the combinations of each fraction to a single unit in a speci-
83
fied order.
Here we will consider designs of this type when each of
the w factors has 2 levels, the 2w series, for w = 2,3,4,5,6.
These
designs are closely related to those described by Patterson [21] and
called serial factorial designs.
Here we shall use this same title
for our designs, and we now describe this serial factorial model.
Denote the factors by capital letters and the entire set by
A = {A,B,C, ••• ,Q}, IAI
scripts 0 and 1.
=w ,
and each factor's two levels by the sub-
Also, letting the subscripts d and r
direct and residual effects respectively, each factor
effects denoted by fdO,fdl,frO' and
represent
f€A has four
f rl • We will use subscripted
lower case letters to represent actual factor levels being applied.
E.g.,
adO denotes factor A occurring at direct effect level 0 and brl
that factor B occurred at level 1 in the previous application, and
thus for the present application residual effect 1 of B occurs.
Rather than estimating both f
and f
dO
dl
interested only in any difference between them.
for each f€A, we are
Thus to reduce the numb-
er of parameters to be estimated we use the restrictions
(6.l.la)
Similarly for residual effects we have
(6.l.lb)
This leaves us with
2w
independent factor effects to be estimated.
We will be interested in testing llypotheses of the form HO: f rl = 0,
Vf€A, and H2 : f rl + f d1 = 0, Vf€A.
For the k-th observation, k=l,2, ••• ,n, on unit j, j=1,2, .•• ,m,
we will represent the levels present by a vector a' k consisting of
~
84
0' sand l' s and having the form
= (ad(jk) ,ar(jk) , bd(jk) , .. . ,wd(jk) ,wr(jk))
a'
-J k
,
(6.1.2)
where
yd(jk)
={
1
if Ydl is present } , y€A ,
0
if YdO is present
and a similar definition holds for yr(jk).
We write the response model
as
(6.1.3)
where II is the overall mean,
O. is the effect of unit j, j=1,2, ••• ,m,
J
N
r
=
{ -1 if
d
1 if
yd(jk) = 0 ,
y,r€A ,
yd(jk) = 1
r d1 is the direct effect of factor
Nr
={
r
-I if yr(jk)
1 if
yr(jk)
at levell, r€A ,
r
=0
=1
rrl is the residual effect of factor
r
at levell, r€A, and the
Ejk are all independent with common distribution N(0,02).
There are 2w + m+l
variance
effects to be estimated in addition to the common
2
0 •
Example 6.1:
As an example of a serial factorial design consider the
simple case where w = 2.
Each unit will receive all four treatment
combinations, and there must be at least two units as the number of
effects to be estimated is 5+m.
ing design.
For m = 2 units we may use the follow-
85
Unit 2
Unit 1
A
B
1
0
£lk
A
B
0
1
a Zk
-------------a 21 = (1,0,1,1)
1 1
a Z2 = (1,1,0,1)
1 0
0 0
a Z3 = (0,1,0,0)
0 1
a 24 = (0,0,1,0)
-------------1
1
all = (1,1,1,0)
0 1
a 12 = (0,1,1,1)
0 0
a 13 = (0,0,0,1)
1 0
a 14 = (1,0,0,0)
Each row for each unit represents an application of a treatment combination and its corresponding a'k'
-J
The first application on each unit is
to be considered as a conditioning treatment and no observation is made
until after the second application.
tions and 7 effects.
estimation of
02
In this design there are 8 observa-
To allow sufficient degrees of freedom for the
we could repeat these series on two additional units.
o
Note that in the example ad (lk) + br (lk)
always occurs with brO
Definition 6.1:
yi(jk)
+
$i'(jk)
= g+g'
I.e.,
a
are unit oonfounded on unit
j
and adO with br1
Yig and $i'g'
for all
= 1,
k, where
Vk.
d1
on unit 1.
i,i'
=r
or d and g,g'
if
=0
or 1.
Definition 6.2:
Yig and
<Piig'
are design aonfounded q times if
Y'1 g is unit confounded with <p.,
, q more times than it is unit con1 g
founded with <p.1 1-"
g where i, i' = r or' d, g, g I = 0 or 1, and
_g' = {O " if g' = 1
1
if g'
=0
.
If the unit confounding occurs equally often with each level we say there
is no design confounding. (If two effects are unit confounded, they are
86
not separately estimable using observations on that unit only.
Two
design confounded effects are separately estimable using all observations unless they are unit confounded on every unit.
Design confound-
ing indicates non-zero covariance between the estimated effects.)
Thus on unit I
adl is unit confounded with bra (and at the same time
adO is unit confounded with brl , but since this confounding always
occurs in pairs we need only mention one of the pair), and on unit 2
adl
is unit confounded with brl , so there is no design confounding
between the direct level of A and the residual level of b. When constructing serial factorial designs we want the amount of design confounding to be as small as possible in order to keep the covariance between estimated effects small.
Example 6.1 (continued):
Using the notation of (6.1.3) we write the
observational equations of the given design as
Yll
Yl2
=~
=~
+
°1
Adl
+
Bdl
+
Arl - Brl
+
Ell
+
°1 - Adl
+
Bdl
+
Arl
+
E12
+
+
Brl
(6.1.4)
o
Serial factorial designs for
following sections.
ZW, W
= 3,4,5,6,
will be given in the
In the final sections we give a general method for
solving the normal equations, as well as some special results for the
given designs with w even.
per experimental unit.
All the designs will have four observations
87
6.2
23 Designs
There are
7+m effects to be estimated, so we need at least three
series of treatments.
Since by the method of selecting factorial frac-
tions there will be unit confounding for each series, some multiple of
three units is required for the design confounding to be in some sense
"balanced."
Thus the simplest possible design with balance requires
only three units.
In addition to this overall balance, we want to select
(
the fractions so that for each unit both levels of each factor will occur
as direct and residual effects twice each.
be orthogonal to unit effects.
Then treatment effects will
We will demonstrate construction of
these designs with two different types of balance, but first we give
another definition.
Definition 6.3:
o and
If on a specific unit a factor is applied at levels
1 alternately, then on that unit it is an alternating faotor.
For the first type of 23 serial factorial design only design con-
founding between direct levels of one factor and residual levels of
another occur.
1)
The procedure followed is:
each unit receives a 1/2 fraction defined by
ad(jk) + bd(jk) + cd(jk} = p ,
where
(6.1.5)
p=O or l,j=unit number, and k=1,2,3,4; and
2)
each unit has a different alternating factor.
(For simplicity, in the sequel we will drop the superscripts in equations of the form (6.1.5) and write a+b+c
Example 6.2:
= p.)
For each unit we give the defining sum, treatment series,
and unit confounding.
88
sum
factor
series
Unit 1
Unit 2
Unit 3
a + b + c =0
A B C
a + b + c =1
A B C
a + b + c =0
A B C
confounding
1
0
1
0
1
1
0
0
1
I
0
1
1
0
0
0
0
0
1
0
1
1
0
0
0
0
1
0
1
1
1
0
1
I
0
0
0
0
1
0
0
0
1
1
1
----_._-
-------
--_ ........ _--
adl - c rO
bdl - brO
a dl - arO
bdl - crO
a dl - brO
bdl - arl
cd1 - a rO
cd1 - brl
cdl - crO
The alternating factors for the three units are
B
f}or unlt
. 1,
A for
Note that each direct level of each factor
unit 2, and C for unit 3.
is design confounded once with one residual level of each factor, and
in this sense the design is balanced.
o
For the second type of 23 design different factors have design
confounding between the same type of effects, i.e., direct with direct
and residual with residual.
Each factor is also confounded between its
direct and residual effects.
Again the method of construction involves
two steps.
1)
Each unit receives a different 1/2 fraction. the defining
sums being a+b = PI' b+c
= P2,
and a+c
= P3 respectively,
where
PI ,P 2,P3 = 0 or 1.
2) The factor not occurring in the defining sum is taken as the
alternating factor.
Example 6.3:
I,
In this design the alternating factors are
A for unit 2, and 3
for unit 3, as given in (2).
C for unit
89
Unit 1
sum
factor
series
confounding
Unit 3
Unit 2
a +b
=0
A
B
C
b +c
=1
A
B
C
1
a +c
=1
'A
B
C
-----
0
0
1
1
0
1
1
0
-----
-----
0
0
0
0
0
1
1
0
0
1
1
1
1
1
0
0
1
1
1
1
0
0
1
0
0
0
1
0
0
1
1
0
1
1
1
0
adl - bdl
adl - arO
adl - cdO
arl - brl
cdl - crO
bdl - cdO
brl - c rO
arl - crO
bdl - b rO
0
Whether an experimenter uses a design of the first or second type
depends on the covariance pattern between effect estimates that he is
willing to accept.
If residual effects are of little interest the first
type of design would be better, but if it is of primary interest to test
for the absence of residual effects the second type would be preferred.
6.3
24 Designs
There are 9+m parameters to be estimated, so we require at least
4 fractions of four treatments each. As in the 23 designs, at least
one factor must alternate for each unit.
Since each time a factor
alternates its direct and residual effects are confounded, we want each
factor to alternate equally often in the design.
These conditions can
be satisfied by a design with a different alternating factor on each of
four units. We give a six step procedure for a class of 24 designs.
90
1)
say.
Separate the factors into two pairs
nl ,n 2
and ~1,~2,
Each of these pairs will be used for defining sums on two units.
" 1 use the e
d· f"lnlng
.
sum
For unlt
2)
two use ~l + ~2
3)
~.l
+
~2
=0
and f or unl"t
= 1.
The sums in (2) leave us with a 1/2 fraction for each of the
first two units, so further restrictions are needed to get 1/4 fractions.
For both units use the same three-factor sum $1
where
i
=1
4)
or 2 and
p
=0
+
~2
+
~i
= p,
or 1.
Order the treatments of the 1/4 fractions defined by (2) and
(3) such that $1
and $2 each alternate in one of the two units.
For units 3 and 4 reverse the roles of
5)
The three defining sums for these units will be
and
~1
6)
+
~2
+
$i
= p,
In addition to
i = 1 or 2 and p
=0
=1,
or 1.
~ and:It each being an alternating factor on
one of the last two units, we must select the order of application to
reduce the amount of design confounding.
After ordering the treatments
for the first two units there will be unit confounding between direct and
residual levels of different factors.
Arrange the treatment order on
units 3 and 4 so that the opposite unit confounding between direct and
residual levels of different factors occurs, and in this way all design confounding between different factors is eliminated.
As a result of this procedure the only design confounding is once
between the direct and residual levels of each factor.
Example 6.4:
Let the factor pairs be A,e and B,D.
91
Unit 2
Unit 1
sums
{
C·
=0
a + c
ABC D
factor
series
=0
b + c + d
a +
III
0
Unit
Unit 4
3
=1
b +d
=I
b + d
ABC D
1 100
ABC D
0
I
1
000
101
o 1 1
0
0
I
I
I
0
1
o
I
0
001
1
000
1
o
1 1 0
I
0
1
1
I
000
I
100
1
0
=0
ABC D
1 000
101 1
o 0 0 0
o I 0 I
1
=0
a + b + c
1
confounding adl - cdl
adl - br1
bdl - a rO
cdl - drl
bdl - crO
dd1 - arl
cdl - br1
ddl - crO
ddl - drO
bd1 - brO
I
1
1
I
I
0
bdl - ddl
bdl - ddO
a dl - brO
bdl - crl
adl - drl
bdl - arl
cdl - brO
dd1 - arO
cdl - c rO
ddl - crl
adl - a rO
c
dl
... drO
The design confounding is given by the last line as a consequence of
(6).
6.4
0
25 Designs
At least 4 fractions of size 4 are required in order to estimate the
ll+m parameters of the 25 factorial. As in the previous designs each
fraction will have at least one alternating factor, so a multiple of five
fractions is needed for each factor to alternate equally often and the
design to remain balanced. We give a procedure for constructing 25 designs with two alternating factors on each of five units, thus having
each factor alternate twice.
independent two-factor sums.
Each 1/8 fraction will be defined by 3
92
1)
From the letters {A,B,C,D,E}
form five sets of triplets such
that every letter occurs in three triplets and each pair of letters
\
occurs in the same triplet either once or twice.
2)
There are 3 pairs of letters in each triplet.
Select two of
these pairs to be used in defining the fraction associated with the
triplet.
Assign to the sum of the levels of the two factors in each
pair the value 0 or 1.
Once these values are given the value assigned
to the sum of the levels of the remaining pair is determined.
If the
first two values are the same the last is 0; if they are different it
is 1.
Of the 10 distinct pairs available from 5 letters, 5 will occur
twice and 5 once in these two-factor sums.
Example 6.5:
We will demonstrate the method described in (2).
the triplet be ABC, and the three pairs are AB, BC, AC.
pairs AB and BC to be independent and use the sums a+b
in defining the fraction.
Then the restriction a+c
for each treatment combination.
Definition 6.4:
Select the
= 1 and
=1
(See also Example 6.6.)
Let
b+c
=0
will also hold
0
For each triplet in (1) there are two letters from
{A,B,C,D,E} which are not present.
We will call this pair the aeeoa-
iate paip of the triplet.
3)
Assign to each triplet its associate pair.
These pairs will
be the sarneas those in (1) that occur together twice in the same triplet.
(To see this, let AB be such a pair.
There are 3 remaining
triplets, one of which contains A but not B, and one which contains
B but not A, by (1).
This leaves one triplet containing neither A
nor B, and its associate pair must be AB.)
Now assign values as in
(2) to the sum of the levels of the associate pair.
Do this in a way
93
such that the three values assigned to this pair in (2) and (3) are
not all the same.
4)
We now have 5 sets of 4 two-factor sums, three of the four
being independent.
Each of these sets defines a 1/8 fraction, and the
treatments of each fraction are ordered so that the associate pair will
be alternating factors.
As a result of (2) and (3), each factor's direct level is design
confounded
o~ce
residual levels.
I
with each other factor's direct level and similarly for
The only design confounding between direct and residual
levels occurs with the alternating pairs.
From (4) each factor's direct
level is design confounded once with the residual level of two other
factors and twice with its own residual.
In this sense the design is
balanced.
Once all the two-factor sums of the design have been written down
the design confounding is determined in the following manner.
i)
For the two factor sums that occur once.
If the assigned value is
0, direct level 1 of the two factors
are confounded, and similarly for residual levels.
If the assigned
value is 1, direct level 1 of a factor is confounded with direct
level 0 of the other, and similarly for residual levels.
ii)
For the two-factor sums that occur thrice.
Confounding between direct levels and between residual levels
is determined as in (i), except that the value assigned twice is
used rather than the single assigned value.
Confounding between direct and residual levels is determined using the
value assigned to the sum of two factors when they occur as associate
pairs and the reverse of the method for finding
dir~ct
with direct
confounding in (i). (That is, if the sum is I, direct level 1 of a
factor is confounded with residual level 1 of the other factor, and
if the sum is 0, direct level 1 of a factor is confounded with residual level 0 of the other.)
Example 6.6:
We give the step by step construction of a 25 serial
factorial design.
1)
Let the five triplets be ABC, ABD, ACE, BDE, and CDE.
2)
For each triplet the three two-factor sums are given, the
first two of which are taken to be independent.
triplet
sums
ABC
a+b=O
b+c=l
a+c=l
ABD
ACE
BDE
CDE
a+b=l a+c=O b+d=O c+d=l
b+d=l c+e=l d+e=O d+e=l
a+d=O a+e=l b+e=O c+e=O
Note that the third sum for each triplet can be gotten by adding the
first two (mod 2).
Associate pairs and sums.
3)
pair
DE
sum
d+e=O
4)
factor
series
CE
BD
AC
c+e=l b+d=O a+c=l
AB
a+b=O
Treatment series with associate pairs alternating.
Unit 1
ABC D E
1 1 000
Unit 2
ABC D E
10011
_... _---
-----
1 101 1
00100
00111
1 1 000
1 0 1 1 0
o 1 001
o11 00
10011
Unit 3
ABCDE
o1 0 1 1
Unit 4
ABC D E
1 000 0
Unit 5
ABC D E
000 1 0
11 0 1 1
01111
1 0 00 0
1 1 101
00101
1 1 010
00010
----- ----o 0 001 00100
11110
10100
o 1 011
-----
95
The five pairs that occur once in (2) and (3) and their assigned values
are:
AO-O,
AE-I,
BC-I
J
BE-O,
CO-I.
The five pairs that occur thrice in (2) and (3) and the values assigned
to them twice are:
AB-O,
AC-I,
BO-O,
CE-I,
OE-O
Using (i) and (ii) on these results we get the design confounding
between direct levels.
(Confounding between residual levels is the
same.)
adl-bdl ' adl-c dO ' adl-d dl
J
adl-edO
bdl-c dO ' bdl-ddl ' bdl-edl
cdl-ddO ' cdl-edO
ddl-edl .
The design confounding between direct and residual levels is derived
from (3) and {ii).
adl-b rO ' bdl-a rO ' adl-c rl , cdl-arl , bdl-drO ' ddl-brO
ddl-erO' edl-d rO ' cdl-e rl , edl-c rl •
o
If the experimenter has 5 additional experimental units available
and wants to eliminate design confounding between different factors,
the same two-factor sums as in (2) and (3) may be used to define the
new fractions, but with all the assigned values switched. (I.e., I
replacing 0 and 0 replacing 1.)
In this new design each factor's
direct level is design confounded 4 times with its own residual, but
all other design confounding is eliminated.
96
26 Designs
6.5
There are l3+m parameters to be estimated, so at least 5 fractions
of size 4 are required.
If each unit has two alternating factors, as in
the 25 designs, the number of units must be a multiple of 3. Thus 6
I
is the fewest number of units we can use for a 26 serial factorial. We
give a procedure for constructing these designs on 6 units, two a1ternating factors per unit, such that the only design confounding is twice
between each factor's direct and residual levels.
1)
From the letters {A,B,C,D,E,F} select three disjoint pairs to be
used in two-factor sums for defining fractions.
2)
Select one letter from each pair in (1).
These three letters
will be used in three-factor sums.
3)
Divide the units into 3 groups of 2 units each.
Each group uses
a different pair from (1) as alternating factors on its two units.
The
two-factor sum for this pair is assigned the value 0 on one of the units
of the group and 1 on the other.
4)
Each pair from (1) occurs in two groups in which its factors are
not alternating.
In one of these groups its two-factor sum is assigned
the value 0 and in the other 1.
5)
The three-factor sum is used on every unit for defining the fract-
ion and is assigned a different value for each unit of a group.
6)
On each unit the direct levels of each non-alternating pair are
unit confounded with the residual levels of the other non-alternating
pair.
The order of treatment application to the two units of a group is
-
arranged so that this confounding is for opposite levels on the different
units.
(I.e., if
confounded with
~dl
~r1
is confounded with
on the other.)
~rO
on one unit, then it is
97
7)
As a result of (3-6) the only design confounding is twice between
the direct and residual levels of each factor.
Example 6.7:
Choose as the three disjoint pairs AB, CE, and OF, and
choose as the triplet of (2) BCF.
For groups 1,2, and 3, let the altern-
ating pair of factors be CE, OF, and AB, respectively.
We give the
groups and the sums that define the fractions for each unit.
Group 1
Group 2
Unit 1
Unit 2
Unit 3
Unit 4
a+b=l
c+e=O
d+f=l
b+c+f=l
a+b=l
c+e=l
d+f=l
b+c+f=O
a+b=O
c+e=l
d+f=O
b+c+f=O
a+b=O
c+e=l
d+f=l
b+c+f=l
Group 3
Unit 5
Unit 6
a+b=O
c+e=O
d+f=O
b+c+f=O
a+b=l
c+e=o
d+f=O
b+c+f=l
Note how the values assigned to the sums agree with (3-5).
6.1 are the treatment series derived from the above sums.
In Figure
The only
design confounding is twice between direct level 1 of each factor and
its residual level
O.
o
98
Figure 6.1
Group 1
Group 2
Group 3
Unit 1
Unit 3
Unit 5
ABC D E F
ABC D E F
1 101 1 1
ABC D E F
1 1 101 0
o1
0 1 0 0
o1
1 011
10000 1
101 1 1 0
-----o 0 001 0
00111 1
110101
001 101
111 000
o0
0 0 0 0
010 1 0 0
1 101 1 1
1 1 101 0
Unit 2
ABDCEF
o 1 001 1
Unit 4
ABC 0 E F
1 1 100 1
Unit 6
ABC D E F
-----o 1 1 100
101 1 1 1
-----o 1 1 010
------
100 1 1 0
001 1 0 0
00001 1
10100 1
o 1 001 1
1 101 1 0
1 1 1 001
1 000 0 0
o1
0 1 0 1
101111
6.6 General Analysis of 2w Designs
For the designs given in the previous sections each experimental unit
receives a series of five treatment combinations.
The first application
is considered only as conditioning, and observations are made after each
of the succeeding applications so that there are 4 observations per unit.
Because each unit receives a defined fraction of the entire factorial,
each level of direct and residual effects for each factor affects 2
observations per unit.
We used w units in each of the given
2w
designs, but they are not all practical because of the small number of
degrees of freedom available for estimating
02•
This drawback may be
99
overcome by replicating a design a certain number of times, otherwise
extending the design as suggested for the 25 designs, or by constructing
more complicated designs.
Because of these possible extensions we give
the analysis for the general case of n observations on each of m units.
From model (6.1.3) it is seen that there are
effects to be estimated:
~,
2w+m+l independent
OJ' j=1,2, ... ,m, r d1 , and rr1'
r~A.
In
order to rewrite (6.1.3) in matrix form we relabel the observations.
will be denoted by Yn(j-l)+l' Yn (j-l)+2'.·.'
Ynj , and there are a total of N = mn observations represented by the
Those made on unit
j
vector
(6.6.1)
Now rewrite (6.1.3) as
~ = ~~
+ G'i + D'r +
where G' is an Nxm matrix with elements
R'~
+
(6.6.2)
€
g! .
1.)
given by
,
if observation i is made on unit j
1.)
otherwise ,
goo
i
= (ol,02,···,om)' is the m-vector of unit effects,
D' is an NXw matrix
with elements d!k
given by
.
1.
if the k-th factor occurs at direct level
1 for observation i
if it occurs at direct level 0 ,
r= (Ad1 ,Bdl , •.• ,Odl)' is the w-vector of direct level 1 effects,
R' is an Nxw matrix with elements r~k given by
100
if the k-th factor occurs at residual level 1
for observation i
if it occurs at residual level 0 ,
~
€
= (Ar1,Brl, ••• ,Qrl)
is the w-vector of residual level 1 effects, and
is an N-vector with NN(Q,cr 2 I N) distribution.
The expected value of £ may be written as
.
(6.6.3)
El=(l~.:G':D':R')
.
r
which is in the form of the general linear model.
meters
e
To solve for the para-
we have the usual normal equations
AA'§..= Af...
(6.6.4)
We will examine the two sides of this equation separately.
The matrix M' may be written
l'
-N
---G
(~ :G'
D
--R
l'G'
l:N.4l
:D' :R)
=
G.!N
D~
R~
I'D'
1'R'
-=N
-=N
~
GG'
DG'
RG'
GD'
DD'
GR'
DR'
RR'
RO'
(6.6.5)
This matrix is symmetric, so we need only examine the products on and
above the main diagonal.
.!N' iN = N
•
lNG' is an m-dimensional row vector of column sums of G'. The sum of
the j-th column of G'
which is n.
is the number of observations made on unit
j,
101
= nl' m •
11, G'
-1'1
(6.6.6b)
-
!NO' is an w-dimensional row vector of column sums of 0'.
The sum of
the k-th column is the number of times the direct effect of factor
k
occurs at level 1 minus the number of times it occurs at level 0, which
is O.
I'D'
-=oN
where
~
= -w'
0'
(6.6.6c)
is an w-vector of zeros.
l'R' = 0'
-w
=N
(6.6.6d)
for the same reason as (6.6.6c) holds, with residual levels taking the
place of direct levels.
The (i,p) element of GG'
N
l
gkigkp
k=l
=number
of times an observation occurs on both
unit i and unit p
=
GG'
is given by
{nO if i F P
if i = P
= nIm
(6.6.6e)
The (i,p) element of GD'is given by
N
L gk·dk = number
k=l
1
P
of times the direct effect of factor
P OCCUTS at level 1 on unit i minus the number
of times it occurs at level 0
=0
(6.6.6f)
Similarly,
GR'
= 0mxw
(6.6.6g)
102
The (iIP) element of DO'
is given by
N
r dk·dkP = atnumber
of times the direct effect of factor i occurs
the same level as the direct effect of factor P minus
k=l
1
the number of times they occur at different levels.
o
if i t- P and no design confounding between
these effects
~ if .i 1- P and these effects are design confounded
q tJJnes at the same level
=
- ~ if i ~ P and these effects are design confounded
2 q times at different levels
DO'
(DO')
where
=P
if i
N
= NI w +
(DO')
is DO'
with the main diagonal replaced by zeros.
(6.6.6h)
I
The (iIP) element of DR' is given by
N
r
d ' .r ' = number of times the direct effect of factor i occurs
k=l k 1 kP at the same level as the residual effect of factor P
minus the number of times they occur at different
levels
01
=
DR'
where
(DR')
{
-nt
~
I
if i 1- PI as above
if i = P and t is the number of times
each factor alternates
=- ntI w +
is DR'
or - ~
(DR')
I
with the main diagonal replaced by zeros.
The (i,p) element of RR'
is given by
(6.6.61)
103
N
l
k=l
rkirkp = number of times the residual effect of factor i occurs
at the same level as the residual effect of factor P
minus the number of times they occur at different levels
= 0, ~ , - ~ , or N as for DD'
RR'
where
(RR')
= NI w +
(RR') ,
is RR'
with the main diagonal replaced by zeros.
(6.6.6j)
Using these results we write
N
M'
=
nl'
~
n!m nI m
0
-w
°wxm
0
0
-w
0'
0'
-w
-w
°mxw
°mxw
NI w + (DD')
-ntI
wxm
w
+
(DR ')
-ntI
NI
w
+
(DR')
w + (RR ')
(6.6.7)
The sYmmetric matrices (DD'), (DR'), and (RR'), represent the design confounding.
From the pattern of this matrix it is seen that the estimates
for factor effects are orthogonal to those for the mean and unit effects.
The right hand side of (6.6.4) can be written as
1'
-~-
l.
=
T
G
G
D
D
R
R
(6.6.8)
where T is the sum of all observations,
,
G = (G I ,G 2 ,··· ,Gm) , and Gi is the sum of observations on unit i,
D = (D l ,D 2 , • ~ . ,Dw) , and Di is the sum of observations for direct level I
of factor i minus the sum of observations for direct level 0 of factor
i , and
104
is as for
D.1
but with residual levels
replacing direct.
Combining (6.6.7) and (6.6.8) the normal equations (6.6.4) become
N
-m
l'
0'
-w
0'
-w
1.1
nl m
°mxw
0mXw
<5
1
-m
T
=
G
0
-w
0
wxm
NI +(OD')
w
-ntI +(ORi)
w
r
D
0
-w
°wxm
-ntI +(OR ')
w
NI +(RR ')
w
~
R
m
Using the constraint
I o.
j=l J
=
o,
(6.6.9)
the first m+l equations of (6.6.9) give
the estimates
'"
1.1
..
'"OJ
= Gi/n
(6.6.10)
TIN
(6.6.11)
- TIN .
The estimates of factor effects are given by
-ntI w + (OR')J - [_~_]
NI w + (OD')
-R
\
(6.6.12)
(Note that (RR') has been replaced by (OD') since they are identical
matrices.)
If we let the generalized inverse in this equation system be
B, then the estimated covariance matrix of the factor effects is
Bcr 2 , where 02
I'[I -A'(AA'jA]x.
N
To test whether or not there is a
N-2w-m
difference between using level 0 and level 1 of a particular factor, we
::;
-
test
(6·6·13)
105
If there is a difference between levels it will be indicated by a
rejection of HZ' as a result of restrictions (6.l.la-b).
test is given by rejecting H with significance level
2
"
Ify
a
The usual t(0 < a < 1) if
A
+
\pyl
---------~-----~------------ ~ t(1-a)/2,u
&2(b yy + bw+y,w+y',
+ 2by,w+y)
(6.6.14)
where t(1-a)/2,u is the lOO(l;a)% point of the t distribution with u
degrees of freedom,
u
= N-2w-m,
and B = (bee)'
Tests of the form (6.6.14) for different
independent.
y are not necessarily
For the designs given in this chapter the tests are inde-
pendent for w even but not for w odd.
6.7 Simplified Analysis for w Even
Because the designs given for
w even have no design confounding
between levels of different factors, we can derive simplified formulas
for (i) estimating direct and residual effects, (ii) estimating direct
effects under the assumption that residual effects are absent, and (iii)
testing for the absence of all residual effects.
In fact, these results
hold for any design with no design confounding between factors.
Suppose a particular design for w even is repeated s
that m = sw, n
and (DR')
= 4,
and N = 4sw.
times so
In (6.6.lZ) all the elements of (DDt)
are 0, so we write
H-] =
4swlw
[ -4tI
w
4tl w ] -1
4swI
w
[_~_]
-R
Note that the generalized inverse will be a true inverse here.
of Kronecker matrix products this inverse is
(6.7.1)
In terms
106
-t
-1
sw]
(6.7.2)
Placing this result in (6.7.1) we get for factor YEA
(6.7.3)
Tests of hypotheses of the form (6.6.13) are independent for different
rand (6.6.14) becomes
(sw+t) (Oy+Ry)
i_t
4 (s2 t
.
2)
/
2(sw+r)
2 2 2
4(s w -r )
A
0
2
>
-
t(1-a)/2 u
,
or
10
+R
I
~2 Y ~t(l-a)/2,u' u=w(3s-2) .
«L7.4)
20
If we make the assumption that no residual effects are present in
our model, (6.6.3) becomes
and
4SW
[ 4.!m
o
4~]-
O(m+l)""
4I m
wx(m+1)
__1__ I
4sw w
(6.7.6)
107
Then the estimate of the direct effect of treatment y is
A
(6.7.7)
If we repeat the analysis assuming now that residual effects are present,
we can get the new direct effect estimates from the previous ones as
(6.7.8)
For the case of w even and testing the hypothesis that no residual
effects are present, i.e., HO:
~
=~ ,
we use the usual
F statistic
1..1 [A' (AA')-A - Ai (A1AP-Alh:.
------------ = F
(6.7.9)
l..[I-A I (AA I) ~ A]l.
which under HO has the
of freedom. Letting
M= [4SW
4.!m
F distribution with wand w(3s-2) degrees
:~
r· [
o
-;n
0' ]
(6.7.10)
~I
4 n
and P be the matrix in (6.7.2), we have the following simplification of
matrices from (6.7.9):
Al (AA')-A = (1=N.:G' :D'
:Rl) [M(m+l)X(m+l)
.
.
O·wx(m+l)
!N
G
D
R
(6.7.11)
108
Ai (A AP-A
1
1
=
(~:GI :0') [:(m+ llX (m+l)
wx (m+1)
= (1=-N.:GI)M(-¥]
G
Using
(IN:G')M[-~~J
'L'
+
_L
D'D
4sw
(6.7.12)
•
= }G'G (6.7.9) becomes
[(D';R')P(_;_) - <dw D'D ]
'L' [IN - }G'G - (D'
:R')P[+)J
'L
=
'L
F.
(6.7.13)
We reject HO if F > F1-a,w,w (3 s- 2)' with significance level a
(0 < a < 1), where F1-a,w,w (3 s- 2) is the 100(1-a)% point of the F
distribution with wand w(3s-2) degrees of freedom.
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