•
SIMPLEX-SUM DESIGNS
A CLASS OF SECOND ORDER ROTATABLE DESIGNS
DERIVABLE FROM THOSE OF FIRST ORDER
G. E. P. Box and D.W. Behnken
This research was- supported in part
by the United States ArmyOronance Deparment
under Contract No. DA-,36-o 34-ORD-2297
.',
Institute of Statistics
Mimeograph Series No. 232
July, ~959
i1:1
It is a pleasure tD acknowledge the assistance provided by
Dr. J. W. Tukey through his helpful suggestions and cr:1t1cal review of
a major port:!:.on of the manuscript. Thanks are also due to Dr. R.J.
Hader for many useful conversations during the latter part of the work
and for h1s editDr:lal'assistance in preparing the final manuscript. The
financial support of the Depar'lment of
~
Ordnance and the Depar1ment
of :bper1m.ental Statistics at Raleigh are also grateft1JJ.y acknowledged,
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
.
•
•
.vi
•
1.0 . INmODUCTION •
2.0
REVIEW OF LITERATURE
2.1
2.2
3.0
0
.
Surface Fitting
Sampling Moments
GENERAL THEORY
•
•
0
•
• 1
•
•6
• 6
0
.10
0
•
•
•
•
Conditions for Rotatability.
•
Notation and Definition. • • • • • •
.Ar:lalogy to Sampling from a Finite Population
Form of Moment Components.
•
•
0
• 011
.11
.12
•
.14
0
0
.16
0
. .
4.0
RADIUS MULTIPLIERS AND ROTATABILITY
5.0
SECOND ORDER REQU'IRE}1ENTS FOR RO~TABJlJ:TY
•
.23
. ..
5.1 Application of Moment Requirements
5.2
5.3
• •
Standard Solution for Radius Multipliers •
Second Order Rotatability for the Case n • 3 •
o
•
THIRD ORDER REQtJIREMENTS FOR ROTATABJlJ:TY
7 .0
SECOND ORDER ROTATA,BLESIMPLEI-suM DESIGNSg THE STANDARD
SOLUTION
•
o
•
o
•
0
8.0
ADDITIONAL SECOND ORDER ROTATABLE SIMPLEX-SUM DESIGNS
•
0
0
•
8.1 Solution Space of Radius Multipliers.
8.2 Specific Solutions. •
9.0 REPLICATION
10 .0
BLOCKING
10.1
10.2
0
•
.•
. . .
•
.
.
.33
.35
•
•
•
.36
.39
.39
.41
0
•
•
.46
o
.48
Orthogonal Blocking • • •
Orthogonal .Blocking - Nearly Rotata:ble Designs
0
•
•
•
o
0
.27
.33
Radius of Experimental Points.
7 .2 Radii for the Standard Solution •
7.3 Singularity and Near Singularity of.
Moment Matrices.
.23
.25
.29
6.0
7.1
.20
0
•
.49
.53
v
TABLE OF CONTENTS (continued)
Page
10.3 Non-Orthogonal Blocking of the Rotatable Designs •
10.3.1 Restrictions on Non-Orthogonal Blocking
Necessary to Retain Rotatabilit y .
10.3.2 Qeneral Solution for Regression Coefficients
and their Variances - Blocks First Order
Rotatable •
11.0 A CONVENIENT, REDUCED DESIGN FOR k • 7 •
0
12.0 SUMMARY.
:LIST OF REFERENCES
•
•
•
•
0
00
o
Introduction • •
Second Order Moments
Third Order Moments
Fourth Order Moments
Higher Order Moments
Low Values of n •
• 68
o
• 78
•
• 82
•
•
• 87
•
•
0
0
0
•
•
• 88
•
• 90
o'
o
o
0
•
• 87
o
0
.
72
• 74-
•
APPENDIX B DERIVATION OF GENERAL M<J1ENT FORMULAS
B.l
B2
B.3
B.4
B.5
B.6
• 62
• 71
0,.
0
.'
APPENDIX A MULTIVARIATE BRACKETS
• 56
• 68
o
11.1 Construction. • • • • • • • •
11.2 Projeption in~ Lesser Dimensionalit,y.
11.3 Relation to 3 Design.
• 54
•
•
93
98
• 100
vi
LIST OF TABLES
la. Summary of general moment components of Ds •
••
lb. Fourth order moment components of Ds for n • 3.
•
2. Radii of experimental points for standard solution
rotatable designs.
•
3. Comparison of
A.t
•
•
.17
•
.16
.35
•
to its singular value
•
.45
4. Radius multipliers for some second order rotatable designs.
5. The standard solution with k • 3 and various
replication patterns.
••
•
•
•
•
.37
.47
6. Summ.ary of orthogonal blocking schemes for rotatable
designs of Table
4
••
•
7. Raclius multipliers arid. center points for orthogonal
pearly rotatable submatrix blocking..
.
.55
8. Non-orthogonality parameters for submatrix blocks.
9.
Seven factor second order rotatable design in
three l e v e l s .
•
•
.67
.70
•
Appendix B
1. Moment components of samples of s £roni zero mean finite
orthogonal populations of n, Cn 2 ..<.)
•
•
.99
1.0 IN'lRODUCTION
The objective of many experimental
pro~~s
is to elucidate the re-
lationship between a measurable characteristic of a material or process
(Y'\.) and a set of continuously variable factors (~,X2'.·J~) suspected
.An
of influencing its behavior.
~sight
into the problem can freqt1ently
be obtained by fitting a hypothesized model
data by least squares.
ll- f(l)
to experimental
The geometric representation of the fitted
function has been called the response surface and much has been written
in recent years describing the philosophy and technique of obtaining an
estimate of the surface and of locating optimum points thereon.
Often lacking a more fundamental model, it is assumed that within
the l:iJn.ited range of the experiment the unknown function can adequately
be represented by a limited number of terms of its Maclaurin's Series
expansion oWe thus assume the approx:iJnation
yt
==
where the
k
k
k
k
k
k
.
J.=
J.=
~ts
are unknown constants proportional to the partial deriva-
~ o + 01J.J.
~~.X. + 01
~ ~~OjXoXj
+ ~ JOOO
~ ~~iji1XoX.Xo
+
oJ. J.
'fI'J.J'fI
=J.
tivas evaluated at the origin.
J.=
. =J. 'fI'= J
...
.A model utilizing all terms through
degree d in the above expansion is called a d-th order model.
Designs used in response surface work are called d-th order designs
if they allow estimation of all coefficients up to and including. those
of a d-th order model
0
In discussing designs it is convenient to think
of the set of k factor levels called for by an experiment as the elements
of a row vector and hence as a point in a k·dimensional factor space.
The set of N experiments called for by a design are then specified by an
N x k design matrix D, each row of which defines the levels of an
2
experiment and is one of the N points in the design configuration in the
factor space.
For convenience and to gener$lize tm· designs the levels
of the independent variables are coded such that the or:l.gin of the new
variables (:1],' x 2' ••• , ~) lies at the centroid of the design points
and a suitable sealing convention is employed to make the design units
uniform for all factors.
From the first convention we have ~ x iu • 0
and the second is usually effected
bY letting
f xi/N •
1 2, Ua constant
for alli ~ ....
Any design providing uniqt1e estimates of the coefficients in the
model could be used but certain configurations of points offer distinct
advantages.
In response surface work the primar,y emphasis is, by defini-
tion,on the estimate of the surface and its variance • However, at the
inception of an exper::1.mental program there is usually no knowledge of the
orientation or shape of the surface relative to the axes of the factors
and it is therefore desirable to use designs which are independent of
this state of ignorance. With this objective in mind designs have been
found for which the variance of an estimate is constant at all points
equidistant from the origin of the design,
independent of direction.
~
for which the variance is
These are called rotatable designs since the
variances of both the estimated response and the individual coefficients
are invariant under orthogonal rotation of the design in the factor space.
In this way the design is unprejudiced by the arbitrary characteristics of
the surface relative to the orientation of the factor axes. While rotatability is a function of the scaling convention adopted, the variance
contours will be spherical in those units appearing to the experimenter
as most reasonable at the time the experiment is conducted, .L.b units
based on the relative experimental variation in each factor.
3
First order rotatable designs are obtained when the columns of the
design matrix D are orthogonal to each other a.nd to a vector of ones.
The first order model, which fits a planar surface to the data, contains
k + 1 constants to be estimated and consequently requires a min:i.rm:lm of
k + 1 points. When this minimum number is used it has been shown by Box
in (5) that the design points will be at the vertices of a regular simplex
in the factor space. When k
1m
2 the points lie at the vertices of an
equilateral triangle, when k
lili
3 at the vertices of a regular tetrahedron
amd in general are one orientation of the unique set of k + I vectors of
equal length forming equal angles with each other in k space.
Second order rotatable designs are obtained when more restrictive
moment conditions are satisfied (where a general moment is defined as
'\
~
-<2
..
~ XJ.u x 2u ••• ~u/N), and in two instances were found to bear an
u
interesting relationship to the simplex designs just described. When
k == 2 the points at the vertices of a hexagon provide the basis for a
second order rotatable design.
Now i f the vectors of the equilateral tri-
angle, forming the first order rotatable design, are added in pairs the
three derived vectors, when taken in conjunction with the original tbree,
will produce the vertices of a hexagon.
If the original vectors are added
three at a time they will provide a center point which is required to
ma.ke the set of points a usable design of full rank.
When k
1=
3 a similar construction exists. A set of six vectors can
be derived from those at the vertices of the tetrahedron by forming
of all possible vector pa.irs.
~
These vectors will pass through the mid ;..
points of the edges of the tetrahedron.
Formation of all possible sums
of three vectors will yield four derived vectors each passing through the
mid point of a face of the tetrahedron.
.Again a center point is obtained
4
by adding all four vectors.
'When the length of the three sets of vectors
are suitably chosen (by scalar multiplication of each set) a previously
derived second order rotatable design is obtained.
This is the composite
design which can be represented, as described above, by the row vectors
~ ·'(XJ.u x
2u x3u ) , u· 1,2, ••• ,14. The design consists of the six
points denoted by (±23/ 4 0 0) , (0 ±23/4 0) , (0 0 ±23/4) and the
eight points at the vertices of a cube which ,can be represented by all
permutations of sign of the vector (±l ±l ±l).
To this basic set of
fourteen a certain number of points at the origin (0 0 0) are usually
added.
If we chose the original tetrahedron to be in that orientation in
which the vertices have the coordinates of a fraction of the 23 factorial
(-1 -1 -1), (1 1 -1), (1 -1 1) and (-1 1 1), then the relationship
is readily seen.
The six vectors corresponding to the a.xi.al points are
those generated by adding the vectors two at a time, while the. vectors
corresponding to the second mating tetrahedron forming the remaining
points of the cube (the second half-replicate) are those obtained by
adding the basic vectors three at a time.
To obtain the proper vector
length for the axial points it can be seen that all six derived vectors
must be multiplied by the constant 2-1/4.
This dissertation is concerned first with establishing the generality
of the connection between first and second order rotatable designs, that
is demonstrating that the design matrix of a first order rotatable design
can always be used to produce the design matrix of a second order
rotatable design (or designs).
Secondly it is concerned with the possi-
bility of generating third order rotatable designs in this manner, and
5
finally with a description of the properties of the designs which evolved.
It was found that a class of second order rotatable designs, labelled
simplex-sum designs can always be generated by taking as design points
the vectors obtained
b.Y forming all possible sums of the k + 1 rows of
any minimum first order rotatable design matrix, and multiplying' the
derived. vectors by suitable constants. Third order rotatable designs
however did not materialize from this straightforward approach.
As a
by-product of the investigation, sampling moments for means of finite
multivariate orthogonal populations were derived which may find use in
other applications
0
The blocking properties of simplex-stUn designs were
"
.
thoroughly investigated and a general theorem proved, for any second
order rotatable design, concerning requirements necessary to retain
rotatability when non-orthogonal blocking is used.
6
2.0 REVIEW OF LITERATURE
2.1 Surface Fitting
The literature on the general subject of response surface fitting
has been reviewed extensively during the past several years, the most
recent complete compilation appearing in a doctoral dissertation by
Carter in 1957 (15). Papers covered there will be but briefly touched
upon here, emphasis being put on the more recent publications.
In 1941 Hotelling (24) devised procedures that had certain optimal
properties for locating the m.a.xim.um of a function within a predetermined
region of one and two dimensional factor spaces.
A sequential scheme
was then provided by Friedman and Savage (22) in 1947 for locating an
optimum, when more than one factor is involved, based on the classical
approach of varYing one factor at a time.
The
Box~Wilson
paper (13) appeared in 1951 as the first of a
sequence of papers by Box and associates and outlined an approach and
viewpoint which set the stage for much of the work that followed.
In
this paper a sequential tlsteepest ascentll procedure is outlined for
locating the region of a maximum and composite designs were then proposed
for estimating
~he
coefficients of a quadratic model within this region.
In 1952 Box (5) showed that orthogonal designs were most efficient for
estimating the constants of a first order model.
A review and discussion of the work published in this general field
by 1953 was given by Anderson (3).
In 1954 several papers were published, one (6) of a general exposi'tory nature by Box on the methodology-and philosophy on surface fitting,
another by Box, Hader and Hunter (9) on the effects of having asstuUed an
7
inadequate model showing biases existing in both the estimates and sums
of squares, and a paper by Box and Hunter (10) concerned with setting a .
confidence region on the solution, of a set of simultaneous equations
with random coefficients (the latter problem being that encountered in
solving for the stationary point of a fitted quadratic surface).
De La Garza (20) also published a paper in 1954 showing that when only
one independent variable was involved, designs for fitting polynomials
of degree d involving more than d+l points were equivalent from the
standpoint of the variance-covariance matrix, to a design calling for
exactly d+l points •
.A paper published in 1957 (12) by Box and Hunter, (having in large
part appeared in 1954 in mimeograph form (11», made explicit the definition of rotatable designs establishing the necessary and sufficient
moment conditions for models of any order.
Second order rotatable designs
were found and blocking procedures provided for them.
.An approximate
confidence region for the maximum was given based on the earlier Box and
"
Hunter paper (10) but which was considerably simplified through the use
of rotatable designs.
The work of Wallace (29) was cited as a means of
finding approximate limits which are easier to compute.
Box and Youle in 1955 (14) indicated how an empirical response
surface appr.oach.coUld'lead to.anunderstanding of the fundamental
theory of a process or reaction.
It is. shown how reduction to canonical
variables can be of assistance in such an investigation.
Third order rotatable designs were discussed in 1956 by Gardiner,
Grandage and l[ader (23) for two, three and four factor experiments.
These designs were based upon the regular figures and several were of
a type allowing sequential fitting of the second and third order model.
8
Attention was given to blocking composite designs by DeBaun in
1956 (17). In a short paper he indicated the biases encountered in nonorthogonal blocking and gave an orthogonal blocking scheme for the five
faotor composite rotatable design.
This material was later covered more
generally in (12).
An
extensive investigation of response surfaoe designs for two
faotors was made by Carter in 1957 (15).
The oonnection between the
individual degrees of freedom of the standard factorial analysis and
response surface coefficients was demonstrated.
First and second order
rotatable designs were given which are not necessarily based on regular
figures.
A procedure was given whereby any: set of points could be
completed into a first order rotatable design by adding two additional
points.
It was also shown how second order designs could be constructed.
from the various classes of first order designs described, by adding
four additional points.
A general theorem was proved showing necessar,y
and sufficient conditions which must be satisfied for a combination of
rotatable sets of order d-l to be a rotatable set of order d.
In 1958 Bose and Draper
(4) found infinite classes of second order
rotata,ble designs for from three to seven factors. Infinite classes
of sequential third order rotatable designs were also described.
All
previous second and third order designs were shown to belong to these
classes.
It was shown that the necessary and sufficient condition for
the moment matrix of second and third order designs to be singular is
for all design points to lie on a hypersphere in the factor spa.ce.
The
designs discussed are constructed from sets of points whose odd moments
vanish and whose combination into rotatable designs depends upon satisfying certain Itexcess'· functions defining the required relationship
"
I
9
among the fourth order moments for second order designs together with
sixth order moments for third order designs.
In 1958 Draper (21) added
a four factor third order rotatable design qy using the same approach.
DeBaun in 1958 (18) and 1959 (19) described three factor three level
second order designs for response surface work which are often of
interest to an experimenter who finds it expensive or inconvenient to
create more than the minimum number of factor levels. With one exception
these designs are in the nearly rotatable class, having variance contours
which approximate spheres •
.A general method for producing three level designs in k factors
was presented qy Box and Behnken in 1958 (7) giving examples of both
rotatable and nearly rotatable second order designs.
These designs are
generated by utilizing the combinatorial properties of partially balanced
incomplete block designs.
A means of characterizing the departure of the
variance contours from sphericity was obtained for nearly rotatable
designs.
Also in 1958 van der Vaart (28) proved some results on estimation
of latent roots of a symmetric matrix relevant to the representation of
a response surface as a quadratic function.
He showed that estimates
of the lowest latent root are biased downward and those of the largest
root are biased upward.
This implies that the latent roots will be of
different signs unduly often and that hence there is a tendency to obtain
surfaces that are of the minimax: type more often than is correct. ,This
result has some experimental verification •
.A different departure on designs for exploring response surfaces
was taken qy Box and Draper in 1958 (8).
Designs were sought which
were optimal when both the random experimental error and the bias
10
introduced by failure of the model were considered.
The paper was
principally concerned with the situation where a first order model was
assumed when the true function was quadratic.
It was found that the
best design under typical conditions was one that min:lJnized bias alone,
ignoring variance.
In this situation, at least, .the opt:lJna.l design is
first order rotatable since it was shown that the requirement. :lJnplies
orthogonality.
any
~ ..th
An interesting general result found here was that for
<
I
order model which was used· in place of the correct model of
order d
the average bias is minimized by making the moments of order
2
~ + d and less equal to those of a uniform distribution over the
2
region of interest.
2.2 Sampling loments
In developing the moments of the experimental designs it was help-
ful to use a tool due to Tukey (26,27) which he entitled bracket notation.
In two papers techniques were developed to simplify the algebra involved
in computing moments and k-statistics of finite populations.
A corres-
pondence is established between k-statistics and brackets and rules for
manipulating the two are provided. While considerable work has been
done by others utilizing k-statistics, only the bracket notation was
applicable here and we will not digress beyond this field of interest.
One example of t.he use of these methods was given 'by. Wishart (31) who
considered univariate populations.
11
3.0 GENERAL THEORY
3.1 Conditions for Rotatability
To formalize the dis:oussion in Seotion 1.0 we shall adhere to the
notation in (12) and define the design matrix D for the k faotors
%:I.,x , • •• ,~ as an N x k matrix whose u-th row
2
defines the coded factor levels to be used in the u-th of N experiments
called for by the design.
The general moment of the design will be
denoted by the symbol [1"'1 2"'2 •••k"'k ] where
... "'k .
~
The sum of the powers, denoted by '" = ~~ will be called the order of
the moment.
Recalling that a design isca11ed rotatable (12) when the function
defining the variance at an arbitrary point
~
in the factor space depends
only upon p = /~t~, the distance of the point from the origin, and that
this function is completely determined by the moments of the design
matrix, it is clear that the problem of finding rotatable designs is in
essence one of finding configurations of points possessing the proper
moments.
It is in fact shown in (12) that when fitting the model
including all terms through degree d, a rotatable design will be obtained
when the moments through order 2d are of the form
12
llll
any ..<.. odd,
0
::L
where A.< is a constant for a:ny design and .<.
3.2 Notation and Definition
If we define the minimum k f'actor first order rotatable design
matrix as D , all n x k matrix (letting n • k+l) possessing the minimum
l
number of' rows needed to estmate the constants of' a f'irst order model,
then the moment conditions will be satisfied if the matrix obtained by
augmenting D on the lef't by a column of' unit elements is proportional
I
to an orthogonal matrix, i.e.,
Starting with D then it is conjectured that rotatable designs may
l
be generated by taking all possible sums of·· the n roW'S of'
X
•
•
f
"""U
•
-n
X I
taken s at a time where s ..: 1, 2, ••• k.
The problem thus reduces to one
of' finding the moments '01' a design matrix D derived in this way. We
shall allow the moments ·to be modified by multiplication of the set of'
13
veotors obtained by taking sums of s rows by a oonstant a
oonstants
~,
a 2,
I
I
I,
~
> O.
s-
These
will be oalled radius multipliers sinoe this
soalar multiplioation in effeot determines the distanoe of eaoh point
from the oenter of the design.
We shall oonsider the derived N by k design matrix D as partitioned
into the k subnatrioes
~ D '
l
-------
-~g-~gD =:
•
•
--_.:_--
a D
s .....
s
-----_.:._-I
I
where Q.I denotes a null veotor, sinoe
for each i,
~.
-.L + ~
-, +
loe
+ -s
x'
= -(x'
x' 2 +
-s+ 1 + . -s+
I ••
+ -n
Xl).
Thus we see that each vector obtained by summing rows s at a time is
the negative of one obtained by summing rows n-s at a time.
The points
in the factor space represented by D are, therefore, reflections through
s
the origin of those represented by D
• (Of course, when n is even,
n-s
n - n/2 • n/2 and half the rows of Dn/ 2 are reflections of the other half.
"'1 "'2
"'k
n-l
Let us define ,the moment component [1
2
••• k ]s as (s)N .
times the specified moment of Ds'
"'1 ."'2
"'k
1
[1
2
• •• k ]s· i
'5
1:.!..,
~ ....
t1~ <u 2
~'(XJ.u + XJ.u +
<us~n
• (X2u + x 2 + ••• + x 2 )
1
u2
Us
"'2
1
2
• •• (~ + x..
1
KU 2
+ ••• +
:lCJ.ro)
"'k
s
Then the corresponding moment for the entire design can be -written
where",
=
k
_
"5; "'-. following the notation adopted previously.
:t=;iJ.
3.3 Analogy to Sampling from a Finite Population
The problem of finding
expressio~s for
[1"'J. 2"'2 ••• k "'k] in terms
s
"'l
"'2
"'Ie
of either the moments of D or of [1
2
••• k ]1 corresponds to that
l
of finding the sampling moments for means (or totals) o:f samples of s
drawn from a k - variate finite population of n elements.
can be derived by a method due to Tukey
Wishart (31) for the univariate ease.
(26~?7)
and elaborated by
This method has been extended
here in part, to cover the multivariate ease.
are covered in the appendix.
Thesemoments
The details of this work
Appendix A extends the definition' of
Itbrackets ll as defined in (26) to the multivariate case.
It is shown
•
15
that these expressions possess the same property as their univariate
Namely, i f we use ltAve ll to denote averaging over all
counterparts.
possible samples of s from a population of n, and
~"denotes
summation
over the unequal subscripts i, j, ••• , q, then
or, using bracket notation instead,
Ave
Pl
P2
x 2 , . • .,
{<JC:L,
Pm,}
/
~
where the vector (~u x 2u ... xmu ) is one of s m - variate observations
drawn from n such and a prime superscript added to a bracket denotes
substitution of population values for sample values.
In Appendix B this property is used to denve the general expressions
for the joint sampling moments of multivariate means.
The moments are
the average values of power products of the k coordinate means,
-
-
-)
(~x2 ••• ~
of samples of s
<
III
s
~l(JeJ..u x 2u ••• ~)
1 ~
S
Ulll
n k - variate vectors drawn from a finite population
of n such vectors.
These values, which can be written in bracket
notation as
Ave
{(
'1> "'l
since
s
(xiJ=
f-i
X.
J.U
u=
s
,
...
Ave
.
(~)
'\.
}
since we can regard the n row vectors of D as a k - variate population
l
and we require the moments of sums of vectors taken 13 at a time. The
factor 1300< is requireq since we want moments of sums, not means, and the
factor (~) to give the total moment of all (~) terms.
These results are
sUIlIIIIaI'ized in general form in the appendix and are given in Table 1 for
the particular case of interest here, that of standardized vector length
equal to
/no
The exception to Table la (p. 17) follows .in Table lb.
Table lb. Fourth :order moment components of DB for n • 3, (n
General Formula
Abbreviation
C (s)
41
[i2j 2 JS
C (s) + C41(s)[i2j2Jl
4
[i4 I s
3C~(s)
<00<)
4
C (13) •
=!i(;)
[2 - 7(s~1)J
3T (;) (s-l)i
.
+ C41(s)[i411
3.4
Form of Moment Components
Table 1 shows the moment components in termS of a notation designed
to simplify their use in succeeding sections, and which makes clear their
general pattern.
The coefficient functions 0(13) are of two types.
Those
with single subscripts are not multiplied by unrestricted moment components
of D and hence they are constants in the moment component equations for
l
a given n and s. The coefficient functions involving two subscripts,
e
e
e
Table la.
Summary of general moment components of D , Cn >..<)
S
General. Formulas
[i]a
0
.
Abbreviationa
[ij]a
0
[i2 ]a
°2(a)
n-2 n
°2(a) .. (a-l)N
[ijk]a .
031(a) [ijkh
°
[Ii] S
031 (s)[ii]1
[i3 ]S
°31(s)[ i3 h
[ijkl]s
°U(s)[ijklh
[ijk2]
041(a) [ijk2]1
a
°
31
(a)" fn-2j)
. n-2
U
(s) .. [(n-2aHn-r) - n{s-1)] (1i-2)
(n-2) n-3}
a-1
[ijhs
[i2
0u(a)[ ij3 h
2
04(a) + 0U(a)[i j 2]1
[i4 ]
a
304(a) + °U(s)[ i4 h
[ijk1m]&
051(a) [ijk1mh
°
[ijkl2]a
051(s) [ijkl2]1 + 052(s)[ijk1212]1
°52(a) ..
[i 5 ]a
051(s)[i5h + 100 52 (a)[ i5 12 ]1
[i512]1 .. [ i3 h
[i2j~2]s
06(s) +
061(s)[i2j~2h + 062(a)[i2j~21211 + 2063(a)[i2j~213]1
n-6 n3
°6(a) .. (a-3)P
[i6 ]a
1~6(s)
+ 061(a)[ i6 h + 15C62(a)[i612h + 10063(a)[i613h
°
l]s
-
04 (a)"
°
51
2
(n-4 n
a-2)Jr
(a)" (n-2a) r(n-~)ffiW-) -~Fll] (n-2)
n- n-3)(
a-I
61
63
(n-2)
a-I
(a)"
fUj)(~j)n;
2
[ijkl 12 ]1" [ijk]1
()
062 a
2
2
n +~n-6sn+6s -4 n-4
.. [D=!i) (n-S)
J (a_2)n
[(n-2a)(n-3s}fn-4s~{n-~
n(a-1)(16n2-79ant-lln+86s2=4s-4) ](n-2)
n-2) _3
) (n-5)
.
&-1
2
2
(a)" [n - n - 4sn + 48 + 4] (n-4)N
(n-4) (n-5J
8-2
[i2j~212] .. [i2lh + [i~2h + [j~2h
..
[i2j~2/3]1 .. [ij211[ik2]1 + [i2jh[jk211 + [i~h[ikh + 2[ijkJi
[i612h .. [ i4 11
[i61311 .. [i3 ]i
~
18
however, are multiplied by Dl moment components such as [i j 2 ]1' or
combinations of Ds moment components and are therefore coefficients of
quantities which will not in general be constant for different choices
of i, j, k, 1, m.
The values taken on by any coefficient function O(s), ·when n is.·
beld constant, possess a symmetry with respect to s as a result of the
renection relationship between vectors in Ds and Dn- s • Since one
matrix is the negative of the other thedr respective components must
differ only by the factor (-1) co< or
From Table 1 we see that in general
where the
b~
values are zero or positive constants varying with the
Particular partition of ~
= (~
.,(2
0
o. " \ ) .
Substituting s and n-s for u and using the equality relating the
respective moment components, we may write
Using the set of simultaneous equations which can be obtained by
considering all moments of, a givan order it can easily be shown that
19
only the trivial solution
C..<::t(s) - (-l)..<c..<::t(n-s) • 0, for all i
is possible, whence
C..<::t(s) • (-l)..<c~(n-s).
This result can also be verified 'by direct substitution.
20
4.0 RADIUS MULTIPLIERS AND ROTATABILITY
Having general formulas for the moment components contributed by
each subnatrix De of a derived design matrix D, we now seek a suitable
set of radius multipliers such that the moments of D will fulfill the
requirements for rotatability.
Or, more specifically, such that the
moments
•••
•••
will be of the form listed under 3.1.
For convenience in describing these requirements we will refer to
even - order moments as those for which .,(. ~.,(i is even and odd - order moments as those for which .,( is odd.
In addition, let us define
those moments for which .!nZ .,(i is odd as odd moments and those where all
.,(i are even as even moments.
Using these definitions, we can say that
for rotatability all odd moments must equal zero and all even moments of
the same order must be specified multiples of each other, i.e.,
k
1T
(.,(.)!
[1.,(1 2""2 ••• k4<:] ... A...< i=1 kJ. -.,(
2""/2
1r
i=l
,
all ""i even
(..1) I
2
where A.,( is constant for a given design and "".
From Table 1 we see that the moments [i], [ij
J
and [i 2 ] of D, will
satisfy the rotatability requirements for any choice of radius multipliers
since the corresponding odd moment components [iJ and [i j] are
s
s
2
identically zero and [i ]s is constant for all i.
The other moments however all involve variable terms C""i (s) [
]1
and only in the case of the even moments is the constant term b~C.,«(s)
21
a.dded to this variable function.
It can be seen that the moment
requirements will be generally satisfied only if' the radius multipliers
are so chosen that each variable term sums to zero in the expression for
all [1
~
-<2
2
-<k
••• k
]. For odd moments this is obviously required.
For
the even moments it would otherwise be impossible to attain the required
constant ratio between moments of the same order since the quantities
[ Jl
in general change in their relationships, from one moment to another.
The only further requirement for rotatability is that the constant terms,
)8),
b~C
are in the required ratios.
In general then since
...
where b~C-«s) does not appear unless the moment is even, we have
...
where we require
i == 1, 2 ••
0
p.
22
Since we have shown previously that O(s) • (-l).,(O(n-s), then for all
odd - order moments 0..d(s) • -0..d (n-s).
We can say further, because of
the factor (n-2s) in all such odd order moment coefficients, that when
.,( and k are both odd, 0..ct. (~) • 0..ct. (~l) • O.
Therefore as long as radius
multipliers are selected such that 'as • a ns all
- 'the odd - order moments
will sum to zero for any value of as'
2
m• k l
Setting m •
2'k when
k is even and
when k is odd it then follows that
k
~
m.,(
~ (a) 0J4(S). ~ (as) [0..<1(s) + 0..ct.(n-s)]
s~
~
s
s~
=0
for alli,
.,( odd.
We will call this type of solution for the radius multipliers,
where a s • a n, a s
symmetric
- , solution.
Having satisfied the odd - order moment requirements for rotatable
designs of
~
order we must now find which synnnetric solutions will
also satisfy the requirements for even - order moments.
23
5.0
SECOND ORDER REQUIREMENTS FOR ROTATABTI,ITY
For a design to be second order rotatable the even moments must
have the following general form
2
[J.. ]
= "\11.2
[i 2 j2} • A.4
4
[i ]
• 3A.4
where 1.. 2 and A.4 are constant for any design
less than or eqwal to four must vanish.
0
The odd moments of order
It may be noted here that the addition of center points to a design
matrix D will not change the general form of these moments since their
only effect is to increase the denominator N.
Hence if the moments of
D satisfy the rotatability criterion we can add center points at will.
5.1
Application of Moment Requirements
As noted previously, the general second order moment [i2 ] places
no restrictions on the choices of radius multipliers since
a constant for all values of i.
We may generalize the expressions of Table 1 for fourth order
moment components by letting' the coefficient \
such that, for"). + .,(2 +
[ i~
j.,(2 k"'3 1..<4]
s
"'3 + 1+ = 4
= b.11.C4(s)
-
vanish for odd moments,
+ C {S)[i.,(l j~ k"'3 l~]
41
1
24
so that
In the previous section we showed that, for rotatability, we must
have
making
~
..(
[i 1 j 2 k
..(..<!+
31
] . bl •
J;I;
4
k
'OS (a
a-=1
a
) 04 (a)
Hence· all odd momenta of order four vaniah with'\. and
[1
2 2
j ] •
k
4
'OS (aa) 04 (s)
a~
• 14
.AIry symmetric aolution for the radiua multipliera such that
k(aa) 4041 (a) = 0 will therefore clearly provide a rotatable design
2
a=l
in that
(a)
all odd order moments will be zero due to symmetry
(b)
the odd fourth moments will equa.l zero
(c)
[1
(d)
2
J will
be the aame for all i
[i4 ] will equal 3[i2 j2], which will not depend on the
choice of i and j
Such deaigna will be called aimplex-sum deaigna
0
25
5.2 Standard Solution for Radius Multipliers
k
The equation
s~ (as )4C41 (s)
III
0 was solved for the symmetric set of
values { as} in the specific cases of k
III
2, 3, and
4. The results
suggested that a standard solution, for any k, might be
as
III
(n-2)- ~
s-l
~
III
1, 2, ••• , n-l
- 1
which we shall henceforth denote by B 4'.
s
- 1
To prove the generality of this solution we substitute Be 'l! in
the general formula,
•••
and show that a rotatable design results.
It is irmnedia.tely evident that all odd moments will be zero since
the set of values, {Be -
kJ,
provide a. symmetric solution a.s defined
earlier. Further
Let u
III
s-l and substitute k
2
and, for each i, [i ] equals
binomial coefficients (n~2).
i
III
n-l, then
times the
SuIll
of the square roots of the
26
II
n-l 2 6
6 2
~ n - sn + s + n
s~
(n-2) (n-3)
II
[i j k IJl
(n-2)(n-3)
II
[1' j k 1]1
(n-2)(n-3)
[i j k IJ
2
{(n-l)n - 6n[~(n-l)
[OJ
II
1
J + 6[~(n-l) (2n-l)] +
(n-l)n1
0 .
Since the zero quantity in brackets is the
express~on~ a~
041(s),
common to all fourth order moments, we can write
[i j k2 J
[i j3]
II
0,
= 0,
2
==
=
n ~n-l)
N
2
n (n-l)
2N
,
II
1\,4'
= 3"\1\,4
e
We have thus demonstrated that sets of points satis:fying the moment
conditions of second order rotatable designs can be derived from those of
first order e
The solution as
= Be -ft
has been shown to apply for any
number of variables greater than two (k
moment formulas hold only for n
2 4,
~
3) since the fourth order
as noted previously.
27
5.3 Second Order Rotatabilit,y for the Case n
=3
Using the formulas for fourth order moments for n
we can show in a similar way that as
also applies for k
= :8; ~
III
III
3 in Table Ib,
(~:i) - t
= 2.
2
[i J.3]
III
~(s_ll)-l 3i
s (3)
s [2 - 7( s-1) ][i j3]1
Sill
This result, although apparently inconsistent with previous results
in that [i j3] is not immediately zero via the route
will satisfy the moment requirements.
C'41 (s) .. 0,
The conditions are satisfied due
to a property of 3 x 3 matrices of the type [1
columns that makes [i j3]1
~ a~
s~
~ ~]
with ortho gonal
= O.
This propert,y can be demonstrated algebraically, but since we have
already shown that a matrix of all sums of rows takens at a time is the
negative of the matrix of sums taken n-s at a time we have Dl == -D a.nd
2
But the general moment formula for n = 3 shows
Hence
and must vanish.
28
Proceeding to the moment [i2 j2],
= [i2
But since [i 2 j2]1
[i
or
2 2
j ]1
2
[i j2
j2]1 - 5[i2 j2]1
= [i2
j2]2
= -5 [i 2
h = ~,
+~.
j2]1 +
*
a constant for any matrix of this type.
2
2
Hence [i j2 J • 2[i j 2]1
=i· ~4·
Similarly the moment [i4] is found to be
and therefore the moments are those of a rotatable design.
This result
agrees with that obtained in (12) and shows that it is a special case of
the standard solution a
k.
• Bs
B
While these designs satisfy the moment requirements for rotatabi1ity,
they yield a singular, or almost singular, moment matrix.
Thus to become
usable designs they require the addition of center points (to be discussed
in Section 7 .3) •
29
6.0
THIRD ORDER REQOIREMENTS FOR ROTATABILITI
To prove that the derived matrix D possesses the moments required
for third order rotatability it is necessary to show that a set of radius
multipliers can be found which will not only satisfy the requirements for
second order rotatability, but will also
make all fifth order moments equal to zero,
(a)
(b)
make all terms in the sixth order moments involving moment
components of Dl other than [i 2 sum to zero
A symmetric solution will again satisfy the requirements of (a)
h
above.
0
From Table 1 we see that part (b) can be accomplished 1£ we find
as such that
where i
III
1, 2, 3.
This result will hold true for all sixth order moments and therefore
we need only find a symmetric solution for this set of three si.mu.ltaneous
equations involving
a~ which
will also satisfy the corresponding fourth
order moment equation involving
a~.
To do this we will solve the sixth order set of three equations and
see 1£ a feasible solution is obtained.
If we let
a~
Il!I
a
s6
we
may
them as a set of homogeneous linear equations for each value of n,
~1II0
where
consider
30
and
A•
C61 (1)
061(2) ••••• 061(k)
062(1)
062(2) ••••• 062(k)
063(1)
063(2)
00'"
063(k)
.As our moment equations do not hold for n
s:
5 we will begin with
the min:i.mmn value of n • 6.
Substituting in. 06i(s) we have
A-
1
-26
66
-26
1
0
1
-4
1
0
0
1
-2
1
0
•
However this matrix is equivalent by row operations to a matrix
A* ==
1
o
o
o
l'
o
1
o
1
o
o
1
o
o
o
Hence the solution for
~ ==
0 demands that
~6 == -&66'
&26 == -&56'
&36 == 0,
which is not & feasible solution sinoe all &i6 =
a~
must be positive
for real radius multipliers. We have thus shown the design matrix D
for n == 6 is not third order rotatable.
31
Similarly we :find that :for the case n
A=
=7
1
-25
40
40
-25
1
0
1
-3
-3
1
0
0
1
-1
...1
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
0
1
1
0
0
,
and
A* •
:for which no :feasible solution can be obtained.
For n .. 8
A-
1
-24
15
80
15
-24
1
0
1
-2
6
-2
1
0
0
1
0
-2
0
1
0
1
0
0
92
0
0
1
0
1
0
-2
0
1
0
0
0
1
-4
1
0
0
1
-23
-9
95
95
-9
-23
1
0
15
-15
-120
-120
-15
15
0
0
10
10
-20
-20
10
10
0
1
0
0
7
7
0
0
1
1
1
0
-5
-5
0
1
0
0
0
1
3
3
1
0
0
and
A*
==
For n = 9
...
A=
and
A*
==
32
Neither of these matrices yield solutions such that all a 6 20.
i
Since a polynomial of degree three has (k ; 3) constants, a third
order design must have at least this many points.
Therefore k IIIIlSt be
at least five before D has sufficient experimental points to be considered
a candidate. We have thus demonstrated that D cannot be third order
rotatable for k So 8.
It would appear impraotioal to consider designs
involving more than 8 variables.
The size of the experiment alone
approaches prohibitive proportions as a minimum of 220 experimental
points is required for k
= 9.
33
7.0
SECOND ORDER ROTATABLE SIMPLEX-8UM DFSIGNSz
THE STANDARD SOLUTION
We have shown previously, in Section
5.2, that the derived design
matrix D can be made second order rotatable by using a set of radius
n-2 -
nmltipliers as • (s-J.)
t•
Before proceeding to show that other specific
solutions are possible in higher dimensionality, we will first investigate
the effect that the choice of radius multipliers has on the geometrio
oonfiguration of the design in the factor space.
It was noted earlier that the points in k space desoribed by any
n by k first order rotatable design matrix, D , are the vertioes of a
l
regu.lar simplex. When we add those vectors two at a time the vector
which is generated will pass through the midpoint of the edge of the
simplex conneoting these two points. Sums of three vectors have a
resultant passing through the midpoints of the faces.
s at a time for s
>3
Whiles1..Ul1S taken
cannot be visualized in as few dimensions, some
concept of the geometry of the design oan be obtained through the
extension of this idea of the average or midpoint vector of s symmetrically
spaced vectors.
To oomplete the feeling for these designs we must find
the relative distanoes along these vector direotions to the specified
design points.
7.1' Radius of Experimental Points
The radius of an experimental point is its distance in the factor
space from the geometrio oenter of the design, which we took as the origin
of 'coordinates.
Considering the coordinates of the point as a vector,
the radius is the length of that vector.
To obtain a general expression
34
for the length of the row vectors in each submatrix D we
s
. 1
u th.row as~,
s ·2
, 3,T
••• , k.hen
XI.
-au
where
• aE.~
asJ. ' asJ. ' "" asJ. is
Inatrix
1
.s
.2
+
Xl
""lu 2
+ •••
may
denote its
Xl
-Us
ths'u-th set of s rows of the first order design
Now since
D •
l
•
•
l~
xl -j
X •
-J..
n-l • k ,
.. -1,
The square of the length of the row vector
L2 ..
s
x' (x'
)'
-au
-au.
.. (x{
~u
can be written
• -au
x' -au
x
+ ~ + ••• + ~)(~ + ~ + ... + ~)
.. s(n-l) + 2(~)(-1)
.. s(n-s).
Thus the radius of the experimental points in' the subnatrix a
of the design matrix D must follow as
2
2
r a2 .. (aaL)
a .. as sen-a)
1
r
s
.. as [s(n-a)]]'. .
D
s s
35
It is evident that r s • r n-s since the points are reflections of each
other through the origin.
7.2 Radii for the Standard Solution
For the particular set of radius multipliers of the standard
solution,
!.:.!., a,s • (n-2)-k
s-l
'
rs •
(::i)- k
the radii will 'be
[s(n-s)]
~.
A summary of the radii for k • 2 through 8 of the standard solution
rotatable designs follows in Table 2.
Table 2.
r
Radii of experimental points for standard solution
rotatable designs
r
r
k
1
-
-
2
1.41
1.41
3'
~.73
1.68
1.73
4
2.00
1.86
1.86
2.00
5
2.24
2.00
1.92
2.00
2.24
6
2.45
2.11
1.95
1.95
2.11
2.45
7
2.65
2.21
1.97
1.89
1.97
2.21
2.65
8
2.83
2.30
1.98
1.84
1.84
1.98
2.30
2
-s
2.83
3.6
7.3 Singu.larity and Near Singularity of Moment Matrices
It is shown in (12) that a set of points can have the moments of a
rotatable design of order 2 but be impractical as a design since it leads
to a singular moment matrix.
The singularity ariseS from a dependency
between the columns in the X matrix for the bO and quadratic terms,
'b:J.l' b 22 , .,.,~. The situation is easily remedied, however, by the
addition of center points to the design matrix, so that the appearance
of this property need not concern us too much.
As shown in (12) the moment matrix will be singular when the
standardized fourth moment constant 1. achieves the value
4
4
1.
1.4
.
2
III
III
.(A 2 )
k
k+2 '
implying that the design points all lie on the
Satn(9
hypersphere (4).
For
the designs arising from the standard solution tor as we have .
4
1.
(n(n- h [n~ <:=il ~]
2
l
6 N
=:=
-2
N
'J
(n-l) (2 n - 2)
I:
6
[
lJ 2
~ (~:i)2
where we have used'N == 2n - 2.
The value for 1.
4 is
equal to the singular :value (k~2) when k
III
2
and remains close to the singular value as k increases, as is shown in
Table 3.
37
Table 3.
Comparison of
k
~~
to its singular value
~~
...L
-
k+2
2
.500
.500
3
.600
0601
4
.667
0670
5
.714
.724
6
.750
.769
7
.778
0811
8
.800
.850
Since the addition of center points has no effect on the moments
except to change N we see that the addition of No center points will
n
change ~~ by a factor of2 - 2 + No' The addition of center points
n
2 - 2
serves other useful purposes however.
First it provides an estimate of
error for testing the adequacy of the model and secondly it affects the
shape of the variance function (12)
0
As is true for other designs
(e.g., factorials) which might be used in response surface experimentation,
the variance of an estimated response is usually relatively high at the
center of the design, decreases to a minim:um and then rises rapidly as
p
=l~t
3S approaches the radii of the outer design points
0
By a.dding
the proper number of center points the variance at the center can be
reduced so that the variance is more or less uniform over what is usually
the principal region of interest.
If the number of center points
required to attain this is not economical a compromise can be reached.
In general it is suggested in (12) that uniform variance can be
appro:x:iJrlated for rotatable designs by adding sufficient points at the
38
center to equate the variance at pliO to that at p • 1 in terms of
standardized units (Le., xi scaled so that
~X~u
II N).
For the
u
unstandardized units usually used in actual designs this implies
equating the variance at pliO to that at p
II
;:r;-.
39
8.0 ADDITIONAL SEOOND ORDER ROTATABLE SIMPLEX-SUM DESIGNS
The' standard solution for as ,affords a set of rotatable designs
> 2. When k >5 however, we shall demonstrate that other
specific solutions for the radius multipliers are possible. Further,
for all k
since the number of experiments required b,y the standard solution soon
becomes excessive we shall seek sets of radius multipliers which include
some zeros thereb,y providing smaller, reduced designs.
8.1 Solution Spa.ce of Radius Multipliers
We have shown in Section 5 that for second order rotatability we
must find values for as' s • 1, 2, ••• , k, such that
and
~ a:
~ a; 031 (s) • 0
041(s) • 0 where 031 (s) and 041 (s) are the coefficients of
the moment components of D (Table 1). When those values are found it
l
was shown that the other moment requirements were automatically satisfied.
To state these requirements in a more convenient form for our present
problem let us define vectors of linear components
!o
!o3
and
~
=
=
(af a~
•
(~ a~
(~ a 2
... as ...
~),
.
...
~)
,
...
•
a4 ... ~)' •
s
a 3s
•
0
.
•
(~3 a 23
(~ a 24
... as3 ... ~3) "
... a.s 4 ... ~)' .
Let us also define row vectors
0'
-.31
=
(031(1) 031(2) ••••• 031(s) ••••• 031(k»,
(041 (1)
°41 (2)
••••• 041 (s) ••••• 041 (k» ,
whose elements are the moment component coefficients.
40
We
~
therefore restate the requirements for second order
rotatability in our present situation as that of finding a set of radius
multipliers such that Q.,31.!3 • 0 and
-
values of
Q.,31!!.3
==
!OS'
o.
~~
• O.
But if we choose symmetric
-
such that as
==
a n- s ' we have shown previously that
Therefore, calling any vectors !o3 and
.!J+
which are derived
from symmetric solutions, symmetric vectors, we may further simplify
our problem to that of finding aJ.l symmetric vectors ,
~l'_· O.
such that
We must also add the restriction of course, that.!4
2' 2,.
The restriction of symmetry on the vector!4 has the effect of
confining its values to an m dimensional subspace where
m•
t, if k even
and
m • ~1 , if k is odd.
This is evident since .!4 has exactly m elements which can be varied
independently, the remaining k - m elements then being determined by
the relationship a s • a n-s • The elements -of -z.j.
~'l are symmetric in a
corresponding way as was shown earlier. Hence for convenience we might
consider !4 and
~l
as two m - dimensional vectors and use
~he
fact that
m-l independent vectors can be found orthogonal to any vector in m-spacs" .
Thus _i f we find m-l independent solutions to the equation
~1!4 ==
0 they
will form a basis for the solution space of all possible vectors satisfying
the equation,
(1.:.!.,
all vectors in the m-l space orthogonal to .~l) •
Since the elements of
~
are of mixed sign it is olear that solution
k
vectors can be found which fall in the positive 2 -drant.
41
8.2 Specific Solutions
We will now obtain the m-l basis vectors 'Y1' 'Y2' ••• "fm-l for
k
a
3, 4, ... 8, selecting them to contain the maximum number of zero
elements possible. Where zero's can be introduced, the equivalent
designs will, involve fewer points than the standard solution since any
submatr.ix with a zero radius multiplier, corresponding to points at the
origin, may be eliminated from D without altering the moments.
All other
designs, resulting from the ortho gonality relationship, can be derived
from these basis vectors by taking linear combinations
where the di'S are any constants such that ~
1Q.
It will be recalled from the discussion of the standard solution
that the two factor design is an anomaly in that its rotatability does
not result from the orthogonality relationship.
For k
Ill!
2,
~1!4
r°
and hence a specific solution does not follow in the usual wa.y.
When k
==
3 m := 2 and hence only one solution, the standard solution,
is availa.ble, ('Yl
= !4) •
Therefore
~l~ ~ (1
-4
1)
1
2'1
=
0,
1
-1
A = (1 2 4 1)'
where
. the k elements of the
of the standard solution.
-.l
= Bs 4 ,
vector~..!- t
- 1
are the B
s
4'
values
42
The one modification of the design possible, of course, is
multiplication by a constant
(1:..2.., , •
dl"1"l) which merely amounts to
a scaling change.
When k ...
4,
again m ... 2 and the standard solution is again unique
£tl~ ... (1
-3 -3 1)
1
1
3'
... 0
1
3
1
and
-a ...
(1
3-
fi-
When k ... 5 m ... 3 and two independent solutions 'YJ. and "1'2 are
possible.
Here, for the first time, we can obtain reduced designs.
Both basis
vectors will be of this type where
"1'1 ...
( l o t 0 1)'
1 0
...
(1
!.
==
(1 0
!.
...
(1
"'2
1 1) ,
'2
and
2-
3-
'2
t
0
1)',
0
2-
t
*
1)'.
The design resulting from "'1 omits D2 and D while the "1"2 design omits
4
])3 from the design.
43
To demonstrate that the standard solution is, spanned by these vectors
1
1
4'
1) t
4'
from which the standard solution is obtained,
!.
-J.
= (1
4 4
- 1
1)
t
Any other linear combination such that
B
•
J.
4'.
!4 ~ 0 would
also provide a
solution as for example
and
-a
• (1
When k • 6, m = 3 resulting in two independent reduced solutions,
£41"'i
=
(1
-1
-8
-8
-1
l)"'i
"'1
=
(1
1
0
0
1
1)',
"'2 =
(1
0
tr
1
0
1) t
1
~
=
0,
and the specific solution can be obtained by taking the fourth root of
each element.
44
When k
2tl"fi
"fl
"(2
"f3
III
8, m
IllI
4 providing three basis vectors
•
(1
1
III
(1
0
III
(1
0
•
(0
1
-25
-25
-
0
0
°
E
0
-9
1
9
!
9
1
1) "f1 • 0,
-9
0
1) "
25
0
0
1) , ,
0
-
1
0) , •
1
1
-9
1
1
9
A fourth reduced design can be derived from the following vector
[0
1
0
1
1
0
2'5.
25
1
0)'.
A sUIl'lIl18.I'Y of the radius multipliers used to obtain the standard
- 1
solution designs (B 4) and the specific solution designs derived
s
from the basis vectors, is given in Table 4. It can be seen that only
the reduced designs will be practical in most· instances when k
N increases rapidly.
>4 since
Also included in the table are the number of center
points required to attain Ituniform varia.nce lt within a hypersphere of
radius
p;::;- .
In order to produce a design using Table
4,
it is only necessary to
select a suitable matrix Dl and by taking all sums of rows s at a time,
for each s of the non-zero as values, generate the required D matrices.
s
Multiplication .of Ds by a s will then give the coordinates of the design
points.
An e:xmnple is given in Section 11.
45
Table
4. Radius multipliers for some seoond order rotatable designs
Rae.ius Mult1nliers
k Design
~
2
Std.
1
1
3
Std.
1
.8409 1
4
Std.
1
.7598
.7598 1
5
Std.
1
.7071
.6389
"'2
1
.8409 0
"'1
1
0
.7598 0
Std.
1
.6687
.5623
"'2
1
0
.5946 . .5946 0
"'1
1
1
0
0
Std.
1
.6389
.;b81
.4729
Yl
1
0
o5'n4 0
1
0
0
.5946 0
"(3
0
1
0
0
Std.
1
.6150
.4671 .•4111
Y2
1
0
0
0
1
o5'n4 0
1
0
S174 0
6
7
"'2
8
"'3
Y1
a2
a
3
a4
'7
a6
a5
No. of Exoer:l.me rital Po1nts a
S:l.mplex-Sum
Composite
Desi,ns
DesiR.': s
Center
Center
Radial
Radial
&8
Points Points b Points Points b
6
8
5
3
14
6
14
6
30
14
24
7
.7071 1
62
24
.8409 1
42
10
32
8
26
6
126
38
1
84
16
1
1
56
13
44
9
.;b81
.6389 1
2~
59
1
128
21
0
1
86
15
0
1
0
56
10
78
14
.4111
.4671
.6150 1
~O
90
0
1 270
26
0
05'174 1
0 240
0
0
05'774
1 186
28
80'
13
.5623
.W.j.72
1
.6687 1
o5'n4 0
.4472 0
0
a The "Composite Designti values refer to the composite second order rotatable designs
derived in (12) and are included for oomparative purposes. Hall' replioates of the cube
portion are used for k • 5, 6 and 7 and one qwarter rep1ioate for k • 8.
b Number of centerpoints requ.ired for "uniform varianoe" within p • .,;-:;:;:
46
It 1t should be des:l.red to replicate certa:l.D meets of the
der:l.ved matrioes th:l.scan easily be dome by :maJdDg Ritable
ad~.
to the rad:lus multipliers. We will oDly oOl'.lllJider the case wb.ere S1JIIlfII!tne
rep3,ioat:tctll:is uBed(i.e.,]1)8 and Dn-s are repl1cated equ].lyh tku
eranr:J.ng
thata~ric
solution fer the radius multipliers can 'be
fcnmc1.
It .e replicate a pazoticuJ.a.T set
ot nlaatrices Ds
'V s
and ~s
times, the elements 0.31(.) Jl C (s), C.31(n-s) ad 0l,l(n-s) will 'be
hl
Dmltlplied by
V8
ud t~ 11lO1IlEmt equtions will become
~.
Vs (as.31
).3 C <s) e
~
'
1II
~
v.(a.)4
e~(.) • e.
The :£'!rat equatiol'.l will still be negatively symmetric ad will there.fore ,
'be sat:i.afied by amy symmetric vector. ·'I'hesecond eqnation wUl he
satisfied i t the new
vs (as )4 equal the (l)ld (a.,4.
,
as (replicated
.
.
~s twa)
Thull
a.('lUlreplieated)
1II
(oJ
)l/L.
:I
a
ud a s1m:iJ.a.r relatioD bolds tor rad1i"
Fer aam,ple,Hnsider the 8'lia:ndardsolution tor k
!llS
patterJ1S o£ replication. ,<We w:Ul alwaysbave oJl *J.4 till 1,
V.3a.34 = l,,) Table 5 shows some re.J:ts.
),
ud, va:ri....
S2~4
III
1/2,
47
Table
5.
The staltda:rdlOlution with k • 3 and
w:rious replieat::ton patterDI
Patt8l'n
Repl::teat:LoDI
ladiul Nultip1::ters
Vl Va V,
a:..
....
1
1
1
1
1
2-1/h. 1
2
2
1
2
2-1/4 2-1/ 4 2-1/u'
3
1
8
1
1
2-1
&3
1
Rad11
;1"1
,:ra
:I":V
1.73 1068 1073
104$ 1.68 104'
1·73 1 ..00 1·7.3
48
10 .0
BLOCKING
When an experiment cannot be run under homogeneous conditions it is
usually desirable tG block the trials in such a way that the coefficients
can be estimated efficiently while the error is confined to the magnitude
of variation within blocks.
We will assame that under the experimental
conditions pecu.liar to any block the relationship of the response tD 'the
factors remains 1Ulchanged with the exception of a shift in level.
Following the development in (12) then we will assume tbe expected value
of the uth experimental observation is represented by the model
= ~
m ~.
~u
w=l
z .
ow wu
+
~ 13.
~. J=J:
~. 13..x1u.x,.
i~u + i==
i=
1.J
JU
'
where 13 ow is the level parameter for the wth block and z......
is a dummy
....
th
variable assuming the value unity when the u
experiment falls in block
w and zero otherwise.
j
~
'We might arbitrarily define
.
...
-
13.0 as the weighted average of the 13ow t s
so that the model above can be rewritten in an equ.ivalent form but which
---
-..
,
is now identical to the model when no blocld.ng is used except :tor the
addition of an incremental block effect.
~u = 13
+
0
~ 131..x.1.U
•
1.=
k
III
~ 13••x.x'. + ~ 0 (3 =z),
j~ 1.J 1.11 JU
w=l W Wtl W
k
III
+
That is
~
•
1.=
where
(Jl
t"o
=
~ ~
1r
~
ttl
I"'ow
,
6
•
W
c:l
I"'ow
_
c:l
"0
,
z.._.
llw
-w N
and. ~ is the number of observations in the wth block (including
centerpoints) and N
=
A
Il.w'
49
10.1 Orthogonal Blocking - Rotatable Designs
It is shown in (12) that orthogonal blooking is obtained when the
within block moment components of the design (denoted by [i'"l j
"'2 JbW
)
bave the following propertiesg
Xlw
~ X:i.u
u
.3. [i
==
w • 1, 2 ... m
0 ,
2
low • 'H1
~
indicates summation over the nw design points within the
where
th
u
w blook.
orthogonal blocking ensures that, in the normal equations, the
portion involving the regression coefficients will be .free of block
effeotsand hence the estimation of the vector of regression coefficients (!!.) proceeds as though there were no blocking.
for est1mating the elements 9:£ ~.
are given
General fornmlas
in (12) together with
expressions for their variances • !he only departure .from the analysis
of an unblocked experiment is the removal of the block sum of squares
.from.
the residual
SWIl
of
sqIl&res
in estimating the error.
Simplex-sum designs le:nd themselves to orthogonal blookingin
most ca$es although not always in manageable block sizes as we shall
see.
From the general formulas for the moment components of the _ matrices
»1'
D2, ... I k we see that conditions 1 and 2 for orthogonal
blocking will automatioally be satisfied i f we use these partitions, or
50
combinations of these partitions, of the design matrix as blocks. For
-
convenience we shall refer to blooldng schemes of this sort as lBUbmatrix blooldns schemes.
Submatrix blocking is desirable in that each
is a first order rotatable design and hence provides an easily identis
fied natval set of blooks for sequential experimentation. tJ'nfortunately,
l)
sub_trices do not in general satisfy the third cond:LtioD
mwever, these
for 0rthogonal blocldng, although as will be shown later, they come qu.1te
c10se.
For a standa:l"d solutioR design the block sec0nd:"m0J1lent component of
D is
s
Now to satisfy conditi0n 3' for orthog0nal blocking we would
2
[i'" ~s
~.
requ1r~
n 2. 1/2
<;i)
.............._--
k
2 1/2
"S'e~-
s~
s-l
)
where n denotes the number of center PCJints added to block a,
.
s o ·
(i.e.,
Jl) ), and III == ""'n
. Since the ratio [i21~s 1 L""2 will in
s
o..c:::i· so
general involve the ratio of irrational numbers, it is clear that
exact orthogonality cannot usually be obtained by considering the
indiVidual submatrices as b10oks. The ome exception is the two factor
hexagon design where, since n
= 3,
the binomial coefficients are 1 and
hence rational numbers are obtaimed. 'rbis proves the simplest case,
however, of a general orthogonal blocldng se_e since whemever k is
even, two Grthogonal blocks are easily formed by combining D1' D2 .•• D
k/2
ill one block and the remaining half of D in the second block. Then since
51
as long as an equal number of oenter points (:l.f' any) il added to eaoh
bloo~ •
However, simee I1.w
= 2k
- 1 + 1\,0 the block size grows qaite
rapidly and .for many situations may be excessive. 1ftlen k is odd monch
simple orthogonal 'blocldng exists since splitting D{~1)/2 into symmetrio
halves results in losing the property [:1.1ow
III
O.
!!:ds same general scheme can be used for specific solution desips
and here, even when k is odd, we ea .fEmll two orthogonal 'blocks as lomg
as k ~ 5 since in all oases reduced desips exist in whioh a(kfol)/2
(Table !,.). Thus, with the exception of the k
III
llII
0
.3 case, s:imple,x...sum
designs are always available which can 'be divided into two orthogonal
blocks and only in the five factor design are we forced to use Gther than
the design requi:ring tl1e fewest mumber of points to obtain orthogonality.
A few other designs which block ort.gomally exist among the
speoifio solutions. In these designs the iDdividual submatrices may be
used as bloaks, but center points nmst beaMed in a specified maner.
:For specific solution desips cond:1.tion .3 can be satisfied whenever the
1MJl-HrOa2 are ratio~ . Referrillg to Section 8.2 it can be se. that
s
among the designs listed nch solution vectors of radius nmltipliers
=6
(Yl ), k • 7 (Y , Y ) and k • 8 (T , T2, T ). For these
l
l
3
3
designs values of nso have been found which satisfy 3 and also satisf'y
e:x:Lst for k
or approxLmate the cOl1ditions forl9tmiform variance" within a sphere of
radius
r-r;.
These blocking schemes are swmnarized in Table 6.
,2
Table 6.
Swmnary of orthogonal blocking schemes for
4
rotatable designs of 'l'able
Number of Po lJoIts
k Design
Bloak
":1.Dl
1
2
3
1
2
5
10
:; '1'2
1
6
15
6
1
2
7
21
'1'2
1
2
7
'1'1
1
2
7
'1'1
1
2
3
4
7
'1'1
1
2
8
'1'3
1
2
'1'1
1
2
3
4
8
std.
1
2
9
'1'2
1
2
1
2
1
2
1
2
3
4
1
2
3
4
1
2
3
4
9
2
etd.
3
none
4
stel.
'7
8
std.
"(3
'1'1
"(2
'1'3
'1'1
2
a~2
tl'
aD
33
Bloak from Su. 2natrix
~D4
a!fJ5
a D
77
aeD8
10
2
2
5
15
15
7
7
22
22
21
21
5
5
26
26
35
7
63
63
19
19
82
82
35
7
42
42
8
8
50
50
7
28
28
6
6
34
7
21
21
7
5
15
6
35
21
35
21
21
7
21
21
56
28
84
56
126
126
36
126
84
9
84
84
84
36
9
9
84
36
84,
126
36
8
126
84
9
8
56
56
36
Total Number of points in Bloak
Sans,
Center
Ch'and
Center
Points
Total (lly)
Points
Added
3
3
3
28
9
a6D6
9
126
9
84
84
36
9
(0)
(14)
(14)
(0)
5
34
7
35
35
7
64
64
10
10
74
74
28
28
5
5
33
33
8
56
56
8
(4)
(4)
(4)
(4)
12
255
25>
45
45
300
300
135
135
120
120
93
93
9
126
126
9
36
84
84
36
9
84
84
9
00
00
12
148
148
120
120
107
107
14
18 19
(9)(10)
126 133
~O) (7)
0) (7) , 126 133
(9)(10)
18 19
84
(48~
84
~g
84
(48)
84
13
13
0
0
14
(4)
(7~
(7
(4)
13
91
91
13
53
The entries in the body of the table represent the n'W11ber of experimental points contributed to the block bY' the particular column heading
in which they fall.
Except for those values in brackets the listed center
poiDts are those required for approximate17 *uniform. variance1t aiKl could
be replaeed bY' any number as long as tbey were evenlY' dist:r:i.lm.ted among
bl..ks.
no.. values
those giving uniform
CllDMt
in brackets, however, are the D'WIHrI closest to
~e
wb:Lch w1l1 give ort_puJ. ltloGkbg. The)"
be altered withoutehecking for
108S
of $rthopnality.
10.2 Orthogonal Blocking -Nearly lbtatable :Designs
Irthogonal su'bmatr1:x: 'blocking can always be obtaiaed bY' acdjustiaC
the radius Imlltipliers and saC:r1£:Le:1.Rg I"OtatabilitY'. Since it will prove
tbat very small adjustments are required, ad hence small departures from
wtatabilltY' are to be exp,cted, this appreaoh has
~uch
However" there is also a major drawback to changing the as values :trw
very geDeral desipsbeing considered here.
t.
tG recommend it.
If the radius multipliers
are adjusted so that the,. remain symmetric, the odd third moments v.i.ll
naish as for rotatable designs; the odd .fovth order moments will not,
however.
Fltrther, as will be recalled :f'roo1 fable 1, aU fourth order
,
mGlI1ms may va'1!T depending
UpGD
the e:b.oice of i, j,
~
to .Hence, the
solution of the normal equ.,tions will in general imrolve the inversion
2
of a matrix having (k + k + 2)/2 elements on a side and not having
sufficient patten to all" a geaeral 1mrerse tG be foud.
Ift special
eases, of oourse, tlds doe. Rot "ear (as for the central composite
design
ot k
== 3 where [1 j k .ell' [1
;j k 2 Jl ,
and [i j3]1 are all zero)
However, since we are dealing witA c0Dlpletely general
lSimple~SUIIl
0
54
designs here, this approach is not too appealing, in that it requires
considerable computation.
In the interests of completeness, however, and in view of the widespread availability of high speed computing machines, the ra.dius multipliers required for OI?thogonal submatrix blocking of the Table
are tabulated in Table 7.
4 designs
These values are based on the number of oenter
points required for uniform variance and will, of course, vary if the
nwnber of center points is changed.
Those designs for which orthogonal
rota.table blocking is available are omitted.
Comparison of these radius multipliers with those of Table
4 will
show that the adjustment has been very small and indicates how surprisingly closely the submatrices approach the conditions for orthogonal
blocking.
In raising these to the fourth power however the difference
becomes more sizeable and is quite effective in causing the unwanted
components of the fourth order moments to be retained.
10.3
Non-Orthogonal Blocking of the Rotatable Designs
A third approach to su1:matrix blocking is to accept the small loss
in efficiency suffered through using these slightly non-orthogonal sets
as blocks
0
The estimates of the regression coefficients will then be
the solution of more complicated normal equations but ones for which the
general form of the solution is readily found as in the orthogonal case ~
The question then arises, under what conditions will the variance contours,!,fan estimated response remain rotatable when the b's are solutions
of the adjusted normal equationso
While the loss of rotatability would be unimportant here, since the
departure would be small, the question is of general interest and worth
e
e
e
Table 7. Radius multipliers and center points for orthogonal nearly rotatable submatrix blocking
k
D
1
Original
D
2
D
4
D
3
Design
-a
3
n
30
a
D
5
n
40
--~
~O-
a
3 Standard
1
2
.8165
2
1
2
4 Standard
1
3
.7638
4
.7638
4
1
3
5 Standard
"{'2
"{'I
1
1
1
4
1
1
.7071
.8238
0
5
4
0
.6583
0
.7868
6
0
6
.7071
.8238
0
5
6 Standard
1
1
6
4
.6679
0
8
0
.5547
05954
5
4'
.5547
.5954
5
-I
i
8
3
.6455 12
0
0
05164
8
0
.4776
1
12
.4690 13
"(2
7 Standard
Y2
8 Standard
2
.6172
n2Q
20
0
4
a
5
6
n
50
a
6
a
D
60
7
1
1
1
4
4
.6679
0
8
0
1
1
05992
3
9
.5164
0
8
0
.6455 12
0
0
1
.4140
0
04140
0
.4690 13
.6172
4
0
D
8
D
7
D
D
70
a8
1'1
80
1
1
6
4
1
8
3
20
1
12
~
56
a short digression to develop the general conditions under which
rotatability is retained.
The development that follows will then
be useful in obtaining the general solution for the regression
coefficients and their variances.
10.3.1
Bestrictions~ Non-Orthogonal
Blocking
Necessary~
Retain Rotatability. Expressing the general model, adopted at the
start of Seotion 10, in matrix notation we have after ordering the
trials by blocks
where
!lis
an N x 1 matrix of expected values ,
X is an N x (k + l)(k + 2)/2 matrix of independent
variables and is of full rank,
!
I 2 22 I.
I
)
= ( ~o I ~l ~ 2 ••• ~k I ~1 ~ 2 ••• ~k I ~12 1313 ~.. ~k-l, k t,
Z*, is an N x m matrix whose column veotors -w
z are
independent of the column vectors of X and are
defined as
-nw -D
1
l
-----c=Q'-n 1
-liT
,~.
1
z =-w
N
,
e-
··
-n
2
ltd
..
-_ _
.!n
- .. _--- ..w
(N .., ~)
·
......
.a
_
,
.!1'l.w
and
is an
~
:x: 1 column vector of unit elements
.§* is an m x 1 column vector of the incremental block
effects.
;r
The columns of Z* are dependent since Z* 1m".Q but it is readily
shown that any m - 1 of them are independent. If we drop one column
from Z*
(.!m
say) the remaining column vectors will be a basis for the
same . space and hence this will have no effect on the estimates of !.
We will denote this N x (m - 1) matrix by Z.
This is equivalent to
letting 6 .. 0 in the model and hence the 6* vector is replaced by an
m
-
-
(m - 1) x 1 vector S whose elements will 'be different but we are not
usually interested in estimating these parameters in any event. Thus
we may rewrite the model, now of fall rank, as
1:L.. XI. + Z! '
and the normal equations become
x.'I!
+ X'~
III
I'Z
Z'X! + Z· z.s .. Z'Z '
where 1. is an N x 1 vector of obserVations -and.2. And!! are the least
squares estimates of ! and ! respectively. Eliminating!! from the
'
least squares equations for! we get the usual adjusted normal equations
[XII - X'Z
[z'zrl
! .. [X' - X'Z [z'zrl zaJ 1. '
[Z. zrl Z'} for simplicity
ZIX}
or letting XI ... [XI - XI Z
.
X~1. ..
Under our assumptions of .full rank above, 1.'1. must be non-singular and
X'I! ..
we have V(E.) ..
[i. irl
. •
conditions [X'
Xr
1
a2 •
It may now be d~temined under what blocking
will be of the fom required to retain rotatability
givan that our original design was rotatable.
It will be convenient to utilize Schlatlian matrices and power
vectors here (1, 2, 30) in a manner similar to that developed in (12).
Letting::' ... [1
:x:t -~ .. :JIkl we
0
denote the derived second degree
'power vector by x,[2J defined snch that x' [21x [21 ... [x l x)2. lie will
-
-
-
--
58
also require the seoond Sohla.flian matrix H[2J whioh is defined suoh
that i f z .. EX then z[21 .. H[21x [21.
- -
-
-
In terms of these matrioes then the marginal model, ignoring blook
effeots (exoept for our definition of po)' may be written
~u .. ~[21 Is
where the elements of As are the same as tho se in
linear and interaotion terms are divided by
!
/2 to
except that. the
oompensate for the
oorresponding coefficients in
~
[21
I
2
2
.. [ l'I:x:t.,
J!2'
•.• Jlk2
I
I
~
{2
J1.'
/1f""'.~
It;.
x 2, ...
It;.
JCk!I
~
{2 J1.~
. eo].
In general we will use the subscript s to denote matrices adjusted for
power veotor oonstants.lfe will denote the moment matrix aris:Lng from
the use of power veotors then by XI X , i.e.,
ss -
x~x. • u~ i,,2 1 ~ [21.
Now for v(y) .. V(x.[21 b ) to be a fUnction of XiX" p2 alone
-
-8
--
X
~1 x[2} a2
s s
-
v(y) .. x,[2} [it
must be invariant under rotation.
R-
H :Ls
xt [2J
-
o~hogona1
- Rx,- where
Henoe, i f z
loD -k
O'~
]
.. o- .- ... --H --"-
f
-k
0
0
(~1)
by (k+1),
.
and.2k demotes a k by 1 veotor of zeros., then
l x[21 .. xt [2] it [2] [VX r l
r
ss
ss
[X'X
==
and sinoe this must hold for all
~[21
Ii [2J x[2j
we may take advantage of the
uniqueness of the inverse of a matrix and write
X'sX s == Rt [21 iii
R[21.
s s
59
Henoe
XIX - XI Z[Z'Z]-l Z'X • R,[2] X'X R[2] - R,[2] X'Z[ZIZ]-l ZIX R[2]
8 s
8
S
8 S
s·
S
•
But since the original design matrix was rotatable this implies
To find the conditions this imposes on the block moment components,
we iwill define the veciior!. [1
t
l
t
2
•••
\ ] and the generating
funotion
where
P
s
==
XI Z[Z'Z]-l Z'X •
s
s
- Rt- and impose the requirement
Then 11' we define u
==
this implies that Qmust be a function of ~'~ ==
1'1 ==
p2 only, since Q
is .invariant in the k dimensional space of t , t , .0., t under a
2
k
l
general rotation to new axes~, u , ... , '1c. Hence, since Q is a
2
polynomial of fourth degree in the t , it must be of the form
i
The conditions on the block moments can now be obtained by expanding
the quadratic form 1,[2} P .s:[2] and equating the coefficients.
s
Inverting ZIZ (25) and performing the necessary algebra, we have
1
(1- .., ..)1
l'
~
-
-
1
- ..
1 l'
.l~ -n..- n
!~-nl-X:l
e
-
-
-
-
J. -
r - - .'. - - - , -
-
...
~
m
--------
u· •• u
o
c_-~~-~-r-~~-~--,-u-~~~_·_-
1
-1 1 1°
-nm~
Nby N
and partitioning Ps into constant, quadratio,-
lin~ar
and interaction
SIlbmatrioes we have
o
Of
0
-k
o
o
o
Qk:
p • ~
s
~
.
-
o
~
P22
~
~.
~
~
:
~
~
D
Qk :
/2 P~l
f2
-
~
P21
~
~
:
~
u
~
ff.
P2 (11)
~
~
~
~
~
~
~
~
~
0
: 2 Pl1
-
: 2 P1 (11)
D O D
~
~
~
~
~
~
~
~
~
~-
~
~
~
~
~
~
~
~
D O D
o
0
-( ~)D
f2
Pi
u
~(11) D
2 pi
1(11)
D
D
2 P(11)(1l)
(~2) by (~2)
61
where the element in the i th row and j th co1um%'1 of the submatrix
1, j == 1, 2, •• " k
i, j == 1, 2, ••• k
i • 1, 2, ... k
jj' == 12, 13, ••• (k-l)k
i, j == 1, 2, .•. k
The factors
{2 and 2 arise in P s since we wUl express the moment
conditions in terms of the
of P rather than of P s •
. elements
.
After expanding (4; and equating coefficients, we fimd that ten con-
ditions must be satisfied. i f rotatable variance contours are to be
retained after using non-orthogonal blocld.ng with any rotatable design.
In the following, all summations are over w == 1, 2, .• " m.
i
1:1
1, 2 ..• k
i • 1, 2 .". k
4·
2
[i.]b.... [j 1,....
~~ ..
= 0,
62
:1. ... 1, 2 ••• k
-
ri2~W [j}bw
6.
~~
7.
~. U'\[~dbW •
2
8. ~
[i 1
I:
2',,~, [i~W",
_
II ,
[if'
~
""OW. O ,
-
~
9· ..&i
10. ~
[ij)bW ,
..&i!),
.
2
[1 JbW [jtlow
= _~
n"
[1j1
f.&n1
~
w
'Ow
[ijlow
..::::;
=0
rit~w ,
Il.w
,
It can be noted that no d:Lrect restrict:1.cms are placed on the
:tnd:tvidllal moment components
(i~w' [i2~w ad
[:I j}bw by these tem
cond:1.tions although they may be implied unless m is large.
However,
it is reasonable in many instances to use blecks lilieh are .first order
1'Gtatable,
!.:!.:.,
f, i
.
blocks f0r which
2)
--Ow
• c0nstant f0ri
[i}
_ bw
:8
0
[lj}bW
:8
0
:=
1, 2, ... k
Substituting these restrletioJils in the ten general c01'1di:tions, we see
that whenever blacks are first order rotatable the use of :BOD-orthogonal
blocking will not disturb the Ntatability of an already rotatable design.
10.'.2 General Solution
Variances - :Blocks
Fir~
!!! Revessioa
Order Rotatable.
Ooefficients
!!!! Their
From the preceding secti&:B
we .:ve the general form. of the left hand side of the adjusted Dormal
63
o
•
The r.t gilt hand side of the equation
equations as XIX. XIX ... P.
I'Z • X'Z- z[z'zfl ' Z'Z is 'easily found
ZrZ'Zl-l Z from 10.3.1)
..
tom
of
to be
{oy}
_
(utilizing the general
..
••
•
,{i:f.y}
·. . . . . . . .
-Nl [ 1
m
i2
bw
Q;lCllO
Yw+ A. 2 {Oy}
e
••
•
X'z. •
o
••
- -•• -- - --- - -- - .. ~ ~
-~
•'
m
{ijy} .. ~ [ij3m,
~ ~
Yw
.•
o
,
where
Yw is
the mean of aJ.l observatio~s in block w~ and we use the
notation
{oy}
N
II!
~ Yu' {~}
u~
N
Ili
~ xiuYu '
u~
N
{ijy}
II!
~ xiuxjuYu •
u";1
Since each subr1at'rix of a sill'lpleJe-SU1l1 design (D ) 1B iDiividua.l1y'
s
first order rotatable, we only require the general solu:tion of these
nomal equations when [i Jm,
..
[ij 3m,
III
0 and [i2
1aw is
constant for all
i, in order to provide solutions for a:xry au'bmatrix blocking schemeS'
2 <. m<. k.
This greatly amplifies the solution sinoethe 8lemc:mts of
2
P22 all become [~ [i ~ In" .. A.~ IN] and the other elements of P
•
vanish. Further, X'z. differs from Xlz. only by' the constant correction
factor
subtra~ted
from the k elements in the quadratic portion 1.80$
~
2
..
[N ~ [1 ]bw yw .. A. 2 {oy
}J.
Hence only the estimates of the qaadratie
(b:f.:L)ll1d correlated b 0 coefficients will be affected by this particular
kind of non-orthogonality.
The estimates of the linear (b ) and inter-
i
...c;tion (b ij ) coe.tficients a:roe unaffectedsinoe they are in fact orthogonal
to blocks, so that the general formulas given in (12) apply.
The general form of the information matrix when blooks a:roe first
order rotatable, in terms of unstandardized units, is given by
o c
c
o
-
-
~ -
-
-
~ -. -
-
-
~ 0
l-I<:a °12 0 11 ••• °12
I
I
°12
...
D
•••
8
--
.~- ~
o
-8- -
-
--
o
o
n
... ell
o
-
o
I
o
••• 0
·0- -
12
0 11
o
o
lJ
1 oiiiik
I
a
D
-- - ---D- - -1- - - -~
~
0
0
a
LI
~
0
I
k
0
I
- - -, -- - - - -- - - - - -- - o
•
8
0
I
~
~I
0
a
.
a
a
U
I·
where
011 • A..( [(k+l)
44 -
1:W
[ i2
°12 == .A..( [1'1 ~ f!.w
A~l
= 2\
4-
[(lct2) 71.
(k-l) 1'1 ~
[i2 )2
f!.wbw 1 ,
- 441 ,
k N~
1.. I
a
I
•
.
I~
[i2~
l\z. 1,
and 0 represents a null submatrix of suitable size.
44 (J;l)
2
65
Since under orthogonal bloc1d:ng
[i2~w· ~
VN
it is readily seen that in this case the information matrix collapses to
the usual standard form (12) for a rotatable design.
The estimates of the regression coeffioients may be written as
..
1
b o • tr [ {oy:; .. 2A..
?1.4~
bii • )("1 A.. [ {tiT) A:.,l +
bi •
k
(f
.
{iiy] .. k N
(N ~
2
[i (
"If. -
~ [ i21bW Yw)l
~
'I.) l' {iiT} - 2\N) [i2V.J.
(~N)-l { iY} ,
b ij • (A4Nr1 {ijy}
0
Their variances and covarianoes
i:n.t'ormation matrix.
are obtained easily from the
The varianoe of an estimated response may be
written as
-
V(;.) • a!Jf'lA.. {
- (k 1/ ~
2(1<>2)A~ -
[i2 )2
"Ifbw
+
2k
V1(~ [i2'
..;. - {) + 2\>;1 [(1<>2) \
2~)J ,,2 +
[(1<>1) \
.
[i2~
- (k-1) 1/ ~ "If wJ
2
A,
The varianoe function defined in (12) as V(p) • (N/a ) V(y) for
standardized
variables,~, ~ ~u
u
fill
H, becomes
,,4 }.
66
To gain an insight into the effect of departures from orthogonalitY'
on the va:r1a.nces of the eBt1mates.9 we maY' define a "non-orthogonalitY'
parameter Vi (11
ill'
) as
l1iw
2
[1 Jbw
~
~
lIB
'"
w .. 1, 2 '.0 III
i .. 1, 2 .. k
r
0
For first order orthogonal blocking l1iw is constant :tor all i in a
block and we may therefore dispense with the su.bscript i and use Aw as
the blook parameter.
Since
we may write
.
(1<>1) \
V(b ii )
- (k-1)
- k
4
(k+2) A4 '" k
~ (II]~ + 1)J
~
~ N ~n;
2
V(bo ) ..
11[(1<>2) \
~
If we let 9 .. ~- , ( 0
ohal.lge o:t
~ (II]~ + 1)
- 2
21.. 11[(1<>2) \
o:t
2
- k
-
< Q)
~ (II]~ + 1)J
.
.
.
and di.f:terentiate to find the rate
~~
V(bii ), :tor example, with respect to Gl we. have
2>V(bi :l.)
c)Q.
.
llB[(k+2)A4
co
k
~
~-(N
.
9'+ l ) r
Henoe, since the slope :l.s strictlY' positive we have that V(b:l.i ) is
m:tnimum at Q .. 0 (i.e., when the blooking is orthogonal) and is
-
monotonicallY' increasing as
~V~o)
CJQ
.. [(k'i-2)A4
Q
increases. Similarly
~~
k ~
co
(N
Q
+ 1) 12
67
(The slopes must remain finite under our original assumption that the
model is ot full rank since the denominator is the same as that of the
elements of the info:rmation matrix.)
Aw
The magnitudes of the
tor the standard solution submatrix blocks
are listed below when center points have been added to approach uniform
variance.
These va.lues will serve to indicate the magnitude of the
parameters s:Lnoe the speo1:f:Lc solution
Aw
are of the same order.
Table 8. Non-orthogonality Parameters for submatr:Lx blooks
k
Al
A2
A3
AlL.
A
2
A7
A6
A8
3
-.0071
.0142
-.0071
4
.0012
-.0012
-.0012
.0012
5
.0021
.0041
-.0124
.0041
.0021
6
-.0012
-.0021
.0032
.0032
-.0021
-.0012
7
.0012
.0003
-.0018
.0006
-.0018
.0003
.0012
8
.0004
.0002
.0003
-.0009
-.0009
.0003
.0002
The tabled values also Ulustrate the properties of the
-
evident :trom the formulas, i.e.,
and ;for s;ymmetric blooks,
Aw
.0004
that are
68
11.0 A. CONVENIENT REDUCED DESIGN FOR k • 7
The design derived from the basis vector, 'Y3 for the seven factor
design in Section 8.2, has several interesting features which will be
discussed here.
Since it requires but 56 points (plus center points) to
estimate the 36 coefficients of a seven factor second degree polynomial,
it is extreme~ efficient.
The comparable central oomposite design (12)
requires 78 points (plus oenter points) •
11.1 Construction
The vector of radius multipliers which defines this design is
~.
(0
1
0
0
0
1
0)'
and thus utilizes the points specified by the matrices D and D on~.
2
6
In seven dimensions it is possible to find a matrix D , giving the'
l
coordinates of a regular simplex, which involves only the two levels -1
Consequently D and D will only involve three
2
6
This design is ,therefore a desirable one from the stand-
and +1, for each factor.
factor levels.
point of the experimenter who, as a result of physical or financial
problems, is often forced to keep the number of different levels of each
factor to a minimum.
orthogonal blocks.
It will also be recalled that D and D6 form
2
69
The 8 x 8 matrix [,1
~1
which can be used to generate this design
is
1
1
1
1
1
1
1
·1
1
1
1
-1
1
";1
-1
-1
1
1
-1
1
-1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
-1
1
1
-1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
1
1
1
-1
Its squared vector length is eight, as required, and all rows and
columns are Qrthogonal.
The denved matrices ~ :02 and ~ :0 6 are shown in Table 9. Since
mult:i.plicati. on by a constant is permissible, we will define our derived
design matr:ix :0 therefore as
T\_
JJ-
70
Table 9· Seven factor second order rotatable design
in three levels
1
1
~ D2
~ D6
0
0
0
0
0
0
1
-1
-1 -1
'0
1
0
1
0
0
0
-1
-1
-1
0
0
0
0
1
1
0
0
0
-1
0
0
0
0
1
0
0
1
-1 -1
0
0
0
1
1
0
0
0
-1
-1
1
1
0
0
1
0
0
0
-1 -1
1
1
1
0
0
0
0
0
-1
-1
1
1
1
1
0
0
1
0
1
0
0
0
0
1
0
0
0
0
·-1
-1
-1
0
0
0
1
0
-1
0
0
0
1
0
1
0
0
0
-1
-1
0
0
-1
0
0
-1
0
1
-1
-1
-1
0
0
1
0
-1
0
0
0
0
-1
-1
0
0
0
1
0
0
0
-1
1
0
0
-1
0
0
1
1
-1
0
-1
0
0
0
-1
1
0
0
0
0
1
0
0
-1
0
0
-1
0
0
-1
-1
1
0
0
0
0
1
0
0
0
0
0
-1
-1
-1
-1
0
0
1
0
-1
0
0
0
-1
·1
0
0
-1
-1
-1
0
0
0
-1
-1
0
0
-1
-1
0
1
0
0
0
-1
..1
-1
-1
-1
-1
0
0
0
0
-1
-1
-1
0
0
-1
-1
-1
0
0
1
1
0
0
0
1
1
0
1
0
0
1
0
-1
1
1
0
-1
1
0
0
1
0
0
0
0
0
1
0
1
1
-1
1
0
1
-1
0
0
1
0
0
1
0
0
-1
0
1
0
0
0
0
1
1
-1
0
0
0
1
1
0
0
0
1
1
0
1
0
0
1
0
-1
-1
0
1
0
0
1
1
0
0
-1
0
-1
0
0
0
1
-1
0
1
0
0
0
1
0
-1
0
0
1
0
-1
0
1
0
0
0
0
0
0
1
0
0
0
-1
1
0
0
0
-1
1
1
0
0
-1
1
0
0
0
0
1
-1
0
-1
0
1
0
0
1
0
i
0
-1
0
0
-1
0
1
0
0
0
1
1
0
-1
0
0
0
0
0
71
Obtaining
)..t
either directly or, as shown below, v:La the general
formulas, we have
2
[i j 2J
.4
• )..4·
~ a~ (~:2)
• "2..
...:::J as
The singular value
""
r ·t
2
2 (n-2) n
s-l 'R
="13
k ~ 2 equals ~ also and hence this design
requires center points in order to make it possible to estimate all
coefficients separately.
The singularity is readily detectable also by
noting that all design points lie on the hypershere of radius
!3.
11.2 Projection Into Lesser Dimensionality
Orthogonal projections of this des,t gn into lesser dimensionality
will also proVide second order rotatable designs.
Pro jections parallel
to one or more design axes provide simple, easily obtained designs.
Taking as an example the pro jections of this type, into three spaoe
we .find that two distinct designs result.
The first is typif'iedby the
projection on the axes of ool'lm'lns 1, 2 and
4 of
D (parallel to axes
3, 5,6, 7) and turns out to be precisely the design obtained in
Section 9.
In this design eight replicates of a 2D2 were taken resulting
in the cube with vertices at all pernmtations of (± 1 ± 1 ± 1) and eight
replioates on each of the faces at (± 1 0 0), (0 ± 1 0) and (0 0 ± 1).
The second projection is typified by taking the coordinates of
columns 1, 2 and. 3 (projection parallel to axes
4, 5, 6,
obtain from a 2D2 the vertices of a cubootahedron,
!..:!.:.,
7).
Here we
the points at
72
the midpoints of the edges of the cube obtained above. Inside this
:figur~,
with vertices at the midpoints of the faces of the cube, are two
!
replicates of an octahedron, and" finaJ.ly, four center points.
These
twenty-eight points from D2 superimpose on those from D6 and hence only
one set is required. A three level second order rotatable design is thus
obtained requiring twenty-eight points.
inherits the same value of
'At but the
.
The projection of the design
moment matrix is no longer singular
.
since k has changed.
'At • ~
I:
;2 =i·
.778
.&>0
11.3 Relation to 37 Design
It is interesting to note that this seven factor design matrix
D I:
1
2 D6
is a piece (or improper fractional replicate, since it is a ~76) of a
.
3
37 design which, after adding sui'ficient center points, gives estimates
of all second order terms. Its projections into lesser dimensionality
will yield 3k designs (k < 7) with similar properties i f orthogonal
projections parallel to sets of design a.xesare used. The matrices D
3
7
and D give pieces of a 4 design (each is a ~ replicate) since the
5
. 4 .
values 3, 1, ...1, ...3 are generated. In this case, however, we have shown
that second order rotatability cannot be achieved without adding D:t.. and
D to the matrix. D.) and D by themselves, however, have orthogonal
7
5
columns and are, therefore, useful designs in their own right.
73
Future work may show this to be a .fruitful way of obtaining useful
pieces of factorial designs.
74
12.0 SUMMA.RY
It has been shown that second order rotatable designs in k factors
can be generated by using the k + 1 rows of any minimum k factor first
.order rotatable design matrix as a generating set. The required .first
order design matrix is easily obtained by taking any matrix with k
orthogonal columns and n • k + 1 rows, normalizing the length of the
column vectors to
of each column.
rn,
and removing the column means from the elements
calling this matrix
D:L'
the additional matrices
D2, 1)3' ••• , Ds ' • eo, Dk can be generated where each Ds is an (~) by k
matrix whose row vectors consist of all possible sums of the rows of D:L
taken s at a time. Defining the N by k design matrix :0, (N = 2n - 2) as
~31.
------
D=
---.
a D
s s
--..--..._it was shown that by suitable choice of the radius multipliers
~,
a 2, •• 0'
~,
the matrix :0 could be made to satisfy the moment con-
ditions for second order rotatability.
A solution for the radius multipliers, called the standard solution,
was found which holds generally for all k and requires that
a
s
= (n-2) -1/4
&-01
'
s • 1, 2, .•• k.
Designs obtained from the standard solution, however, all require 2m - 2
points plus any added center points and hence for most applicatioms the se
75
designs become excessively large as k increases above .five. Add:Ltional
designs, called reduced designs, were then found and tabulated for
k = 5, 6, 7, 8.
These were more conservative in the number of points
required since some of the radius :nmltipliers were zero allowing the
corresponding submatrices to be omitted from the design matrix.
General
expressions for the radius multipliers of reduced designs were not given,
but a s1mple procedure was outlined which oould be used i f designs in
higher dimensions were desired.
for 5 ~ k
~
A means of obtaining additional designs
8 was also provided.
It was found that designs generated by this method did not satisfy
the moment conditions for third order rotatability for k ~ 8 •
.A. means of replicating submatrices arbitrary numbers of times, with-
out destroying rotatabllity, was given.
This provides an alternate means
of obtaining an estimate of experimental error in the event that center
point replication is not desired.
Blocking procedures, both orthogonal and nearly orthogonal, were
provided for these designs. In connection with the latter blocking
schemes, a general theorem was proved showing the block-moment conditions
necessary to retain second order rotatability when the regression
coefficients are estimated by least squares taking into account the Dono
orthogonal block effects. It was shown that if blocks were formed such
that each was itself a first order rotatable design, this would be
sufficient to intfure rotatability.
This implied that any submatrix
asDs of the simplex-sum designs could be used as a block.
General
formulas for the regression coefficients and their variances were found
under general conditions of non-orthogonal first order rotata.ble
blocking.
76
An extremely effioient seoond order rotatable simplex-sum. design for
seven factors was discussed as an illustration requiring only :t':Lfty-s:1.:x:
points and three levels of eaoh factor ° It was shown that the projeotions
of this design produced useful three level designs in lower dimensionalityIn order to obtain general expressions for the- moments of simplex-sum
designs, the moments of each submatr:1.:x: were found as a .function of the
moments of Dlo
elements of'
a
Since the rows of Dl can be regarded as the 1'1 veotor
finite k-variate population nth orthogonal columns, it is
readily seen that the expressions found are in essence sampling moments
and as such may find wider applioation than considered here.
The voluminous algebra involved in deriving these moments was s1m-
plified considerably by using generalized bracket· notation'. In order to
extend the use of this tool to multiva.riate populations it was demonstrated in Appendix A. that the
S8lIe
rules used for taking averages or
-
expectations in the univariate case apply for the multivariate
generalizations.
Additional work is indicated in the JClirection of finding third
order rotatable designs by methods similar to those described here ° A
study of the sixth order moments leads one to suspect that third ord~'
conditions can be met by introducing additional flexLbility into 'the
method by some means.
The introduction of :f\u:other submatrioes by
additional vector summing is.a possible candidate.
In the interest of completeness, the der!vation of the missing fifth
and sixth order moments, which were not required here, would prove useful
in sampling applications as well as in a further investigation of third
order rotatabilityo
77
One other avenue of investigation suggested by this dissertation
was the possibility of generating usefUl portions of faotorial designs
by simple:xi-SUM or related prooedures.
As discussed in the final ohapter,
the fifty-six point design for seven faotors oan be oonsidered as a pieoe
of a 37 faotoioial. Although not the usual sort of traotion it is
extremely effioient in est:Lmating all seoond degree ooeffioients atter
adding oenter points to the basio design.
along these lines are discussed in (7).
Some additional investigations
78
LIST OF REFERENCES
1. Aitken, A. C. 1949.
London.
2.
Determhla.nts and Matrioes.
Oliver and Boyd,
Aitken, A. C. 19490 On the Wishart distribution in statistios.
Biometrika 36~59-62.
3. Anderson, Ro L. 1953. Recent advances in finding best operating
conditions. J. Amer. Stat. Assn. 48:789-798.
e
4.
Bose, R. Co and Draper, N. R. 1958. Rotatable designs of seoond
and third order hl' three or more d1mensions. Inst. of Stat.
Mimeo Series No. 197. University of North Carolina, Chapel
Hill.
5.
Box, G. E. P .1952. Multifactor designs of first order.
39:49-57.
6.
Box, G. E. P. 19540 The exploration and exploitation of response
surfaces 8 some general oonsiderations and examples. Biometrics
10&16-600
Biometrika
7. Box, G. E. P. and Behnken, D. W. 1958. A olass of three level
second order designs for surfaQe fitting 0 Statistical
Teohniques Researc'h Group Technioal Report, No. 26.
Prinoeton University, Princeton, New Jersey.
8.
Box, G. E. P. and Draper, N. R. 1958. A basis for the selection
of a response .surfaoe design. Statistical Techniques Researoh
Group Technical Report No. 2.3. Princeton University, Princeton,
New Jersey.
9.
Box, G.E. P., Hader, R. J. and Hunter, J. S. 1954. The effect
of inadequate models in surfaoe fitting. !nst. of Stat.
Mimeo lSeries No. 91. North Carolina State College, Raleigh.
10. Box, G. ·E. P. and Hunter, J. S. 1954. A confidence region for
the solution of a set of simu!t.aneous equations with an
app1ioation to experimental design. Biometrika 41& 190-199.
11.
Box, G. E. P. and Hunter, J. S. 19540 Multifao.1;orexperimental·designs •
Inst" of Stat. Mimeo Series No. 92. North Carolina State
C6l1e ge, Raleigh"
12.
Box, Go E. P. and Hunter, J. S. 1957. Multi-factor e::x;perimental
designs for exploring .response surfaces. Ann. Math. Stat.
28:195-241.
13.
Box, G. E. P. and Wilson, K. B. 1951. On the experimental
attainment of optimum conditions. J. Roy. Stat. Soc. Sera B
13&1..45.
79
140
Box, Go E. P. and Youle, P. V. 1955. The exploration and
e:x;ploitation of response surfacest an example of the l:U1k
between the fitted surface and the basic mechanism of the
system. Biometrics 11:287...323.
15.
Carter, Ro L. 1957. New designs for the exploration of response
surfaces. Inst. of Stat. Mimeo Series No. 172. University
of North Carolina, Chapel Rill.
16.
David, F. N. and Kendall, M. G. 1949. Tables of symmetric
functions, Part I. Biometrika 36:431-449.
17.
DeBaun, R. M. 1956. Block effects in the detennination of
optimum conditions. Biometrics 12:20-22.
18.
DeBaun, R. M. 1958. .An experimental design for three factors at
three levels
Nature 181g 209-210.
0
19.
DeBaun, R. M. 1959. Response surface designs for three factors
at three levels. Technometrics 19 1-80
20 0 De La. Garza., A. 1954. Spaoing of information in polynomial
regression. Ann. Math. Stat. 25~123-130.
21.
Draper, N. R. 1958. A third order rotatable design in four
dimensions 0 Insto of Stat. Mimeo Series No. 198.
Universi ty of North Carolina, Chapel Hill.
22 0 Friedman, M.· and Savage, L.J. 1947 0 Planning experiments seeking
maxima.. Techniques of Statistioal Analysis, (Eisenhart,
Hastey and Wallis, EeL) MoGraw Hill Book Co., In c ., New York 0
23.
Gardiner, D. A., Granda.ge, A. H. E. and Hader, R. J. 1956.
Some third order rotatable designs. Inst. of .Stat. Mimeo
Series No. 149. North Carolina State. Colle ge, Raleigh.
24.
Rotelling, H. 1941. Experimental determination of the ma:ximum
of a function. Ann. Math. Stat. 12g20=45.
25 • RoY', S. N. and Sarhan, A. E. 1956. On inverting e. class of
.
patterned matrioes. Biometrika 43g 227-231.
26.
Tukey, J. W. 1950. Some sampling simplified.
Assn. 45g501-519o
J • .&ner. Stat.
27.
Tukey, J. W. 1956. Keeping moment-like sampling computations
simple. Anno Math. Stat. 27g37-540
28.
van del' Vaart, Ho R. 19580 Some results on the probability
distribution of the latent roots of a symmetric matrix of
continuously distributed elements, and some applications to
the theory of response surface estimation. Inst. of Stat.
Mimeo Series No. 189, North Carolina State College, Raleigh.
80
L. 1958. Intersection region oonfidenoe prooedures
with an applioation to the looation of the maximum in
quadratic regression. Ann. Math. Stat. 29:455-475.
2~ .Wallaoe, JJ.
30. Wedderburn, J. H. M. 1934. Leotures on Matrioes. Amer. Math.
Soo. Colloquium Publioation Vol. 17.
31. Wishart, J. 1952. Moment ooeffioients of the k-statistics in
samples from a finite population. Biometrika 39::1-13.
APPENDIX
82
APPDDIX A
MULTIVARIATE BRACKETS
Following Mey's notation (26) let us define expressions called
Itbrackets" as follows for the scalar (univariate) case:
where the summation takes place over unequal indices and the denominator
consists of the total nmnber of terms in the numerator.
These expressions
are ftinherited on the averagelt which is to say that their average or
e:xpectation over all samples of size s is equal to the same function of
the n population elements.
In extending this notation to a bivariate population [x y] the
bracket (l 2) for example, can take on several meanings. We shall,
therefore, change our notation slightly such that
83
~
==
~
r xi"j'11'2
(x, xy)
These brackets will be the bivariate equivalent of the univariate (l 2)
bracket. An obvious extension of this principle yields multivariate
'braokets of any desired order.
To demonstrate that these braokets are also inherited on the
average we will first oonsider the bivariate o&se.
Let zi
==
xi + Yi'
then
s
and Ave
r(
Z·)
2
2 +
1
] == ';
s
2
or
13-1
Ave t\ z2) + S
Ave <z,
Z
8-' ( z,
J • S'1 ( Z 20) '+ 7
But
Jif.• ~ ("i :
----{z2-
==
(x,x') ,
Yi)2 •
~xi + ~Y~ + 2 ~XiYi
+ (y ,Y ), + 2 <- x,y ) , •
Therefore
2
Ave [( z) ]
==
1
2
S'(
(x )'
2
(
+ (Y ), + 2( xy ),) +
+
(Y,Y)' + 2( x, Y) ,) •
8;1) ((x, x ) ,
z )'.
84
Alternatively we may expand
<z ) 2
before taking the avera.ge over all
samples
+ 2
<x,y »)
+ (S;l) (Ave [( x,x ) ] + Ave [( y,y )] + 2 Ave [( x,y ) ]) •
Recognizing Ave
[(:xy '»)
..
<:xy) I since we can redefine xiYi as a
new popula.tion whose elements are ui .. xiYi and use the univariate
property of the brackets, we can w.ri te
+ (Y,Y)'.+ 2 Ave [(x,y»).
Equating terms with the previously derived expression for
Ave
[< z2 ») we see
that
A.ve [(x,y)] .. (x,y) I.
By a similar series of operations we can begin with (z)3
am
take
the average before and after expanding in terms of x and y. We will then
have upon equating terms
Ave [(x,x,y)J+ Ave [(x,y,y)] .. (x,x,y)1 + (x,y,y)t
or
Ave [(x,x,y')] -(x,x,y)t + Ave [(x,y,y)] - (x,y,y)I .. O.
Since this equation holds for all values of x and y consider the
two populations of n values
85
(li
iJ •
(~ z)
(.!:L Zl)
•
(2.!:L Zl) •
Substituting in the above for each
Ave
[< XJ.,:x:l'Y1 )
J-
(XJ.'XJ.,Yl )
+ Ave [( XJ.'Yl'Y1 ) ]
f
- (Xl'Yl'Yl)
I
•
0
and
If we define the vector J!. [u u JI where
l 2
we can then consider these equations as two homogeneous equations in two
unknowns
r: :J [~] ·
Clearly, only the trivial solution, J! •
coefficient matrix is non-singular
0
0
2" is possible since the
Therefore we can write
86
This procedure can be extended step by step indefinitely for (z)"<
since the same function of the zits is obtained whether the average is
taken before or after expanding in terms of x and y.
As a consequence,
upon equating terms all will cancel except those (..<-1) terms of order ..<
which equate in total and must then be proven equal term by term.
.A
procedure corresponding to that just discussed for the case ..< • 2 will
lead to a set of homogeneous equations .AJ!. • Q with a non-singular
coefficient matrix. We can therefore say in general that the bivariate
....
brackets are inherited on the average.
To extend to trivariate brackets we let zi • Wi + Xi + Yi and
proceed again in stepwise fashion beginning with (z >3 and proving
Ave [( w,x,Y
>J:oI
(w,y,z
>'.
It is evident therefore that multivariate
brackets in general are inherited on the average since we can proceed in
this manner to any order and any number of variables.
87
APPENDIX B
DERIVATION OF GENERAL MOMENT FORMULAS
B.l Introduction
Using this convenient property of the bracket expressions it is
possible to derive the general moment formulas for D • In finding the
s
sampling -moments of- -means of samples of s drawn from a univariate
population of n we seek
If we consider the n elements in the i-th column of D as the population
l
then the same column in Ds contains all possible sums of s from this
population.
Therefore
• N-ls.,«:) Ave [( x) .,(]
where Ave [(x).,(] must be multiplied by s.,( because [i.,(]s inVolves
SUlTlS,
not means, and by (:) to bring the average value up to the total moment
of all (:) terms.
We may write in general then
where the general row vector of D is considered as the
l
(~u x 2u •••
Xm)
and Ave
[(:x:t)
-<J.
( x2 )
.,(2
k~variate
vector
- .,(k
••• (~>
represents its
]
sampling moments. We denote the order of the moment by .,(. ~.,(i.
88
The bracket notation previously adopted therefore is conveniently
usable in the same context, where
<xi>
denotes summation over an
arbitrary sample of s rows and (xi>' sums over the entire column of
n rows.
Tb find [i1s we proceed as follows:
Ave [( xi) ]
= <xi >' =
~ xiu
=
0,
n
[i]• • O.
B.2 Second Order Moments
Similarly,
89
In all summations that follow we shall sum from 1 to n over eaoh of
the seoond subsoripts, (u, v, t, eto.) unless otherwise indioated.
[ij]s
III
o.
The final step of expanding the braokets into single index summations,
1.:.!., in terms of the moments of Dl' oan be accomplished most easily for
the more complex summations whioh oocur in the higher moments by using
tables of symmetric funotions (16).
These tables provide fonnulas for
the univariate summations only but are extremely helpful for writing out
their multivariate generalizations.
Illustrations of this will appear
in the derivations whioh follow.
Ave [( Xi
>2 J
~l+~2+···+xis2
III
Ave [(
s
III
2 ). + !:!(x x ),
1.(x
S
1
s
i' i
•
) )
90
1
. -n-1'
-
2.
Ave [ ( xi) ] .
s1 - ...!.::Js (n-l •
n-s
s (n-1) ,
2
-1 2 nl
.Jl:!-1 n-2
[i ]s • N s sl(n-s)Ts(n-1)· N (s-2)n.
Or, in the more general case where ~ X~u has not been standardized to
equal n,
B.3 Third Order Moments
The general third order moment [ijkls can be derived in the
following manner
91
Utilizing tables of symmetric functions (16) to expand the brackets
N
• - n(n-l) [ijkJl ,
2N
• n(n-l) (n-2) [ijk]l'
•
N(n-s)(n-2s) [Ojk]
s 2n(n-l) (n-2) ~ 1
0
92
Substituting this result in the equation for [ijk]s
In the preceding derivation we have implicitly assumed that s (and
hence n) is strictly grea.ter than 2 since (Xi'Xj'~) does not exist
for s • 2.
However it can be seen that after taking averages "the 'unwanted
tem involving (Xi'Xj'~) t will vanish if' we let s • 2 (or
its coefficient (s-l) (s-2) vanishes provided s < nand n
> 2.
s • 1)
since
In general
we will follow this procedure where the eJlPansion in terms of brackets
will assume s ~ -< (implying n ~ c.() in order to obtain the most general
expression for the moment
0
The same argwnent hoids however, mutatis
mutandis, so that the formulas are valid for s • 1, 2,
n
->-<.
0 •• ,
n - 1 and
When s • n it is obvious that all moments vanish by the
assumption of zero means
,
0
'
The only moments required here for n <;;{ are
the fourth order moments when n • 3 and these are treated separately.
To find [ij2} we follow the preceding derivation but let k • j.
s
Simplifying and substituting in the equation for
93
Similarly for [i3 ]s let k • j .. i.
Substituting this result we then have
B.4 Fourth Order Moments
The fourth order moments may be derived in a similar fashion by
first obtaining the general result [ijk1]s.
[ijkl]S ..
Ave [( xi) (Xj )
·e
(~>
( XJ. ) J
N-1s4(~)
Ave [( xi
.. ~[( XiXj~XJ.)t
s
>(Xj
+ (s-l) (
)
(~)
<XJ.) ],
(Xi'Xj~XJ.
>t-
94
Using tables of symmetric functions to expand these brackets and
omitting terms containing zero elements of the .type ~ x
( Xi' x j x.Kx.L >'
x ~~
•. ~Xiun(n-l)
ju
u.
-1\1
n(n-l)
iu
[ijkl]l
(all permutations of i, j, k, 1 yield an identical result),
(for all permutations of subscripts),
D
nCn-ifCn-2) [ijkl]l
(for all permutations of subscripts) ,
D
- 6N
nCn-l) (n-2)Cn-3) [ijkl]lo
Sums of the type
~XiUX;U~~:lCJ..u were written down, even though
equal to zero in this case, because for some other fourth order moments
they will be
non-zer~.
95
• L[ (n-s ) fn 2 +
s3
2
n - 6sn + 6s ) ][ijkl]
n n-l'(n-2'(n-3'
1
.. N(n1!,}[(n....2s) (n-3s)- n(s-111[ijkl]
'"'"S3n (n-l)(n-2)(n-3)1
0
Upon substitution in. the . equation.£or [ijkl]s we have
In order to obtain [ijk 2 ]s let 1
2
[ijk JS
•
==
N-ls4(~)
k in the foregoing derivation.
Ave [( xi> (X j >
<~) 2]
0
Upon substituting for the braokets in Ave [( xi >(X j ). (~ ) 2] it is
olear that the same ooeffioients will be obtained as in
Ave [ (xi ) ( x j ) ( ~)
the e:xpressions
<X:L )]
0
No other term will .be introduoed sinoe
~XiUXjU ~~ • ~xiu~ ~XjU~
• 0 as before.
Therefore referring to the expression for [ijkl]s we oan write
To obtain [ij3] we let k • 1 • j in the first derivation.
s
.,
Then
Inspection of the- expanded brackets listed for the [ijkl]s moment,
keeping in mind the revised subscripts, again shows that the coeffioients
96
will be unchanged.
~ xiuxju ~ X~u • 0, hence no
The expression
Utilizing the previous algebra we can write
additional term appears.
Another fourth-order moment is [i2 j 2]s which can be obtained by
letting k = i and 1
.
=j
in the derivation of [ijk1]·.
s
2 2
Again the same coefficient will be obtained for the [i j ]1 term
in Ave [(
X~ ) ( X~ ) ] as was obtained in the three previous derivations
for the corresponding terms.
However a new tem will be added to the
moment· due to the failure of the following terms to go to zero in the
expansion of the brackets.
!?g?ression
Added Term
~xiu~X~u
n (n-1)
n
•
n:i'
The additional term to be a.dded to Ave [( xi) 2 ( x ) 2] therefore
j
follows as
97
Therefore
2
The lastn 2 in the above expression can be replaced b.Y N2[i ]1[j2]1
for the more general case where the column vectors have not been
standardized.
Substituting Ave [:( xi>
2
2
2 2
(Xj ) ] in the expression for [i j ]s
and simplifying we have
t
• [(n-2s Cn-?s) - nCs-l)](n-2)[i 2j 2] + (n-4)rl
[ i 2j 2].
s·
n-2 (n-3)
s-l
1
s-2 N
The final fourth order moment, [i4 ]s is obtained easily from the
2
result for [i 2j ]s
The added terms in this case become
0
kpression
The term
Added Term
(x~,xi) I appears three times
in the expansion of Ave [( Xi )
4].
This can be readily seen by letting all subscripts equal i in the expansion
of Ave [( xi) ( x j ) (~)
(XJ.l ].
In the same way we see that
98
(Xi'Xi'X~)' appears six times and (xi,xi,xi,xi )' once. The coefficient
of the additional term is therefore emctly three times that obtained for
the previous case and
Ave [(x. )4]
~
= !L3[~.::;.:;.r.~~_~~[i4]
s
1
+ 3 (n-s){s-l)(n-s-l) n
2
n(n-1Y (n-2Y (n-3)
Therefore
As in the preceding moment, the more general eJq)ression would be
obtained b,y replacing the last n2 b,y N2[i 2 ]i for the case where
~xiu rj
n.
B.5
Higher Order Moments
In order to determine whether a derived design is third order
rotatable it is necessary to have its moments through the sixth order.
However, due to the property demonstrated above, it is not necessary
to obtain all the moments of any order to determine at least some multiple
of the coefficients that are involved. For the 'purposes of this paper,
then, it was unnecessary to denve all fifth and sixth order moments; in·
fact only three of the fifth order and two of the sixth order moments
were obtained.
The derivation of these moments will not be given as the
procedure has already been outlined and the magnitude of the algebra
makes it :iJJIpractical to do so •.Sixty.,.six different multivariate brackets
are required to expand Ave [( Xi ) 2( X ) 2 (~ ) 2 ] for example.
j
These results and those derived above are summarized in.the following
table.
•
e
e
Table 1 Moment components of totals of samples of s from
. zero mean finite orthogonal populations of n, (n >..<)
0
-
fils
"0
[ij]s
.. 0
2
[i ]s
.. (n-2)[i2 ]
s-l
1
[ijk]s
.. ~~~}(n-2)[ijk]
n-2 s-l
1
[ ij 2
.. fn-2j}(n-2)[ ij 2]
n-2 s-l
1
Js
3
[i ]s
.. fn-2j}(n-2)[i3 ]
n-2 s-l
1
[ijk1J
S
.. [(n-2s)(n-~~ - n(s-l)] (n-2) [ijk1]1
(n-2 n-3)
s-l
2
[ijk ]s
.. [(n-2s}~n-3s~ - n(s-l)] (n-2) [ijk2 ]1
n-2) n-3)
s-l
[ ij3
..
1s
lJs
[(n~2s) ~n-~~ - n(s-l)] (n-2) [ ij3
. n-2 n-3)
s-l
l]1 +
[i ]s
(n-4)[i2 ] [ i ) N
s-2
1
1
.. [(n-2s) fn-inl - n(S-l)] (n-2) [i4 ] + 3 (n-4)[i2 ]2 N
1
s-2
1
Ij-2 n-3J
s-l
[ijk:Jm]
s
.. (n-2s)[(n-~)(nm) - ~S-1)] (n-2) [ijklm]
n-2 n-3H. )
s-1
1
[ ijk12] s
.. (n-2S)[(n-3s){n-4s 2 - ~S-1)] (n-2) [ijk12 ] + ~n-t»(n-4) [i"k] [12 ] N
(n-2)(n-3Hn)
s-1
1
ns-2
J 1
1
[ i 5 ]s
.. ( -2s}[(n-i)(nfts)~- ~S-1}J (n-2) [i5 ] + l.Ofn-f»(n-4) [i3 ] [i2 ] N
n
n-2 n--,)(n)
s-1
1
ns-2
1
1
[ i2
4
..
[
(n-2s)(n-3sl - n(s-l) (n.-2) [i2
(n-2) n-3)
] 8-1
h
-nfs-1tf16n2-79Sil+lln+86s2_4s-4>(n-2)[i2ik2] + ~n2+$6s2-4J{n-4)([i2j2] [k2 J + [i;2] [lJ + [j~2] [i2 ] }ll
[ "2j~2]S .. [(n-2s)(n-3s)(n-4s)~n-~~
n-2 n-3 (n- (n-5)
s-1
1
n
n-5)
s-2
1
1
1
1
1
1
J.
+ 2r n
6
[i ]S
2
2
;n-~~4s,J<+4](~~)([ij2h[ik2h+
2
[i jh[ji]1 +
[i~]1[j2kh
~
+ 2[ijk]i)ll + (S_3)[i2h[j2h[k2]1
If-
2
2
2
2
.. [(n-2s)(n-3S)(n-~Hn-rn)-n(s-1~(16n -Z9sn+lln+86s -4s-4) ](n-2)[i6 ] + 15[n +!f: - 6sn + 6s - 4](n-4)[i4 ] [i2 ] II
n-2) n-3) (n=4) n-5)
s-1
1
(n ) (n-5)
s-2
1
1
2
+ l.O[n
2
-!!If
Mn + M + 4](n-4)[i3 ]2 N + 15(n-6)[i2 ]3 w2
(n ) (n-5)
s-2
1
s-3
1
'0
'0
100
B.6 Low Values of n
The fourth order moment fo:rm.ulas obviously do not hold for n • .3.
This results from the use of terms suoh as (xi,xi'xi'xi) in the expansion
of Ave (
xi) 4] and other oomparable four subscript summations in the
other moments",
Eliminating the contributions of these brackets, and letting n • .3
we have
3! (~)([2 - 7(s-1)] [i2 2]1 + (s-l) N[i 2 ]1 [j2]1)
4
4
4
[i ]s • TI (~)([2 - 7(s-1)] [i ]1 + ,3(s-l) N[i Jl )..
2
[i j 2]S •
j
Utilizing the 'results of seotion 5.,3 it is easily shown that these,
mq be written simply as
[ij.3]l • [ij.3 J2 • 0,
2
2
[i j 2]1 • [i j 2]2 •
4
4
[i ]1 • [i ]2 •
~
f [i2 ]1 [j2]1'
[i
2
]i"
S:im.ilarly the fifth ana sixth order moment formulas do not apply
for n< 5 and n<6 respectively"
However, these formulas were not required
for such values of n in this dissertation"