One at a time plans for 2p factor sequencing designs
by James Leonard Hansen
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Mathematics
Montana State University
© Copyright by James Leonard Hansen (1974)
Abstract:
This thesis examines 2^p factorial experiments where the order of the application of the factors may be
significant. In the experiments considered, the low level of a factor is the absence of a factor, while the
high level of the factor is the presence of a factor. The effects of the factors are permanent and each
unit may be tested at least p+1 times without the test affecting the unit. The assumptions which relate
order effects are examined and a system with algebraic properties is proposed to assist the experimenter
in estimating and interpreting order effects. A design and analysis are presented which allow for the
estimation of order effects in addition to the usual main effects and interactions. The system for order
effects is used to construct one at a time plans which allow for the estimation of order effects and
factorial effects in experimental situations where the experimenter can get quick results with random
error small in comparison to the effects which are to be estimated. ONE AT A-TIME PLANS FOR 2P FACTOR SEQUENCING DESIGNS
"by
- '
James Leonard Hansen
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
DOCTOR OF PHILOSOPHY
in
Mathematics
Approved:'
Heu._,
-
MONTANA STATE UNIVERSITY
Bozeman5 Montana
June5 1974
ill
ACKNOWLEDGEMENT
The author wishes to express his gratitude to the
chairman of his graduate committee5 Dr. Kenneth J . Tiahrt5
for suggesting this thesis problem, and for his guidance
in the preparation of this thesis.
The author is also grateful to Professors Martin A.
Hamilton, Richard E. Lund, Byron L. McAllister, -Franklin S .
McFeely and William R. Taylor for serving on his graduate
committee.
iv
TABLE OF CONTENTS
CHAPTER
I.
II.
III.
IV.
V.
VI.
PAGE-
INTRODUCTION .....................................
. I
A SYSTEM FOR EXAMINING ORDER EFFECTS
5
Preliminary Considerations
...... .......
Assumptions Regarding Binary Operations In
Order Effects ...... .............................
5
12
ESTIMATION AND INTERPRETATION OF ORDER EFFECTS . . 16
Design and Model ....................... .
Analysis ................. .......... ....... .
Examples ................. ............ ......... 37
Fractional Replications of 2P FSD Designs .....
l6
28
ONE AT A. TIME PLANS FOR THE 23 F S D .............
46
Preliminary Considerations ...... ........;....
One at a Time Plans for Case I .. . . ...........
■ One at a Time Plans for Case 2 ...... '.........
Examples ................... ....... . .........i.
46
50
6l
44
65
ONE AT A TIME PLANS FOR 2P FSD WITH p > 3 .....
Tl
I n t r o d u c t i o n ....... ...................... .
An Example of a One at a Time Plan for a 2^ FSD
Example .......... .......... ................... ..
Tl
72
78
SUMMARY AND EXTENSIONS ..... ....... .
86
BIBLIOGRAPHY
APPENDIX
___ ....... . ........ .......... ..........
88
,89
V
ABSTRACT
This thesis examines 2^ factorial experiments where
the order of the application of the factors may be signifi­
cant.
In the experiments considered, the low level of a ■
factor is the absence of a factor, while the high level of ■
the factor is the presence of a factor.
The effects of the
factors are permanent and each unit may be. tested at least
p+1 times without the test affecting the unit.
The assump­
tions which relate order effects are examined and a system
with algebraic properties is proposed to assist the experi­
menter in estimating and interpreting.order effects. A
design and analysis are presented which allow for the esti­
mation of order effects in addition to the usual main ef­
fects and interactions.
The system for order effects is
used to construct one at a time plans which allow for the
estimation of order effects and factorial effects in
experimental situations where the experimenter can get
quick results with random error small in comparison to the
effects which are to be estimated.
CHAPTER I
INTRODUCTION
Factorial experiments are useful when the researcher
is investigating the effects of each of a number of factors
on the response of some variable.
Usually all factors are
•
applied simultaneously to the experimental units and the
response is recorded.
This is especially true in agri­
cultural experiments where different levels of the treat­
ments may be applied, simultaneously and the response is
observed and recorded.
This type of experimentation was
developed.by'R. A. Fisher [4] in the 1920's and early 1930's.
Factorial experimentation is very efficient because every
observation supplies information about each factor included
in the experiment.
In many industrial experiments where the factors are
environments, the factors can not be applied simultaneously,
but may be applied in any order.
For example, in testing
electrical switches or relays, one factor may be vibration
and another mechanical shock.
In this instance, the factors
must be applied sequentially and the order of the application
of the factors may be important.
R. R . -Prarie and W. J .
Zimmer treated, this problem in two papers published in 1964
and 1 9 6 8 in the Journal of the American Statistical Society.
2
The type of sequential experimental design referred to
in the Praire and Zimmer papers is related to the order of
the application of the factors.
This is different from the
definition of sequential experimental designs, where
observations are obtained in sequence in time and it is to
be decided at each point in time whether the experiment is
to be continued and possibly what treatment combination is
to be used.
Therefore, Prairie and Zimmer .termed their,
designs Factor Sequencing Designs
(FSD) to distinguish them
from Factorial Designs or from Sequential Designs.
In [9] 5 Prairie and Zimmer considered 2^ experiments
which apply to the situations where:
a)
Each unit may be tested p+1 times without the
test itself having an effect on the same unit.
b)
There can be no trend effect with successive
tests on the same unit.
c)
The high level of a factor is the presence of
a factor and- the low level is the absence of
the factor.
d)
The effects of the factors are permanent.
The experimental designs discussed in this thesis will
always be assumed to satisfy this same set of assumptions.
3
The usual 2P factorial designs require -2^'units with
one test per unit to estimate all factorial effects.
The
Factor Sequencing Designs (FSD) developed in [9] require
pI units and p+1 tests per unit.
In [9], the design and
the analysis for full FSD are presented, and in [10],
fractions .of -the full FSD experiment are presented.
The
purpose, of the FSD experiment as presented by Prairie and
Zimmer is to determine the importance of the order of
application of factors.
If order is not important, the
factorial effects are estimated in the usual way.
But if
order is important, they state that .the interpretation of
the factorial- effects is. difficult.
If the factors cannot be applied simultaneously, they
must be applied in sequence one at a time.
Many scientists
do their experimental work in single steps, and strive to
learn something from each trial or^ run.
These scientists
can react quickly to the results of individual runs ; however,
in order to achieve good results, the experimenter should
have effects which are at least three or four times as
large as his average random error per trial.
In [3], C. Daniel proposed, one at a' time plans to
produce data of greater value to the experimenter than the
sequences of one at a time trials previously used.
He
4
indicated that this type of experimentation is economical,
but may give biased estimates.
In his paper he described many of these biases as two
factor interactions, and then provided sequence's of one at
a time runs to separate main effects from these two factor
interactions and gave methods of estimating each two factor
interaction separately.
Factor Sequencing Designs were designed to estimate the
effect of the ,order of application of treatments.
But each
2p FSD requires pi units and (p+1) tests per uni t5 there­
fore, one at a time experimentation can be used to determine
as quickly as possible if the order of application of the
factors makes a difference.
The purpose of this thesis is
to present a model which will yield the same tests for
orders as those presented in [9 ] and. [1 0 ] and in addition,
will allow for the estimation of the order effects.
When
one at a time plans are.used, the proposed model will also
enable the experimenter to interpret his results if order
effects are present.
If the sequence of one at a time runs
is incomplete, the model allows for estimation of the re­
sulting biases in the order parameters.
The one at a time
plans are constructed to yield maximum information about
order effects as early as possible in the experiment.
CHAPTER II
• A.SYSTEM FOR EXAMINING ORDER EFFECTS
Preliminary Considerations
To facilitate the study of order effects, a model
analogous to the usual mixed model will be used with random
unit effects and fixed treatment effects.
The model used
to represent a response for the 2p FSD is
Yij(fV
*
m + ui + rIf1 , ... , f ^ ^ + 6 I. . .j-l + e ij
j-l)'
where
m = the general mean.
U^ = the effect of the ith unit.
' • -p
= the effect of the factors
1 I " * * * 5 j- 1
Tif.
f^,...fj_^, in any order, for j=l the symbol used
is Tj «
0^
= the order effect of applying, factors
f^,...,f ._^ in the specific order . f ...,f
e^. = the random error associated with the jth test on
the.ith unit.
For example, in the response
Y ij(fl"f2 )
m + u. +
i + ^ f 1 3 F2 .+ 6 12 + G ij
the term 0 ^ 2 indicates the order effect resulting from
applying treatments F^ and F2 in that specific order; first
6
If1 and.then fg.
Similarly O21 would indicate the order
effect resulting from applying treatments in the order f2
and then f^ •
The model will be discussed in more detail later.
The
present section is involved with developing a system to
assist is the discussion of order effects.
Throughout the
discussion,, reference will be made to properties of a binary
operation.
The properties used are the following:
Commutative Property:
A binary operation * will be commu­
tative if and only if a * b = b * a.
Associative Property: A binary operation * will be assoc­
iative if and only if ( a * b) * c = a * (b * c).
To motivate the discussion, consider the following
responses
hi(i)
Y12(A)
y IS^fV
, m + u ^ + r j ^ + O 11
m + uI
n + 1Qf
I
+ 01 + e 12
I
f2 ) - m + U1 + rIf^f0 + 612 + 613
1*2
'
f
+ 6 1 2 3 + 6 14
'l*-2*-L3
The response Y 1 1 (I) is a preliminary test before any
Y l4^fl ,f2^f3^
m + U1 +
factor has been applied.
The response Y 1 2 (T1) is the
response after one factor has been applied, and the symbol
G 1 is not subject to an order interpretation.
In the
7
response Y 13 (
f
) , e 12 could be.denoted by S 1 * e2
where the star would define an operation on the order
effects.
Hence,
and indicates the order effect of applying treatment f^
first and then treatment fg second.
Gl * Gg * G 3
Similarly,
^12 * 83
0 -123.'
■ The operation * is well-defined,- but is not commutative,
If * were commutative,
0 l2 =
* 0g
= Gg * G^
G 215
which would imply no two factor order effects were present.
Thus .the assumption of commutivity would eliminate precisely
the property of the model which the FSD is used to determine.
Associativity would imply
6 123 = 9 1 2 * 03
.
= Gi * Gg * G3
= 9 I * 9 23*
Although this concept does not contradict the basic
design assumptions as commutivity does, the expression
8
0 I * e23' 18 difficult to. interpret.
However, the assumption -'
of associativity simplifies the discussion which follows
and for this reason will be included in the definition of *.
The following definition is based on the preceding
discussion.
Definition 2 . 1 :
The associative binary operation *
relating order effects is defined by
0I * 0 2 = 0 1 2 *
The operation is extended to three or more effects by
repeated operation on the right.
For example,
0 12 * e'3 = 0 123
The following lemma is derived from the definition.
Lemma 2 . 1 :
Let P and Q, represent two permutations of
k factors, then if 6 p = 6 q , then 0 pj = 0 ^. for any factor
j not a member of the original k factors.
Proof:
If 0p = 0 q , then by Definition 2.1
"•
-
6P * 8J = 6Q * 6J
or
8Pj = 6Qj1
.
9
For example, if the two order effects
and ^ 21
are equal, i.e. 6^23 = Sg2 1 , then If factor f^ is applied.
Lemma 2.1 with P = 1 2 3 ,
Q .= 321, and j = 5 implies
6 1235 = 9 3215*
The above Definition and Lemma will be used to develop
a system which aides in the interpretation of results from
FSD experiments where order effects are not negligible and
assists in developing one at a time plans for these designs.
To motivate the discussion,, consider the following example
of a 2 U experiment.
Example 2 . 1 :
factors f f g
The experiment consists of applying three
and fg each at two. levels where the low level
indicates no treatment is applied and the high level
indicates, the treatment is applied.
Assume there are six
units available and the following experiments are performed.
Unit
I
2
3
4
Application Sequences
•flf 2
• fIf 3
f2 f l
f2f3
5
f3f l
6
f3f2
10
In the array given above, the notation f ^f2 represents an
application sequence.
The symbol T f 2 indicates the unit
has been tested three times at this stage of the experiment:
first, prior to any treatment ; second, after factor f^ was
applied; and third, after the application of factor fg.
A similar interpretation follows for application sequences
of more than two factors.
Then suppose the appropriate contrasts have been tested
and the following equivalences are determined from the
experimental data.
612
921
=
'
'
923 = ^32 ,
e31 = e13’
By applying Lemma 2.1, the-following equalities can be
derived.
9123 = 0213
923I = 9321
9132
=
'
9312
'
The above relationships are intuitive because if it is
assumed that experimental units are not affected by order
after the .applications of two factors, .the' two units should
be the same except for experimental error.
■Therefore, the
11
same response is expected from the two units after applying
the same third treatment to each.
H owev e r , t h e r e is no reason to assume a priori that
0 12 3 = e 2 3 1 or 9 1 2 3 = 0-i3 2 *
is possible that the three
sets could all be non-zero quantities, and in this instance
the interpretation would be that three factor order effects
occur, but there are no significant two factor order effects
Prairie and Zimmer, in their 1 9 6 8 paper, presented some
fractions, of the full PSD design.
Some of these fractions
involved testing only all possible two factor order com - '
binations by an. F test.
The above example is intended to
illustrate the possibility of a higher order effect, in
this case a three factor order effect, even though the
test for two factor order effects would not be significant.
12
Assrunptions Regarding Binary Operations In Order Effects
Example 2.1 indicates a need to examine the assumptions
underlying order effects.
The following discussion will
consider two situations and each one will he examined by
relating the binary operation to the types of properties
it satisfies.
Case I:
The first case considered is the situation
where it is known that the set of all order effects of k
or fewer factors is insignificant.
This prior knowledge
gives no information about the set of order effects when
more than k factors have been applied.
This is equivalent
to assuming those properties on the binary operation given
in Definition 2.1 and Lemma 2.1 for combining order effects.
A situation where'this occurs would be as in Example 2.1,
and in this instance there could possibly be three distinct
three factor order effects for the 2 , the effects related
to (I) O 12^.and 0 2 1 3 ? (2 ) e 2 3 1 and 6321 > and (3) 0 i32 and
S3 I2 , even though two factor effects are absent.
Lemma 2.1
gives partial commutativity for effects judged insignificant
using experimental data.
This means .symbols related to
two factor order effects which, have been determined to be
insignificant can be commuted only when they appear in the
left-most positions in the sequence.
13 .
•5
For example, consider a 2 U FSD and assume the
appropriate two factor order effects have been tested and.
based on these observations the experimenter assumes
6
12
O2
0 1 , O 10 ^ O 01 , and e,
0.
1 5 13 ^ w31’
23 -.*32'
Lemma-2.I indicates
the following relationships concerning three factor inter­
actions,
■6123 = 0213 and 0231 = 0321'
Based on this information the experimenter can plan the
remaining experimentation to get the most information with
the least expenditure of resources.
Using the assumptions of Case I, the experimenter can
use this system to develop relationships.among the order
effects.
Case I could be referred, to as the case where two
factor order effects judged insignificant commute only on
the left of the application sequence.
Case 2: ..The next situation discussed is one where the
symbols related to two factor order effects determined to
be insignificant can be commuted anywhere they appear -in
the complete order sequence.
This is equivalent to assuming
the binary operation is commutative for the symbols judged
insignificant as well as associative for all symbols.
The
following definition is based on the preceding discussion.
14
Definition 2 . 2 :
if 0
= 6
The binary operation * is commutative
then 0 p-j_jQ = 0PjiQ for any two sequences- of
factors P and Q.
Applying the definition with i = I, j = 2, P = 4 and.
Q = 3 5 , and if
= 0 12 j then
6 4l235 = 6 42135*
This Definition can be used to determine the sequence
of factors to be run to yield maximum information.
Por
example, consider a 2U PSD design where it has been
determined that
=
and e 2 3 = 0 3 2 5
that
^ 0 g^.
Since G ^2 ~ @21* Lemma 2.1 implies 6-^23 = 0 213*
addition,
and- Definition 2.2 implies 0 ^ 2 3 = ®132 '
Hence,
0.
'123
6,
"213
0n
132*
Similarly, Lemma 2.1 and Definition 2.2 imply
e312
e321
e231*
The two sets can be defined as .equivalence classes of
order effects.
The result is intuitive in that if there is
only an order effect related to factors A and C, i.e.
0 13 ^ ®31J •then the position of B in the treatment sequence
has no effect, only the relative positions of A and C.
15
If the experimenter can assume the conditions of Case
2 , he. can find more relationships among the order effects
than he could in Case I.
Case 2, could be referred to as
the case where two factor order effects judged insignificant
commute anywhere they appear together in the application
sequence.
Even if Definition 2.2 can not be assumed, a priori,
the experiment can be run as in Case I where the results
of Case 2 are possibilities to be tested for experimentally.
If.there.is a set of equivalence classes of order
effects, the sets can be helpful in interpreting and applying
the results of the experiment when order effects are present.
For instance>'if an FSD is being used to determine factor
effects for some industrial process, the order effect could
indicate a type of catalytic effect related to the order of
application of factors, and the desired results might be
achieved only through a specific order of application.
When implementing a production process, the. knowledge that
order effects have been separated into equivalence classes
allows for.the selection of any particular factor applica­
tion order sequence from an equivalence class on the basis
of optimization in terms of criterion such as production
time or cost.
CHAPTER III
ESTIMATION AND INTERPRETATION OF ORDER EFFECTS
Design and Model
The model used is similar to the one discussed by
Prairie and Zimmer [9]j and is the one stated in equation
2.1.
The main difference is the addition of specific
representations of order parameters and assumptions con­
cerning these parameters to allow for their estimation and
an interpretation of these order effects.
Each factor f\(i = l,...,p) has a high level (f^ has
been applied) and a low level (f^ has not been applied).
Because the effects of the factors are assumed to be perma­
nent, a unit which has received factor f^ at some point must
be considered as having the high level of that factor from
that time on.
In addition, in a complete experiment, every
unit will receive each of the p factors in some order and
will be tested (p+1 ) times with the first test occurring
before the application of any factor and each succeeding
test occurring after the application of each of the p factors
The design considered in [9] was one Where pi groups of
r units each were subjected to. exactly one of the possible
pi orders.
A notation for the model was given in Chapter II ■
and is restated here.
17.
(3.1)
Y±j
f '
■j-l; - ;m + u i + 1If
1 I 5 *•'5lJ-I
+
+ eIj
I — l^eee^^.p*
j — lj««°jP+ I
where
denotes the response from the jth
test on the i.th unit resulting from the j-I factors applied
in the application sequence f^...f
defined as they were in Chapter II.
The parameters are
In addition, it is also
assumed that
(I)
eIj ~ "ID(
(2)
U1 ~ NID( 0,a® )
(3)
There are no interactions among the 9 1s or
0,a2
e )
between the 6 1s and the -q «s.
(4)
Let A k be the set of all permutations of k
then
(3.2)
Z
IeAk
9L = °9
L
for 2 < k < p .
Example 3 .I:
If the experiment being run is a
2~> F S D 3 then the set of order effects is:
{9 1 2 5®2 1 5®1 3 5®3 1 5®2 3 5^1 2 3 3^1 3 2 5®2 1 3 5®2 3 1 5^3 1 2 5^321 ^
18
and. condition (4) would imply
612 + S21 _ ° 5
9 13 + 031
■
0 23 + Sg2 ~ Oj
and
9 123 + 9 132 + 9213 + 9231 + 0 312 + 9321
Conditions
(I) and (2) are usual assumptions for mixed
models where the population of inference is infinite.
Con­
dition (3 ) indicates an assumption of independence between
the order effects and the factor effects.
The order effects
are a special type of interaction and to assert they interact
with the factor effects, would be a redundancy.
To assume
they interact with each other would imply that orders inter­
act with orders which is also redundant because the test is
for order effects and the 0 *s as defined are order parameters
Condition (4) is imposed to provide a full rank model.
How­
ever, the assumption does not seem unreasonable because if
there is for example an order effect related to f^ and fg,
the assumption S 12 + S121' = 0 would indicate one order pro­
vides an increase to the general mean while the other yields
a decrease in the response from the general mean*-
Similar
interpretations hold for order effects for more than two
factors, some will decrease the response level and some will
, .
19
increase I t 5 but the deviations from the general mean will
add to zero.'
Using matrix notation, the model stated- in equation 3.1
may be written as
(3-3)
Y = X*(™») + gu + e
where g is the r(p+l)pl x I column vector of observations,
X* is the design matrix with r(p+l)p! rows and
I + 2^ +
P
^
pl/(p-i)l columns, p* is column vector of
1=:2
factor and order parameters with 2 ^ +
P
^
p!/(p-i)I. rows,
1=2
u is the rp! x I column vector of unit parameters, ¥ is
the r (p+l)pl x rpi matrix, whose cth column is a column of
zeros except "for ones in the [(c-1 )(p+l)+l]th' row through
the [c(p-hl) ]th row and e is. the r (p+l)pl x I column vector
of random errors.
To illustrate the model the responses from a 2V are
20
=
m
+
U1 +
Ypg(A)
-
m
+
H
2
Y 1 ^ f A , B)
=
m
+
U1
=
m
+
m
+
U 2. +
Y 2 4 (A,C,B)
=
m
+
+
m +
e 12
+
i
1I
1^A, B
+
tiI
U2
+
6 12
9 123
+
(Tl
H
-Fr
=
^A
C + '9 1 3 2
^
g 24
n A , B,,C +
+
OO
I—I
U)
Ygf(I)
+ G 11
+
Y^fA,!)^)
^l
+
Y 1 1 (I)
6 21
+
+ !^,2 , 0 +
+ =54
This model is different from the one proposed in [9] in
that Prarie and Zimmer did not include explicit parameters
for.order effects.
However, the model presented in 3.2 has
the same number of linearly independent observation vectors/
hence their result concerning the rank of.the design matrix
holds for X*.
The rank of X* is
P
(3.5)
g PlZ(P-I)I
i=0
For example, if unit parameters are ignored (the vector
P* does not include unit parameters), then for one repli- .
cation of a 2^ PSD, there are 6 units which have T^, two
21
units each for T]^
t)q 5
one unit for each of
and its
order parameter and one unit for l
H 1 Jk with the appropriate
order parameters, hence there are
3
Z 3 1 / (3-i)I = ! + 3 + 6 + 6 =
1=0
16
linearly independent column vectors in X* for a 2 .
. The argument can he extended and therefore the rank
of X* for arbitrary p is as given in Equation 3.5»
■ The rank of X*'X* is the same as the rank of X*, and
the size of the X*'X* is equal to the number of columns in
X*.
The following discussion will show that a series of
constraints' on the model parameters will lead to a new
model of full rank.
A reparameterization of the factor effects formed by
subtracting H 1 from each factor parameter and deleting the
column of zeros will, reduce the dimension of X* by one
without changing the rank.
Using condition 4, the order
effects can. be -reparameterized to form a full rank model.
The following, lemmas show that the reparameterization based
on condition 4 is sufficient for a full rank model.
Lemma -3.1;
In a 2^ FSD experiment,' there are
22
P
■Yi
1=2
p l/(p-i) I order effects.
Proof:'
For a fixed I. the number of order effects f o r '
I factors is the permutation of p factors taken I. at a
time, i.e. pl/(p-i)l.
For a 2^ FSD order.effects are
possible if i = 2,...,p.
So there are
P
Yj
1=2
pi/(p-i) I possible order effects.
This -completes the
proof of the lemma.
Lemma 3.1 implies there are
3 .
Z
1=2
31/(3-i)I = 12
order effects for the 2U FSD.
.The twelve effects were
enumerated in Example 3.1.
The following lemma will show how many constraints are
imposed by condition (4).
Lemma 3 . 2 :
For a 2^ FSD experiment condition (4)
implies
constraints are imposed on the model defined by Equation 3.1.
23
Proof: •For a fixed, set.of I factors 2 < ± < p, there
are
possible combinations of factors each of which
provides ohe constaint of the form
Z
eL = 0 .
LeA1 L
Consequently, the total number of constraints is given by
This completes the proof of the lemma.
Lemma 3.2 implies that four constraints are imposed on
the 2~> F S D e .' The four constraints were given in Example 3.1.
If the number of constraints is subtracted from the
number of order effects, the number of parameters present
after imposing the constraints is given by Lemma 3«I and
Lemma 3.2.
P
I
i=2
PV'(p-i)
The next lemma indicates the number of order parameters
to be estimated after imposing the constraints equals the
number of degrees of freedom for order.effects in the 2^
FSD design, ■
24
In a 2.^ FSD experiment there arq
Lgmtna 3 » 3 :
L
1=2
degrees of freedom for order effects.
'
Proof: 'The number of degrees of freedom is equal to
the number of linearly independent comparison's which can be
formed between order effects of the same set L of i factors.
Hence, each of the
combinations can be permuted, in il
ways which implies there are (il-l) independent comparisons
which can be formed and for fixed i there are ^?J^ii-l
degrees of freedom.
Since 2 < i < p, the total degrees of
freedom for order, effects in a 2p FSD design is
P
?YiJ-lJ.
This completes the proof of the lemma.
i=2
Lemma 3.3 is illustrated using the o3
2 FSD experiment
described in Example 3.1.
of two factors.
There are ('3'' = 3 combinations
They are {12, 13, 23}
can be permuted in 21 = 2
.
Each combination
ways, and. only one linearly
independent comparison of these two order effects can be
obtained.
Thus there is (21-1) = I .independent effect for
each combination.
Similarly, there is (^j = I combinations
of the three factors, - and there are 3 ! = 6 order effects.
Among these six order effects only five linearly independent
25
comparisons can be made.
•
3'
Therefore there are
.
J 2 ( i X 11 -1 ) = 3 - 1 + 1-5
=
independent order effects for the
"■
8
-Q
FSD.
The rank of X* can be decomposed as follows:
P
. E
Z
i=0
P
Pi/(p-i) I =
Z
i=0
(f) ii
P.
Z ■ ^i
i=0
+
Z
i=0
il-l
Because the first expression on the right equals 2^ and the
first two terms of the second expression are zero,
. Z
1=0
PV(p-i) I = 2P +
Z (±
1= 2 Xly
P
I + (2 P -1 ) +
Z
i=2
(?
i )( 11 -I
where the expressions on the right correspond to the degrees
of freedom for the mean, factors and orders respectively.
The above discussion implies that X* can be
reparameterized to form a full rank model.
The reparam­
eterization can be accomplished by multiplication on the.
.
26
right by a matrix M.
The design matrix X used will be one
which contains the appropriate contrasts for factor effects„
The factorial representation and nomenclature for 2^
experiments is given in many tests, eg., .[8 ] .■ For a
2p
factorial experiment, the treatment combination can be
represented.- as an n-tuple,
X^ = + I.
(X^,Xg,...,X^) where each
Thus the factorial representation for the result
Q
of a single run of in a 2^ may be written as
jm + 1 / 2 [AXi + BX2 + CX3 + (AB)X1X 2 + (AC)X1X 3 + (BC)X2X 3
+ (ABC)X1X 2X 3 ^
where
m = the general mean
X i = -I at the low level of A, B or C for
i = I, 2 , 3 , respectively.
= + 1 at the high level of the corresponding
factor.
The symbols (AB), et cetera are not products, they represent
interactions0
The parenthesis will usually be omitted.
This representation simplifies the discussion of one at a
time plans in Chapters IV and V e
A matrix M such that X*M = X exists by the results
concerning generalized inverses of Chapter I of Searle [11].
27
After .the reparameterization the model becomes
(3.6)
Y-Xf-)
\P
+ Wu + e .
/
rv/
For an example of X from a 23 FSD with r = I, see the
'Appendix.'
mV ■
pj
Also5
■
'
- (IHjA jB5C3AB,ACjBC,ABC,S12^e13,S23,S123^e132,S213.
e231’e312) •
Examples of responses are:
' Y n (I) = m + l/2{ -A - B - C +■ AB + AC + BC - ABC)
.+ U i + G11
Y1 2 (A) = m + l/2{ A - B - C - AB - AC + BC + ABC)
+ uI + g12
Y13 (A5B) = m + l/2( A + B; - C + AB - AC - B C + 0 1 2 + Uf + e I3
ABC)
■
Y3 2 (B5A) = m + l/2{ A + B - C + AB - AC - BC - ABC)
- O 12 + U 3 + G 32
The model used is analogous to the mixed model of ran­
dom unit effects and fixed treatment effects.-
The analysis
of the design will be explained in the next section.
28.
■ Analysis
The analysis of the design given in the previous section
is based on the analysis presented in [9].
The procedure
used was to transform the model by a,linear transformation
and then to analyze the transformed model using the method
of least squares.
The same technique is used in the section
which follows.
In the discussion which follows, it will be desirable
to use a method of multiplication of two matrices which is
different from the usual matrix multiplication.
This method
called the direct product is very useful when working with
blocks of submatrices.
The following definition is given by
Graybill [5].
Definition
——
... .. 13.1:
''
Direct Product:
Let ~
P be a m0C X n0C
matrix and let Q1 be an m n x nn matrix; then the direct
^
J_
J-
product of P and Q written P ® Q is a matrix T of size
ralm2 x n in2 - defined by
T = [\j] =
The symbol I will always denote an identity matrix and
J will always denote a matrix with every element equal to
one.
The model defined, by equation 3.6. is of full rank, but
the Gauss-Markov Theorem does.not apply because the Y 1s are
29
The non-independence is a result of the
not independent.
following theorem.
Theorem 3 . 1 :
For the model given by Equation 3,6,
(!)
e
(S)
Tar(Y) = c h + ^ [ J ® I]
Proof.:
(1)
U ) = x(g)
.
Applying conditions I and 2 of expression 3.2,
B(Y) = E(x(™) + W u + e)
= B x(“) + E(Wu) + E(e)
= 2(g) + Fte)
- - zlg)
2
(2)
Var(Y) =
.
E(Y - X 0 )
■
= EfWu + e)2 = E(e2 ) + E(Wue) + E(eWu)
+ W' E(uu' )W'
2
2
= CT^I
+ a,
WW'
VW
TJ1
VWVW
2
2
= »e I + = u e ®
where the I associated with a
2
I)
is the identity of order
r(p+l)pi, and [J (g I] is the direct product of a J matrix
of size (p+1) X
Hence,
(p+1) and I is the identity of order rpl.
[J ® I] is a square matrix of order r(p+l)p! which is
30
p
the same dimension as the identity associated with a
The Gauss-Markov Theorem for least squares estimation
can be applied to a transformed vector Z = TY if
Var(Z) = a®I.
T h e r e f o r e t h e transformation must satisfy ■'
Var(Z) = Var(TY)
= T Var(Y)T
= T [ct2I + C
t 2W
}T'
= CT2TT + tC 1
^TW(TW) *
- aL 1
Hence the matrix T must satisfy the following two conditions
(1)
TI" = I
(2)
(TW)(TW)1 = 0
or equivalently
TW = 0.
If r = I, a matrix which satisfies the above conditions
is
(3.7)
T = [H® I]-
where T is a rectangular matrix of dimension p p I x (p+l)p I
and. H is a.p x P+1 matrix which is a Helmert matrix with the
31
first row deleted (see Searle [11], page 33).
1/V2
-IXzrP
0
1/V6
1X/T5
-2X/6
..•
0
-i
0
H
■
I
I
I
; -P
VxP(P+ I) y p (p+ i) V F T p + I)
;
... V p (P+ I)
•'
For r replications of the FSD experiment, the desired matrix
is.
■
T=
[T ®
j 1 ]■
where T is defined as above and j ' is a r x I column vector
of ones.
2
3
A n example of a matrix T for one replication of a
'
is the Appendix.
The following theorem follows from the definition of T.
Theorem 3 . 2 :
For Y defined by Equation 3.6, and T
defined Equation 3.7.
(I)
If Z = TY, then
Z is normal
(£) >(!> = e (p )
(3)
Var(Z) =
Proof:
The proof follows from Theorem 3.1 and a well
known result from multivariate analysis (see Anderson Theorem
32
2.4.5) which states if X is N(jj.,y) and. if Z = DX then
X is N ( D ^ D W ) ' ).
Hence, Y normal implies Z = TY is normal
and
B(Z) = E(IT) = IB(Y) = IK(™)
and
Var(Z) = T Var(Y)T.'
= fT[
o-2I +ct 2T
aTW1]T'
v .£
U.~
«"W
2
2
= c £ TT'
+ a"jj_,TW(TW)'
fw
zvrrv» 'rv<x»^
= V 1
6~
.
where I is of order rp.pl x rp*pl.
This completes the
proof.
The first row of TX is all zeros because the trans­
formation removes the mean effect as well as the unit
effects.
A full rank model for Z will result if the first
column of TX is deleted, and the parameter m deleted from
the parameter vector.
Set R equal to the matrix TX with
the first column deleted, then
(3.8)
where R is a
Z = Rp
+ e
33
p
rppi x
2
i=l
Ip -TJT
matrix of full column rank, p is a
I
■ill X p W
x I
x
1
column vector of parameters, and e .is a rppi % I vector of
random errors.
Because the mean and unit effects have been removed,
all that remains for degrees of freedom are those for
treatments,.orders and error.
O
distributed with Var(Z) =
By Theorem 3.2, Z is normally
therefore, by the Gauss-
Markov Theorem the best linear unbiased estimate of {3 is
(3.9)
P = (R1R)-1R 1Z
If order is neglected, a new parameter vector is formed,
composed only of factor effectss
P
— (A,B,C,AB, .. .)
Corresponding to this parameter vector, a design matrix R^
of order rppi
(2^-1) can be found by
(3 .10)
where
is an augmented matrix with I the (2^-1) x
identi' by and 0 is a null matrix of order
(2^-1)
34
■P
2
■1=1
- (2p -l))x (Sp-I).
(P -i)
The vector of factor" effects P 1 is estimated "by
(3-11)
Il = (KlRi)-1JiZ
For a
FSD with r = I, the matrices R, (R1R ) R
#<#
/X/
#\# (
I
and (R!^R^) ^ are given in the Appendix.
A partition of the total sum of squares Z 1Z is
Z'Z = Z 1 [I - R ( R 1R)-1R 1 ]Z '+ Z' [R(R1R)-1R' - R 1YR' R 1 )"1R' ]Z
/n/
#X/
#X# rv
^
*V>
rxz
yx#
y\yJ. 'yx>Jp^* J.
«%#J_
+ 5' [ K i ( K i K f 1Ri)S
where all of the matrices in the brackets are idempotent.
Using Equation 3.10, it can be shown that all cross products
are zero.
The first term of the partition is the error sum
of squares associated with fitting the full model.
The
second term is the sum of squares associated with fitting'
order effects only and the third term is the sum of squares
for factor effects. .
The degrees of freedom for factor effects are 2^-I5 and
by Lemma 3.3, the degree of freedom for order effects are
P
2
i=2
The total degrees of freedom are 'rppi, and by subtraction.
those for error are
a.
35
rpp! - (gP-l) The preceding discussion can he summarized in the following
Analysis of Variance table.
Table 3.1
Source
(SV)
Degrees of Freedom
(DF)
Total
rpp!
Z'Z
r—I
%
Factors
Z 1B 1 (EiS1 ) "1Bliz
Orders
Error
*
Sum. of Squares
(SS)
.
Z t [R(R1R)-1R - R 1 (R'Rn )"1R* ]Z
rpp! - (2^-1) -
:J 2 ( a —
Z ' [I - R ( R 1R)-1RjZ
)
The expected mean squares corresponding to the sums of
squares of Table 3•I are
CTe + E f E S f e P / D F
'
“ e + [6' (s'5)g - ei(Ei5i)6iJ/EI?
...
for factors, orders and error, respectively.
36
Because the matrices in the quadratic forms for the
sums of squares are idempotent with cross product zero,
under the null hypothesis of no factor and. no. order effects
P
the sums of squares are independently distributed as a
times a %
2
variable.
Thus the F-test can be used to test
the hypothesis that order effects are all neglible, also
the factorial effects can be tested by the appropriate
F-test.
In the situation where the order effects are negligible,,
the factorial effects would be estimated by
If the
A
order effects are significant, p provides an estimate of
the factorial and order effects e ■ In this situation, the
discussion of Chapter Two applies and aides in the interpre­
tation of the. order effects.
Two examples are presented to
illustrate the analysis and to illustrate the use of this
design in interpreting factor and order effects when order
effects- are significant.
37
Examples
In [9]j examples are provided to illustrate the model
presented in that paper.
The same data is used in the
following two examples to show that the model and design
presented in this thesis leads- to the same F tests for order
effects and f o r .factorial effects.
Example 3 . 2 :
The following set of observations is ob­
tained from the data of Table II of [9]•
Y' = (56.258, 56.579, 52.661, 51.315, 55.500, 57.461,
.57.475, 50.396, 58.515, 56.323, 62.023, 61.673,
56.583, 56.924, 56.111, 62.085, 54.217, 55.914,
53.974, 49.754, 56.034, 57.895, 55.440, 62.863)
The model used is the one given by (3.3)•
Now using
the transformation Z = TY , the vector of transformed observations Z is
z ! = (-0.227, 3. 068, 3.335, -1.387, .-0 .812, 5.556,
1.550, -3. 759, -2.355, -0.241 , 0.524, -4.803,
■-1.200,
0.
891, 4.285, -1.316, 1.245, ■-5.548).
The best estimate of g given Ny (3. 9) is
A
P1 = (.62, -I. 19, .27 , .60, - .65, -.39,' .47, -3.66, .88,
-.23,-4 •7, -5.157, 3.62 , 4.71, -4.59).
38
By neglecting- order the best estimate of f3^- given by
(3 .1 0 ) is
Bi = (3.06, -3 .3 9 , .0 1 9 , .58, -.76, -.26, .47).
In Table 3*2, the analysis o f variance corresponding to. '
Table 3.1 is given.
Table 3.2
I
88
DF
SV
ITotal
18
MS
F
.P-value
154.18 '
Factors
7
5 6 .9 9
8.14
10.71
.0 3 8 7
Orders
8
94.91
11.86
15.61
.0229
Error
3
2.28
.76
S
The test of significance indicates that the order effects
are significant.
used.
Table 3 . 3
Therefore the model given in (3.9) is
lists the parameters, estimates, standard
errors and tests of significance for this model.
The t
entry in the table is the usual t-test of significance where
.t - =
' estimated effect
n
standard error of estimate
The estimates of the variances are given by
VarIci1B) = k (R1R)
For this example,
(R1R)
I
'ov .
is given in the Appendix.
39 .
Table 3.3
Parameter
A ■
t S
P-value
.570
1 .0 8
.36
-1.19
.‘570
-2.08
.13
C
.27
.570
• .47
.67
AB
. .6 0
.445
1.35
.27
AC
-.65 -
.445
1.46
.24
B C
-.39
.445
.88
.44
.47
.399
1 .1 8
.32
.8 1 5 ’
4.49
.02
1 .0 8
.36
ABC '
a
■
-3.66
CO
H
CD
.88
00
H
Ul
h
Standard Error
.62
B-
*
. Estimate
CO
CXJ
CD
-.23
■ .815
. .28
.80
-4.70
1.04
-4.52
.02
-5.67
i.o4 '
-5.45
.01
6 123
8132
e2l3
CD
UO
H
PO
e231
3.62
1.04
3.48
.04
4.71
i.o4
4.52
.02
-4.59
i.o4
-4.4i
.02
Using Condition (4) of the model, ^he.additional
estimates are obtained.
02i = 3.66 '
9 31
“ '•88
'
4o
Ggg
-
'23
/N
G3 2 I “ ^ '^ 2
This example is given to illustrate possible interpre­
tations when order effects are determined, to be significant.
If the standard factorial analysis had been performed, the
A and the B main effects.would have been judged significant
at the 10$ level on the basis of the.results of the 24 tests.
However, the FSD indicates the observed effects are partly
due to the order of the application of the factors.
Significant effects include the B main effect (p-value = .13),
the two factor order effect 8
and. the three factor order
effects.
The estimates of the three factor order effects form
two equivalence classes
{0123 = -4..70,
= "5.67,
O ^ 12
= -4.59}
and
{e2i3 = 3.62, P231 = 4.Tl5 Gggl = 6'°62} .
Since 0^2 was the only significant two factor order effect,
the system developed in Chapter II could have been used to
predict the possible existence of these classes.
4l
AsSLime this experiment was designed to test a
sequential- process for cleaning grease and particles from
transparent circuit boards.
ments
There are three possible treat­
(cleaning techniques)5 a chemical treatment (A)j a
vacuum cleaning (B), and a second chemical treatment (C),
The board is tested for cleanliness by measuring the
diffusion of light passed through the b o a r d . , Lower response
values (little light diffusion) indicate cleaner circuit
boards„
The previous analysis indicates there are no significant
factor effects; however5
vacuum
treatment B seems to be the
only one significantly reducing diffusion.
There also seems
to be a catalytic effect by doing the vacuum treatment B .
after treatment A.
The experimenter probably would recom­
mend one of the three treatment.sequences related to the set
of effects,' '
{©ABC* 6A C B 5 6 CAB^
or he could conduct further experiments specifically designed
to see whether treatment C was really necessary.
42
Example 3 « 2 r
The following set of observations is
obtained, from Example 2 of [9]*
Y'
= (9:219,
1 5 .0 6 1 , 1 4 .9 5 0 , 1 7 .5 5 0 , 1 1 .0 9 5 , 15.964,
15.732, 14.033, 9.541, 8.517, 15.213, 13.922,
9.104,-11.484, 9.109, 14.153, 9.045, 9.173,
1 4 .7 0 1 , 12.596, 9 .2 2 6 , 10.591, 15.806).
The transformation Z = TY yields,
-1.798,
Z 1 = (-4 .1 3 1 , -2 .2 9 4 , -3 .8 7 4 ,. -3.443,
0.200,
0.724, -5 .0 4 9 , -2 .4 5 2 , -1 .6 8 3 , 0.968,. -3 .6 8 4 ,
-0.091, -4:676, -3.190, 2.383, 0.261, -4.332).
The best estimate of P given by (3.9) is
P
' = (5.78,
.5 0 , -1.21,
-.16,
.02, .14, 3.16, -1.75,
.44, -.17, -.14,
.3 0 ,
-.73,
-.52,
-.1 5 ).
By neglecting order the best estimate of P , given by
(3.10) is
p = (5.54,
.56, -.12,
.2 9 , -.75, -.33,
-.14).
A
The first seven entries of p 1 estimate the same param~
A
eters as the corresponding entries in p .
By inspection the
estimates appear to be approximately the same.
Intuitively,
it appears that order has little or no effect for this
experiment.
This conjecture is verified by constructing the
analysis of variance table corresponding to Table 3.1. ■
43.
Table 3.4
DF
'
Total
Factors
Orders
Error
-
.
SS
MS
F
9.769
-=t
O
SV
.666
.7 1
18
158.395
' 7
141.202
20.172
8
10.999
1.375
3
6.194
2.065
P-value
The F test for order effects indicates order of
application is insignificant.
Therefore, the model given
in (3.10) is used.
Table 3•5 lists the parameters, estimates, standard
errors and.' tests of significance for this model.
Table 3.5
Parameter.
Standard Error
tS
P-value
.006 .
.80
6.93
.29
.80
.36
C
-.75
.80
-.94
AB
-.33
.69
. -.47
.56
.69
.80
BC
-.12
.69
■ -.17
.88
ABC
-.14
.65
-.22
.84
A
. 5.54
B
.74
.42 •
.6 7
CO
,
''
-d*
' AC
I
Estimate
The only significant effect is the main effect A and
its significance level is less than .01.
44
Fractional Replications of 2^ FSD Designs
The purpose of. Prairie and Zimmer's 1 9 6 8 paper [10]
■Q
U
c
was to present some fractional designs of 2 , 2 , 2 ^ and
2
6
FSD designs.
The designs were constructed to satisfy the
following characteristics.
(a ) . All main effects and two factor interactions are
estimable.
(b)
For a given design the variances of all main
effects are equal and the variance of all two
factor interactions are equal.
Also, the
variances of all k-factor interactions are
equal for fixed k.
( c)
The importance of some order effects can be tested
by a test of significance.
The purpose of this section is to introduce the notation
and terminology invented by Prairie and Zimmer.
The designs
are of two basic types,
,( a)
l/r x 2 ^ designs
( b)
s/p x (l/r x 2 P ) designs„
The l/r x 2^ designs are fractions of the 2^ FSD
requiring (l/r)pI units where- each unit is tes.ted p+1 times.
45
All factorial effects are estimable and information is
sacrificed only on order effects„
In the s/p x
(l/r x 2^)
designs5 each of (l/r)pl units is subjected to s factorS5
s < p 5 and s+1 tests. .
The analysis of these designs is similar, to the analysis
presented, in this chapter.
referred to [10].
For more detail the reader is
The terminology presented in this section
will be utilized in the development of one at a time plans.
CHAPTER IV
■ ONE AT A TIME- PLANS FOR THE 2 3 FSD
Preliminary Considerations
The plans proposed in this section are designed to
provide estimates of order parameters as soon as possible in
the experimental sequence.
The main purpose of the FSD is
to decide whether order of application of factors is
important. ■. By the time enough trials are run to provide
estimates of order effects, main effects and two factor
interactions are estimable.
Usually the three factor inter­
action is estimable or is estimable .with the addition of one
more run.
The estimates used for the one at a time plans will d e ­
constructed to remove the unit effect, but will not neces­
sarily be of minimum variance.
The estimates used are linear
combinations, of the observations and therefore, they are ■
simple and.easy to calculate.
The ease of calculation and
simplicity will be useful for the one at a time experimenter.
The variances of the estimates given are compared to t h e '
variances for the best linear unbiased estimators for the
full 24 runs as found in Chapter III.
The variances of the
estimating contrasts follow readily from Theorem 3.1.
4?
Lemma 4 . 1 :
If
. are
are two responses from the ■
same unit i , then
Var(Yl j - Y lfc) = S a e
2.
Proof:
By Theorem 3.1,
Var(Y^)-=
-Ij
^
^cl CovfY^j, Y^) =
Therefore,
Var(Tlj - Ylk) = Var(Yl j ) + Var(Ylk) - 2 CovfYl j , Ylfc)
= (°e + °u) + (°s + au) " 2au
.
■.
Lemma 4 . 2 :
= 20EIf Y^^, Y ^ are two responses from unit i,
and Ygm , Y^n are two responses from unit £> then
Var[(Y1 j - Ylfc) - ( Y ^ - Yj n )] = 4a2 .
Proof:
Again applying Theorem 3.1,
V a r f t Y y - Ylfc) - (Y^ - Yin)] = Var(Yy ) + Var(Yy )
' + Var (Yim) + Yar(Yin) - 2 CovfYlj, Yy )
- 2 CovfYy ., Yim) + 2 CovtYlj, Yjn)
+ 2 CovfYy , Yim) - 2 Cov (Yy , Yjn)
' '
- 2 ^ v ( Y i m , Yi n )
48
= 4(0^ + a 2^ _ ga^ - 0 +
0 + 0-
0-
2a^
= 4a J .
This completes the proof of the lemma.
The notation used in the one at a time plans will be as
follows.
T h e .individual trials will be called runs.
For
each run the unit and the number of the test the unit has
been subjected to will be given.
This labeling will make it
easier to relate the one at a time plans to the model
presented in Chapter III.
The plans presented are runs from
Q
one replication of a 2
FSD.
Therefore, there must be six
units available, one for each specific order of application.
The six .possible application sequences will be assigned to
the units randomly; however, the units will be labeled to
correspond to the alphabetical order of the application
sequence as follows:
Unit
Application Sequence
I
abc
2
acb
3
bac
4
bca
5
cab
6
cba
49
This is the same convention used in constructing the matrices
in the Appendix.
The sequence coding specification will refer to treat­
ments that have been applied before a specific test.
The
specification will always be denoted by lower case letters
with the notation (1)^ for no treatments on unit i.
Thus
(1 )2| will mean that no treatment has been applied on the
fourth unit3 and (ba)^ will indicate that the third unit
has been subjected to the two treatments b a n d a in that
specific order.
It is not necessary to subscript ba with
the unit number since the factors are applied in this se­
quence only on unit 3•
The notation for this observation
for the model given in (3.1) is Y ^ ( B 3A).
The specification
ba is another simpler symbol for this same response.
One at a time plans are developed for both of the cases
considered
in
Chapter II.
The systems developed for order .
effects for each case are used to augment an initial set of
runs to achieve unbiased estimation of order parameters as
soon as possible in the sequence.
According to Daniel [3] •> the one at a time experimenter
who achieves good results
looks
for effects three or four
50
times the magnitude of the experimental error.
If the
experimenter has some prior knowledge of the magnitude of
the random error in his experiment, he will he able to look
for these large effects without first estimating cr^.
One At a Time Plans for Case I
This case is the one where symbols related to insignifi­
cant order effects commute only on the left of the appli­
cation sequence.
The first nine runs will be on units
designated .1 , 4, and 5 with application sequences abc, bca,
and cab respectively.. This will enable the experimenter to
get biased estimates of all main effects, two factor inter­
actions and two factor order effects.
The experimenter may
get some information from these estimates for use in plan­
ning the next sequence of runs.
The first nine runs are:
51
RUN UNIT TEST S E Q .
NO . NO.
NO, SPEC.
I
I
I
2
I
2
3
I
•' 3
4
4
I ’
5
4
2
. ESTIMABLE FUNCTIONS
(at run indicated)
1I
a
4
3
7
5
■I
8
5
2
9
5
a-1!
A-AB-AC+ABC
ab
14
.b
B-AB-BC+ABC
2 (AB-ABC)+S12
'6
' ESTIMATORS
ab—a—b+lj^
be
1S
C-AC-BC+ABC
C
ca
3
C_15
2(B C -ABC)+S2 3
bc-b-c.+lr5
2 (AC-ABC)-S13
ca—c—a+l1
Using the model. given in Equation (3.6),
E (a — 1I J •= {m + 1/2(A - B - C - AB - AC + BC + ABC)}
— (m + 1/2(-A - B - C + AB + AC + B C - ABC}
. ■'= A - AB - AC + ABC.
Similarly, ■
E(ca - c) = A - AB + AC + ABC and by subtraction,
,E(ca
c - a + I1) = 2 (AC + ABC) - S13.
52
By applying 'Lemmas 4.1 and 4 . 2 5 the variances of the
estimates are
Var(a - I1)'= 2a^
2
V a r (ca, - c - a + I1 ) = 4c^.
The other estimates are estimated by the appropriate
contrasts and can be.derived similarly using (3.6).
The
variances of these estimates can also be calculated using
the. appropriate result, from Lemmas 4.1 and 4.2.
At this stage of the experiment, main effects and
order effects are confounded with two and three factor inter­
actions. .Many one at a time experimenters assume that three
factor interactions are insignificant, and under this
assumption,.estimates of two factor order effects confounded ■
with a corresponding two factor interaction can be found.
The probability that these two effects are offsetting is
very small so if the estimate of 2AB + 0 ^
is "small”, the
experimenter may conclude after five runs that there is no
significant two factor order effect related to treatments .
A and B (and also ho AB interaction).
Based on the assumptions of negligible three factor
interactions and of the small probability of offsetting
53
effects the experimenter can conclude that all or some of the
two factor interactions are insignificant.
There are four
possible situations which can result depending on how many■
of the effects can be assumed insignificant.
They are:
(1)
all three' two factor order-effects insignificant/
(2)
two of the three two factor order effects
insignificant,
(3)
one of the three two factor order effects
insignificant,
(4)
none of the three two factor order effects
insignificant.
If a scientist was unwilling to make the above
assumptions concerning three factor interactions and off­
setting effects, he would need nine additional runs to get
unbiased estimates of the two factor order effects.
This
situation is experimentally identical to the fourth possi­
bility just listed.
Each possibility will now be examined
and sequences of runs will be given for each situation.
I)
Suppose all of these biased estimates of two factor
order effects are small, leading to the conclusion that two
factor interactions and two factor order effects are
assumed to be insignificant.
Thus, since '
54
912
e21f 9 13
9S l 5 9 2 3
93 2 .
by Lemma 2.1 the system for order effects Implies
9123 = 9 213^ 9 132 = 9 31 2 5 9231 = 9 321'
Therefore, 'three more runs are needed, to test for equality
of these classes of three factor order effects..
RUN UNIT TEST SEQ'.
NO. SPEC.
NO.
NO.
10
I
4
abc
11
4
4
bca
12
5
■ 4' ■ cab
ESTIMATORS
ESTIMABLE FUNCTIONS
(at run indicated)
abc-l^-bac+1^
9 123 “ 6231
abc-l^-cab+lp.
9 123 " 0 312 .
e231 " 0 312
•
■ bac-l^-cab+1 ^
Each of the differences is estimated by the indicated.
contrast, and the variances of the estimates follow from
Lemma 4.2 and. are equal to 4a£ .
■
The estimators of these
differences based on twelve runs compared favorably to the
least square's estimators from the full F S D .
Using the
matrix (R1R ) - 1 found in the Appendix, the variance of the
least.squares estimator for one replication requiring 24
2
runs is 3 .5 o"e-•
■
If the estimated contrasts from runs .10,'11 and 12 are
small, the one at a time experimenter w o u l d .assume there
55
were no significant three factor order effects because he is
looking for effects approximately three or four times the
magnitude of his experimental error.
O
This sequence of 12 runs is the (1/2 x 2J) fraction of
the FSD given in [10].
For this fraction there are no
degrees of freedom for error; if the experimenter wanted an
estimate of error, an additional four tests, on another unit
could be run to yield a (2/3 x 2^) FSD with one degree of
freedom for error.
2)
Suppose two of the three estimates for two factor
order effects are small compared to the assumed magnitude of
the random error.
Then two of the three two factor order
effects are judged insignificant and assume 6
effect which may be significant.
is the
Because the estimate was
biased, the significant response may have been a result of
the interactions which biased the estimate.
assumed that
■G
’
^31
0 23
6-32 ^
^
the system developed for Case I implies
G
— ^312
^231
^ 321'
Since it is
.56'
.
In this situation the. following additional runs are
suggested.. The first three to provide an unbiased estimate
of 0^25 and the last four to estimate the three factor order
effects.
RlM UNIT TEST SEQ.
NO. NO. NO. SPEC.
10
3
•I
11
3
,2
12
3
3
■13
3
■4
b:ac
14
I
' 4
abc
15
4
4'
bca
16
5
4
cab
ESTIMABLE FUNCTIONS
(at run indicated)
ESTIMATORS
■
1S
b
ba
ab-I1-LaRl^
2012
abc-l^bac+lg
0 123 " 0 213
0 123 ■“ 0231
■
0123 " 0 312
abc—11-bca+l^i
abc-l1-cab+l^
The unbiased estimate of 012 is
S12 = 1/2(ab - 1^ - ba + l^).
Applying Lemma 4.2,
Var(G12) = l/4(4o2) = ( A
This compares favorably to the minimum variance, estimate
2
based on '24 runs which has variance equal to .875crG
After 16 runs have been completed, two of the two
factor order effects have been judged insignificant, an
57
unbiased, estimate of
exists, and estimates of the three
factor order effects have been obtained.
Hence, this one at
a time plan of 16 runs allows the experimenter to determine
if order effects are significnat.
The 16 runs constitute
O
a (2/3 X 2J ) PSD with one degree of freedom for error.
3)
The third possibility occurs when only one of the
three two factor order effects is judged insignificant on the
basis of the biased estimates from the first nine runs.
Assume the insignificant effect is G2^, than Lemma 2.1 implies
S23I = 0 3 2 1 -
The following six runs will produce unbiased
estimates of 0 no and 0
§
RUH UHIT TEST S E Q .
HO.
HO. SPEC.
10
3
I '■
11
3
2
b
12
3
3
ba
13
2
I
4
14
2
2
a
15
2
3
■ac
23'
ESTIMABLE FUHCTIOHS
(at run indicated)
ESTIMATORS
1B
•ab—I^-ba+!^
2612
ac~l2 ~ca+l^
2 6 I3
A
A
The two. unbiased estimates 0 ^ 2 and 0 ^
2
equal to ae. as indicated, earlier.
have variance
If one or both of the
two ■factor order effects related, to these estimates are
58
judged insignificant, the estimates of three factor order
effects can be found by augmenting with the proper three
factor sequences previously discussed in possibility one or
two.
However, if both are judged significant, runs 16-20,
given below, are necessary for the estimation of three
factor order effects.
RIM UNIT TEST S E Q .
NO.
NO. SPEC.
NO.
16
I
.4
17
2
' 4.
acb
18
3
4
bac
19
4
4v
bca
20
5
'4
ESTIMABLE FUNCTIONS
(at run indicated)
ESTIMATORS
■ abc
cab
abc-l^-acb+lg
8l23 ” 6 132
- abc-l^-bac+lg
6 123 ™ e2l3
8123
0231
abc-l-^-bca+1 ^
■
abc-l^-cab+1 ^
8 1 2 3 " 6 312
As before , the contrasts, constraints and system yielded
estimates of two and three factor order effects as well as
the usual factor effects while allowing two degrees of
freedom for error.
PSD. .
4)
O
This design of 20 runs is a (5/6 x 2'5)
I
.
The fourth possible situation occurs when the
experimenter does not want to base his judgements on biased
estimates "or if all of the biased estimates indicate that
all two factor order effects may be significant.
The
59
following nine additional runs provide unbiased estimates of
two factor order effects..
RUN UNIT TEST SEQ.
NO. NO. NO. SPEC.
10
2
'I
11
2
'2
a
12
2
3
ab
13
3
I
14
3
2
b
15
3
3
ba
16
6
■I ■
17
6
'2
18
6
ESTIMABLE FUNCTIONS
..(.at run indicated)
.ESTIMATORS
*
26I3
ac-Ig-c'a+/
•
/3
:
C
3 .'cb
ab-l-^-ba+1
26IS
be—lj|-cb+l|
^23
These 18 runs form a 2/3 x ( l x 2^) FSD.
If any of the above
estimates are judged insignificant, these tests can be
augmented, b y the appropriate runs where three factors have
been applied by using the appropriate situation from the
first three possibilities presented.
If all two factor
order effects appear to be significant, each unit must be
tested again.after application of the last treatment.
completes a full FSD which can be analyzed using the
techniques developed in Chapter III.
This
6o
These four situations consider all possibilities for
examining one at a time plans' assuming only the basic
ordering operation (Definition 2.1) and commutativity of
insignificant order effects only on the left (Lemma 2.1).
If in addition one assumes commutativity of any factors
previously judged insignificant (Definition 2.2), then the
second case of Chapter II is encountered.
6l
■One At A Time Plans For Case 2
This situation is where the symbols related to insig­
nificant effects commute anywhere they appear in the
sequence.
This additional assumption reduces the number of
runs required to estimate the order effects.
The first nine runs are the same as those for Case I 3
and if the experimenter is willing to make the same assump­
tions for determining if two factor order effects are insignificant3 then the same four situations that occurred in the
plans for Case I will occur for Case 2.
I)
All three biased estimates are small in magnitude
compared to the size of the a priori random error.
In this
Situation3.the three two factor, order effects are determined
to be insignificant.
The system for order.effects developed
for Case 2 implies
='0213
= 0231
=
8321
= 0 312
= S132
by Lemma 2.1 ■
by Definition 2.2
by Lemma 2.1
by Definition 2.2
by Lemma 2.1
Therefore3 all of the three factor order effects are
equal.
The fourth model assumption (3*2) states that the
62
sum of all of the three factor order effects is equal to
zero.
Therefore, the system for order effects implies there
are no significant three factor order effects.
One more run
is required to have unbiased estimation of all factorial
effects.
RIM UNIT TEST SE Q .
NO.
NO.
NO. SPEC.
10
2)
.I
4
abc
ESTIMABLE FUNCTIONS
(at run indicated)
All factorial effects
The. second possibility,occurs if one order effect
is possibly significant.
Assume 9
the order effect in
question. ■ Then the system implies there are possibly two
equivalence classes of order effects.
{©123^ 6^32^ ®312^
® 213 3 e2315 ® 321^
The following five runs will provide estimates of these
order effects.
RUN UNIT TEST S E Q .
NO.
NO. SPEC.
NO.
10
3
"I
11
3
'2
12
3
13
I
■’ 4 '• abc '
l4
3-
; 4
'3:
ESTIMABLE FUNCTIONS
(at run indicated)
' ESTIMATORS
1S
b
ba
bac
2612
6 123 " e2l3
ab-l^-ba+l^
abc-l-^-bac+lg
63
At this stage of the experiment, estimates of the
factor and. order parameters can be found using the con­
straints, the estimable functions, and the equivalence
classes.
3)
The third possibility is when two of the two factor
order effects may he significant.
the order effects in question.
Assmue 0^2 anc^ 0 i3 are
In this instance, the system
implies there may be four equivalence classes of three factor
order effects.
{0123, 0132^ 5 te23l> 9321$ 5
132} 5 ^ 231}
The first six additional runs will provide unbiased
estimates of the pair of two factor order effects under
investigation.
The next four runs will provide contrasts
which will enable the experimenter to estimate the three
factor order effects.
64
RUU UNIT TEST S E Q .
NO.
NO. NO. SPEC.
10
3
I
11
3
2
12
‘3
3
13
2
14
2
'2
15
2
3
16
I
-4
abc
17
2
-4
acb
18
4
19
5
ESTIMABLE FUNCTIONS
(at run indicated)
ESTIMATORS
• 1S
'b
ba
2 0 12
ab-l^-ba+l^
I ... I2
4'
• a
ac
bca
. 4 ; cab
2013
0 123 “ 0132
. 0 123 " 0231
0 123 " 0312
ac-lg-ca+l^.
abc-l^-acb+lg
abc-l^-bca+1 ^
abc-In -cab+lr1
5
Again, the contrasts, the conditions of the model, and
the. equivalence classes provide estimates of the order
effects." Using these the experimenter can get estimates of
factorial effects.
4)
Possibility four is identical to the fourth situ­
ation discussed in the plans for Case I.
The experimenter
does l8 runs to estimate two factor order effects and then
one to six more runs are necessary to estimate the three
factor order effects.
65
Examples
-.1 :
If the data from Example
time analysis, the first nine
RUN
S E Q . SPEC.
I
R
56.258
2
a
56.579
3'
ab
52.661
14
56.583
5
b
56.924
6
be
56.111
"4
7
■
OBSERVATION
54.217
1S
8
C
55.914
9
ca
53.974
The following "biased estimates of two factor order
effects are obtained.
After run f i v e :
2 (AB - ^BE) + G
= 52.661 - 56.579 - 56.924 + 56.583
-4.259
66
After run eight:
2(BC
Sc)
+ S2 3 = 56.111 - 56.924 - 55.914 + 54.217
= -2.510
After run n i n e :
2(£c - ABC) + G13 = 53.974 - 55.914 - 56.579 + 5 6 . 2 5 8
= 2 .2 6 1
If the experimenter has prior knowledge of his experi­
mental error, he would be able to make decisions concerning
the possible two factor order effects.
In particular, for
A
this experiment, it is known from Example 3.2 that
= .8 7 .
Consider the. following two instances; one where the expert.‘
1 A
menter assumes a
a
G
■
*
is about . 5 and the other where he assumes
is about .1.
A
I)
For. cre = .5 5 the experimenter would decide after
nine runs that all of the effects were possibly significant
and would do the following additional- runs..
67
RlM
10
SEQ. SPEC.
OBSERVATION
V
58.515
.11
b
56.323 .
12
ba
62.023
13
1E
55.500
a
57.461
ac
57.475
' 14
15
16
'1S
56.034
17
C
57.895 .
18
cb
55.440.
The following unbiased estimates are obtained.
After run twelve:
8l2 = 1/2(56.661 - 56.258 - 62.023 4- 58.515)
=-3.553.
After run fifteen:
= 1/2(57.475 - 55.500 - 53.900 + 5.4,217)
= ’1.146.•
After run eighteen:
S23 = 1/2(56.111 - 56.583 - 55-440 + 56.034)
= .0 6 l.
68
The experimenter would decide that only
was
significant, and then do two or four more runs depending on
whether he was using Case I or Case 2 of the system for
order effects.
Suppose he is operating under Case I
requiring four more runs.
RUN
S E Q . SPEC.
OBSERVATION
19
ah c
.51.315
20
bca
.62.085
21
cab
49.754
22
bac
61.673
The following contrasts among three factor order effects
can now he formed.
After run twenty:
(S1 2 3 r S231) = 51.315 - 5 6 . 2 5 8 - 6 2 . 0 8 5 + 56.111
= -10.917
After run twenty-one:
"8312) = 51.315 - 56.258 - 49.754 + 54.217
. = — .48
After run twenty-two:
(9213 " ^ 3Ig) = 61.673 - 58.515 - 49.754 + 54.217
= 7.621
69
Also after run twenty-two:
62.085
(^231 ~ ^ 2 1 3 )
56.111 - 61.673 + 58.515
= 2.8l6
The above differences imply the existence of two equivalence
classes of three factor order effects
61235
9132’
e312^
and { S2 13, d 2 3 1 ’ 9 32 J '
If the- assumption of the system for Case 2 (symbols
commute anywhere) had been made, the above classes would
have been verified by runs' 1 9 and 20.
2)
Oe' = I.
In this case, the experimenter would have
assumed, after the first nine runs, that
insignificant.
and S2 3 were
He would then do runs 10 to 12 and determine
that 0 ^ 2 was significant.
After these twelve runs, he would
then complete the experiment by testing for three factor
interactions'in exactly the same way as in the previous
situation where 0
E
was assumed to be .5 .
That is, after
completing runs I through 1 2 , he would immediately do only
four additional runs, I9 through 22.
13 through'-18, would not be needed.
The.other six runs,
Conclusions regarding
significance of order effects are identical to those
obtained when a
E
was assumed equal to .5.
7P
The problem of estimating a
G
is difficult, and the
experimenter should be aware of the consequences of a wrong
estimate.
The more conservative approach is to under­
estimate and as a result more runs may be necessary to
determine whether or not order effects are significant.
CHAPTER V
ONE AT A TIME PLANS FOR 2P PSD
■, .
Introduction
In the previous chapter, one at a time sequences for
the 2
PSD.were examined.
Using the systems for order
effects to form equivalence classes, the number of runs
necessary to estimate order parameters was reduced from
twenty-four to as few as ten.or twelve depending on whether
or not the.experimenter was using Case I or Case 2 of the
system for order effects developed in Chapter I I . ■ When
p >
3j> the number of units and number of tests required
increase rapidly.
For example, a single replication of a
h
full 2
PSD requires 24 units and 120:tests.
The use of
equivalence classes is very helpful in reducing the number
of runs required for estimation of order effects.
In the' previous chapter, all possible instances of a
2^ PSD were given to illustrate the procedure for deciding
on additional sequences of' runs on the basis of the infor­
mation derived from the funs already completed.
enumeration of distinct possibilities for the 2
not be given in this thesis.
A complete
4
PSD will
However, a,sample situation
is presented to illustrate the procedure for deciding on
the runs necessary to estimate the order effects of inter­
est.
The. situation is presented without data and at suc­
cessive stages of the experiment, some effects are arbi­
trarily assumed to be significant in order to illustrate
the use of the system in determining whether order effects
are present or not.
■ h
An Example of a One at a Time Plan for a 2 FSD
V
When planning one at a time plans for a 2
.FSD, the
experimenter has considerable latitude in selecting the
initial sequence of runs.
There are ( ^ ) = 6 sets of two
factor sequences from which the experimenter could form
. • ■■■
*
■
preliminary .estimates of two factor order effects. Be­
cause there are only four factors, two must be replicated
in order to get the first eighteen runs..
These replications
will provide greater precision in estimating these two
main effects as well as provide two degrees of freedom for
an estimate of experimental error.
Suppose'.A and B are the
main effects which are most important, then the first
eighteen"runs are:
.. -
73
ROT UNIT TEST S E Q .
NO.
NO. 'NO. SPEC.
ESTIMABLE FUNCTIONS
(at run Indicated)
' ESTIMATORS
e 12+2(AB-ABC-ABU+ABCO).
ab-a-b+lg
C
6.23+2 (BC-ABC-BCD+ABCD)
bc-b-c+lg
-6 13+2 (AC-ABC-ACD+ABOD)
c a— c—a+1^
-9g^+2(CD-ACD-BCD+ABCD)
dc-d-c+l^
I
I
■ I'
2
I
'2
a
3
I
3
aU
4
2
I
5
2
2
6
2
3
7
3
.1
8
3
'■ 2
9
3
3
ca
IO
4
■T
1||
ll
4
. 2
d.
12
4
■ 3
dc
13
5
l4
5
,2
15
5
'3
16
6
I-
17 .
6
2
b
18
6.
3
bd
I'.
1I
R
"be
1B
1S
? .
_
.
'
a
ad.
9 ^ + 2 (AD-ABD-ACD+ABCD)
ad— a—d+1^
8 2 h+2 (BD-ABD-BCDtABCD)'
bd-b-d+1^
1G
74
After these l8 runs, the experimenter has available
estimates of all two factor order effects.
As in the 2^
F S D , the estimates of two factor order effects are biased,
but if there is some prior knowledge of the random error,
some of the order effects and the corresponding interactions
can be judged insignificant.
Suppose that
and
are possibly significant on .the basis of the first
eighteen runs .
Then the following nine runs should be
completed. to find unbiased estimates of e 1 2 5 ©2 4 and ©3 ^ ®
RUN UNIT TEST S E Q . .
NO. SPEC.
NO. NO.
19
7
.I
20
7
. 2-
21
7
■3
ba
22
8
I
1S
23
8
2
d
24
8
3
Sb
25
9
'I
26
9
2 .
C
27
9
3
cd
ESTIMABLE FUNCTIONS
(at run indicated) .'
ESTIMATORS
4
b
^12
.
ab-l-^-ba+l^
2e24
bd-lg-db+lg
34
cd-l^-dc+1 ^
1S
20
After these runs, assume the estimates of
0 2 4 are significant, but that S ^2 is insignificant.
and .
75
Up until this point in the example, the 2 7 runs completed
are suitable for either Case I or Case 2 assumptions.
The
remainder of the discussion will assume Case 2, where non­
significant order effects may commute regardless of where
they appear.in the application sequence.
Thus, using
Lemma 2.1,- and Definition 2.2, the following equivalence
classes of three factor order effects can be derived assum­
ing @24 ^ £ 4 3 and 8 2 4 7^ djjg, ^ut that all other effects are
equal.
I9-1239 6 1325 6 3125 0 321> e2 3 1 5 9 213^ •
{0 ^ 4 , 8 3 1 4 , 0 3 4 %}, t0 i4 3 5 9 4135 9 431 ^
{0124i 62l45 024l^ 9 (9l42* 94125 9421-^
^ 92 3 4 9324^9 ^ 9432ji 9423^9 ^9342^ 5 ^ 9243^
Similarly, the following equivalence classes of four factor
order effects can now be derived, using only information about
two factor order effects.
{9 12345 9 1324^
9 312459 32l45 9 3241' 9 2134'
92314)
{9 1423' 9 4l23'
9 4l32' 9 4312' 9 4213' 9 4231'
9 l432)
^91243' 9 2l43'
924l3' 9243l)
{0 34l2' 8 3 4 2 1 '
93i42' 9 1 3 4 2 )
76
These equivalence classed may be further condensed using
information concerning significant three factor order ef­
fects.
All. six three factor order effects with subscripts
which are permutations of the symbols I 5 2 5 and 3 appear
in the same equivalence class.
The fact that all of these
effects can be considered equal5 in conjunction with the
model condition (3 .2 ) that all of these effects must sum
to zero5 implies that these six order effects are insignifi
cant.
Therefore the following runs sould be performed in
order to estimate the remaining order effects of three and
four factors.
RUN UNIT TEST SEQ.
NO. NO. .NO. SPEC.
cad
5
4 ' adc
30
I
4
abd
'8 " -4
dba
2
• 4
bed
33
4
4
deb
34
9
4
cdb
35
6
4
dbc
9 314
cad-I0-adc+Ic
3
9
cad-lg-abd+1^
6124 " 9421
abd-1-^-d.ba+lg
cad-lg-bcd+lg
- 0432
bed—Ig-dcb+l^i
bcd-lg-cdb+lg
CO
-dCM
CD
I
-dCO
CM
CD
32
6 314 “ 9143
-3-
29
CD
I
3
-=jCO
CM
CD
I
-3i
—i OO
CO CM
CD CD
28
31
ESTIMATORS
. .i
CD
UO
-Pr
PO
’■ 4
ESTIMABLE FUNCTIONS
(at run indicated)
. bcd-lg-bdc+lg
9234
77
RUN UNIT TEST SEQ „
NO.
NO.
NO. SPEC.
36
3
5
cadb
37
5
5
adcb
38
I
5
abdc
39
2
5
bcda
ESTIMABLE.FUNCTIONS
(at run indicated)
ESTIMATORS
c adb-1 0 - adcb+lr3
5
cadb-lg-abdc+1 ^
6 3 142 " 9 1432
63142 “ 6 1243
93142 ™ 9 234l
■cadb-lg-bcda+lg
T h e ■contrasts formed from runs 28 to 39 9 the equiva­
lence classes,and the assumptions of the model can then he
used to form estimates of the order parameters.
If the less restrictive assumptions for the system
of order effects for Case I (symbols commute only on left)
.had been used, more equivalence classes of order effects
would have been formed and therefore, more runs would have
been required to estimate the order effects.
But the tech­
nique of examining two factor order effects for significance
and then forming equivalence classes of higher level order
effects is the same for both cases.
illustration of this technique.
This example is an
One at a time plans for a
2^ PSD can be developed by using the system to form equiva­
lence classes of effects, and then estimating one effect
from each class.
78
Example
Example 5 . 1 :
If the data from the example of a
1/2 x 2^ FSD appearing in [10], is subjected to a one at a
time analysis assuming a
£
= I 3 the first 1 8 runs are as
follows:
RUN
I '
S E Q . SPEC.
'
1I
OBSERVATION
8.51 .
2
a
15.67
3
ab
13.68
4
1S
10.77
5
b
10.08
6
be
7
8.77
9.86
8
h
C
11.04
9
ca
15.39 -
10
14
11
d
12
dc
'
9 .6 6
1 0 .1 5
8.97
10.13.
13
14
a
15.53
15
ad
13.66
79
RUN
S E Q . SPEC.
16
16
OBSERVATION
IO.9 6
17
b
1 1 .0 6
18
bd
10.15
The following estimates are formed.
After run 5:
S12 .+ bias = 13.68 - 1 5 . 6 7 - 1 0 . 0 8 + 10.77
= -1.30
After run 8:
.
+ b i a s = 8.77 - 1 0 . 0 8 - 11.04 + 9.86
= -2.49
After run 9: ■
- 0 1 3 +'bias
15.39 - 11..04 - 15.67 + 8.51
-2.8l ■
After run'.-12:
-0 ^ + bias = 8 . 9 7
- 10.15 - 11.04 + 9.86
= -2. 3 6 '
After run 15:
®l4 + '
= 13.66 - 15.53 - 10.15 ■ 9.66
= -2.36
8o
After run 18:
+ . M a s = 10.15 - 11.06 - 10.15 + 9.66
= -1.40
If the one at a time experimenter Is looking for ef­
fects three or four times the magnitude of the experimental
error, none would appear to be significant '
0 ^2 *
To be safe. do three runs to estimate
RUN
S E Q . SPEC.
OBSERVATION
19
I7
7.97
20
a
16.32
21
ac
16.63
The following unbiased estimate of 9 ^
13'
found after
run 21,
8^2 = 1/2(16.63 - 7.97 - 15.39 + 9.86)
= 1.565
The conclusion based on the assumption a£ = 1
insignificant.
is the 8^^ is
If Case 2 of the system for order effects
is assumed, no two factor order effects would imply no
■ ■
4
order effects for the 2 FSD.
For any other two factor
order effect which might be significant, an additional
three runs will be sufficient to obtain an unbiased estimate.
8l
Prairie and. Zimmer in [10] ran a 1/2 x 2
Zi
fraction
of 6 0 runs to arrive at the same conclusions reached after
only 21 runs using the one at a time plan under Case 2
assumptions.
The assumption of Oe = 1
could have been re­
placed b y an estimated value of 1 . 6 obtained from runs
I, 2 5 4 5 5.5 13 5 l 4 5 1 6 , and I/.
If this estimated value
had been used, the experimenter would have had no reason
to conduct more than the first 1 8 runs.
If this example is considered under assumptions of
Case I then 52 runs are required, which is a saving of
only eight runs (or tests).
Using Case I, the following
equivalence classes for three factor order effects are ob­
tained.
{01235
6213^ 5f0i3256312^ 5 t63215 0231^
{0124.5
e2 l 4 ^ ^0 l425 04 l 2 ^
{0 1 3 4 ,
^ 64215 9 24l^
5 {0 ^ 3 5 8 4 1 3 ] 5 ^ e34lj’ 6 4 3 1 ^
C 8 2 3 4 ^. 0 3 2 4 ^ ^ ^■0243-’0 4 2 3 ^ 503 4 2 04 3 2 ^
Because there are twelve classes of three factor order
effects, ■twelve units will need, to be tested.
Only seven
units were used, initially3 therefore, five additional units
must be tested which incidently, give unbiased estimates of
82
order effects and a total of
error.
RUN
SEQ . SPEC .
OBSERVATION
22
abc
■ 14.25
23
bca
16.13
24
cab'
16.27
25
deb
11.03
.
26
adb
14.79
.
27
bdc
9-03
28
acd
14.18
29
1S ■
30
b
10.37
.
Si'
ba
15.74
■
32
■ bad
14.99
33
4
34
C
35 ’
cb
36
cbd
37
1IO
38
C
39
cd
4o
c da
8.01
10.06
10.98
8.18
.1 0 .9 4
9.78
10.21
9.92
15.28
'
83
rim-
41
S E Q . SPEC.
OBSERVATION
8 .6 0
1Il
9.43 -
42
d
43
da
14.86
44
dac
13.87
45
R 2
9.32
46
d.
10.99
47
dh
11.86 .
48
dha
16.08
The following contrasts are formed.
After run '24:
(8123
e2 3 l) ^ 14.25
8.51 - 16.13 +
10.77
8.51
1 6 .2 7 >
9.86
8.01
14.79 + 10.13
8 .0 1
1 6 .0 8 + 9 .6 6
= .38
After run 25:
^123 ~ 8 312)
14.25
-.6 7
After run.32:
(e2l4 “ ®l42) == 15.74
= 2.97
After run 48:
(e2l4 ■“ 0421^ = 15.74
= 1.31
84
After run 44:
(9134 " e4l3^
14.18
7.97 - 1 3 .8 7 + 8.60
.94
After run 40:
(0134 “ e34i) ~ l4.l8
7.97 - 15.28 + 9.78
= .71
After run 3 6 :
(9324 " 8 2 4 3 )
10.94
10.06 - 9 . 0 3 + 10.96
1.91
After run 3 6 :
( 8 3 2 4 "'8 4 3 2 ) ^ 10-94
10.06
- 1 1 .0 3 + 9 .6 6
= - .4 9
The contrasts5 the equivalence classes, and the con­
ditions of the model imply there are no three factor order
effects.• Four equivalence classes of four factor order
effects result by appling Lemma 2.1 and the fact that three
factor order effects are insignificant.
To estimate the
possible significance of order effects from these four
equivalence classes, an additional four runs are necessary.
85
RUN
S E Q . SPEC.
OBSERVATION
49
ABCD
15.02
50
BADC
13.23
51
ACDB
14.20
52
CBDA
1 5 .5 6
After run 50:
(61234
S2l43^ = 15.02
8.51
13.23 + 8 . 0 1
8.51
14.20' + 9 . 9 7
8.51
1 5 .5 6 + 1 0 .0 6
= 1 .2 9
After run 51:
Ce 12.3k ~ 61 3 4 2 ) = 1 5 *0 2
=
2.28
After run 5.2.:
(01234 “ ^324%) ^ 15.02
=
1.01
These contrasts together with the conditions of the
model and: the equivalence classes imply no order effect.
Also5 there are eight degrees of freedom for error which
give an estimate based on the replications of each of the
four factors for the first treatment on each unit.. The
estimate of
= . 6 5 or a
= . 8 0 is close to, the initial
A
estimate5 cr = 1 .
G
These are the same conclusions con-
cerning order effects that Prairie and Zimmer found in [10].
CHAPTER VI
SUMMARY AND EXTENSIONS
In experiments where the order of the application of ■
factors may he important, the 2p Factor Sequencing Design
proposed in [9] is of use in determining order effects.
The
design considered for 2p factorial experiments where the
factors are applied sequentially, the effects of the factors
are permanent, each unit may be tested p+1 times without
the test itself affecting the unit, and the low level of a
factor is the absence of the factor.
In this thesis, a model with explicit representation
of the order parameters is formulated.
In Chapter II, the
underlying assumptions regarding order effects are examined
and a system' with algebraic properties is developed to
assist the experimenter in the estimation and interpretation
of order effects.
The model used is a mixed model with random unit
effects and fixed treatment effects.
The design and
analysis of this model is presented in Chapter III along
with examples to supplement the text.
87
Because the number of runs required, is (p+l)p'3 the
size of the experiment becomes prohibative for p > 3.
Methods for reducing the number of runs without losing the
ability to estimate order effects are discussed in Chapters
IV and. V. ' The one at a time plans proposed in those sec­
tions use the systems developed in Chapter.II to form,
sequences of runs which allow for the estimation of members
of equivalence classes of effects.
These plans are very
useful in situations where the experimenter can achieve
quick results with small error.
The possibility of extending these designs to factors
with more than two levels is open to further study.
88
BIBLIOGRAPHY
1»
Anderson, T. W. An Introduction to Multivariate
Statistical Analysis. John Wiley and Sons,
New York.
1958.
2.
Cochran, W. C. and Cox, C. M. Experimental Designs.
Wiley, New York.
1957.
3.
Daniel, C. "One - at- a- Time Pla n s. 11 Journal of the
American Statistical Association, '
6 8 (1 9 7 3 ) : 353-360.
4.
Fisher, R. A. The Design of Experiments.
Boyd, Edinburgh.
1947•
5.
Grayhill, F. A. Introduction- to Matrices with Appli­
cations in Statistics. Wadsworth Publishing Company
Inc.-Belmont, California. '1 9 6 9 .
6.
Graybill, F. A. An Introduction to Linear Statistical
Models, Volume I. McGraw-Hill Book' Company, In c .,
New- YorkV 1961.
7.
Hunter, J . S. "A Sort of Least Squares Estimation After
Each Run." Technometrics, 6(1964): 41-58.
8.
Kempthorne, 0. The Design and Analysis of Experiments.
John Wiley and Sons, New York.
I9 6 7.
9.
Prairie, R. R. and Zimmer, W. J . "2^ Factorial Experi­
ments with the Factors Applied Sequentially."
Journal of the American Statistical Association,
59(1964): 1205-1216.
Oliver and
10.
Prairie, R. R . .and Zimmer, W. J . "Fractional Repli­
cations of 2'P Factorial Experiments with the Factors
Applied Sequentially."
J ournal of the American
Statistical Association, 63(1968X7 644-652.
11.
Searle, S . R. Linear Models.
New York.
1971.
John Wiley, and Sons,
APPENDIX
90
The matrices which appear on the following pages are
Q
for one replication of a 2 ^ Factor Sequencing Design.
The
matrix formulation of the design and analysis is discussed
in Chapter III.
'
Equation (3.3) which appears on page I9 is:
Y = X* Zm X + Wu + e.
Equation (3.6) which appears on page 27 is:
Equation (3.7) which appears on page 30 is:
T = S ®
I].
Equation (3.8) which appears on page 32 is:
Z = Rg
+ e.
Equation (3.9) which appears on page 33 is:
I
0
91
Matrix X*
I
0
0
0
0
0
0
0
0
0
I
I
0
I
0
I
0
I
0
0
0
0 '0
I
I
0
0
0
0
0
0
I
0
0
0
0
0 .0
0' 0
0 0
0 0
I. 0
0
0
0 ■0 ■ I
0 0 0
I
I
I .0
I I
I 0
I 0
I 0
I
I
I
I
0
0
0
0
0
0
I
0
0
0
0
O- 0 0
I 0 0
0 0 ■0
I
I
I
0
I
0
0
0
0
I
I
I
0
0
I
0
0
0
I
0
0
0 .0 0
0 .0 .0
0 I .0
0 0 0
0 0- '0
0 0 0
0 'I 0
I
0
0
0
0
0
I
0
0
Q '0
0
0
0 0 0
0 0 0
0 0 I
•0 0 0
0 0 0
I 0 0
0 0 I
0 0 0
'0 0 0
0■0 0
0 0 I
0 0 0
0 0 0
0 I 0
0 0 I
0 0 0
0 0 0
I 0 0
0 0 ■1
0 0 0
0 0 0
0 I 0
0
0
0
0
I
0 0
0 0
I 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 I
0 0
0 ■0
0 0
0 0
0 0
0 '0
0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
0 0' 0
0 0- 0
0 0 0
0 0 0
0 0 0
I 0 0
0 0 0
0 0 0
0 '0 0
0 0 0
0 0 :o
0 0 0
0 0 0
0 0 I
0 0 0
0 0 0
0 0 0
0 I 0
0 0 0
0 0 0
0 0 0
0 0 0
O
O
O
0
0
0. 0 O O O O O
0 0 O O O O O
0 0 O O O O 0'
0 .I O O O O O
0 0 O O O O O
o. 0 O O O O O
0 0 O O O O O
0 ■0 I O O O O
0 0 ■ o. O O O O
0 0 O O O O O
0 0 O O O O O
0 0 O I O O O
0 ;o O O O O O
0 ' O'- O O O ■ O O
0 ■0. O . O . O O O
0 O O O I O O
0 '0. O O O O O
0 O .O O O O O
0. O O O O O O
0 '0 'O O O I O
0 O .0 O O O , O
0 O O O O •O O
I -O O O O O O
0 O O O O O I
92
Matrix X
I
-.5
-.5- ;-.5
.5
.5
.5
-.5
O
.0
0
0
0
O
O
O
I
.5
-.5 '-.'5
-.5
-.5
.5
.'5
O
0
0
0
0
O
O
O
I
»'5
.5-
.5
-.5
- .5
-.5
I
0
0
0
0
O
O
O
I
.5
.5'
.5
.5
.5
.5
O
0
0
I
0
O
O
O
I
-.5
.5
•5
.5
-.5
O
0
0
0
0
O
O
O
I
.5
-.5
-.5
-.5
-.5
.5
o5
O
0
0
'0
0
O
O
O
I
.5
-.5
-.5
•5
- .5
-.5
O
I
0
0
0
O
O
O
I
.5
. *5
•5.- .5
.5
.5
.5
0
0 .0
I
O
O
O
I
-.5
.5
•5
.5
0
0
0
0
O
O
O
I
-.5
-.5
..5 ■ O
-.5 O
-.5
.5
-.5
o5
O
0
0
0
0
O
O
O
I
.5
.5
•5
-.5
-.5
-.5
-I
0 '0
0
0
O
o-
O
I
.5
•5
.5
.5
.5
.5
O
0
0
0
0
I
O
O
I
-.5
-.5.
-.5
.5
.5
.5
o5
-.5
O
0
■0.
0
■0
O
O
O
I
-.5
.5
-.5
-.5
•5
-.5
•5
O
0
0
0
0
O
O
O
I
-.5
.5
.5
-.5
-.5
.5
-.5
O
0
I
0
o'
O
O
O
I
'.5
.5
■■ .5'
.5
,5
.5
.5
O
0
0
0
O
O
I
O
I
-.5
-.5 ’- .5
•5
•5
.5
- .5
O
0
0
0
O
O
O
O
I
-.5
-.5
•5
c5
-.5'
-.5
<>5
O
0■ 0
0
O
O
O
O
I
.5
-.5
.5
-.5
.5
- .5
-.5
O -I
0 '0
O
O
O
O
I
..5
.5
.5
.5
.5
.5
.5
O
0
0
0
O- O
O
I
I
-.5
-.5
.5
.5'
.5
-.5
O
0
0
0
O
O
O
O
l'
-.5
-.5
-o5 . .5
•5
-.5
-.5
•5
O
0
0
0
O
O
O
O
I
-.5
.5
.5
-.5
-.5
•5
-.5
O
0 -I' .0
O
O
O
O
I
.5
•5
' .5
' .5
•5
.5
»5
O
0
-.5
.-.5 . -.5
•5'
-.5
■-.5"
' -.5
0 -I -I -I -I -I
Matrix T
0
.7071
-.7 0 7 1
.4082
.4082
-.8 1 6 5
.2887
.2887
.2887
0
0
o
0
0
- .8 6 6 0
0
0
0
0
0
.7071
-.7 0 7 1
.4082
.4082
- .8 1 6 5
.2687
.2887
.2887
0
0
0
0
0
O
0
O
O
O
O
O
O
0
O
O
O
O
O
O
O
O
O
0
0
O
O
O
O
O
O
O
O
0
O
O
O
O
O
O
O
O
O
0
O
O
O
O
O
O
O
O
O
O
O
O
O
O
0
0
0
O
O
O
0
, O
P
O
O
O
O
O
P
O
O
0
0
O
O
O
O
O
O
O
O
O
O
O
0
O
O
O
O
O
O
O
P
O
O
0
0
0
O
O
O
O
O
0
P
O
O
O
O
0
0
0
0
Q
0
0
0
0
0
0
0
0
O
.7071
-.7 0 7 1
O
O
O
O
O
O
O
0
0
0
0
0
0
0
0
O
.4082
.4082
-.8 1 6 5
O
O
O
O
O
P
O
O
O
0
O
O
0
0
0
0
Q
0
0
O
.2887
.2887
.2887
- .8 6 6 0
O
O
O
0
O
O
O
P
O
O
O
O
0
0
0
0
0
0
0
0
O
O
O
O
.7071
-.7 0 7 1
O
O
O
O
P
O
O
O
O
O
0
0
0
0
0
0
P
0
O
O
O
O
.4082
.4082
-.8 1 6 5
O
0
0
O
O
0
O
O
O
0
0
0
0
0
0
0
O
O
O
O
O
.2887
.2887
.2887
- .8 6 6 0
d
O
O
O
O
O
O
O
0
0
0
0
0
0
P
0
O
O
O
O
0
O
O
P
.7071
-.7 0 7 1
O
O
O
O
O
O
0
0
0
0
0
0
0
O
O
P
O
O
O
O
.4 0 8 2
.4082
-.8 1 6 5
O
0
O
O
O
0
0
0
0
0
0
0
O
O
O
O
0
0
O
O
0
O
O
.2887
.2887
.2887
- .8 6 6 0
P
O
O
O
0
0
0
0
0
0
0
O
O
O
O
O
O
0
0
O
O
0
0
0
P
P
0
O':
O
O
O
O
O
O
O
O
O
0
0
0
0
0
O
O
O
P
P
O
O
O
O
O
O
0
- .8 6 6 0
0
O
O
0
.7071
-.7 0 7 1
O
O
O
P
O
.4082
.4082
- .8 1 6 5
O
O
O
O
P
.2887
.2887
.2 8 8 7
- .8 6 6 0
Matrix R
.7 0 7 1
0
.4082
- .8 1 6 5
.2 8 8 7
- .5 7 7 4
.7071
0
.4082
0
.2 8 8 7
0
- .8 6 6 0
0
.7 0 7 1
0
- .4 0 8 2
.4 0 8 2
- .2 8 8 7
- .5 7 7 4
.7 0 7 1
.7 0 7 1
- .8 1 6 5
.4 0 8 2
- .4 0 8 2
- .5 7 7 4
- .5 7 7 4
- .2 8 8 7
- .8 6 6 0
0
0
0
.8 1 6 5
.4 0 8 2
- .2 8 8 7
- .5 7 7 4
0
0
0
8165
0
0
0
2887
0
0
0
0
- .7 0 7 1
0
0
0
0
0
0
0
0
8165
0
0
0
2887
0
0
0
0
0
0
0
0
0
0
0
0
0
0
- .8 1 6 5
0
0
0
0
0
.2 8 8 7
- .8 1 6 5
.4 0 8 2
.8 1 6 5
- .4 0 8 2
.4 0 8 2
- .2 8 8 7
- .5 7 7 4
- .5 7 7 4
- .2 8 8 7
- .2 8 8 7
- .5 7 7 4
0
0
0
.8 6 6 0
0
0
- .4 0 8 2
0
0
- .8 6 6 0
0
0
- .8 6 6 0
0
0
0
0
0
0
0
0
.8 1 6 5
0
0
0
0
0
- .2 8 8 7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
- .7 0 7 1
.7 0 7 1
.7 0 7 1
- .7 0 7 1
0
0
- .4 0 8 2
.8 1 6 5
- .4 0 8 2
.4 0 8 2
.4 0 8 2
0
- .2 8 8 7
- .2 8 8 7
- .2 8 8 7
- .5 7 7 4
- .5 7 7 4
0
.7 0 7 1
.7 0 7 1
- .7 0 7 1
0
0
0
0
0
.7 0 7 1
- .7 0 7 1
0
.4 0 8 2
- .7 0 7 1
0
0
- .7 0 7 1
0
0
0
.7 0 7 1
- .5 7 7 4
- .8 6 6 0
0
0
- .5 7 7 4
.5 7 7 4
0
0
.8 1 6 5
.8 1 6 5
0
.2 8 8 7
- .2 8 8 7
0
0
0
0
- .2 8 8 7
0
0
0
- .5 7 7 4
- .2 8 8 7
.8 6 6 0
0
0
- .2 8 8 7
.5 7 7 4
.7 0 7 1
0
0
0
.8 1 6 5
0
0
0
.4082
0
0
- .8 1 6 5
.7 0 7 1
0
0
0
-.4 0 8 2
0
0
.4 0 8 2
0
0
0
- .8 6 6 0
.8 1 6 5
0
0
0
0
- .4 0 8 2
0
0
- .7 0 7 1
- .7 0 7 1
- .8 6 6 0
0
- .7 0 7 1
- .8 1 6 5
- .4 0 8 2
.8 1 6 5
.4 0 8 2
- .4 0 8 2
.4 0 8 2
0
0
.8 1 6 5
- .5 7 7 4
- .2 8 8 7
- .2 8 8 7
- .5 7 7 4
- .2 8 8 7
- .5 7 7 4
0
0
- .2 8 8 7
.8 6 6 0
.8 6 6 0
.8 6 6 0
- .8 6 6 0
.8 6 6 0
0
0
0
- .8 6 6 0
.8 6 6 0
Matrix (R 1R )"1
.4271
-.1510
- .1 5 1 0
-.0 1 5 6 2
-.0 1 5 6 2
.03124
-.04168
-.1510
.4271
- .1 5 1 0
-.0 1 5 6 2
.03124
-.0 1 5 6 2
-.04168
-.1 5 1 0
-.1 5 1 0
.4271
.03124
-.01562
-.0 1 5 6 2
-.04168
-.01562
-.01562
.03124
. 26 o4
-.06772
-.0 6 7 7 2
- .0 1 5 6 2
.03124
-.0 1 5 6 2
-.0 6 7 7 2
.2604
.03124
-.01562
-.0 1 5 6 2
-.06772
-.06772
- . 04 168
-.04168
-.04168
.1563
.1563
0
-.1 5 6 3
0
0
0
-.1 5 6 3
-.0 9 3 7 7
.1563
- .1 5 6 3
-.0 9 3 7 7
.2500
- .0 6 2 5 3
-.1 8 7 5
.02085
.2500
- .1 8 7 5
-.06253
0
-.06253
.2500
- .1 8 7 5
-.1 0 4 2
.02085
0
-.09377
0
.09377
-.1042
.02085
-.0 8 3 3 4
0
.1563
.1563
.1563
0
0
.2500
.2500
.1563
- .6 2 5 3
-.1 8 7 5
-.1 8 7 5
-.0 6 2 5 3
0
- .1 5 6 3
- .1 5 6 3
0
-.0 9 3 7 7
-.0 9 3 7 7
- .6 7 7 2
0
-09 3 7 7
0
.09377
.2604
0
.09377
.09377
0
0
.2084
0
0
.09377
0
.8750
.06252
.09377
0
.06252
0
0
.02085
-.1 0 4 2
.08334
0
.8750
.06252
.1250
• 3751
.8750
.06251
.06251
.3751
.1250
0
1.417
.08334
0
.1250
.3751
0
-.0 8 3 3 2
-.1042
-.1 0 4 2
.08334
.02085
-.06253
-.1 8 7 5
.2500
-.8 3 3 4
.02085
-.1042
0
0
0
.375
0
.1250
0
0
0
.1250
0
-.06253
.08334
.3751
0
.2500
.02085
-.0 6 2 5 2
- .0 8 3 3 4
-.1 8 7 5
-.1042
- .3 7 5 1
0
0
-.06253
.2500
-.1875
.02085
.08334
-.1 0 4 2
0
-.3 7 5 1
0
- .1 8 7 5
.2500
-.0 6 2 5 3
- .1 8 7 5
- .0 6 2 5 3
.2500
-.1 0 4 2
.08334
.08334
.02085
6
- .1 2 5 0
0
.02085
-.1042
0
0
-.3751
.1 2 5 0
.3751
-.0 8 3 3 2
-.3 3 3 4
- .3 3 3 4
-.3334
1.417
- .3 3 3 4
- .3 3 3 3
-.3334
-.1 2 5 0
.1250
- .3 3 3 4
-.3334
1.417
-.0 8 3 3 2
-.3333
.3751
- .3 3 3 4
-.3333
-.0 8 3 3 2
1.417
-.3 3 3 4
-.1 2 5 0
- .3 3 3 4
-.3334
-.3333
-.3 3 3 4
1.417
'96
Matrix R n
~ J-
0
■.7071"
.4082 - - . 8 1 6 5
.2887
-.5 7 7 4
.7071
.4 0 8 2 .
.2887
' 0
0
0
-.8 6 6 0
0
0
-.8 6 6 0
0 •
-.8 1 6 5 '
-.5 7 7 4
.7071
4.4082
■-.2887
.7 0 7 1 .
.4082
-.5 7 7 4
.8165
.■ - . 7 0 7 1
T.4082
.5 7 7 4
-.2887
■0
0
-.07071
.-.4082
-.8 1 6 5
.4082
.8660
-.2887
-.5 7 7 4
-.5 7 7 4
0
0
-.8 6 6 0
0
0
0
-.7071
.8165
0
-.4 0 8 2
-.8 6 6 0
.5774
0
0
-.2887
-.8165
-.7 0 7 1
-.4 0 8 2
. 8 6 6 0 .- . 5 7 7 4
-.2887
0
...
.7071
-.4 0 8 2
-.2887
.7071
0
.8165
-.2887
0
.8165
-.2887
.7071
.4082
-.5 7 7 4
.7071
-.4082
-;2887
0
.8165
-.2887
0
.8165
-.2887
.7071
-O4082
-.2887
.7071
■ .4 0 8 2
-.5774
0
-.7 0 7 1
.8165
-.2887
-.5774
0
-.7 0 7 1
.8165
-.2887
.4082
.4082
-.5 7 7 4
.7071
.4082
-.7 0 7 1
-.5 7 7 4
-O5774
. .7071
-.4 0 8 2
-.7 0 7 1
-.2887
-.5 7 7 4
.4082
.4082
.7 0 7 1 ’ - . 7 0 7 1
.4082
.4082
-.5774
-.5 7 7 4
.7071
-.4082
-.2887
-.7 0 7 1
.4082
-.5 7 7 4
4
D378
E198
Hansen, James Leonard
cop. 2
One at a time plans for
2^ factor sequencing
designs
^AMt ANP AbowEBa
»Lwz