•The Use of Orthogonal Polynomial Contrasts in the

Confounding of Factorial Experiments

by

Dirk van der Reyden

,.' Institute of Statistics,{

t Mime0 Series No. 181~""

June, 19.5'1

r'

TABLE OF CONTENTS

List of Tables

CHAPTER

iv

Page

vii

1

2

3

4

Introduction and Review of Literature

1.1 Introduction

1.2 Review of Literature

1.2.1 Symmetrical Factorial Design1.2.2 Asymmetrical Factorial Design1.2.3 Fractional Factorial Experiments1.2.4 Orthogonal Polynomials

1.3 Notation To Be Used

PART I. THE GENERAL CASE OF ORTHOGONAL OONrRASTS

Matrix Notation

1he Effects of Confounding a Single Contrast

3.1 Theorems

3.2 Bundles of Contrasts

The Effects of Confounding Two Contrasts Simultaneously

PART II. THE ORTHOGONAL POLYNOMIAL SYSTEM OF CONTRASTS

The Orthogonal Polynomials

5.1 Introduction

5.2 Partial and Complete Confounding

5.3 Partitioning of the Polynomial Values

5.4 Bundles of Homogeneous Order

1

1

5

79

1214

16

19

26

26

34

36

46

46

47

50

53

CHAPTER

v

Table of Contents (Continued)

Page

6 The Confounding Pattern of a Contrast

6.1 Bundles of Heterogeneous Order

6.1.1 Confounding on (AuBu.)6.102 Confounding on (AuBe )6.1.3 Confounding on (AeBe )6.104 Summary

56

56

56596162

6.2 Rules for Obtaining Confounding Patterns 64

6.2.1 Introduction 646.2.2 Polynomial Confounding Patters for

an m x n Experiment 656.2.3 A General Rule for Obtaining the Con-

founding Patterns of Any PolynomialContrast in an m x n Factorial Experiment 67

6.2.4 Polynomial Confounding Patterns for anm x n x p Experime~t 68

7

6.3 Determination of the Coefficients of theBlock Constants

The Confounding Pattern of Two Contrasts

7.1 Simultaneous Partitioning of Two OrthogonalPolynomials

70

73

73

7.2 The Generalized Interaction 75

7.3 Reduction Formulae for Products of OrthogonalPolynomials 76

7..4 The Status of the Contrasts in the LinearCombination 78

7 ..5 Confounding Patterns of Two Contrasts ConfoundedSimultaneously 79

8 Confounded ]OCperiments With Equal-sized Blocks

8..1 Confounding a Single Highest Order InteractionContrast

84

84

CHAPTER

8

vi

Table of Contents (Continued)

Page

(Continued)

8.2 Simultaneous Confounding of Two Highest OrderInteraction Contrasts

8.3 Experiments in Equal Blocks With Main EffectsUnconfounded

8.3.1 Two-factor Experiments8.3.2 Three-factor Experiments

89

94

9495

8.4 Confounding of Linear Combinations of Contrasts 99

8.5 Balanced Confounding 101

9 Summary, Conclusions and Suggestions for FurtherResearch

9.1 Summary and Conclusions

9.2 Further Research Needed

LITERATURE CITED

104

104

105

109

Table

1

2

3

4

5•

6a

6b

7

8a

8b

9

10

11

12

vii

LIST OF TABLES

Page

The orthogonal contrasts of the m x n x p factorialexperiment" 24

Conformable submatrices of (A B Ct ). 28r s

Signs of the conformable parts of (A B Ct)' (A~B Ch ),(A Be) r s r-g I.,

v Vi Z .. Lj...L

Signs and absolute values of the orthogonal polyriomials. 48

Numerical values of pl =AP • 49r r

Partitioning of the orthogonal polynomials. 52

Sums of elements of conformable parts. 53

Allocation of contrasts to homogeneous bundles. 55

Blocking pattern due to confounding of (.AuBu) whenm andn are odd. 57

Sums of elements of AE I(AuBu) (y = 1) correspondingto blocking pattern in Table 8a. 58

Blocking pattern due to confounding of (!nBe ) andcontrast totals. 60

Blocking pattern due to confounding of (AeBe ) andcontrast totals. 61

Confounding of general contrasts when specific ABcontrasts are confounded.. 63

General procedures for determining all confoundedcontrasts using polynomial confounding in a two-factor experiment. 66

13

14

15

16

Generalization of the parts of Table 12"

Confounding- pattern for any polynomial contrast inan m x n x p factorial experiment.

Confounding pa.tterns of the four basic contrasts ofhighest order of an m x n x p factorial experiment.

Products of polynomials"

68

69

77

'fable

17

18

19

20

21

22

24

List of 'fables (Continued)

Form of generalized interaction contrast.

Rule for obtaining homogeneous bundles in thegeneralized interaction of the contrasts (rs,fg) ofan m x n experiment

Examples of contrasts which satisfy condition (8.3),with at least two contrasts having zero elements.

Frequencies of block indicators in the orthogonalpolynomials.

Size of blocks when (.A:rBs,AfBg ) is confounded undercertain conditions.

Signs of the elements of the orthogonal polynomialvectors.

Confounding contrasts of highest order not confoundingmain effects and yielding four equal blocks when takenin pairs.

New three-factor experiments in four equal blocks withmain effects unconfounded.

viii

Page

78

82

86

88

90

93

97

99

Errata

The Use of Orthogonal Polynomial Contrasts in theConfounding of Factorial Experiments

Page 7, line 4: for sn/k read en/k.

Page 10, line 13: for electrotonic read electronic.

Page 16, line 13: for at read of.

Page 19, line 3 of second paragraph: for very read vary.

Page 23, third line from bottom: for (J~, J~)' read (J~, J~).

Page 25, line 1: for (J , J') read (JI, J').- + - +

Page 29, equation (3.2):

Page 30, equation (3.3):

read (A B Ct ) •r s +

read (A B Ct ) •r s +

for (tr b ) read (tr b ).e- -s-

for (A B Ct )r sfor (A B Ct )r s

(3.3.1), fourth line:Page 32, equation

Page 37, third line from bottom: for ar read a-+ r-+

Page 39, line 15: for were read where.

Page 43, first line of Theorem 4.1: delete T in (ArBsCtT,AfBgCh)'

Page 47, line 3: tor Pri read Ipril.

Page 48, N even, P3

: for 3(N2.4) read 13(N2-4)\ •

Page 50, third line from bottom: tor (p ,P , ••• ) read P2 P4 .,,)., ,Page 54, third line from bottom: for Hnece read Hence.

Page 57, title of Table 8a: for Blooking read Blocking.

Page 58, Table 8b, last column: tor sE read -sEao ao

Page 65, line 11: for othogonal read orthogonal.

Page 70, line 11: for J'a b c J read J'a b 0tJ.r ePage 89, line 6: for afibgj ) read (afibgj ).

Page 90, Table 21, fourth line: for n + n read n++ + n++ +-

Page 91, line 8: for Ph+ =Ph- read Ph+ =Ph.'

Page 92, equation (8.8): for (r - r ) read (n - n ).g+ g- g+ g-

Page 102, sixth line from bottom: tor A2B2Cl read A2B1Cl •

CHAPTER I

Introduction and Review of Literature

1.1 Introduction

Factorial experiments are used in experimental stiuations that

require the examination of the effects of varying two or more factorso

The various representatives of each factor are designated as levels, re~

gardless of whether they are continuous or discreteo In the complete

exploration of such a situation all combinations of the different factor

levels must be examined in order to elucidate the effect of each factor

and the possible ways in which each factor may be modified by the variation

of others.

When a factor is investigated at two levels, its main effect is

uniquely defined as the difference between the mean of the results at the

higher level and the mean of the results at the lower level. Its main

effect can thus be expressed as a unique contrast between the higher level

mean and the lower level mean. Interaction effects between two or more

factors, each at two levels, are also uniquely defined by similar con­

trasts. Thus if, in a 23 factorial experiment, the factors are denoted

:1

w~ere the subscript 0 refers to the lower level, and if Yijk denotes the

response to the treatment combination aibjck' the contrasts can be pre­

sented as follows:

2

Main

Response effects Interaction effectsA B C AB AC BC ABC

Yooo -1 -1 -1 +1 +1 +1 -1

-1 -1 +1 +1 -1 -1 +1YOOl

Y010-1 +1 -1 -1 +1 -1 +1

:r0 11 -1 +1 +1 -1 -1 +1 -1

Y100+1 -1 -1 -1 -1 +1 +1

Y101+1 -1 +1 -1 +1 -1 -1

Yuo+1 +1 -1 +1 -1 -1 -1

Y11l+1 +1 +1 +1 +1 +1 +1

It will be observed that any two of the contrasts are orthogonal;

ioeo, the sum of products of their corresponding coefficients is zeroo

This property of the contrasts makes for easy analysis, and ensures that

all the main effects and interactions can be independently estimated..

When the number of factors increases, the complete factorial design may

become too large to be accommodated under uniform experimental conditionso

The totality of treatment combinations is then subdivided into smaller

groups, each group being assigned to a given block of experimental units,

the grouping being made so that block effects will not affect the main

effects of the factors and those interactions regarded as being of impor-

tance 0 This process is termed "confounding 0 tl

Confounding of a contrast in the 2n series is accomplished by placing

these treatment combinations corresponding to the positive coefficients in

one block and those corresponding to the negative coefficients in another

blocko These parts of the contrast are now non-estimable from the data,

3

and we say that the contrast is completely confounded with block effects,

or in the usual parlance, with blocks ..

Even though the device of confounding may take care of the hetero­

geneous material or conditions, with a large number of factors the total

. number of observations may become prohibitive.. Under certain conditions

it is then possible to examine the main effects of the factors and their

more important interactions in only a fraction of the number of treatment

oombinations required for the complete factorial experiment. This type of

design is known as a fractional factorial and is always equivalent to one

block in a system of confounding. In the 23 experiment, for instance, the

block containing the treatment combinations corresponding to either the

positive signs of ABC or to the negative signs of ABC would be a one-half

replicate ..

With factors at three or more levels, their main effects are no

longer uniquely expressible as contrasts between the different levels ..

It is well-known that the infinite number of contrasts that can be set up

between N observations can be divided into an infinite number of subsets

consisting of (N-l) orthogonal contrasts, with the property that the sum

of squares of each of the contrasts in the subset add up to the total sum

of squares of the observations, oorrected for the mean. In this thesis

exclusive attention will be paid to the subset of orthogonal contrasts

based on the orthogonal polynomial values tabulated by Fisher and Yates

(1938) and other authors.. Apart from satisfying the orthogonality oriteria,

these contrasts have usefUl interpretational value for an experiment in­

volving factors with continuously varying levels; these being eqUally

spaced in the experiment.. The tables of the orthogonal polynomial values

have been set up in such a way that any value is either a positive whole

number, a negative whole number, or a 2rero ..

4

If a particular contrast is to be confounded with blocks and if it

contains no zero elements, a blocking procedure identical to the one for

the 2n experiment is applicable" If the contrast contains zero elements,

a third block is obtained by placing those treatment combinations corres­

ponding to the zero elements in yet another bl9ck"

In general,\l when the number of levels exceed two,jl complete confound­

ing does not occur with orthogonal polynomi8J... contrasts. Even though the

treatment combinations corresponding to the negative coefficients of, say

the linear contrast, are put in one block and those corresponding to the

positive coefficients are put in a second block, it is possible to esti­

mate a linear contrast in each block based, of course, on only one half

the number of observations. These two estimates can then be pooled to

give a single estimate of the linear effect, adjusted for blocks. If

there are no block effects,Il the variance of the adjusted linear contrast

will be much larger than one not adjusted for blocks, because of the re­

stricted range of the independent variable in each block" The actual

efficiency of the adjusted estimate will depend on the reduction of error

variance due to blocking.

A complete discussion of partial confounding in this sense should

include a method of estimating all pertinent contrasts and their standard

errors. This thesis, however,ll will be concerned only with the blocking

procedure5 •.

Although use has been made of orthogonal contrasts to obtain con­

founded designs, in other than the 2n series,ll no systematic study appears

to have been made either on the theory of general orthogonal single con­

trasts, or on the special system of orthogonal polynomial contrasts.

5

Whatever number of levels the factors may have, in many experimental

situations practical interest is mainly centered on the estimation of the

main effects and the lower order interaction contrasts, especially those of

second degree (linear x linear). One reason for this interest is the dif­

ficulty of interpreting higher degree interaction contrasts. Most designs

appearing in the literature were developed from the point of view of ob­

taining balanced sets of confounded blocks. It therefore appears to be of

interest to investigate the possibility of obtaining designs with the more

limited objective in mind. Considering the m x 19. x p experiment from this

point of view, one would be satisfied if a confounded design could be found

in which the main effects and most, if not all» of the linear by linear

interaction contrasts were unconfounded or only slightly confounded.

Simple a.nalyses and relatively precise estimates of these contrasts can be

obtained from a balanced set of replicates, but in many practical situa­

tions the total number of observations thus required would be prohibitive.

With this background the objectives of this thesis may be summarized

as follOWS: (i) To investigate the theory of confounding general orthogonal

contrasts; (ii) to establish mathematical procedures and rules to obtain

the confounding patterns of higher order interaction contrasts, based on

the orthogonal polynomial system.

102 Review of Literature

Prior to 1926, when Fisher (1926) first suggested confounding, the

factorial experiment was known as the ueomplex tl experiment.; indeed, after

many years it was still referred to as the complex experiment. This type

of experiment had been used by the Rothamsted Experimental Station on

wheat experiments at Broadbalk since 1843 and on barley experiments at

Hoosfield since 1852; the randomization element, however, was lacking in

6

these early years 0 Fisher and Wishart (1930) gave a detailed explanation

of a confounded experiment, and Yates (1933) discussed the principles of

confounding, giving different types of confounding and setting out methods

of analysiso Later, Yates (1935) gave some more illustrations of confoundei'i

designs and discussed their relative efficiency compared with randomized

complete blocks o

Ever since the first appearance of the book liThe Design of Experi­

ment n by Fisher (1935) and the monograph "The Design and Analysis of

Factorial Experiments" by Yates (1937), experimenters have used factorial

designs in a great variety of situations in many fields of researcho This

has led to a voluminous literature on all aspects of the subject. Mathe­

matical statisticians became interested and found new explanations for

existing designs in terms of the results of combinatorial mathematics.

The utilization of these results, in turn, led to the development of new

designs. Special mention will be made of the work of Bose and his co­

workers who gave the complete solution of the symmetrical factorial design.

Recently Binet et al (1955) re-examined most of the factorial plans

published to determine what effect the confounding had on the orthogonal

polynomial contrasts; in recent years these contrasts were found to be

important in certain fields of chemical and industrial researcho The

authors also presented new single-replicate plans, some of which were ob­

tained by confounding of the higher order high degree orthogonal polynomial

contrasts. Anderson (1957) reviews the whole field of complete factorials,

confound and fractional factorials and explains the interrelationship of

these subjects 0 The present thesis is an attempt to formal ize some of

the techniques of Binet and his co-workers, and can hence be regarded as

providing the theoretical background for some of their practical findingso

7

1 0201. Symmetrical factorial designs

A factorial design is symmetric when each of the n factors is at s

levels, where s = pm and :£ is a prime number. Confounding occurs if each

complete replication is allocated to b =- sn/k blocks of ! plots each,

where ! divides sn 0 The problem that arises is how to allocate the treat-

ment combinations to the various blocks in such a way that only certain

desired treatment contrasts become non-estimab1eo Bose and Kishen (1940)

and Bose (1947, 1950) showed that by identifying the! levels with the

elements of a Galois field GF(pm), any treatment combination can be repre­

sented by a point in finite Euclidean space EG(n, pm)o If any linear

homogeneous form U = aJ.X1. + aax2 + ••• + anxn is considered, where .the ! IS

belong to GF(pm) then U = c, where £ is a constant, will represent a flat

space of (n-1) dimensions. If the constant varies over all the elements,

U = c represents a pencil P(aJ.,aa, •• o, an) of parallel (n-l)-flats. To

each flat of the pencil corresponds a set of (n-l) treatment combinations,

and the contrasts between these ! sets carry (s-l) degrees of freedom •.

Thus the (sn_l ) degrees of freedom can be split into (sn_1 )/'(s_1) sets of

(s-l) degrees of freedom each, such that each set is carried by one parallel

pencil, and the degrees of freedom belonging to different sets are orth-

ogona10 If 1 particular coordinates of the penoi1 P(aJ.,a2,···,an ) are

nonzero, then the (.s-l) degrees of freedom carried by that pencil belong

to that particular (t-1)th order interaction. Thus if only one coordinate

of the pencil is nonzero, the (s-l) degrees of freedom will belong to that

particular main effect.

In practice these pencils can be obtained by means of the orthogonal

latin squares, since Bose (1938) showed that a latin square of prime order

8

can be regarded as the addition table of the roots of the cyclotomic poly­

nomial, these roots being the elements 'of a Galois field.

It follows that when s=2, and the eXperiment:is confounded into two

equal blocks, a single contrast only will become nonestimable. When, how­

ever, two contrasts of the 2n experiment are confounded, Barnard (1936)

showed that this implies the automatic confounding of another contrast, the

generalized interaction. He also gave various systems of confounding,

using factors up to 6 in number.

As far as symmetrical experiments are concerned, Yates (1937),

in his comprehensive monograph, gives designs, analyses and examples for

various confoundings of the 2n , 3D and 4n experiments. When s= 3, he

makes use of the combinatorial properties of the orthogonal latin squares

to define his orthogonal components 1. and:1., each with two degrees of free­

dom. In terms of Bose's development these components correspond to the

pencils of parallel 2-flats. For s ::: 4, Yates derived confounded experi­

ments by defining single degree of freedom contrasts, based on two pseudo­

factors of two levels each, and blocking according to signs:",-similarly for

s = 8. He also suggested a routine of calculation for experiments with

several factors at two levels each, this depending only upon a succession

of additions and subtractions of pairs of numbers.

Nair (1938, 1940) discussed balanced confounded arrangements for

the 4n and :f experiments. Fisher (1942) established the connection be­

tween the theory of finite Abelian groups and the relations recognizable

in the choice of interactions for confounding contrasts of the 2n experi­

ment. He showed that, using blocks of 2r plots, it was possible to test

all combinations of as many as (2r -1) .factors in such a way that all inter­

actions confounded shall involve not less than three factors each. He also

9

supplied a. catalog of systems of confounding available up to 15 factors,

each at two levels. Fisher (1945) generalized this proposition to factors

having s = pm levels, where E is a prime number.

Ke1'l'lpthorne (1947) introduced a new notation for the different

sets of degrees of freedom in the sn experiment to simplify the enumeration

and choice of systems of confounding. Comparing his notation with that of

Yates and Bose, we have, for instance, for the 32 experiment,. the following

components, each with two degrees of freedom:

Factor Yates Kempthorne Bose

A A A P(l,O)B B B P(O,l)

AB(I) AB P(l,l)AB AB(J) AB2 P(1,2)

Cochran and Cox (1950) survey the field of factorial experiments

mainly from an applied point of view, discuss detailed examples, give eom-

plete plans of those e~eriments most likely to be generally used in practice

together with details on the confounding and the relative information. In

his book Kempthorne (1952) devotes considerable space to a discussion of

the methods of confounding, the statistical models, methods of estimation

and the assumptions involved in the various confounded designs he presents.

Binet et al (1955) used the Bainbridge extension of Yates t (1937) technique

to examine the effects of the confounding systems proposed by the previous

authors on the orthogonal polynomial contrasts.

10202. Asymmetrical factorial designs

A factorial design is asymmetric when the ~ factors are respectively

Since G~lois fields do not exist for products of prime numbers, no single

10

mathematical system is available to deal with the confounding of the asym­

metrical designo Of course, if the levels of a certain number of factors

are the same prime number or power of a prime number, the general theory

can be applied to that part of the experimento Thus in a 5 x 32 experi­

ment the 32 part can be confounded as usual. In general, however, the

single replicates of such confounded designs are not very satisfactory

from a practical point of view, since low order interactions will be con­

founded. This difficulty can be overcome by obtaining balanced sets of

replications, but such a procedure usually requires a triplication or more

of the total number of observations o

With a general mathematical theory lacking, much of the success in

obtaining satisfactory confounded designs will depend on the ingenuity of

the designero With the availability of electrotonic calculators, it is

conceivable thatgood practical designs may be obtained mechanicallyo

As far as asymmetric designs consisting of combinations of smaller

number of factors of 2, 3 and 4 levels are concerned, it would appear

that Yates (1937) and Li (1944) have supplied most of those likely to be

required in practice 0 The designs presented by these authors are balanced

with respect to the partial confounding of interactions, and more than one

replicate will usually be requiredo

When the number of levels of each factor is either 2, 4 or 89 Yates

(1937) employed the signs of contrasts with single degrees of freedom as

a basis for confounding o Consider the case of a factor A with four levelso

Since 4 :: 22, the factor A can be regarded as consisting of two pseudo­

factors X and Y, each at two levels9 and hence the pseudo-effects are:

11

x :: (x-l)(y+l) = xy + x -y - (1)

Y :: (x+l)(y-l) :: xy.- x + Y (1)

XY :: (x-I) (y-l) :: xy - x - y + (1)

Let ao :: (1), a1. :: x, aa :: y:; a3

:: xy. Then

Ai :: a3

+ aa a1. ao:: Y

All :: a3 - aa a1. + ao :: XY

A"! = a3 - aa + a1. - ao

:: X

and it will be seen that

If the levels are equally spaced and if Al , Aa and A3 denote the linear,

quadratic and cubic orthogonal polynomial contrasts, then

~:: 2At -+ Alit

Aa :: A"

A3:: _Ai + 2AIII

For confounding purposes it should be noted that there is a one-to-one

correspondence between the signs of Ai and A1.' All and Alia' Alii and A3 •

Hence, since the confounding of any contrast proceeds on the basis of

the signs only, identical blocking arrangements will result with both

systems of contrastso

Employing Yates! techniques, Li (1944) constructed confounded

designs for 10 types of the asymmetrical factorial experiment with the

purpose of filling some of the gaps existing among the plans previously

available 0 Nair and Rao (1948) developed a set of sufficient combinatorial

conditions for the balanced confounded designs, and discussed the

12

requirements for optimum designs& Thompson and Dick (1950) showed how to

obtain factorial experiments in blocks of equal size, equal to or less

than the number of levels in the leading factor from the orthogonal latin

squares & Thompson (1952) presented the layouts and details of statistical

analysis for two and three-factor designs, the number of plots per block

in no case exceeding five.

Binet et al (1955) re-examined the asyrmnetric designs proposed by

Yates and Li for the effects of the confounding on the orthogonal poly­

nomial contrasts. Detailed tables are presented, showing which orthogonal

polynomial contrasts are confounded, and the size of the coefficients of

the constants of the block contrasts appearing in the expected values of

the treatment contrastse Using either single orthogonal polynomial con­

trasts of high order interaction, or the methods of Yates, as a basis for

confounding, they present new single-replicate designs with confounding

patterns & Their investigation is entirely of a practical nature. The

present thesis supplies the theory of confounding a single orthogonal

polynomial contrast and the simultaneous confounding of two or more poly­

nomial contrasts.

le2.3 Fractional Factorial Experiments

Referring to the mathematical summary at the begi~ning of Seotion

le2.1, fractionalization of a factorial experiment wi~l occur when in­

stead of taking all the £ blocks, we take only ~ blocks where ~ is less

than b. We then have a Bib fraction of a complete replicate. Bose

(1950) summarized the mathematical procedures to obtain fractions of

factorial experiments, when the number of levels is either a prime number

or a power of a prime number. He showed that contrasts are no longer

separately estimable, but that the sum of the contrasts belonging to any

13

complete alias set is estimable. He pointed out, however, that i£ all

contrasts o£ any alias set with the exception o£ one contrast are o£ su£­

£iciently high order to be negligible, then this one contrast is estimable.

Bose and his co-workers have recently expanded the original theory and de­

veloped a whole series o£ new designs; these are to be published by the

National Bureau o£ Standards.

Finney (1945) originally developed the principles o£ £ractional

replication with particular re£erence to the 2n and 3n £actorial experiments

in terms o£ £initeAbelian groups. Finney (1946) presented a popular ex­

position o£ these ideas, together with some plans and examples; and Chinloy,

Innes and Finney (1953) gave an interesting example o£ the use o£ a one­

third replicate o£ a 35 experiment on sugar cane manurin~. An interesting

generalization o£ Yates' (1937) routine o£ calculation was included in

this paper.

Plackett. and Burman (1946) discussed what they call optimum multi­

£actorial experiments which Kempthorne (1947) showed to be £ractionally

replicated designs. In his book, Kempthorne (1952) treats the general

case o£ £ractional repl~cation, and describes a one-ninth replicate o£ a

36 design applied to measurements on the consistency o£ types o£ canned

rood. Brownlee, Kelly and Loraine (1948) enumerated subgroups suitable

£or high-order £ractional replication £or 2n designs o£ practical interest.

Davies and Hay (1950) gave special attention to the use o£ £ractional

£actorial designs in sequence with special re£erence to industrial ex­

periments. Kitagawa and Mitome (1953) and Davies et al (1954) compiled

plans £or £ractional designs o£ the 2n and 3n series. Daniel (1956)

discussed £ractional replication in industrial research.

14

Box and his co-workers [see Box (1952, 1954), Box and Wilson (1951),

Box and Hunter (1954), Box and Youle (1955)] have recently opened new

fields in which complete and fractional factorial experiments are fruit-

fully employed. Their work is devoted to the problem of determining opti-

mal factor combinations in chemical and industrial research, and to describe

the response surface in the neighborhood of this optimum. These procedures

are sequential in nature and have been described and reviewed by Anderson

(1953) and Davies et al (1954).

No work, as such, haS been done on the fractionalization of the

asymmetrical factorial experiment.

1.2.4 Orthogonal polynomials

The orthogonal polynomials used in statistics are discrete cases of

the well-known Legendre polynomials in mathematics 0 The latter are usually

defined between the limits of -1 and 1 by the formula of Rodrigues

Using results from the finite calculus, one can show that the discrete

orthogonal polynomials for n observations between the limits a and b

can be written as

r thwhere D denotes the r advancing difference, and or is an arbitrary

constant.

The orthogonal polynomial systems proposed by various authors for

the equi-distant case differ only in the choice of the arbitrary constant

cr ' this choice being dependent on the purpose the author had in mind.

15

Tchebycheff (1859) first gave the theoretical basis of the orthogonal

polynomials of least squares" He treats the nonequidistant as well as the

equidistant case, and for the latter derived a reduction formula" Poincare

(1896) developed similar polynomials for the nonequidistant case. Gram

(1915) developed orthogonal polynomials for the purpose of smoothing em-

pirical curves" Jordan (1929, 1932) simplified Tchebychefffs methods for

practical a.pplication, presented tables and explained the method of suc-

cessive summation of the observa.tions. He chose the arbitrary constant in

such a way that SSP (x) = n, where SS denotes the sum of squares.r

In the system proposed by Esscher (1920) the value of c r was de-

termined by the convention that the coefficient of xr shall be unity.

In his later system Esscher (1930) chose the same convention as Jordan.

A similar choice of arbitrary constant was made by Lorenz (1931), who de-

veloped his system of orthogonal polynomials-by means of determinants.

Fisher (1921) derived the polynomials in which x is measured from

the mean and adopted the convention that the coefficient of xr shall be

unity" Allan (1930) derived the general expression of this polynomial.

In his studies on the regression integral, Fisher (1924) presented orthogo-

nal polynoIllials such that SSPr(x) equalled unity" Fisher and Yates (1938)

present standard tables of the polynomial values for ~ through 52, and

r through 5. Anderson and Houseman (1942) extended these tables to n

through 104. DeLury (1950) presents values and integrals of these

polynomials up to n = 26 for all degrees.

Aitken (1933), by deriving new forms for the orthogonal polynomials

showed in a remarkably lucid way the relationship between the theory of

interpolation and the theory of orthogonal polynomial curve fitting" The

choice of the arbitrary constant equal to unity results in obtaining

16

integers throughout the standard tables presented by Van der Reyden (1943).

for Eo through 52 and !. through 9. Pearson and Hartley (1954) reproduced

part of these tables in slightly modified form for Eo through 52 and I'

through 6.

Due to the arbitrary nature of the constant cr , it follows that if

the greatest common factor is removed from the numerical values of a poly­

nomial in any system described above, the resulting figures would be

identical for all systems.

1.3 Notation To Be Used

A factor and contrast

a element of coefficient matrix of factor A

B factor and contrast

b element of coefficient matrix at factor B

C factor and contrast

c element of coefficient matrix of factor C

D diagonal matrix

E or e even

Fr vector of tr !r; FR vector of tr !a

f subscript of A contrast

Gs++ matrix of tr E.s ; GS matrix of tr E.sg subscript of B contrast

Ht vector of tr !:!.t; HT vector of tr !:!.T

h subscript of C contrast; polynomial coefficient, Section 5.1

I identity matrix

i level of factor A

J vector

j level of factor B

confounded

k level of factor C; polynomial coefficient, Section 501.

L linear combination

M Ipl

m number of levels of factor A

N total number of yields

n number of levels of factor B

o zero

P polynomial

p number of levels of factor C

Q N/2 or (N-1)/2

R subscript of A contrast

r subscript of A contrast; degree of polynomial

S subscript of B contrast

s subscript of B contrast

T subscript of C contrast

t subscript of C contrast

U or 'U odd

17

v

w

x

x

z

subscript of contrast A (generalized interaction)

subscript of contrast B (generalized interaction)

level of general treatment

times; I-X

yield

subscript of contrast C (generalized interaction)

sign a ; sign P • coefficient 1 Section 702r r~

sign b ; sign P • coefficient~ Section 702s r~

sign ct

sign af

f sign bg

"l.. sign ch

If =..(b

~ polynomial multiplier

18

19

PART I. THE GENERAL CASE OF ORTHOGONAL CONTRASTS

CHAPTER 2

Matrix Notation

Let y. Ok be the yield resulting from the applioation of the treat­J..J

thment oombinations oonsisting of the i level of the faotor A with m levels,

the jth level of the factor B with n levels, and the kth level of the

factor C with p levels.

Arrange these yields into a. (mnp x 1) oolumn veotor, in whioh the

Yijk are ordered in the following special manner: first put both the sub­

scripts i and j equal to zero and very the subsoript k from 0 to (p-l) to

obtaiil p oombinations; then, keeping the subsoript i fixed at 0, vary the

subsoript j from 1 to (n-l) for each of the p variations of k; finally,

vary the subscript i from 1 to (m-l) for all variations of the subsoripts

j and k.. In transposed form the subscripts of the vector y will thus read

as

OOO,OOl,.··,OO(p-l); OlO,Oll, .... ,Ol(p-l); 020,021,··.,02(p-l); •• ·,O(n-l)O,

O(n-l)l, ••• ,O(n-l)(p-l); 100,101, ••• ,10(p-l); 110,111, ••• ,ll(p-l); 120,121,

••• ,12(p-l); ••• ,1(n-l)0,· •• ,l(n-l)(p-l); ••• , (m-l)(n-l)O, (m-l) (n-l)l, ••• ,

(m-l )(n-l )(p-l)

If I denotes the identity matrix, !'tr" the trace (the sum of the

diagonal elements), and SSa = np(a2 +a21+···+a2 1)' let a = a. (mnp x mnp)r ro r r,m- r r

be the diagonal matrix

a I(np x np)ro

1ar1I(np x np)

o

o

20

, r=1,2,··· ,m-l

a lI(np x np)r,m-

such that tr ar = 0, and tr araf = 0 when r 1= f and tr araf = SSar when

r = f. If D denotes a diagonal matrix, let

and

t,hen tr a = np(tr a ) and tr a a", will equal zero when r is not equal tor -r -r-J.

f, and will equal SSa when r equals f, where SSa = np (SSa ).-r r . -r

Let b = b (mnp x mnp) denote the diagonal matrixs s

bsoI(p x p)

b s1 I(p x p) o

1II

o

bsoI(p x p)

bs1I(p x p)

o

o

o

s=l 2··· (n..l), . , ,

obsoI(p x p)

bs1I(p x p) o

o

such that tr b = 0 and tr b b . will equal zero when s 1=' g,' but will equalssg

SSbs when s = g, where SSbs = mp(b~o+b~1+ ••• +b~,n_l). Let

21

Then tr b =mp(tr b ) and tr b b = 0 when s f g, but equal to SSb when5 -5 -s-g -s

5 = g where SSb = mp(SSb ).5 -8

If St. is the diagonal vector D(oto,ct1,···,Ct,p_l), let Ct denote

the diagonal matrix ct (mnp x mnp)

C-t

C =to

•

•

o, t = 1,2, .. ·, (p-l)

suohthat>t3:' ct = O. and tr ct ch equals zero when t f h, but equals SSct

when t = h, where SSct = mn(c:O+c~1+"·+C:,P_l) = mn(SS£~).

Let Jt be the (1 x ~p) row vector, Jt = (1,1,···,1). Then

and hence

Jt&rbsctafbgChJ = tr(ararbsbgCtCh)

= (tr !.r!f)(tr £~g)(t r .£..tE.h)

This expression will equal zero when one or more of the following is

true: r f f, B , g, t f h; or when either r = 0, or s = 0, or t = O. It

will equal (SSa )(SSb )(SSo.) when r = f, s = g, and t = h •. As a special. ...or -s -v

case, ao = bo = Co = I(mnp xmnp).

A contrast between the Yijk will be (ArBsCt ) = JtarbsctY' when not

all three r, s, t are simultaneously zero. This will be a contrast, since

if Yijk = 1 for all i,j,k then

22

In the above expression the notation (y :::: 1) is used rather than (Yijk ::: 1)

to simplify typography..

Define respectively

A ::: (A B C ) :::: J'a b coY'r roo r 0

B ::: (A B C ) ::: JiaobscoYs o s 0

and

Ct ::: (A B Ct ) ::: J'a b ctY,o 0 0 0

where r::: 1,2,""', (m-l); s ::: 1,2, ••• ,(n-l); and t ::: 1,2,···,(p-l) as con-

trasts of the main effects A,B,C; alternatively these contrasts could be

designated as contrasts of the zero order interaction.

Define respectively

(A B ) :::: (A B C ) ::: J1a bscoY'r s r s 0 r

(ArCt ) ::: (A B Ct ) ::: J1a boCtYr 0 r

and

(B Ct ) ::: (A B Ct ) ::: J1a b ctY;s 0 s 0 s

r,s,t having the same ranges as before, as contrasts of the two-factor

inter~ctions AB, AC, and BC; alternatively these could be designated as

contrasts of the first order interaction ..

Define

(A B Ct ) ::: J~a b ctY,r s .. r s

r,s,t having the same ranges as before, as the three-factor interaction

contrasts} alternatively these could be designated contrasts of the second

order interaction ..

23

At times these contrasts will be indicated only by their subscripts,

as for instance in Table 1. All these contrasts are orthogonal to each

other by virtue of the definitions of the coefficient matrices a,b,c. The

orthogonal contrasts of an m x n x p factorial experiment can be displayed

as in Table 10

The major purpose of this thesis is to investigate the consequences

of using incomplete blocks in such factorial experiments. When all treat-

ment combinations are not included in each block of experimental units,

certain of the treatment contrasts become confounded (mixed up) with block

oontrasts. If a given contrast is orthogonal to block contrasts, it is

said to be unconfounded. We will consider confounding systems based only

on the signs of the elements of a contrast (plus zero, if zero elements

occur). When any contrast in Table 1 is confounded, the problem arises to

determine the effect this confounding has on the other contrasts.

From a matrix point of view, confounding amounts to a process of

partitioning of matrices. Consider, for example, the main effect contrast

A = J1a bey, and assume that no zero elements occur in the diagonalr roo

of a. Arrange the coefficients of the matrix a and the correspondingr r

elements of y so that the first npm elements in the diagonal of a will- r

have negative signs and the next npm+ elements in the diagonal will have

positive signs. Let a denote the (npm x npm ) matrix of negative ele-r- --

menta, and ar + the (npm+ x npm+) matrix of positive elements" It is well-

known that when a matrix in a product of matrices is partitioned, the other

matrices have to be partitioned conformably in order that the product may

retain its meaning. Accordingly, let (J~, J~) I and (y~, y~) I denote the

conformable partitionings of the vectors JI and y, so that we can write

the contrast as

24

Table 1 0 The orthogonal contrasts of the m x n x p factorial experiment.

A

B

C

AB

AC

BC

ABC

Total

Contrast

~ = 100

·•Am-I = (m-1)00

~. = 010

·•Bn_1 = 0(n-1)0

CJ. = 001·Cp_1 = OO(p-l)

~J.BJ. - 110

J\Bn_1 = l(n-I)O•

~a~ = 210

·AaB 1 = 2(n-l)0• n-·0Am_IBJ. = (m-1)10•

Am-1Bn_l = (m-1)(n-1)0

AJ.CJ. = 101•·0Am_1Cp_1 = (m-1)0(p-l)

BJ.C:J. = 011•··Bn

_1Cp_1 0(n-1)(p-l)

A:J.~Cl - III0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

mnp-1

25

ar-

Ar

o

:: A + Ar- r+

where A :: J-' a Y is the value of the contrast A in the block contain-r- - r- ~ r

ing those Y values corresponding to the negative values of ar and

A + :: J I a v+ is the value of the contrast A in the block containing ther . + r~ r

Y corresponding to the positive elements of a •r Placing Y :: 1, we have

Since

A (y :: 1) :: Jfa J = tr a f 0r- - r- - r-

Ar+(y :: 1) :: J~ar+J+ = tr ar + f 0

= 1) = tr a = np(tr a ) = 0,r -r

and since

it follows that

Since any contrast is of the form J1arbscty where either n?ne, or

one, or two of the subscripts r, s, t are zero, it follows that partition-

ing of anyone, or any two, or of all three of the matrices ar' bs ' ct

will require conformable partitioning of the other and y. The confounding

of any particular contrast will thus partition the coefficient matrix: of

every remaining contrast of the experiment, but this partitioning mayor

may not confound the remaining contrasts with block effectso Problems of

this type will now be investigatedo In succeeding chapters, r, s and t

will be used to denote specific contrasts and R, Sand T general contrasts.

26

CHAPTER 3

The Effects of Confounding a Single Contrast

3,,1 Theorems

Theorem 3010 The confounding of any contrast belonging to any lower order

interaction does not confound any contrast belonging to any higher order

interaction 0

Consider any contrast, (ARBSCT

)

Suppose we select any element of vector!T' say cTk' ioe", set p = 1 0

Then

Since this is true for all coefficients cTk' ~BSCT will remain a con­

trast regardless of how the levels of C are assigned to each block"

Hence it is possible to confound on any C-contrast, say Ct9 because there

is a one-to-one correspondence between the elements of !t and !T"

Symbolically this may. be expressed as

The above result can be extended to confounding on any BC~contrast,

say BsCt , assuming all A-levels are assigned to each block. Symbolically

Hence

27

since there is a one-to-one correspondence between bSj and bSj and between

0tk and cTk• By permutation of the letters of the contrast (~BSCT)' the

above results follow immediately for all permutations; hence the theorem

is proven.

Theorem 3:2 ~ When m, u, p are even numbers and none of vectors er, ~_S' or

.2.t ~ontains zero elements, and when the vector.~ has the same number of posi­

tive,as negative elements, then, when (A B Ct) is confounded, (AB ), A andr s r s rB will be uneonfounded.s- ..

Suppose that the first m coefficients of a are negative and the-r

next m+ are positive,; that the first n_ of Es are negative and the next n+

are positive.; and finally, that the first p_ of .2.t are negative and the

next p+ are positive. Then ar , bs ' and ct can be written as

ar = D(ar _, ar +);

bs = D(b ,b+,b , b +);s- s s- s

ct = D(ct_,ct+,···,ct_,ct +)

where D denotes a diagonal matrix and the elements of the above matrices

have orders that will now be explained.

Consider some negative element a . in the (npm x npm ) matrixrl - --

a " This element will occur successively np times on the diagonal.r-

Since

we can say that the element a . will first occur in n n successive posi-rl ~-

tions, then in n-p+ successive positions, then in n~_ successive positions

and finally in n~+ successive positions on the diagonal of ar _. The

positive elements of ar + will be similarly arranged on the diagonal. That

is, a can be regarded as a (4 x 4) diagonal matrix having submatricesr-

28

as diagonal elements, respectively with orders (m_ny_) 2, (m..ny+)2,

(m_n~_):a, (m_n+p+)2 e Likewise ar + can be regarded as a (4 x 4) diagonal

matrix with diagonal elements having as orders the same expressions as for

a ,but with m+ substituted for m e Consequently the matrix a can be re-~ - r

garded as the (8 x 8) diagonal matrix given in Table 2 0

Table 2.1 Oonformable submatrices of (AB Ct. ) .r s

Y' = (y' y' I y'. y' y' y' y'i y' )--" --+' -+-' -++' +-' +-+- ++-,. +++

a = D(a , a , a , a , ar+,ar +, a ar +)r r- r- r- r- r+'

b = D(b , b b b b b b bs +)S s- s-' s+' s+' B-' s-' s+'

ct= D(ct _, ct +' ct _, ct +' ct_,Ct +' ct _, ct +)

lrheorder of the 8 respective submatrices are(m n p )2, (m n-p~)2,(m..n+p_)2, (m_n+p+)2, (m+ny_)2, (m+ny+)2, (m+n:p:)2, (m+n:p+). -

Examining ct

in a similar way, it can be expressed as the (8 x 8)

matrix shown in Table 2. Arrange the elements of y in such a manner that

y can be regarded as an (8 x 1) vector, shown in Table 2, where for example

the subveetor y denotes the m_ny_ elements of y that correspond with

the m_ny_ diagonal elements of the product ar_bs_ct_o

Placing those yields that correspond with positive values of

(a b ct ) into one block, and those that correspond with negative values intor s

another block, the constituents of the respective blocks will be given by

,Block 1: (y' y' y' y')'+++' +--' .-+-' .-+

Block 2: (y~+_, yL+, y~++, Y~_)'

,To examine the confounding pattern for contrast (A B C

t) considerr s

29

Jfa b c J = [Jf abc J + Jt, abc J + Jt abc Jr s t --+ r- s- t+ --+ -+- r- s+ t- -+- +-- r+ s- t- +-

+ J~. abc J ]+ [J' abc J + J'_++ar_bs+ct+J_+++++ r+ s+ t+ +++ --- r- s- t- ---

say

where J_+ denotes J(m_ny+:x: 1) and where: (ArBsCt )+ indicates the value

of the contrast in the block containing the positive elements of (a b ct).. r s

Consider (A B )(1' = l)I(A B Ct )+, which is'the value of (A B ) whenr s r s . . . r s

Yijk = 1 in the positive block of the confounded contrast (ArBsCt ).

Let t = 0 in the first square bracket of (3.1). Then

(ArBs)(Y = 1)1 (ArBsCt ) = (tr !r_)(tr £s-)p+ + (tr !r_)(tr £s+)p-

... (tr !r... )(tr £s-)p- (3.2)

+ (tr !r+)(tr £s ... )p+ 000

When p_ = p..., (302) becomes

(ArBsHY = 1) \(ArBsCt )... = (tr !r- + tr !r+)(tr £s- + tr £s ... )p+

= (tr a )(tr b )P+ = 0-r -s

since (A B ) is a contrast. A similar result can be derived forr s

(ArBs)(Y = 1) in the negative block of ArBsCto

30

Consider the value of-the contrast A (y = 1) in the positive blockr

of A B Ct 0 Let s = 0 in (J. 2), thenr s

(Ar)(y = 1) I (ArBsC t ) = (tr !r)(p+n_ + p_n+)

.. (tr !r.. )(p_n_ + p+n+)

(A )(y = 1) I(A B Ct ) = (tr a + tr a +)np+ = (tr a )np+ = 0r r s.... -r- -r -r

Similarly

(B~)(y = 1) I(ArBsCt ).. = (tr .£s)(P+l'n_ + p_m+)

+ (tr E.s+)(p_l'n_ + P+ill+)

= 0 when P_ = P+

Hence the theorem is proved.

Corollary 3.2.1. The confounding of (A B Ct.) will not confound (1) thoser s

contrasts obtained from (ArBsCt ) by putting anyone of the subscripts r,s,t

equal to zero if the vector whose subscript is equated to zero has the same

number of positive as negative elements; (2) those contrasts obtained from

(A B Ct. ) by putting either r and s, or rand t, or s and t equal to zero,r sif either one of the two vectors whose subscripts are equated to zero has

the same number of positive as negative elements.

This corollary leads to the following practical situation: when

contrast (A B Ct.) is confounded, the folloWing contrasts will be uncon­r s

founded if the condition(s) given on the right is(are) satisfied.

31

(ArBs) when P_ = P+

(ArCt ) when n_ = n+

(BsCt ) when m_ = m+

Ar when either n_ = n+ or P_ = P",

Bs when either ~ = m+ or P_ = P+

Ct when either m_ = m+ or n_ = n+

These conditions lead to

Corollary 3.2.2. When each of the vectors !r' '£8' oSt, has the same number

of negative as posItive elements, the confounding of (ArBsCt ) will not

confound any lower order contrast obtained from (A B Ct· ) by placing eitherr s

one or two of the subscripts r,s,t equal to zero.

When the contrast (ArBsCt ) is confounded, and we would like to

know under what conditions the lower order contrasts with sUbscript(s)

identical to one (or two) of r,s,t is(are) confounded, the results of

Theorem 3.2 apply. This theorem, however, states nothing about the con-

founding or not of lower order contrasts with subscripts different from

r,s,t.Also Theorem 3.2 dealt only with even m,n,p.

To investigate these points, consider the confounding of contrast

(ArBf) of an m x n experiment. We consider a two-factor experiment to

simplify typography: with a three-factor experiment we would obtain 27

submatrices instead of the 9 in the present case. Partition the vector

!r into a negative part !r- with m_ elements in the diagonal, a positive

part a + with m+ elements, and a zero part a with m elements. Parti--r· -ro 0

tion the vector b into a negative part b with n elements, a positive-s -s--

part b + with n+ elements, and a zero part b with n elements.-8 -80 0

Placing those yields that correspond to posi-

32

As before, if the matrix ar is expressed in terms of the diagonal

matrices a , a +' a , the matrix b can be expressed in terms of ther- r ro s

diagonal matrices b , b +' b , conformable to the parts of a. The ele-. s- s so r

menta of y can then be arranged into submatrices conformable to the products

of the parts of a and b •r s

tive values of (a b ) into one block, those that correspond to negativer s

values into another block, and those that correspond to the zero values

into yet another block, the elements of the respective blocks will be as

follows:

Block 1: (y~+, y~- ) I

Block 2: (yl_, y' ).-+

Block 3: (Y~o' y~o' y~+, y~, yl ) I00

The values of (A B )(y = 1) in the respective blocks will be givenr s

by the expressions

(ArBs)+(y = 1) =

(ArBs)Jy = 1) =

(ArBs)o(y = 1) =

(tr !r+)(tr ~s+) + (tr !r)(tr ~s-)

(tr !r+)(tr ~s-) + (tr !r_)(tr ~s+)

(tr a+)(tr b ) + (tr a )(tr b )"""I' -so -r--so

+ (tr a )(tr b +) + (tr a )(tr b )"""I'O -s . """I'o s-

+ (tr a )(tr b )-ro -so

parts of a..... conformable to a ,a +' a ,and similarly for b.c •-ft. -r- -r -ro -u

= 0

Consider contrast (AaBS)\(ArBs ). Let !aI' !a2' !a3 denote the

The

values of (~BS)(Y = 1) in the three blocks of (ArBs) can be obtained

by substituting the parts of ~ Iar and bsl bs in (3.3.1).

33

(AaBS)(y = 1) \(ArBs )+ = (tr !R2)(tr E.s2) + (tr !Rl)(tr E.sl)

(~BS)(Y = 1) I(ArBs )_ =(tr !R2)(tr E.sl) + (tr !.J.U)(tr E.s2) (0302)

(~BS)(Y = 1) I(.ArBs)o = -(tr ~3)(tr .£s3) (for R,S,>l)

Next consider contrast (~)I(ArBs)oThevalues of ~(y = 1) in the

first two blocks of (A B ) can be obtained by letting S == 0 in (3.302);r s

for the zero block one must rework the relationship from (303.1).

(~)(y == 1) I(ArBs )+ = (tr ~2)n+ + (tr ~l)n_

(~)(y =1)1 (ArBs )_ = (tr !a2)n_ + (tr ~l)n+

(L. ) (y = 1) I(A B) == (n-n )(tr B..-.3

)-11 r s 0 0 -Ii

(BS)(y = 1)1 (ArBs )+ = (m+)(tr E.s2) + (m_)(tr E.sl),

(BS)(y = 1) I(ArBs )_ == (m+)(tr 12S1 ) + (m_)(tr £S2)

(BS)(y == 1)1 (ArBs)o == (m-mo)(tr E.s3)

Having the above relations, we can now prove the following theorems.

Theorem 3030 In an m x n experiment if contrast (ArBs) is confounded,

contrast (~BS) will also be confounded unless one or both of these situ­

ati ons prevail:

(i) ~IAr or Bsl Bs or both are unconfounded

(ii) tr !a3 = 0 and tr E.sl = tr E.s2; or·

tr E.s3 = 0 and tr 2:Rl = tr !.R2"

It is easy to show that, if (i) or (ii) prevails, (~Bs)I(ArBs)

is uneonfounded" What happens if one of these conditions is not .f'ul-

filled? It is obvious that either tr !a3 or tr E.s3 must be zero in order

to have (~BS) unconfounded"

34

If tr !R3 =0, tr !Rl =-tr !R2' and

Unless (i) is fulfilled, so that tr !R2 = 0, or tr £Sl = tr £S2 = 0,

or (ii) is fulfilled so that tr £S2 = tr £Sl' (AaBS) I(ArBs) will be con­

founded with blocks o

Corollary 3.3.1. If (ArBs) is confounded, contrast Aa will also be con­

founded unless one or both of these situations prevail:

(i) !alAr is unconfounded

(ii) tr !a3 = 0 and n+ = n_.

A similar statement can be made for BS•

This corollary follows immediately from (3.3.3) and (3.3.4). One

can rephrase this corollary as follows:

Corollary 3.3.2. If ~\Ar is confounded, Aa\(ArBs ) will also be confounded

unless tr !a3 = 0 and n+ =~. Similarly if BS\Bs is confounded,

(BS)I(ArBs) will also be confounded unless tr £03 I: 0 and m+ = m_.

30 2 Bundles of Contrasts

In terms of Theorem 3.3 we may define a bundle of contrasts of

homogeneous order to be that set of contrasts with properties such that

when any single contrast belonging to the set is confounded, all other

contrasts belonging to the set will also be confounded. In other words,

in terms of confcunding, any contrast will always possess a homogeneous

bundle. A homogeneous bundle of a contrast (ArBsCt ) or (rst) will be in­

dicated by [rst].

35

In terms of corollary 30301 we may define a bundle of heterogeneous

order to be that set of bundles of homogeneous orders with properties such

that whim any contrast belonging to the homogeneous bundle of highest order

is confounded, all other contrasts ,in the bundle of highest order and all

contrasts belonging to all the other homogeneous bundles in the set will

be confounded. In terms of confounding, any contrast of higher order will

always possess a heterogeneous bundle, but a main effect contrast cannot

possess a heterogeneous bundle.

CHAPTER 4

The Effects of Confounding Two Constrasts Simultaneousll

We now investigate the case when two contrasts are confounded s:i1nul-

t aneously• For the sake of simplicity of representation, we are assuming

that none of the contrastshas zero elements, otherwise 93::: 729 submatrices

will result from the conformable partitionings. In the present case the

more manageable number of 64 will occur. This assumption is one of eon-

venience only, and similar procedures and results will hold for contrasts

with zero elements.

Consider the two contrasts (A B Ct ) and (AfB Ch ) where at least one, r s g

of the following inequalities holds: r 'I f, s 'I g, t 'I h; and a contrast

(A B C ) derived from the two previous ones in the manner shown below.vwz

a ::: D(a I, arlI, ... , a I)r ro r,m-l

af ::: D(a.foI, aflI, •• 0, af 11),rn-a ::: D(a I, avlI, ....., a I) ::: arafv vo v,m-l

where I ::: I(np x np) and a .'" a .af ...V1. r~ ~

Originally the row vector at = (a. ,a l,···,a 1) has m negative-r ro r r ,m- - '_

elements followed by m. positive elements (m_ + m+ ::: m). Corresponding

to the m negative elements of aI, there are a certain number of negative- -r

and a certain number of positive elements of the row vector

~ ::: (af ,afl,···,af . 1)· Hence a' = (a ,a l,···,a 1) would consist-.L 0 ,m- '-'V vo v v,m-

of a sequence of alternating positive and negative elements. Let us

rearrange a' so that it has a single sequence of negative followed by a-vsingle sequence of positive elements; then rearrange a' and a1..

-r -.L

37

Accordingly !~ will have (m_+ + m+_) negative elements where there are

m elements of at which are the products of negative elements from at and-+ -v -r

positive elements from !f' and m+_ elements which are the products of posi-

tive elements from a' and negative elements from ar'. Similarly there· are-r -

+ m++) positive elements of !~o Hence the (!;)I(!~) row vector has a

and !f '" (+5,-1,-4,-4,-1,+5).

at '" (-25,+3,+4,-4,-3,+25).-r

sequence of m_+ negative elements, m+_ positive elements, m negative

elements, and finally m++ positive elements. [If there were zero elements

in at and aft, there would be (m + m + m + m + + m ) zero elements-r - -0 +0 0- 0 00 .

in !~.] Similarly (~f)I(:.~) has a sequence of m_+ positive elements,

m+_ negative elements, m__ negative elements, and finally m++ positive

elements.. It is important to remember that the first subscript refers to

the Ar contrast and the second to the Af contrast. Extensions to more than

two contrasts can be made without too much difficulty.

As an example, consider the two row vectors a' '" (-5,-3,-1,+1,+3,+5)-r

Before rearrangement, the vector at will be-vAfter rearrangement we have the following

a' '" (-25,-4,-3,+3,+4,+25)-v

a ll '" ( +5,-4,-1,-1,-4, +5)-fat '" ( -5,+1,+3,-3,-1, +5)-r

In the diagonal matrices each element of the vector is repeated

•np times .. The matrices, rearranged according to a , are indicated asv

a la '" D(ar ~,a + a ,a + )r v -~ r -, r-- r +

aflav '" D(af_+,af+_,af__ ,af ++)

a /a '" D(a a a a )v .V v-+' v+-' v--' v++

38

where the first sign in the subscripts refers to the element in a , ther \

second sign to the element in af , and the sign in av is the product of the

two basic signs. The respective orders of the diagonal submatrices are

(npm_+ x npm_+), (npm+_ x npm+_), (npm_ x npm__ ), (npm++ x npm++).

A similar procedure can be followed for the B-contrasts and the

C-contrasts. As seen in the original definition of b ,we need consider8

only one of the submatrices here and then repeat the rearranged submatrix

m times to form the rearranged b matrix. Let us consider one of these

submatrices b(np x np). The two basic submatrices will be designated as

b~ and b;, and their product as b~. Hence

b·lb·s w

b"lb·g w

b¥-Ib.w w

= D(b· +,b.+ ,b* ,b·++)s-· s - s- s

= D(b~ +' b fIo+ , b- , b'lll++)g- g - g-- g

= D(b:_+,b;+_,b';_,b:++)

where the first sign in the subscripts refers to the element in b'lll and thes

second sign to the element in b"', and the sign in b* is the product of theg . w

two basic signs. The respective orders of the diagonal matrices are

(pn_+ x pn_+), (pn+_ x pn+_), (pn__ x pn__ ) and (pn++ x pn++). The cor­

responding b-matrices are simply D(b" , b"" , 0 0 0 , b"') where b'f is repeated

m times.

In the c-matrices we need study only the diagonalized c-vector

c·(p x p) = D(Cto,Ctl,···,Ct,p_l). We will consider the two basic matrices

ct and c: with product c: 0

clltle.....t z

c.. \ c*h Z

c*IClf.Z Z

= D(ct_+"c!+_,ct_,ct....,)

= D(c~_+,c~+_,ct_,c~++)

= D(c:_+,c:+_,c:__,ci++)

39

where the signs in the subscripts have the same meaning as before, and the

respective orders of the diagonal matrices are (p_+ x p_+), (P+_ x P+_),

(p__ X P_ ), and (p++ x p++) • The corresponding c-matrices are simply

D( " lj!r *') . 't . dc ,e , ••• ,e where e :LS repeate nm times.

Sinee we have the relations

m =: m + m + m + m..L.L_+ +_ "'-T

n =: n_+ .... n+_ ... n ... n++

p=:p +p +p +p-+ +- -- . ++

where each subscript (~,~, ••• ,~) is either - or +. Hence there will be

64 terms in the sum. The elements of the y matrix can now be arranged

into 64 subvectors conformable to the products of the parts of

Thus

Y "" (y i Y i ; y' • •• Y i ) i. '-+-+~+'-+-++-' ~O~t~~' , ++++++

were the subvector y~o~~'t1\j.has m~6n~~'p~"l. columns.

Observe again that the first subscript of m or n or p (i.e., ~ or

~ or~) indicates the sign of the a or the b or the ct

in the relevantr s

part, and hence the product (~~r) will indicate the signs of the elements

of the contrast (ArBsOt ) in that part. The second subscript of m or n or

p (i.eo, 0 or e or~) indicates the sing of the af or the bg or the ch in

the same part as before, and hence the products (S~~ will indicate the

signs of the elements of the contrast (AfBgCh ) in that part. The products

lD "" ~6 or "'t. ::,!t or " =: 1"1. indicate the sign of a or b or c in the1" v w z

same part as before, and hence the products 'f"X..'Y "" ~6~tn will indicate the

signs of the elements of the contrast (A B C ) in that part.vwz

40

The sequence ..<, b, 'of, 13, to, 'X., 0 , "'l ' 'fI is thus the key to the study

of the confounding situation o When all those elements of y which have"

positive signs in (ArBsCt ) and (AfBgCh ) are placed in one block, those which

have positive signs in (ArBsCt ) but negative signs in (AfBgCh ) in another

block, those which have negative signs in (ArBsCt ) but positive signs in

(AfBgCh ) in yet another block, and those which have negative signs in both

(ArBsCt ) and (AfBgCh ) in the final block, then we will obtain the blocking

arrangement due to the simultaneous confounding of the two contrasts.

Table 30supplies the 64 possible combinations of the sequence; it

can be seen at once that if (A B Ct ) and (AfB Ch ) are confounded simul-r s g

taneously, (A Be) will be confounded automatically 0 (A B C ) can bevwz vwz

termed the generalized interaction of (ArBsCt ) and (AfBgCh ) and written

symbolica.lly as

(A B C ) ~ (A B Ct) x (AfB Ch )vwz rs· g

When, e.g., f ~ 0, s ~ 0, h = 0, (A BC ) ~ (A Ct) x B = (A B Ct) thevw z r. g r g

well-known Uinteraction l1 of (A Ct ) with B. It should be emphasized that. r -g

(A B C ) is a contrast, but in its generalized form will not be orthogonalvwz .

to all of the basic contrasts in the experiment.

Let (ArBsCt , A:rBgCh) denote the simultaneous confounding of the con­

trasts within the brackets. Then

(ArBsOt)(Y =1)1 (ArBsOt.,A-.RgOh) = Jfa ~b c J.".r r..<u s j3f. tit

where the sum contains 64 terms. Similar expressions hold for (AfBgOh )

and (A B ° )when (A B Ct ) and (AfBCh ) are simultaneously confounded.v w z r s . g

Let (ArBsCt,AfBgCh)++ denote the block obtained by selecting those

treatment combinations that have a positive sign in A B Ct. as well as inr s

41

Sign of

+ 01- + + 01- + + + + (+) (+) (+) 01- - - + + + + + + ++ + + + + + + + + - - + + + - - -+ - -++ + + + + + + ... + - - +- +- + -+ - - (+) (+-) (+)+ +- + + + + - + - + + - - + + + - + - - ++ + + - - + + + + + + - - - - + + + + - ++ + + - - + + (+) (+) (+) + - - - - + - - + ++ + -+ ... + -+ + - - - - + + _. - - ++ + + - - + - + - + - -+ - - + - + - (+) (+) (+)+ + + + + + + + - + - - + - - + + + (+) (+) (+)+ -+ + + - - - + + + - - + - - - - + - ++ + + + - - + (+) (+ ) (+) + - - + - - + - -. -++ + + 01- - - - + - + + - - + - - - + - - ++ + + - + - + + + + + - - - + - + + -+ - ++ + + - + - - + + + - - - + - - - -+ (+) (+) (+)+ + + - + - + + + - - - + - + - - - -++ + + - + - - + - (+) ( +) (+) -+ - - - + - - + - +

+ + + + + + + ... - + - + + + + + -+ - +-+ -+ -+ + -+ (+) (+) (+) - -+ - + + + - - -+ -++ + -+ + + - + + - + + ... -+ - - - ++ + + + - ... - -+ + - + + + - + - (+) (+) (+ )

- - + - - + + + + (+) (+) (-+) - + - - - + -+ -+ -+ ...+ "" -+ -+ - + - - - -+ - - -+ - ...+ - - + "" ... -+ - - - ... ... - - (-+) (... ) (-+)-+ - - + - -+ - + + - - - -+ - -+ - - -+

- - + + -+ + + + - - + - + - - -+ + ... - -++ + - - + + + - + - - - - -+ (+) (+) (+)+ + - - + "" - + - + - - + - - - ++ + - + - (+) (+) (+) - + - + - - - + - ++ - + - + + + + - - + - - + - + + + (+) (+) (+)+ - + - + ... - - + - - + - - - + - ++ - + - + (+) (+) (+) - + - - + - + - - ++ - + + - + - + - - + - - + - - +

( ) indicates parts of (++) b1ocko

If F~ denote the (1 x 4) row vector (tr !r++,tr !r+_,tr !r_+,tr !r--)

and G the (4 x 4) matrixS't+

42

tr b tr b tr b tr b-s++ -s+- -s-+ -s--

tr b tr b tr b tr bG -s+- -s++ -s-- -s-+=s++

tr b tr b tr b tr b-s-+ -s-- -s++ -s+-

tr b tr b tr b tr b-s-- -s-+ -s+- -s++

and Ht, the (1 x 4) row vector (tr .£.t++,tr '£'t+_,tr .£.t_+,tr -St__ ),then, using

Table 3 to select those treatment combinations that have a positive sign in

(4.2)

Similar expressions hold for (AfBgCh)(Y = 1) and (AvBwCz)(Y = 1) in the

(++) block. To derive the value of contrast (~BSCT)(Y = 1) in that block,

let ~22 denote the part of !R conformable to ~r++ and afT"'; i.e., those

elements of !a for which the corresponding elements in both ~r and ~f are

positive. Similarly ~R12 denotes a vector for which the corresponding ele­

ments of ~r are negative and of ~f are positive, etc. Similarly for the

parts of Es and.£.To Since

!It = ~ll + !It12 + ~21 + ~22

= !Rl' + ~2o

=~ol + !R.2

with similar results for £s and .£.T' we have a symbolic one-to-one corres­

pondence between (ARBSCT) and (ArBsCt ) if 1 in the subscript of a vector

of the former is substituted for - in the latter and 2 for +. Hence

where

43

(4.4)

tr E.s22

tr E.s2l

tr E.s12

tr E.sll

tr E.s21

tr £022

tr E.sll

tr E.s12

tr E.s12

tr £011

tr £022

tr E.s21

tr E.sll

tr £012

tr E.s21

tr E.s22

(4.6)

Theorem 4.1. If (~BsCT)I(ArBsCt) is confounded, (AaBsCT)\(ArBsCtT,AfBgCh)

must also be confounded; where R, S and T can be any values except that all

three cannot be zero.

This is an obvious result, which can be simply proven as follows.

Since (~BsCT)I(ArBsCt) is confounded, (~BSCT)(Y = 1)\ (ArBsCt )_ = WJ. "I 0 0

But subsequent confounding with respect to (AfBgCh ) subdivides the above

block into two blocks (-- and -+). It is impossible to subdivide a non-zero

quantity into two parts, each of which is zero, the latter being necessary

if (~BSCT) is to be unconfounded in these two blocks 0 A similar statement

holds for the (ArBsCt )+ block. It is equally obvious that this theorem

holds when contrasts containing zero elements are considered.

Corollary 4.101. All contrasts belonging to the heterogeneous bundle of

(A B Ct ) will be confounded when (A B Ct,AfB Ch ) is confounded. Similarlyr s r s g

for the heterogeneous bundle of (AfBgCh)o

Theorem 4.2. The simultaneous confounding of (ArBsCt ), (AfBgCh ) in an

m x n x p experiment, where m, n and p are even, will not confound

44

(i) (AaBS) if p++ = p+_ = p_+ = p__

(ii) Aa if either p++ = p+_ = p_+ = P__, or

n = n = n :: n or both systems of relations hold ..++ +- -+ --'

Consider (4.3) and let T = 0. Then H~ = (p++,p+_,p_+,p__ ) and hence

equal to Jlp++ if the condition in (i) is satisfied. Hence

(4.7 )

Let S = 0; then (4.7) becomes

A:R,(y = 1) \(ArBsCt'AfBgCh)++ = FIJnp++ = °Similarly for the second part of the theorem if n++ = n+_ :: n_+ = n_

Theorem 4.3. When two contrasts (A B Ct ) and (AfB Ch ) are confounded simul-. r s g

taneously so that (ArBsCt ) x (AfBgCh ) :: (AvBwCz ), and if (~BSCT) is a con-

trast not orthogonal to (AvBwCz ) then (AaBSCT) will also be confounded.

When (ARBSCT) is not orthogonal to (A B C ) thenv w z

(tr a...a )(tr bC'Q. )(tr CT·C ) ., 0.-It-V -u-W - -z

Hence tr !R; ., 0, tr~ ., 0, and tr ~TE.z ., 0. If!v and !R are parti­

tioned as before, then

(4.8)

Since the left-hand side of (4 .. 8) does not equal zero, it follows that at

least one of the traces of the diagonalized 8ubvectors will not equal zero.

A similar result can be established for tr £s£w and tr ~T~' and hence the

theorem is proved. If (~BSCT) is orthogonal to (AvBwCz )' it mayor may

not be confounded.

Corollary 40301. If (A B C ) can be expressed as some linear combinationv w z

45

of contrasts, and if one of the contrasts contained in the linear combina-

tion, (~BSCT) is not orthogonal to (AvBwCz )' then (~BSCT) will be

confounded when (ArBsCt ) and (AfBgCh ) are confounded simultaneously.

46

PART II. THE ORTHOGONAL POLYNOMIAL SYSTEM OF CONTRASTS

CHAPTER 5

The Orthogonal Polynomials

501 Introduction

If we have N yields Yo'Y1,···,YN-l corresponding to N equi-spaced

levels Xo,X1,··0,XN_l of a treatment, and we wish to fit a regression curve

of the polynomial type

y = k + k X + k X2 + o. 0 + k XN- lo 1 2 N-l

it is advantageous to employ orthogonal polynomials instead of the power

polynomials, and write

y = h + h P + h P + 000 + hN lPN' 1o 11 22 - -

where, in the Fisher (1921) 'system, or Fisher and Yates (1938)

Letting x = X - I, where I = (N-l)/2, we have for the first few

values of r

P2 = x2 - (1/12)(N2_l)

P3 = x 3- (1/20)(3N2-7)X

P4 = x4 - (1/14)(3N2-l3)x2 + C3/560)(N2-l)(N2-9)

P5

= x 5 - (5/l8)(N2-7)x3 + (1/1008)(15N4-230N2+407)X

47

Let Pri denote the numerical value of the orthogonal polynomial of

degree r at the level 10 With the object of illustrating confounding

prooedures, write P . = ~ .M ., where M . ::: P. and ~ . will be eitherrl rl rl rl rl rl

- or + or 0 0 Some of these values are presented in Table 4 for the first

four orthogonal polynomials, when N is even and when N is odd. In the

table a oommon factor has been taken out to simplify typographyo Note that

when N is odd, the central element of M when r is odd is unspecified.r

This does not matter in the confounding, since the blocking will proceed

only on the signs. Those elements of ~ . given explicitly in the table,rl

retain their signs independent of the size of N; the others depend on the

size of N"

Table 5 supplies the values of P r. ::: }..P i for N through 8" Fromrl, r

the table it can be ascertained that for N even, the elements of P arer

symmetric about the origin; for N odd, the central element is 0 and the

other elements are symmetric about this central element"

5.2 Partial and Complete Confounding

Let P ::: ~ M denote the diagonalized vector-r -r-T

D(~ M ~ M H. ~ M )ro ro' rl rl' 'r,m-l r,m-l

::: D(~ ~ • •• ~ )D(M M " •• M ).ro' rl' , r,m-l ro' rl' 'r,m-l'

let

p =.~ M = D(~ M ~ M 0 ,," ~ M )-a -s-s so so' sl sl' , s,n-ls,n-l

::: D(~ ,~ l'o •• ,~ . l)D(M ,Ml,···,M 1);so s s,n- so s s,n-

let

!:t = ~t~t ::: D("\oMto'''\lMt 1'''· ·,o<.c,p_lMt,p_l)

"" D("\0'''\1'''· "'''\,p-l )D(MtoilMtl' " "",Mt ,p-l)·

48

Table 4.. Signs and absolute values of the orthogonal polynomials ..

N even

x :x: M3

+ 3 I( 2S_N2 )1

o -(N-l)/2 - (N-l) + 2(N-l)(N-2) + 8(N-l)(N-2)(N-3)(N-4)

Is (N-2) (N-3)(N-4) (N-2l)1

- 2(N-l)(N-2)(N-3)

12(N-2) (N-3)(3-N)1

\3 (JN2-52 )J... 3(N2-4)

- 13(4-N2 )1

3(52-3N2)

\2(N-2) (N-7)1

1(28-N2)\

_ (N2_4)-l..5 3

'-0 .. 5 1

1 -(N-3)!2 - (N-3)

N-l (N-l)j2 + (N-I) + 2(N-l)(N-2) + 2(N-I)(N-2)(N-3) '" 8(N-l) (N-2)(N-3)(N-4)

2 12 40 560

N odd

P1 P2 P3 P4

--X X ~ M1 ..(2 Ma ..(3 M,3 ..(4 M4- -

o -(N-l)/2 - (N-l) ... 2(N-l)(M-2) - 2(N-l)(N-2)(N-3)

1 -(N-3)/2 - (N-3)

... 8(N-l)(N-2)(N-3) (N-4)

-2/2 2 1(13-N2 )1

0 o unsp (N2_l)

2/2 + 2 I(13-N2 )1

N-l (N-l)/2 + (N-l) + 2(N-l)(N-2)

13(N2--9) ,(N2-4l )L-

o unspecified +, 3(N2-1)(N2_9)

6(N2-9) 13(N2-9)(N2 -4l )1

... 2(N-l)(N-2)(N-3) + 8(N-l)(N-2)(N-3) (N-4)

2 12 40 560I I denotes a.bsolute value.

Table 50

N "".3p' pe p'

(') 1 2

+1 -1 +1.+1 0-2+1 +1 +1

11.3

N "" 2

p' p'(') 1

+1 -1+1 +1

1 2

Numerical values of p' ",,}.pr r

N "" 5Pb Pi P~ P; Pl

+1 -2 +2 -1 +1+1 -1 -1 +2 -4+1 0 -2 0 +6+1 +1 -1 -2 -4+1 +2 +2 +1 +1

1 1 1 t*

N "" 4pi pe P' pio 1 2 :3

+1 -3 +1 -1+1 -1 -1 +3+1 +1 -1 -3+1 +3 +1 +11 2 1 10

3

H "" 8pi pi

6 7

N =: 7

P6 Pi Pk P1 Pl P~ P~

+1 -.3 +5 -1 +.3 -1 +1+1 -2 0 +1 -7 +4 -6+1 -1 -3 +1 +1 -5 +15+1 0 -4 0 +6 0 -20+1 +1 -3 -1 +1 +5 +15+1 +2 0 -1 -7 -4 -6+1 +3 +5 +1 +3 +1 +1

17 7, IT1 1 1 'bl225b"O

N =: 6pI pi pi pi p' pi

o 1 2 :3 4 5

+1 -5 +5 -·5 +1 -1+1 -.3 -1 +7 -3 +5+1 -1 -4 +4 +2 -10+1 +1 -4 -4 +2 +10+1 +3 -1 -7 -3 -5+1 +5 +5 +~ +1 +1

3 L 211 2 2' '3 12 10

49

+1 -7 +7 -7 + 7 - 7+1 -5 +1 +5 -1.3 +2.3+1 -.3 -3+7 - .3 -17+1 -1 -5 +3 + 9 -15+1 +1 -5 -.3 ... 9 +15+1 +3 -3 -7 - .3 +17+1 +5 +1 -5 -1.3 -2.3+1 +7 +7 +7 +. 7 + 7

2 ~ 71 2 1 '3 l2 10

+1- 1-5 + 7+9 -21-5 +35-5 -.35+9 +21-5 - 7+1 + 1

Then, as before in the m x n x p factorial experiment, any contrast

between the Yijk can be written

(A B Ct) '" JiP P ptYr 8 r s

"" J'..<. -< ~.M M Mtyr sur 8

when not all three r, 8,tare s:i.mult aneously zero 0

If we wish to confound the contrast ArBsCt the confounding will be

based on whether ..<. ,..<. ,~. is -, or +, or 0, from the following factorr s u

combinations.

A B C

ijk 0 1 2 ... (m-l) 0 1 2 ... (n-l) 0 1 ... (p-l)

rat -< ~l "\.2 ..<. ..<. ..<.81 "<'52 ..<.s,n-l ..<. -<tl -<ro r,m-l so to t,p-l

Any treatment combination ijk· (i "" 0,1,·." ,m-l; j =: 0,1,.·. ,n-l;

k =O,l, ••• ,p-l) ~asblock indicator ..<...<. "<'t."" -, or +, or 0, depending on- rs·

the sizes of m, n, p and the degrees r,s,t. If M "" M =: Mt '" I, contrastr s

(A Be) will be completely confounded with blocks based on the above oon­

founding scheme; otherwiseg it will be partially confounded. Note that Pa

for 4 levels and P1 for 3 levels are the only eases within the range of

Table 5 that will allow complete confounding.

5.3 Partitioning of the Polynomial Values

Inspection of Tables 4 and 5 reveals that for even N, the elements

of any even degree polynomials (P ,P , .•• ) will consist of N1, duplicated

negative elements (a total of :ft :0: 2N1. negative elements) and Na dupli­

cated positive elements (N+ "" 2Na ); hence 2N1. + 2N2 "" N" These

polynomials will be denoted by Pe when the elements are rearranged so that

the first 2N1 elements are negative and the last 2N2

are positive.. The

vector of these elements is

P~ = (el1,e11,oo .. ,elN1 ,elN1.; 1921, 1921' n .,e2Na,e2Na)

= (P~-; P~)

where eli denotes a negative element and e2i a positive element ..

The odd-degree polynomials (P1'P3

' ..... ) will consist of Q = N/2

negative elements and Q positive elements, the absolute value of each

negative element being matched by a duplicate positive element" These

polynomials will be denoted by Pu if the elements are rearranged so that

the first Q elements are negative and the last Q are positive. The vector

of these elements is

P~ = (-U1,-U2,···,-uQ; u1,ua,···,uQ)

= (P~-; P~+)

Note that for P , it is unnecessary to use a subscript notation to denoteu

whioh elements are negative and which positive. If the observations are

ordered according to either a P or a P , the other even degree polynomials19 u

when ordered conformably will be indicated as PElpe or PElpu ; similarly

the other odd degree polynomials when ordered conformably will be indicated

as pulpe or Pu/pu • E includes e and U includes u, unless otherwise speci­

fied. These vectors are indicated in Table 6a"

In Table 6a the element s of PEIP19 and pul P19 have the same subscript

notation as for Pe; similarly PE/PU

and pulpu have the same subscript

notation as P. Each pair of equal elements in any P will correspond tou e

a pair of equal (in absolute value) elements of any Pu; similarly each pair

Table 6a Q Partitioning of the orthogonal pOlynomials.1I52

Pe

••Q

81N1

elN1

~eo

e2a

·""e2Na

e2Na

(1-6)eo

Q

Q

Q

•."E2Na

E2Na

(l-S)Eo

If N = 2Q + 1

-Un

....(>

~a1

-U21

"."U2Na

-U2Na

o

..

(>

Q

"

o

"

Eo

".."

o

Yeli is negative and e2i is positive; ui is positive" Ei and Ui are

negative or positive elements in PE and PU' corresponding to elements

in P and P" If N = 2Q, omit the last line and the e lines. Ife u 0

N = 2Q + 1, a negative e belongs to the 81 group and a positive 8 too 0

the ea group; this is indicated by Be where ~ = 1 if e is negativeo 0

and 8 = 0 if e is positive.o

53

of equal elements of PE will correspond to a pair of equal (in absolute

value) elements of any P. Hence if one partitions P' into the two partsu e

P~_ and P~+, pul Pe will be unconfounded; similarly for PEIpu if P~ is par-

titioned into P~_ and P~+. PElpe and Pu \pu will be confounded under these

conditions. These results are summarized in Table 6b.

Table 6b. Sums of elements of conformable parts.

N = 2Q N = 2Q + 1

B~ock PE1Pe PulPe PEIPu pulpu PEIPeY pulpe PelPE PulPu

1 (-) 2~li 0 0 -Eri 2~li + bEo 0 -E /2 -Eri0

2 (+) -2L~i 0 0 Eri -2~li - SEo0 -E /2 Eri0

3 (0) E 00

Ys = 1, if e is negative; 0 if e is positive.·00

For odd N (N = 2Q + 1), the odd degree polynomials have a zero

element and the even degree polynomials an extra negative or positive ele-

ment; the latter is indicated by eo or Eo. Henoe

pt = (Pl. pI ) ! P' = (pI pI 0)e e-' e+ , u u-' u+'

where either pI or PI+ has Q + 1 elements, depending on whether e is-e- e 0

or +. Hence confounding based on these subvectors will result in unequal

sized blocks. In this case PElpu will be confounded, but Pu fpe will still

remain unconfounded. These results are also summarized in Tables 6a and6b.

504 Bundles of Homogeneous Order

A bundle of homogeneous order was previously defined as that set of

contrasts with properties such that when any single contrast belonging to

54

the set is confounded, all other contrasts belonging to the set will also

be confounded.

The theorems in Chapter 3 could be extended to study the confounding

of (AnBSCT) \(ArBsCt ); however, the results in Table 6b can now be employed

to exaIlline the confounding of (~BSCT)I(ArBsCt ) for different values of

m,n,p and the subscripts. For example, i~ A1 B1C1 is confounded for

m :: n :: p = 4, then (A1 B1 C3), (~B3C1)' (A3~Cl)' (A1BsCa), (AaB1 C3),

(A3BsC1 ), (AaBsCa)1lill be confounded automatically. Hence the homogeneous

bundle of contrasts corresponding to the contrast (Ill) will be (111,113,

131,311,133,313,331,333), indicated as [111].

Oonsidering m x n x p factorial experiments, Table 7 shows the bundles

of contrasts of homogeneous orders obtained in the 4 x 4 x 4 and 4 x 3 x 3

experiments. The second-last column in the table gives the bundles obtained

by the application of the theory of Galois fields, where these exist.

A Galois field can exist only for a prime number or a power of a prime

number. Hnece the six contrasts of the interaction AB in the 4 x 3 x 3

factorial cannot be split into bundles according to Galois theory, since

6 is neither a prime number nor a power of a prime number.

55

Table 70 Allocation of contrasts to homogeneous bundles.

Effect Constituent contrasts of bundles

Number ofcontrastsper bundle

Number ofcontrastsper Galois

bundledf

4 x 4 x 4 factorial experiment

1,2

3,3,3,3, 273,3,3,3,3

A

B

C

IE

AC

Be

ABC

300200 100

030020 010

003002 001

330310

230 320 130220 210 120 110

303301

203 302 103202 201 102 101

033031

023 032 013022 021 012 011

333331313133

233 323 332 311231 321 312 131

223 232 322 213 123 132 113222 221 212 122 211 121 112 111

1,2

1,2

1,2,2,4

1,2,2,4

1,2,2,4

1,2,2,2,4,4,4,8

3

3

3

3,3,3

3

3

3

9

9

9

4 x 3 x 3 factorial experiment

A 300200 100

1,2 3 3

020020 010

2 2

nonexistent 6

nonexistent 6

AB

AC

002002 001

320 310220 210 120 110

302 301202 201 102 101

1,2

1,1,2,2

1,1,2,2

2 2

42,2

nonexistent 12

1,1,1,1

322 321 312 311 1,1,1,2,222 221 212 122 211 121 112 111 1,2,2,2

022022 021 012 011

ABC

17If 020 is confounded, 010 will be unconfounded; if 010 is confounded,020 will also be confounded. Similarly for C and 011 of BC 0

56

CHAPTER 6

The Confounding Pattern of a Contrast

601 Bundles of Heterogeneous Order

Inspection of Table 5 reveals that when n = 2, 4, or 8 the same

number of negative as positive elements occur in both the even and the odd

degrees g when n = 6, only in the odd degreeso Hence, in these cases by

Theorem 30 2 and its corollaries, the confounding of any higher order inter-

action contrast will not confound any lower order interaction contrast.

In the other cases the heterogeneous bundle must be determined for each

particular contrast confounded.

6.1.1 Confounding on (AuBu).

2s + 1)0 Let Yij = 10 Then

= 1) are given by the elements inthe elements of the expression (A B ) (yuu

Consider contrast (A B ) (u = odd) of an m x n factorial experimentuu

when m and n are both odd (m =2r + 1, n =

the body of Table 8ao Any element there is obtained by multiplication of

the appropriate marginal elements, where the marginal elements are those

respectively of the orthogonal polynomials A and B au u

Confounding of (A B ) means blocking the elements in the pattern shownuu

in Table 8a: the positive block consists of the elements in the upper left-

hand corner and the elements in the lower right-hand corner; the negative

block consists of the elements in the upper right-hand corner and lower left-

hand corner9 the zero block (omitted if both m and n are even) consists of

the elements in the central "crosso"

Consider AU (y = 1 \(AUBU

) 0 From Table 8a we see that the rs ele­

ments in the upper left-hand corner have the same absolute values as the

57

Table 8a. Blcokin~ 'pattern due to confounding of (AuBu ) when m and nare addb'.

e , t e ze a row~,e1e erIf n is e e ,is removed.

A~ -ubI -ub2... -ubj

...-~s ° ubI ~2

... ubj• 00

ubs

-ual-ua2

• (-,- ) :::: + (-,0) (-,+) :::• -• ==-u . 0a.J..•••

-nar

0 (0,- ) ::: 0 (0,0) (0,+) ::: 0

ual,!

ua2e (+,-) :: (+,0) (+, +) :::: +• -e

==uai 0•e

•uar

11 vn th z a co umn is r moved· if In 's ev·n h r

rs elements in the lower right-hand corner, but will have opposite signs,

and will accordingly cancel in the positive block; similarly for the negative

and zero blocks; hence Aul (AuBu ) will be unconfounded. In a similar manner

~\ (AuBu ) can be shown to be unconfounded. These results are valid for all

values of In and n.

Next consider AE I(AuBu ). For In and n both odd, the sum of the ele­

ments in the various sectors of Table 8a are presented in Table 8b, using

the notation of Tables 6a and 6b. Since AE (y ::: 1)I(AuBu ) f 0, this con­

trast is confounded with blocks. Similarly BE I(AuBu ) is confounded with

blocks when In and n are both odd. If In is even and n is odd, the row of

the zero block is removed in Table 8b and E == 0; hence ~-I(A B ) is·ao -"E u u

58

unconfounded with blocks; however, BE \(AuBu ) will still be confoundedo

Similarly, if m is odd and n is even, BE I(AuBu ) is unconfounded with blocks,

but L I(A B ) will still be confounded (E r 0), although the column of~ u u ao

the zero block is removed) 0 If both m and n are even, both ~I(AuBu ) and

~ j(AuBu ) are unconfounded with blockso

Table 8bo Sums of elements of ~ (AB ) (y = 1) corresponding to blockingpattern in Table 8ao u u

EalEa2ooo

·oo

Ear

Eao

&

o·oo·Ear

-(sE )/2ao

-sEao

-(sE )/2ao

-(E )/2ao

Eao

-(sE )/2ao

sEao

-(sE )/2ao

Finally let us consider the two-factor interactions. From the work

on homogeneous bundles we know that any odd order contrast (.AuBU) I(AuBu

)

will be confounded for any m and no That is, the contra5ts (00) belong to

[uu], the homogeneous bundle of the contrast (uu)o The amount of confound-

ing is easily determined from Table 6b o

59

(AUBU) (y :: 1) !(AuBU

)+ :: 2(L.uai )(LUbi )

(Au11J) (y :: 1) I(AuBu )_ ::" -2 (LUai)(IUbi )

The same applies to (~~) I(AuBu ) for both m and n odd; however, if either

m or n is even, (~BE)I(AuBu ) is unconfounded with blocks. Finally we find

that (~Bu)I(AuBu ) and (Au~)I(AuBu ) are not confounded with blocks for any

m or n. For example

(~Bu) (y :: 1) \(AuBu )+:: (-Eao/2)(-L..ubi) + (-Eao/2)(L.Ubi ) :: 0

6.1.2 Confounding on (!aBe)

As indicated in Table 6b, the blocking pattern for B will have onlye .

two parts, regardless of whether n is even or odd. If n is even, there are

n_ :: 2n1 negative and n+ :: 2n2 positive elements; if n is odd and the middle

element is negative, n_ :: 2n1, + 1; if n is odd and the middle element is

positive, n+ :: 2n2 + 1. The blocking pattern based on confounding of

(A B ) is presented in Table 9.u e

Using Table 9, we see that

Au (y:: 1)\ (AuBe )+ :: (2n2 ­

Au (y :: 1) \(AuBe )_:: (2n1 ­

Au (y :: 1) \(AuBe)o :: 0

Hence Au I(AuBe ) will be unconfounded with blocks only when n is even

(51 :: 82 :: 0) and n1, :: na ; that is, n_ :: n+; this is true for all mo

From Table 6b we see that BU (y :: 1) \Be :: 0 for all sectors; hence

Bul(AuBe ) is unconfounded with blocks for all m and n. The sum of the

elements of BE (y :: 1) \(AuBe ) has the same absolute value but opposite

signs for the two parts of each block. Hence BE I(AuBe ) is unconfounded

with blocks for all m and no AU I(AuBe ) will be unconfounded with blocks

60

Table 90 Blocldnv·attern due to the confounding of (A B ) and contrast1 u etotals.

(y = 1) = -(2n.a + ba)E /2ao

BE (y = 1) = -r(2L.~1.i+5J.~o)

~- (y = 1) = -(2na + ba»)U .-lJ L;, aJ.

~

Au (y = 1) = -(2nJ. + oJ. )L.Uai

~ (y = 1) = -(2n1 + bJ.)Eao/2

BE (y = 1) = r(2L.~1.i+SJ.~o)·•

-uai

-uar

o

~: set r = 1 above

Au = OJ ~ = (2na+Oa )Eao

BE: set r = 1 above

u Au (y = 1) = (2n1 +~1 )LUai Au (y = 1) = (2na+'8a )LUaial•·· (y 1) AE

(y 1)u ~ = =ai as in (- -) sector in (- +) sector; as

uar BE (y = 1) BE (y = 1)

lIOmit the zero row if m = 2r If n = 2s, abo does not exist; hence, set

~J. = 8a = O. If n = 2s +1 and ebo is -, 81 = land Sa = 0; if ebo is +,

~1 = 0 and O2 = 1 0

only if m is even (so that E does not exist) for all values of nj if mao

is odd, ~\(AuBe) will be confounded with blocks for all values of n.

Since Eo (y = l)\Be = 0 for all sectors in Table 6b, (AEBu)I(AuBe )

and (AuEo) \(AuBe ) are unconfounded for all m and n. As for (~BE)' we

note that

(~~) (y = 1) \ (AuBe )+ =(L~li+OI\O)(-·(Eao/2)+(Eao/~))= 0

and similarly for the other blocks. Hence (!EBE)\(AuBe) is also unconfounded

with blocks for all m and no Finally we note that

61

(~~) (y = 1)\ (AuBe )+ = -2(2L~lj+(\Ebo)LUai

(~BE) (y = 1) I(AuBe )_ = 2(2L\lj +01\o)L:.Uai

(AuBE) (y = 1) \(AuBe)o = 0"

Hence (~BE)I(AuBe) is confounded with blocks for all m and n; i"e", (UE)

belongs to rue].

The blocking pattern here can be set up as in Table 9 by using

ebl and eb2 for Be and eal and ea2 for Ae; this is done in Table 10"

Table 10" Blocking pattern due to the confounding of (A B ) and contrasttotals •.!! e e

If m 2r, eao does not ex~st (Oal 0a2 0), s~~larly for ObI and bb2

if n = 28. If m = (2r+l) and e is -, 0 1 = 1 and B 2 = 0; if e is +,ao a a aoSal:: ° and ba2 = 1; similarly for bbl and 0b2 if n = (2s+l).

BA e

eb1n1 sb1n1 bbleboebll ebll... eb2l eb2l

... e e ~ ee b2na b2na 2 bo

ealleall ~(y=l) = (2n], +Obl ) (2~ali ~(Y=l) = (2n2 +Sb2 )( 2LEali""

+ b IE )" + b IE )eam1 a ao a ao

e BE(y=l) = (2~+bal )(2L~lj BE(y=l) =-(2m1 +bal ) (2lEbljam],o e

+ bbl\O) + °bll\o)al ao

ea2l ~(y=l) = -(2n], +obl)(2IEali ~(y::l) :: -(2n2+~b2)(2LEaliea2l + 0 IE + 8 E· a ao al ao·· (2m2 +8a2) (2Ll\lj :: -(2m2+8a2)(2L~lje BE(y=l) = ~(y::l)a2m2e

+ °blEbo) + 8bl~O)a2m2o ea2 ao

y.:: = :: • .

62

From Table 10 we see that

This expression equals zero only whenn is even (6b1 = bb2 = 0) and n1 = na,

so that n_ = n+, but m can be any value. Similarly

hence 11: \(AeBe ) is unconfounded with blocks only when m is even and m1 = ma

(so that m_ = m... ), but n can be any value. As in system 6.1.2, Au \(AeBe )

and 11T\(AeBe ) are unconfounded with blocks for all m and n.

As for interactions, those involving Au or 11T will be unconfounded;

however, (AEBE)\(AeBe ) will be confounded with blocks for all m and n:

Hence (EE) belongs to fee].

6.104 Summary

Table 11 summarizes all the findings on polynomial confounding of

main effects and two-factor interactions when specific two-factor inter-

actions are confounded. The results in Table 11 can be verified by app1y-

ing the conditions of Theorem 3.3, Corollary 3.3.1, and Table 6b. Consider,

for instance, the value of Au (y = 1) I(AuBe ), when mis odd andn is even;

since tr ~3 = 0, AuI(AuBe ) will be unconfounded when n+ = n_.

63

Table 110 Confounding of general contrasts when specific AB contrasts areconfounded.Y

Specific confounded effect

K denotes confoundingo *' lndlcates an lmposslble set of m and n, for

example m must equal mfor a.ll odd contrasts, A 0 All main effects- + u

and two-factor interactions omitted from this table are not confounded

for any m or no

,,

(A B. ) (A B ) (A B ) ! (A B )Value of uu u e e u e e

m andn ~ BE (AuBU) (AEBE) ~ AE (AuBE) Bu BE (AEBu)i AE BE (AEBE)I!

Both odd K K K K ,K K K IK K K K K K

m odd n =n K 0 K 0 0 K K K 0 K 0 K K- . +

n even n:,.fn+ *' *' *' *' K K K *' * * IK K K

Im even m =m 0 K K 0 K 0 K 0 K K ! K 0 K

- + ,odd m_fm+ *' *' * *' *' *' *' K K K i K K Kn

m =m- + 0 0 K 0 0 0 K 0 0 K 0 0 Kn =n- +

m =m- +

*' *' * *' K 0 K * * *' Kn_fn+K 0

m even -n even m_fro",

*' * * *' *' *' *' K 0 K 0 K Kn =n

- +

m.Jm+*' *' ;Eo *' *' * *' *' *' *'n fn K K K

- +

11 . . .

64

6.2 Rules for Obtaining Confounding Patterns

6.201 Introduction

When more than two factors are involved in an experiment, the previous

method to obtain the heterogeneous bundle of a particular contrast becomes

unwieldy, and mistakes can very easily be made. As a sUbstitute, a symbolic

multiplication process will now be derived which allows easy generalization.

Suppose we have an m x n experiment in which A B , denoted by (ee),e e

is confounded with blocks so that m+ f m_ and n+ f n_. We introduce the

syinbol (00) to denote the "dU1lJIllyll cont-rast AoBoo Consider the following

scheme:

(a) (EO) (00) (eO)

(b) (OE) (00) (De)

(c) (EE) (EO) (OE) (00) (ee)

The scheme is to read as follows:

(i) ~ the condition for confounding (EO)\(eO), row (a), and the

conditions for confounding (OE) I(Oe), row (b), we imply the

confounding of (EO) Ieee) and (OE) Ieee), row (c), by Corollary

30302 because m+ f m_ and n+ f n_o

(ii) The confounding of (ee) also results in the confounding of

(EE), row (c)o

Hence the contrasts on the left side of row (c) are then confounded

when (ee) is confounded. The contrasts left of the vertical line of row

(c) are found by multiplying those of row (a) to the left of the vertical

line by the corresponding contrasts in row (b); and similarly for the

right side. In actual practice, one starts with (ee), then constructs

65

(Oe) and (eO), then the left side of (a) and (b) and finally the left side

The introduction of the dummy contrast is an all-important step,

since this fictitious contrast allows us to obtain the heterogeneous

bundle of a higher order interaction contrast from the homogeneous bundles

of those lower order interaction contrasts, whose symbolic product is equal

to the higher order interaction contrast o

The above scheme gives the heterogeneous bundle of the contrast (ee)

only when mo+- t m_ and no+- t n_o Generalization to other values of m and n.

and other polynomial contrasts is required o It will be shown that by using

the properties of othogonal polynomial contrasts and by adopting appropri-

ate conventions with respect to the dummy contrast, the heterogeneous

bundle of any higher order interaction orthogonal polynomial contrast can

be obtained from the symbolic multiplication schemeo This is accomplished

by the provisions of Corollary 30302 when constituting the heterogeneous

bundle 0

6.202 Polynomial confounding patterns for anm x n experiment

One would prefer to substitute the bundle notation for the contrast

notation in Section 60201 and thus generalize these results to both odd

and even contrasts 0 If all confounding contrasts were of even degree,

[eO] = (EO) for all m; hence the use of the bundle notation would be satis­

factoryo However, [uO] = (UO) for even m and [uO] = (UO) + (EO) for odd m;

a general bundle notation becomes unsatisfactory 0 In Table 12 we extend

the scheme of Section 60201 to each type of polynomial confounding pattern,

for any m and no In order to fulfill the provisions of Corollary 303 0 2,

(00) is used or omitted in different situations o Also one will have

Table 120 General procedures for determining all confounded contrasts,using polynomial confounding in a two-factor experiment.

(rs) = (uu)

66

(a) (EO) (00) (UO) (uO)Omit when m is even

(b) (OE) (00) (OU) (Ou)Omit when n is even

(c) (a) x (b) (UU) (uu)

(rs) = (ue)

(a) (EO) (00) (UO) (uO)Omit when m is even

(b) (00) (00) (OE) (Oe)Omit when n+ ::: n_

(c) (a) x (b) (a) x (b) (ue)

(rs) ::: (eu)

(a) (00) (00) (EO) (eO)Omit when m+ = m_

(b) (OE) (00) (au) (Ou)Omit when n is even

(a) x (b) (a) x (b) (eu)

(rs) = (ee)

(a) (00) (00) (EO) (eO)Omit when m+ = m_

(b) (00) (00) (OE) (Oe)Omit when n+ ::: n_

(00) (a) x (b) (ee)

67

separate. sections for (E) and (U) contrasts; the use of these will be

dictated by the absence or presence of each in the homogeneous bundles of

the confounding contrasts. The :multiplicative schemes are simply conveni-

ent arithmetical devices to produce the results of Table II. A separate

Bcheme is constructed for each of the four general two-factor confounding

contrasts (ra) = (uu), (ue), (eu) and (ee).

6.2.3 A general rule for obtaining the confoundingpatterns of any polynomial contrast in anm x n factorial experiment

In previous chapters (RS) has referred to any two-factor contrast

with (ra) being a given confounding contrast. If we now use the restricted

Rf to refer to only odd degree contrasts if r is odd and to even degree

contrasts if r is even, and similarly for S· and s, we can generalize the

four parts of Table 12 into a sin~le part, as in Table 13.

Table l3.Y Generalization of the parts of Table 12 ..

(a)

(b)

(EO)Omit when r = eOmit when m is even

(OE)Omit when 13 .. eOmit when n is even

(00)

(00)

(00)Omit when r = uOmit when m..... m_

(00)Omit when 13 .. uOmit when n..... n_

(rO)

(as)

(c) (a) x (b)

Y(R-O) .. {(UO) if r .. U and. (EO) if r .. e

(a) x (b)

(OS') I: i(OU) if 13 .. u and. I(OE) if s .. e

(rs)

6.2.4 Confounding pattern of .any polynomial contrastin an m x n x p factorial experiment

Since all the theorems that affect the confounding pattern of any

particular contrast of an m x n experiment are embodied in this rule, ex-

tension of the rule to an m x n x p or larger experiment will be valid.

68

Such extension is straightf'orward and given below to obtain the confounding

pattern of any contrast (rst) of an m x n x p experiment.

Table 14. Confounding pattern for any polynomial contrast in an m x n x pfactorial experiment.!!

(a) [eOO] (000) (000) (R100) (rOO)Omit when r is even Omit when r is oddOmit when m is even Omit when m.., = m_

(b) [OeO] (000) (000) (OS"'O) (OsO)Omit when s is even Omit when s is oddOmit when n is even Omit when n. = n_

(c) (OOe] (000) (000) (OOT"'') (Oot)Omit when t is even Omit when t is oddOmit when p is even Omit when p. = p_

(d) (a) x (b) (a) x (b) (rsO)

(e) (d) x (c) (d) x (c) (rst)

Y See footnote to Table 13. T~ is treated as R~ and S~ in that table.

In practice is is simpler to first obtain row (d) and then row (e).

6.2.4 Polynomial confounding patterns for anm x n x p experiment

Table 15 supplies the confounding patterns of the f'our basic con­

trasts of highest order of an m x n x p experiment: (uuu), (uue), (uee),

(eee). Any other contrast can be found by permuting factor letters; i.e.,

(ueu) is obtained by interchanging B and C in (uue). As before, K denotes

the contrast belonging to the heterogeneous bundle of' the confounding con-

trast.

Table 15. Confcunding patterns of the four basic contrasts of highestorder of.an m x n x p experiment.

69

(uuu) (eee)

mjm_ m.jm_ mjm_ m =In+ -

All odd All n.jn_ njn_ n =n n cnp even n,p even even + - + -

pjp- P =p P =p P =p+ - + - + -

(EOO) K K K K(OEO) K K K(OOE) K K K

(EEO) K K K(EOE) K K K(OEE) K K K K(EEE) K K K K K(UUU) K K K K

(uee)

m odd m evenn)n_ njn_ n =n nJn_ n)n_ n =n

+ - + -pjp- P =p P =p pjp- P =p P =p

+ - + - + - + -

(EOO) K K K(UOO) K K

(UEO) K K(UOE) K K K K(UEE) K K K K K K

(uue)

m,n odd In odd, n even m,n even

PjP- p =p pip_ P+=P- pjp- p =p+ - + -

(EOO) K K K K(OEO) K

(EEO) K K(000) K K K(UUE) K K K K ·K K

70

6.3 Determination of the Coefficients of theBlock Constants

Whenever a contrast is confounded with blocks, its expectation will

contain some linear combination of the block constants. The coefficients

of this linear combination may be obtained by application of the Bainbridge

technique (Binet et aI, 1955), or by any easy extension of the rule used to

derive the heterogeneous bundle of the contrast confounded.

Consider first contrast (A B Ct ) of the m x n x p experiment.r s .

When a , b , ct are partitioned into negative and positive parts, ther s

expression J' arbsct J can be written as

J1a b c J ~ (tr!r_)(tr E.sJ(tr 2.t ... ) ... (tr !r_)(trE.s... )(tr '£t-)

• (tr !r.)(tr E.sJ(tr 2.t-) (tr !r+)(tr E.s+)(tr .sc.,)+ (tr !r_)(tr E.s_)(tr '£t-) (tr!r_)(tr E.s+) (tr '£t ... )

... (tr !r... )(tr E.s_)(tr '£t ... ) + (tr !r+)(tr E.s ...)(tr .£tJ

= (ArBsCt )+ + (ArBsCt )_

The values (ArBsCt )+ and (ArBsCt )_ will respectively be the coefficients

of the block constants in the expectation of the confounded contrast

(A B Ct. ).r s

In any particular case, when the traces of the parts of the vectors

are lmown, the block coefficients may be obtained by inspection of the

expression

Consider, for example; the confounding of the contrast (A4B2C1 )

of the 5 x 3 x 2 experiment.

71

Since

!4 ~ D(+1,-4,+6,-4,+1),

~2 .. D(+1,-2,+1),

S ~ D(-l,+l)

tr a .. -8 tr b = -2 tr c = -1-4- -2- -1-tr 2:4+ = +8 tr b2+ ~ +2 tr £1+ = +1~

Consider the expression (+8,-8)(+2,-2)(+1,-1); the coefficient of the posi-

tive block becomes

(8 x 2 x 1) ... (8 x -2 x -1) + (-8 x 2 x -1) ... (-8 x -2 x 1) = 64;

that of the negative block, -64. From Table 15 the heterogeneous bundle ­

of contrast (421) ·is (421,221,021,401,201,001) ~ The contrast (221) in

this bundle is the contrast (221)\(421) which, since the b and ccoeffici­

ents are the same in (221) and (421), may be written (214)(2)(1)~ Since

2:2 = D(+2, -1 ,-2, -1, +2 ), it follows that tr c'!:21I2:4) = -2 and

tr (2:22\2:4) .. +2~ To obtain the coefficients of the block constants in

the expectation of the contrast (221), we have to examine the expression

(+2,-2)(+2,-2)(+1,-1)~ Comparing this expression with that obtained for

contrast (421), it follows that the ooefficients will become 64/4 = 16, and. . a .

-64/4 ::: -16. For the contrast (021) we examine the expression

(+2,-2)(+1,-1), and find the coefficients of the block constants to be

4 and -4. Similar procedures are applied for the contrasts (401),(201),

(001) ~

This work can be systematized as follows:

In 5 x 3 x 2 experiment,

!4 = D(+1,-4,+6,-4,+1)

!2 = D(+2,-1,-2,-1,+2)

.£2 = D(+1,-2,+1)

72

Confounded contrast: (421)

Heterog. bundle

(421) I(421)(221 )I (421)(021) I(421)(401) 1(421)(201 ) t (421 )(001) 1(421)

Partitioned vectors

(+8,-8)(+2,-2)(+1,-1)(+2,-2)(+2,-2)(+1,-1)(+2, -2 ) ( +1, -1 )(+8,-8)( +1.-1)(+2,-2) (+1,-1)(+1,-1)

Block coefficients

(64,-64)(16,-16)( 4,-4 )(16,-16)( 4,-4 )(+1,-1 )

CHAPTER 7

The Confounding Pattern of Twd Contrasts

73

701 Simultaneous Partitioning of TwoOrthogonal Polynomials

We now investigate further partitioning of the polynomial elements

as given in Table 6a. First we will consider the results for even N. As

before, Pe denotes an even polynomial with 2N1 negative elements followed

by 2N2 positive elements (2N1 + 2N2 := N). Let Pellpe denote another even

polynomial with its elements arranged so that the first 2N11 elements are

negative, the next 2N12 are positive, the neri 2N21 are negative and the

final 2N22

are positive (N11 + N12 =NJ.and N21 + N22 = Na). Finally,

let pu\(Pe,Pel ) = pulpel\Pe be any odd polynomial with elements corres­

ponding to those of Pellpe. Each pair of equal elements in Pellpe will

correspond to a pair of equal (in absolute value) elements of any PUJ

ioe., there will be a +U and -U for each pair of elements in Pellpe.

Hence Pu \(Pe'Pel) will not be confounded with block effects.

If N is odd there will be a central element OfPel\PeJ this

central value will be either negativa or positive • However, the corres­

ponding central value of pu\(Pe'Pel ) = pulpellpe will be zeroo Hence,

whatever the value of N, Pu I(Pe'Pel) will be unconfounded with blocks.

It can readily be verified that this is the only partitioning, in terms

of the simultaneous partitioning of two orthogonal polynomials, that will

yield parts whose elements sum to zero.

Let us now consider the general case of an m x n experiment with

contrasts (ArBs) and (AfBg ) confounded simultaneously (in 4, 6, or 9

blocks). Using the notation of previous chapters,

74

(~BS) (y = 1) \ (ArBs,AfBg )++ :: tr ~22 tr 12.s22

+ tr !R2l tr E.s2l

+ tr ~12 tr E.s12

+ tr ~ll tr £S22

(~BS) (y :: 1)1 (ArBs,AfBg )+:- "" tr !R22 tr £S2l

... tr !R2l tr £S22

+ tr !R12 tr £Sll

... :tr .!all :tr E.s12

and similarly for the other blocks. For main effects

Aa. (y = 1) '(ArB$,AfBg~++ :: n++ tr ~22 ... n+_ tr !R2l

+ n-+ tr !:.R12 + n__ tr ~ll

(7.3)

and similarly for BS by replacing n by m and ~ by £S. If R is odd and

both rand f are even in (7.)),

tr~22 "" tr !fl.2l :: tr ~12 = tr ~ll "" OJ

hence Aa is unconfounded with blocks. If r :: f, the right-hand side of

(7.3) becomes

In addition to the above solution, tr ~r2 =tr !fl.l = 0 if m is even, R

is even and l' = f is odd. Hence if m is even, R is even and r "" f is

odd, A.a is also unconfounded with blocks. These results are summarized

as follows:

Theorem 7.1. In an m x n experiment, .t\I(ArBs,AfBg ) will be unconfounded

when either (i) R is odd and both rand f (including l' ::: f) are even, for

all m and n, or (ii) R is even and r = f is odd, for m even and all n.

Afurther examination of (7.1) and (703) indicates the following

Theorem 702. In an m x n experiment, when contrasts (A B ) and (A,&'B ) arers .L.g

confounded simultaneously, contrast (~BS) will be unconfounded when either

~ \ (ArBs,AfBg ) or BSI(ArBs,AfBg ) is unconfounded; ~I(ArBs,AfBg ) will be

unconfounded if ~I(Ar,Af ) is unconfounded and similarly for BS.,

If .Aa is unconfounded, tr ~22 = tr ~21 = tr ~12 = tr !all = 0;

similarly if BS is unconfounded. We have been unable to prove that the

conditions in these theorems guarantee that all other situations produce

.confounded effects; experiments fulfilling theconditions in these theorems

result in unconfounded effects, but other situations may also prevail in

which effects could be unconfounded.

7.2 The Generalized Interaction

'When two contrasts (A B ) and (A,&'Bg ) of an m x n experiment are con-o r s .L

founded simultaneously, then, as shown in Chapter 4, their generalized

interaction (A B ) will be confounded automatically. The usual notationvw

for the generalized interaction of two contrasts is

It is important to observe that this is a symbolic relation only, not an

algebraic one; algebraically, however, av =a at and b =b b 0 Ther w s g

notation is meant to imply that when a new contrast is formed out of the

coefficient matrices of two given contrasts, this new contrast will be

confounded, whenever the two given contrasts are confounded simultaneously.

Assume the matrices a and b are expressible as linear combinations ofv w

other matrices, say

a = araf = ar +f + 0(2ar+f_2 + = L(ar +f ), sayv

b = b b = b + ~2ba+g_2 .... ... = L(bs +g ), sayw s g s+g

where 0(2 and ~2 are constants. In this case,

avbw = L(ar+f)L(bs +g)

= ar+fbs+g + ~2ar+fbs+.g_2 +. o(aar+f_2bs+g + •••

... L(a +fb + )r s g

and

(AvBw) = (Ar+fBs+g) + ~2(Ar+fBs+g_2) + o(a(Ar+f_2Bs+g) + •••

= L(Ar+gBs +g )

Hence, when the generalized interaction of the two contrasts (A B ) andr s

(AfBg ) is expressible in the form of sfinear combination of contrasts,

this linear combination will be confounded when the two contrasts are

confounded simultaneously. It does not follow, however, that each con-

trast in the linear combination will be confounded (see Corollary 4.3.1).

7.3 Reduction Formulae for Products ofOrthogonal Polynomials

The recursion formula of the orthogonal polynomials in the Fisher

system can be written as

It might appear that we would require generalization of this formula to

the case Ps x Pr for the purpose of expressing the generalized inter­

action as a linear combination of contrasts, but this is not necessarily

76

so. Consider the product P x P when there are N observations, ands r

77

suppose that s = r = (N-l). Then P x P will equal some linear combina­s r

tion of polynomials such as P2(N-l) + ..<P2(N_2) + •••• Since there are only

N observations, all polynomials with degrees higher than (N-l) will vanish,

and the number of terms in the linear combination is greatly reduced.

The number of levels of anyone factor being usually small in

practice, the direct calculation of the reduction formulae, for those cases

likely to be useful, seems indicated. Table 16 accordingly supplies the

reduction formulae of the products for N through 5:

Table 16. Products of polynomials.

N == 3 N == 4 N = 5

P1 X P1. = Pa + 2Po

P1 X Pa == P3 + ~P1

_ 36P1. x P 3 - P4 + 35Pa

P1. x P4 == ~P3

P", X Pa == P + ~ +]#p'" 4 7 a :J 0

P P • 2~ + 36pa x 3 -S.r 3 . 25 :L

10 14lLPa x P4 == '"'TP4 ,+ ~a

P3 x P3 == --ftPa + /:oPo P3 X P3 == -~P4 -172?2 + f~~o

P 3 X P4 == -~P3 + i~~1.

P4

X P4

== 108p _ 288p + 288p245 4 343 a 175 0

P1. X P1 == Pa + tPo

P1. X Pa ;'; P3 +~P1

P1 X P3 == loPa

-1 2P x P == --P + -P Pa x Pa == Poa .2 '3 2 -9 0

Since in practice we use P' == AP, the relations in Table 16 should

be adjusted accordingly before application to a specific problem. In this

chapter we use Table 16 in a symbolical sense only.

78

7.4 The Status of the Contracts in the1inear Combination

1et u and u1

denote odd and e and e1 even confounding contrasts;

ua and ea odd and even contrasts in the generalized interaction. There are

four types of confounding contrasts, depending on the odd-even character

of (rs) and (fg). If r and f are both odd or both even, we call them

similar contrasts; if one is odd and the other even, they are dissimilar

,contra.sts; likewise for sand g. The four types are

Charaoter of contrasts

~ r and f s and g Example

I Similar Similar (ue) x (U1

81

)

II Similar Dissimilar (ue) x (u1 u1 )

III Dissimilar Similar (ue) x (e1 e1 )

IV Dissimilar Dissimilar (ue) x (e1u1 )

Based on the results in Table 16, the form of the generalized

interaction 1((r+f)(s+g)] is given in Table 17, for each of these types.

Table 17. Form of generalized interaction contrast, L[(r+f)(s+g)]Y

Type r r fs t g

r = fs t g

r f fs = g

r = fs = g

I 1(e2e2 ) L{eaea,Oea)

II L(eaua) L(eaua,Oua )

III 1(ua6:a) ~

liT L(uaua) ~

17~ indicates this combination cannot exist ..

For example, in a 5 x 4 experiment: (12) x (31) = L(43), where

(12) = (ue), (31) = (u1u1 ); hence, from Table 16

79

This is a Type II contrast with r r f and s f gj hence L(43) is a sum of

even-odd contrasts. Each individual contrast in L(43) [(43), (41), (23)

and (21)] is confounded. However, it should be emphasized that when the

contrasts (RS), (RO), (OS) belong to the linear combination L[(r+f)(s+g)],

the automatic confounding of this linear combination will not necessarily

confound the contrasts (RS), (RO), (OS). For example, consider the con-

founding of contrasts {32) and (33) of a 4 x 4 experiment. Here

Note that the complete compounding of the subscripts does not hold in

this case, since the sum of the corresponding degrees for each factor

cannot exceed the number 3 = 4 - 1, where 4 is the number of levels.

By Theorem 7.2, (23) 1(32,33) and (21)\(32,33) will be uncOnfoundedj

however, (03) \(32,33) and (01)\ (32,33) will be confounded.

7.5 Confounding Pattern of Two ContrastsConfounded Simultaneously

Consider the simultaneous confounding of contrasts (A B ) andr s

(AfBg) of an m x n experiment. By Corollary 4.1.1, all contrasts belong-

ing to the heterogeneous bundle of (ArBs) [if (ArBs) has a heterogeneous

bundle], and all contrasts belonging to the heterogeneous bundle of

(AfBg ) [if (AfBg ) has a heterogeneous bundle] will be confounded when

(ArBs) and (AfBg ) are confounded simultaneously. Hence the first step

to obtain the confounding pattern of (ArBs,AfBg ) will consist in applying

the rule developed in Chapter 6 to each of the two contrasts separately

(alternatively, Table 15 can be used).

80

Let us now denote the generalized interaction (rs) x (fg) by L(rts'),

where L(rts t ) is a linear combination of the contrasts

(rts t ), (r t (s'-2)), «r'-2)s'), «r'-2)(st-2))···

The expression (RtSt) will refer to anyone of the contrasts in L(rts t ).

According to Table 17, when both r f f, s i g, contrasts of the type

(RtO) and (OS') will not appear in L(rts t ); when r = f and s i g, con­

trasts of the type (RtO) will not appear; when r If and s = g, contrasts

of the type (OSt) will not appear. Let [rtst] denote the homogeneous

bundle of the particular contrast (r'st), which can be found in Table 13.

If r t = e' and st = u t , for example, [rts t ] will be the homogeneous bundle

[etu t) = (EU). It follows by the results of the partitioning of the

orthogonal polynomials that if contrast (e tu') is confounded, all contrasts

belonging to the homogeneous bundle [e tu t] will be confounded. Similarly

let [r'O] denote the homogeneous bundle of (rtO) and [Os'] the homogeneous

bundle of (Os').

When both r I f and s f g, L(rts t ) will be a linear combination of

the contrasts belonging to the homogeneous bundle [rtst]; when r = f and

s f g, L(r'st) will be a linear combination of the contrasts belonging to

the homogeneous bundles (rts t ] and (Ost]; when r f f and s = g, L(r'st)

will be a linear combination of the contrasts belonging to the homogeneous

bundles [rts t ] and [rtO]. For example, in a 5 x 4 experiment:

(12) x (31) = L(43), where L(43) is a linear combination of the contrasts

belonging to the homogeneous bundle [43], !!!., (43,41,23,21); in a

4 x 4 experiment: (32) x (33) = L(23), where L(23) is a linear combination

of the contrasts belonging to the homogeneous bundle [23], viz., (23,21)

and to the homogeneous bundle [03], viz., (03,01).

81

By Theorem 7.2, contrast (r's') will be unconfounded by (rs,fg)

when either r'l (r,f) or s'l(s,g) is unconfounded; (r'O) will be uncon­

founded if r' \(r,f) is unconfounded and (Os') Vlill be unconfounded if

s'\(s,g) is unconfounded. Hence, under the same conditions, [r's'], [r'O]

and [Os') will also be unconfounded by (rs,fg).

In terms of the theoretical development above, the following stages

can be distinguished in the procedure for obtaining the confounding pattern

of (rs,fg) in an m x n experiment:

(i) Obtain the confounding pattern of the contrast (rs).

(ii) Obtain the confounding pattern of the contrast (fg).

(iii) Determine the linear combination L(r's') = (rs) x (fg) from

Table 16.

(iv) Obtain the homogeneous bundle [r's'] of the contrast (r's')

when r f f,s ; g; the homogeneous bundle [O's) of the con­

trast COs') when r = f,s r g; the homogeneous bundle [r'O]

of the contrast (r'O) when r f f,s = g.

(v) Determine the confounding status of the homogeneous bundle

[r's') when r r f,s r g: of the homogeneous bundle [Os']

when r = fjs r g; of the homogeneous bundle [r'O] when

r rf,s = g.

In actual practice, stages (iii), (iv) and (v) can be combined into the

SYmbolic multiplication scheme given below, which embodies the consequences

of the theorems stated above 0

The part to the right of the double vertical line is first written

down; then, [r'O] in row (a) and [Os'] in row (b) are obtained from the

symbolic products (f x r) and g x,s) respectively in Table 16, and the

confounding according to Theorems 7.1 and 7.2, remembering that r' is

82

even if r and f are both even or both odd. Depending on the conditions

given in the scheme, multiply the elements in row (a) by the elements in

row (b) to obtain the elements in row (c), subject to the restrictions that

mu~tiplication across vertical lines is not permitted.

Table 18. Rule for obtaining homogeneous bundles in the generalizedinteraction of the contrasts (rs,fg) of an m x n experiment.

(a)

(b)

(c)

(00)Omit when r 1= f

(00)Omit when s 1= g

[r'O]Omit when r = f isodd for m even

[Os']Omit when s = g isodd for n even

(a) x (b)

(fO)

( Og)

(fg)

(rO)

(Os)

(rs)

When the confounding patterns of contrasts (rs) and (fg) are un-

known, the last two columns of Table 18 can be enlarged to accommodate the

scheme of Table 13.

As an example, consider the confounding of the contrasts (12) and

(31) of a 5 x 4 experiment. Since r f f,s 1= g, (12) x (31):L(43). The

scheme for the generalized interaction becomes

[40]

[03]

[43]

(0)

(01)

(1)

(10)

(02)

(12)

The homogeneous bundle of contrast (43) is (43,41,23,21). From Table 13

the heterogeneous pundles of contrasts (31) and (12) are respectively

01,33,11,13,20,40) and (12,32,20,40). Hence the confounding pattern of

contrasts (31) and (12) is

(33,31,13,11,40,20,43,41,32,12,23,21).

83

Generalization of the rule to experiments with more than two factors is

immediate, as was done in Table 14.

The result s in this chapter presume that the only contrasts which

are confounded when (ArBsCt ) and (AfBgCh ) are simultaneously confounded

are: (i) those contrasts confounded by either (ArBsCt ) or (AfBgCh ),

(ii) possibly some or all of the contrasts in the linear combination of

the generalized interaction (A B C ). We have been unable to prove thatvwz

no other contrasts orthogonal to the confounding contrasts can be confounded;

however, we believe that this is the case, and recommend the above pro-

cedures be used in determining which contrasts are confounded.

84

CHAPTER 8

Confounded Experiments with Equal-sized Blocks

8.1 Confounding a Single Highest OrderInteraotion Contrast

Consider the confounding of contrast (ArBsCt ) of an m x n x p ex­

periment. As before, let m_, m+" mo denote the frequencies of the negative,

positive and zero elements respectively of the vector a ; n ,n ,no those-r - +

of the vector .£s; P_'P+'Po those of the vector .£t,. Schematically we have

Blockindicator

+

o

Total

Frequenciesa b 2t-r -s

m n p--m+ n+ P+

mo no Po

m n p

Confounding (ArBsCt ) consists of placing the treatment combina­

tions corresponding to a - sign in one block, those corresponding to a

+ sign in another block. Confounding a single contrast, therefore, im-

plies the arrangement of the treatment combinations into 3 or 2 blocks,

depending on whether the 0 block indicator is present or absent.

Let N_, N+, No denote the number of treatment combinations in the

negative block, the positive block, and the zero block respeqtively.

Then, from the above scheme

85

N = m n n + m nn + m n p + m n n_ _ ',.."F_ _.p- + + _ _ + ¥-_

N = mnp - (m-m )(n-n )(p-p )o 000

= mnp - (m+ + m_)(n+ + n_)(p+ + p_)

If No = 0, we need only have N+ = N_ to have equal-sized blocks (there

will be only two blocks in this case). The condition for N+ = N is

(m - m )(n - n )(p - p ) =0+ - + - + -

(8.1)

(8.1) is satisfied if and only if one or more of these conditions hold:

(802)

If N > 0, (8.1) must still be met; however, a second condition is alsoo

imposed:

N+ + N - 2N = 0; or,- 0

3(m-m )(n-n ) (p-p ) = 2mnp000

(8.3)

It is not possible to derive a simple condition, such as (8.2), which will

satisfy (803). One must determine suitable combinations of (m,n,p) for

given (m ,n ,p ) or vice versa. For example, if m = n = 0 and00000

p > 0, P = 3p , and m and n can be any value. In general, if only twoo 0

of the three main effect contrasts have no zero elements, the number of

levels of the other factor must be three times the number of zero elements;

in a.ddition, at least one of the three contrasts must have the same number

of positive as negative elements to satisfy (802)0

Similarly, if m = 0 and n = 1,o 0

3(p-po)n = p _ 3p0 and p =

3(n-l)po ;n - 3

86

hence, n > 3 and p > 3Po· If m = 0 andn = 2,0 0

6(p-po) 3(n-2)pand 0n p = 6 ;p - 3Po n -

hence n > 6 and p > 3p. These results can be extended to any n , and cano 0

be interchanged to give formulas for the situation where any main effect

contrast contains no ~ero elements and each of the others has at least one

zero element. Of course, condition (8.2) must also be met.

3(m-l) (n-l)p = •mn - 3(m+n-l)'

hence m > 3 and n > ~(m-l)/(m-3). If mo = no = 1 and Po = 2,

6(m-l) (n-l)p = mn - 3(m+n-l) ;

hence m > 3 and n > 3(m-l)j(m-3), as before. These results also can be

extended to other values of mo' no and Po' and can be easily interchanged

to give results for any situation with every main effect contrast having

at least one zero element.

Table 19. Examples of contrasts which satisfy condition (8.3), with atleast two contrasts having zero element s.Y

m n Po m n P0 0

0 1 1 Any 4 90 1 1 Any 5 60 1 2 Any 7 90 1 2 Any 9 81 1 1 6 9 10

1 1 1 7 8 9

YThese are main effect contrasts, the 3-factor con-trast elements consisting of all possible productsof the main effect elements; m ,n and p are thenumber of zero elements in theOcoRtrastsOand m, nand p the total number Of elements.

87

Table 19 presents specific examples of the results mentioned above

for number of levels less than 10. These results can be generalized by

interchanging any of the contrasts. For example, one design presented in

Table 19 is mo = 0, no = Po ::: 1, n = 5, p ::: 6 and m any value. Similarly

one can have m 0, n = p ::: 1, n ::: 6, p = 5 and m any value;o 0 0

m ::: p ::: 1, n ::: 0, m ::: 5, p ::: 6 and n any value, etc. It should be em­000

phasized again that these designs must also have at least one contrast

with equal numbers of positive and negative elements.

When one compares Table 19 with Table 5, hs notes that no orthogonal

polynomial contrasts meet the requirements of Table 19. Only odd-level

contrasts have, zero elements; hence, none of the results in Table 19 with

the number of zero elements being 0 or 1 can be used (in all cases at

least one of the values m, n or p is even). For p ::: 2, we note thato

p ::: 9, whereas the only case in Table 5 is p = 7. If one had an experiment

with one factor having as many as 15 levels, he could use p~ for p ::: 7

in Table 5, because ifm /= 0, n ::: 1 and p ::: 2, (8.3) is satisfied if000

n ::: 15 and p ::: 7. Based on these conclusions, it appears that, in most

situations, at least two of the contrasts must have no zero elements if

(8.3) is to be satisfied. If the third contrast has one zero element, the

contrast must be the linear contrast for a three-level factor; no orth-

ogonal polynomial contrast meets the requirement for more than one zero

element (p = 3p ).o

These conditions lead to the formulation of the following rules

for orthogonal polynomial contrasts:

RULE 1. When a 'highest order interaction contrast is confounded, two

equal blocks will result if (i) no zero elements appear in any coefficient

vector of any factor in the confounding contrast, and (ii) the coefficient

88

vector of at least one factor in the confounding contrast has the same

number of positive as negative elements.

RULE 2. When a highest order interaction contrast is confounded, three

equal blocks will result if (i) one of the factors has three levels and

the coefficient vector of this factor in the confounding contrast is of

the first degree, and (ii) no zero elements appear in any coefficient

vector of any remaining factors in the confounding contrast.

Table 20 supplies the frequencies of the block indicators in the

various degrees of the orthogonal polynomials for n through 8.

Table 20. Frequencies of block indicators in the orthogonal polynomials.

n Degree n n+ n n Degree n ~ n0 0

2 1 1 1 0 1 3 3 12 3 2 2

3 1 1 1 17 3 3 3 1

2 1 2 0 4 2 5 05 3 3 1

1 2 2 0 6 3 4 04 2 2 2 0

3 2 2 0 1 4 4 02 4 4 0

1 2 2 1 3 4 4 0

5 2 3 2 0 8 4 4 4 03 2 2 1 5 4 4 04 2 3 0 6 4 4 0

7 4 4 01 3 3 02 2 4 0

6 3 3 3 04 4 2 05 3 3 0

89

802 Simultaneous Confounding of Two HighestOrder Interaction Contrasts

Consider first the simultaneous confounding of the two contrasts

(A B ) and (AfB ) of an m x n experiment. The block indicator of ther s g

treatment combination y .. will be the combination of the sign of (a ib .)~J rSJ

with the sign of af.b .). The following nine combinations are possible:~ gJ

(--),(-+),(+-),(++), (-0),(+0),(0-),(00), where the first "signt' refers

to (a .b j) and the second "sign" to (af.b .). Sincer~ s ~ gJ

(a .b .)(a~ibh) = (a .bh)(a~.b .),-rJ.-sJ -oJ.;s -rJ.-g -J.J.-sJ

the blocking arrangement resulting from the simultaneous confounding of

A Band AfB will be identical to',the blocking arrangement resultingr s g

from the simultaneous confounding of A B and AfB 0r g s

The blocking arrangement resulting from the simultaneous confound-

ing of (A B ) and (AfB ) can be obtained by first confounding (A B ) andr s g r s

then (AfB ); that is, the blocks obtained by confounding (A B ) are theng r s

divided into as many sub-blocks as would be required by the subsequent

confounding of (AfBg ), or vice versa. Alternatively, by our result above,

(A B ) could first be confounded and then (AfB). This procedure is use-r g s

ful in determining whether nine equal blocks can result from the simultane-

ous confounding of two highest order interaction contrasts in an m x n

experiment.

By Rule 2, the confounding of (ArBs) will yield three equal blocks

if (A B ) = (A1- B ) when m = 3 and s is even, or if (A B ) = (A B

1) whenr s ' s r s r

n = 3 and r is even, but not if (ArBs) = (J\B1 ) when m = n = 30 Since

confounding of (AJ.~) x (AfBg ) or (AJ.Bs ) x (AfBJ.) implies a 3 x 3

e:xperiment, there is no point in examining the confounding pattern;

90

obviously no one will plan a 3 x 3 experiment in nine blocks. Hence the

only possible useful set of nine equal blocks would be based on confound-

ing (-\Bs ) x (-\Bg ) or (Ar~) x (A~J.). Consider the former. As before,

a. = (-1, 0, +1); b and b are restricted only to have no zero element·s.-.1 -s-g

Let n~E. (~ = ... or -; t. == ... or -) denote the number of elements with a

j3-sign on b and an ~-sign in b 0 The size of each of the nine blocks-s -g

is given in Table 21, in the second column from the right.

Table 21. Size of blocks when (A B ) x (ArB ) is confounded underrt . dit· r s gce aln can lons.

Block sizem = 3,

n = n. so go

Block(A B HAfB )r s g

m == 3, r = f = 1n = n = 0so go

n ... n+- -+

Impossible

Impossible

Impossible

Impossible

2n

r - 1, f = 2= 0; n + = ng g-

n + n+_ = n- g-

n_+ + n...1o == ng+

n..._ + n =ng-

n++ ... n_1o == ng+

n + n == n- +- g-

n+... + n_1o = ng+

Impossible

Impossible

Impossible

We note that only five blocks are possible and that the (00)

block is by far the largest. A similar statement holds when A and B are

interchanged. Hence one cannot construct a design of this kind with nine

equal-sized blocks.

Next let 11s investigate if we can construct a design with six

equal-sized blocks 0 In this case we consider (~Bs) x (A2 Bg ), where

neither b nor b have zero elements and n + = n (no other restrictions~ -g g ~

91

on £.s) & The block sizes are given in the right-hand column of Table 2L

Since n = n ,the six blocks are of equal size & A and B can be inter-g+ g-

changed in the results &

In an ~ x n x p experiment, we require the following to have six

equal-sized blocks:

(i) m = 3, r =1 and f = 2

(ii) no other vectors have any zero elements

(iii) ng+ = ng_ or/and Ph+ = Ph-.

Obviously A, Band C can be interchanged in these results.

RULE 3: When two highest order interaction contrasts are confounded simu1-

taneous1y, six equal blocks will result if (i) one and only one of the

factors has three levels, and the coefficient vector of this factor is of

the first degree in one of the two confounding contrasts and of the second

degree in the other contrast; (ii) with the exception of the first degree

coefficient vector of the factor with three levels, no zero elements appear

in any coefficient vector of any factor in the two confounding contrasts;

(iii) for at least one of the other factors in the contrast involving the

second degree main effect in (i), the number of positive elements equals

the number of negative elements in this coefficient vector&

Consider now the simultaneous confounding of the contrasts (ArBs'

AfBg ) of an m x n experiment, when none of the coefficient vectors contain

zero e1ements& Schematically we ha.ve

Blockindica.tor

-+

+-

++

Total

m n

m_+ n_+m n+- +-

m++ n++m n

92

The sizes of the blocks will be given by the relations:

Block sizes will be equal if the following equations are satisfied simul-

taneously:

(N++ + N+_) ­

(N++ + N_+)

(N++ + N )

(N_+ + N ) = 0--(N+_ + N ) = 0-(N+_ + N_) = 0

or; after substitution, if

(m + - m )(n + - n ) = 0r r- s· s-

(mf + • mf_)(ng+ - ng_) = 0

(m++ - m+_ - m_+ + m_)(n++ - n+_ - n_+ + n ) = 0

When three-factor contrasts are considered, these equations become

(mr + - mr_)(ns + - ns_)(pt + - pt -) = 0

(rof + - mf_)(rg+ -rg_)(Ph+ - Ph-) = 0

(m++-m+__-m.+ -+m__ ) (n++-n+_-n_++n__ ) (P++-P+_-P_++p_) = 0

(8.4)

(8-5)

(8.6)

(8.7)

(8.8)

(8.9)

If m++ = m+_ = m_+ = m__ then roT+ ::: mr _ '" rof + ::: mf _, and hence (8.4) to

(8.9) will be satisfied. Similarly for the partitioned n and p values.

These conditions lead to Rule 4.

RULE 4: When two highest order interaction contrasts are simultaneously

confounded, four equal blocks will result (i) if no zero elements appear

93

in any coefficient vector of any factor in the two contrasts; and (ii) when

the first coefficient vector of the first contrast is partitioned according

to the first coefficient vector of the second contrast (or vice versa),

and the same process applied to the two corresponding coefficient vectors

of the other factors in the two confounding contrasts, the parts of at

least one of these partitioned coefficient vectors contain the same number

of elements.

Table 22 supplies the signs of the elements of the orthogonal poly-

nomial values for n = 2, 4, 6 and 8.

Table 22. Signs of the elements of the orthogonal polynomial vectors.

n = 2 n = 4 n = 6 n = 81 1 2 3 1 2 3 4 5 1 2 3 4 5 6 7

+ + - + - + - + ++ + + + + + - + - -+

+ + + - + + -+ + + + - - + + + + +

+ + + + - -+ + + + + + - + + +

+ + - - - - -+ + + + + + +

If the elements of each degree are partitioned in terms of the elements

of each of the other degrees, the relation n++ = n+_ = n_+ = n__ holds

when n is equal to 4. When n is equal to 8, the relation does not hold

for the partitionings 11 3, 115, 317, 51 7, but is true for all the other

partitionings. When n is equal to 2 or to 6, the relation does not hold.

It is also possible to obtain four equal-sized blocks in an

m x n experiment by the following procedure:

In general one can select any factor from (8.7) and a different

(a)

(b)

(c)

mr += mr _ and ng+ = ng_

ID++ = m_+ and ID+_ = m or n++ = n+_ and n_+ = n_

94

(8.10)

(8.11)

....--

one from (8.8) for (8.10) and then apply (8.11) accordingly. Hence one

can set up rule 4a.

RULE 4a. When two highest order interaction contrasts are s:ilnultaneously

confounded, four equal-sized blocks can also result if (i) no zero elements

appear in any coefficient vector; (ii) equality of positive and negative

elements in the original coefficient vectors follows (8.10); (iii) equality

of the m~B or n~~ follows (8.11); (iv) one can interchange factors as sug­

gested in (c) above.

Rule 4aenables one to construct 6 level designs in four equal blocks.

8.3 Exper:ilnents in Equal Blocks With MainEffects Unconfounded

8.3.1 Two-factor experiments

Examination of Table II reveals that those two-factor experiments

in which the confounding of any highest order interaction contrast does

not confound main effect contrasts, also satisfy the rule for obtaining

two equal blocks. In all cases the number of levels of each factor must

be even; it any factor has an odd number of levels, at least one main effect

contrast will be confounded. In certain cases the number of positive ele-

ments must also equal the number of negative elements in the main effect

vector, as indicated below~

Confamdingcontrast

(uu)

(ue)

(eu)

(ee)

Requirements so that no main effectcontrasts will be confounded

m and n even

m even; n+ = n

n even; m.... = m

m =m-n =n+ -'.... -

95

/

Consider next the simultaneous confounding of any two highest order

interaction contrasts with corresponding subscripts not alike _ If

m = n = 0 and m = m and n = n (m andn both even), their generalizedo 0 + - +-

interaction will either be a single contrast of the same order or a linear

combination of contrasts of the same order (see Table 17) and hence no

main effect contrasts will be confounded because of this generalized inter-

action. By Rule 3, four equal blocks will result_ The homogeneous

bundles for various confounding patterns are indicated below_

Homogene aus bundle Sets of contrasts confounding this bundle

[EE] (uu,ul u1.); (ue,u1e], ); (eu,elul ); (ee,e1.e1.)

[EU] (uu,u],e1 ); (ue ,u1.ul ) ; (eu,e1 e1.); (ee,el u1.)

rUE] (uu,el u1.); (ue,elel ); (eu,ulul ); (ee,ule],)

[00] (uu,e], e1 ); (ue,elul ); (eu,ul el ); (ee ,ul U l )

8.3.2 Three-factor experiments

Examination of Table 15 reveals that those m x n x p experiments,

in which the confounding of any higher order interaction contrast does not

confound main effect contrasts, also satisfy the rule for two equal blocks.

The combination of factor levels in the confounding interaction contrasts

which will not result in confounding main effect contrasts are presented

below.

Confoundingcontrast

uuu

uue

uee

eee

Requirements so that no main effectcontrasts will be confounded

m, n and p even

m and n even

two or three of these: m. = m_,

96

The interaction contrasts which are confounded are presented in Table 15.

Consider finally the simultaneous confounding of any two highest

order interaction contrasts. From the left-hand column of the extension

of Table 18 to three factors, we see that no more than one pair of the

corresponding subscripts (out of the three) can be alike and still not

confound a main effect because of the generalized interaction.

Table 23 supplies the confounding contrasts of highest order of

an m x n x p experiment such that the simultaneous confounding of any

two--with not more than one pair of corresponding subscripts alike--wi11

yield four equal blocks without confounding main effect contrasts. To

distinguish between an odd and an even number of levels: affix an asterisk

to each of the letters corresponding with factors with an even number of

levels, but the number of positive and negative elements differ for at

least one contrast; affix an additional asterisk to the same letter if

the coefficient vector of the factor corresponding with that letter con-

tains the same number of positive as negative elements for all contrasts.

For instance, p will denote an odd number of levels; p* an even number of

levels, but some p+ f P_; p~ an even number of levels such that all

97

Table 23. Confounding contrasts of highest order not confounding main l/effects and yielding four equal blocks, when taken in pairs.=t

euu

eue

m x n~ x It*'

euu

eue

eeu

eee

euu

ueu

uue

uuu

m* x n~ x p*:*

euu

eue

uue

ueu

uee

uuu

17Notmore than one pair of corresponding subscripts shoud be alike;e.g., for the first column, A I A or BIB .

e e1 u U1

giNo asterisk denotes odd number of levels.* denotes even number, but n+ I n_ for at least one contrast.,. denotes even number and p+ = P_ for all contrasts.

Details for the ~ x nD x p** and m** x n*"* x p'** are not given in

Table 23, since in these cases all eight possible confounding contrasts

(uuu,uue, ••• ,eee) can be used.

Four equal blocks in which main effect contrasts are not confounded

will also be obtained when an appropriate three-factor and an appropriate

two-factor interaction contrast, with corresponding subscripts not alike,

are confounded simultaneously. Four equal blocks in which no main effect

contrasts are confounded will finally result when two appropriate two-

factor interaction contrasts, with all corresponding subscripts not alike

when both are not zero, are confounded simultaneously.

When two three-factor interaction contrasts are confounded simul-

taneously, the requirement of at least two pairs of corresponding sub-

scripts to be unlike cannot always be met in practice, with the result

that main effect contrasts will be confounded. In such cases it becomes

necessary to confound either one three-factor and one two-factor contrast

98

or two two-factor contrasts. Consider, for instance, the confounding of

(euu,e1 u1 e1 ) in an m x n4 x p~ experiment. If m :: 3, n* :: 2, p*"* :: 4,

the confounding contrasts will be (2lu,2le) and the generalized interaction

L( 20u ' ,0Ou ' ); by Table 18 [oro] will be confounded. If, however,

(21u,01e) or (20u,01e) are chosen as confounding contrasts, no main effect

contrast will be confounded.

By' Rule 2, three equal blocks can only result from the confounding

of a highest order interaction contrast when one of the factors has three

levels, and the coefficient vector of this factor in the confounding con­

trast is of the first degree. If such is the case, however, Tables 11 and

15 show that main effect contrasts will be confounded. Hence it is im­

possible to confound a polynomial oontrast and obtain experiments in three

equal blooks for whioh main effeot oontrasts are unoonfounded. Similarly

for the oase when six equal blooks are desired) here again, by Rule .3, one

of the faotors must have three levels.

From a praotioal point of view, one would be interested in oon­

founded experiments in which the equal blook sizes are not too large, and

in whioh main effect oontrasts as well as first-order first degree inter­

aot~on oontrasts and possibly seoond-order first degree interaction

contrasts (e.g., (110) or (111)] are unoonfounded.

As oan be expeoted from the extensive investigations undertaken by

Yates (19.37), Li (1944), Kempthorne (1952), Binet et al (1955) and other'

authors, most of the two-factor and three-faotor experiments whioh have

some or all of these desirable properties are already available ~n the

literature. The following designs, however, could not be found in the

literature and appear worthy of tabulation. Two of these experiments

are compared with ones suggested by 1i (1944) and analyzed from the degree

point of view by Binet et al (1955).

99

Table 24. New three-factor experiments in four equal blocks with maineffects unconfounded.

Type of Confounding Contrast confounded?experiment contrasts 110 101 011 III

4x 2 x 2 (201,210) No No Yes No

6x 2 x 2 (ell,u1 10) Yes Yes Yes No

8 x 2 x 2 (ell,e1 10) No No No No

4x4x2(2ul,220) No No No No(201,u1 20) No No No No

6x4x2 (e21,u1 01) No Yes No No

8 x 4 x 2 (eel,e1u1 1) No No No No

4x 2 x 2 1st repl- Yes Yes No NoLi(1944) 2nd repl. Yes No No Yes

3rd repl. No Yes No Yes

4x4x 2 1st repl. Yes No No NoLi(1944) 2nd repl. Yes No No No

3rd repl. No No No Yes

8.4 Confounding of Linear Combinations of Contrasts

In the previous section it was shown to be impossible to obtain

three equal blocks with main effects unconfounded when confounding a

single orthogonal polynomial contrast. With a 3 x 3 experiment, for

instance, the confounding of (A2 B1 ) will yield three equal blocks but

will confound B1 and Bli!; the confounding of (A1 B2 ) will yield three equal

blocks, but will confound A:L and Alii. The question arises whether it would

be possible to obtain a contrast equal to some linear combination of

orthogonal polynomial contrasts such that its confounding will leave main

effects free.

100

From the scheme below it can be seen that if such a contrast exists,

the signs of its elements must follow some such pattern as given in the

last two columns.

Alternative

Treatment Elements of contrasts (y = 1) signs of

combination ~B;a A2 B]. A:aB]. + A].B;a A;aBJ. - A].B;a AxBx

00 -1 -1 . -2 0 001 +2 0 +2 -2 +02 -1 +1 0 +2 0 +

10 0 +2 +2 +2 + +11 0 0 0 0 0 012 0 -2 -2 -2

20 +1 -1 0 -2 021 -2 0 -2 +2 +22 +1 +1 +2 0 + 0

It can be seen at once that our requirements will be met when

or

(8.13)

Confounding of (8012) will yield the I-component design, while con­

founding of (8013) will yield the J-component design of Yates (1937) ..

These designs do not confound main effect contrasts ..

A similar investigation of the 33 experiment shows that the res-

pectiva confounding of the follOWing linear combinations of contrasts will

yield the replicates of Yates' (1937) balanced design in bocks of nine.

Again main effect contrasts are unconfounded.. The replicate numbers cor-

respond to the tabulation by Cochran and Cox (1950).

101

Replicate I: (AJ.BaCa ) + (AaB1C2 ) - (AaBaC1 ) + 3(A1B1C1 )

Replicate II: (A1BaCa ) + (AaB1Ca ) + (A B C ) - 3(,\B1C1 ):2 a 1

Replicate III: (~BaC:,) - (AaB1Ca ) + (AaBaC1 ) + 3(~BJ.CJ.)

Replicate IV: -(~BaC2) + (AaB1Ca ) + (AaBaC1 ) + 3(~~C1)

Similarly experiments of the type 33 x q¥f and 32 x p~ X q*" can be

arranged into six equal blocks without confounding main effects by the

simultaneous confounding of the linear combination (8.12) or (8.13) and

an appropriate two-factor interaction contrast. Thus, for instance, we

have the following sets of confounding contrasts for certain experiments.

3x3x 3 x q'* : (XXOO, 002u )

3x 3x 3xq~~ · (xXOO,002u) or (xXOO,002e)•

3x 3 x p'*' x q* · (xXOO, OOuu )•

3x 3 x p~ x q~ · (XXOO, oOnu ) or (xXOO, OOeu )•

Binet et al (195'5') devote considerable space to the 32 x 22 experi-

ment in blocks of 12; 12 new plans were developed and the confounding pattern

for a design of Li (1944) is presented. A design in six blocks of six units

each, obtained by simultaneously confounding the contrasts (xXOO) and (0011),

is identical to the one developed by Kempthorne (195'2) and confounds,

inter alia, ~B]., C]. D]., A].B]. 01D].. From a degree point of view, this design

appears to be useful; when the experimental material is rather variable;

it may be better than the design in blocks of 12.

8.5' Balanced oonfounding

Consider the 3 x 2 x 2 experiment. By Rule 1 and Table 14, oonfound-·

ing of the oontrast (211) will yield two equal blocks; contrast (011) will

be confounded, but no others. This blocking arrangement is identical to

102

Replicate II of the plan given by Oochran and Oox (1950). The confounding

scheme for this blocking arrangement is given in the first line in the

body of the scheme below:

Oonfounding A B 0contrast 0 1 2 01 01

211 + - + + +

xlI - + + + +

XII + + - - + - +

Let (xlI) and (XII) be contrasts, the signs of which are as in­

dicated above. The confounding of (xlI) yields Replicate I, and the con­

founding of (XII) yields Replicate III of the balanced design supplied by

Oochran and Oox (1950). Let x' be the row vector (-2,1,1) and X' the row

vector (1,1,-2). If P denot es the orthogonal polynomial of degree r then,r

when x and X are expressed in terms of P1 and Pa for m = 3, we find

x = (JP1 -P2 )/2

X = -(JP1 +Pa )/2

Hence

(xlI) = (J~B101 - A2~01 )/2

(XII) = -(3A1B101 + AaBa01 )/2

This result shows that a balanced group of sets for the 3 x 22

experiment can be obtained by confounding the required highest order

interaction contrast in one replicate, and those linear combinations of

highest order interaction contrasts derived from the signs of the required

contrast in two other replicates.

103

In practice one would obtain this results as follows: interchange

levels ° and 1 of the first factor in Replicate II to obtain Replicate I,

and interchange levels 1 and 2 of the first factor in Replicate II to

obtain Replicate III. Similar results can be derived for a 3 x 2n experi­

n-l Gment in equal blocks of size 3 x 2 • eneralization is possible. Hence

we have shown that linear combinations of orthogonal contrasts play an

important part in obtaining balanced sets of treatment combinations; how-

ever, this subject is not pursued further in this thesis.

104

CHAPTER 9

Summary, Conclusions and Suggestions for Future Research

901 Summary and Conclusions

The survey of literature reveals that for all factors of a factorial

experiment 'With the same prime number (or power of a prime number) of levels,

results from Finite Group Theory can be employed to confound the experiment

into equal blocks, or into fractions, leaving main effects and some lo'Wer

order interactions unconfounded. For factors with different numbers of

levels, no such general theory exists.

In many experimental situations it 'Would be desirable to confound

higher degree rather than higher order contrasts. A general mathematical

technique, based on conformable partitioning of the coefficient matrices

of orthogonal contrasts, was developed to obtain confounded designs for

both the symmetrical and asymmetrical cases. This approach not only leads

to designs required from the degree point of view, but reproduces most of

the designs based on the confounding of order components. By defining the

positive and negative signs of the 'matrix elements and the zero elements

of a contrast as block indicators, it 'Was shown that confounding of a

contrast with blocks corresponds mathematically to a partitioning of the

coefficient matrix of the contrast according to the block indicators.

The study of the confounding pattern of a contrast was thus reduced to a

mathematical examination of the values of the traces of conformably par­

titioned coefficient matrices.

Using this technique, certain general theorems were derived in

Part I. Considering the confounding of a single contrast at a time, it

was shown that the confounding of a contrast of lower order does not con­

found any contrast of higher order. Conditions were established for the

105

case in which the confounding of a contrast of higher order does not con­

found any contrast of lower order, and for the case in which the confound­

ing of a contrast of given order does not confound contrasts of the same

order. These theorems led to the definition of the heterogeneous bundle

of a confounding contrast, such a bundle consisting of all the contrasts

confounded by that contrast. When two contrasts are confounded simul­

taneously, it is shown that their symbolic product yields another

heterogeneous bundle, that of the generalized interaction.

In Part II these general results are specialized to the orthogonal

polynomial contrasts. Tables are presented of the heterogeneous bundles

of confounding contrasts of highest order. An easy rule for determining

the heterogeneous bundle of any given contrast was found, this rule being

a convenient arithmetization of the consequences of the various theorems.

A rapid procedure to determine the coefficients of the block constants in

the expectation of the confounded contrasts is demonstrated. The general­

ized interaction of two polynomial contrasts was reduced to a linear

combination of polynomial contrasts, and theorems were developed to

determine the confounding status of these contrasts. The rule to deter­

mine heterogeneous bundles was generalized. Conditions necessary to

obtain equal blocks, when one or more contrasts of highest order are

confounded, but main effects are not confounded, were established.

These conditions ll?ad to new single-replicate three-factor experiments

in four equal blocks in which main effects are unconfounded.

10.2 Further Research Needed

Since this thesis dealt with a field of research rather than with

a single topic, many single topics require further elaboration.

106

1 0 A complete discussion of partial confounding (in the sense of

this thesis); including a method of estimating all pertinent contrasts

and their standard errors.

20 A systematic search for new four- and five-factor confounded

designs in equal blocks with main effects unconfounded, using the short­

cut methods developed in this thesis.

30 A .fuller treatment of the theory of confounding linear combina­

tions of orthogonal contrasts, along the lines proposed for single

orthogonal contrasts.

40 The use of unequal-sized blocks.

5. The development of a comprehensive theory of fractional repli­

cation, based on signs of contrasts, with special emphasis on the asym­

metrical case.

6.. The extension of the present theory of confounding based on

the signs only of contrasts, to one based on both the signs and types of

elements of the contrasts; hence, the extension of the system of block

indicators (-,+,0) to something like (-e,+e,-u,+u,O), where e denotes

an even element and u an odd element.

107

LITERATURE CITED

Aitken, A. c. 1933. On the graduation of data by the orthogonalpolynomials of least squares. Proc. Roy. Soc. Edinburgh 21: 54-78,

Allan, F. E. 1930. The general form of the orthogonal polynomialsfor simple series with proofs of their simple properties.Proc. Roy. Soo. Edinburgh 2Q: 310-320.

Anderson, R. L. 1953. Reoent advances in finding best operatingoonditions. Jour. Amer. Stat. Assn. 48: 789-798.

Anderson, R. L. 1957. Complete faotorials, fractional factorialsand confounding. Proc. Symp. Design of Industrial Experiments,North Carolina State College (in press).

Anderson, R. L., and Houseman, E. E. 1942. Tables of orthogonalpolynomial values extended to n =104. Iowa State College,Agr. Expt. Sta. Res. Bull. 297.

Barnard, M. M. 1936. An enumeration of the confounded arrangementsin the 2 x 2 x 2 •.• factorial designs. Jour. Roy. Stat. Soo.Supple 2: 195-202.

Binet, F. E., Leslie, R. T., Weine~, S., and Anderson, R. L. 1955.Analysis of confounded faotorial experiments in singlereplications. North Carolina Agr. Expt. Sta. Teoh. Bull. 113.

Bose, R. C. 1938. On the application of the properties of Galoisfields to the construction of hyper Graeco-Latin squares.Sankhya 2: 323-338.

Bose, R. C. 1947. Mathematical theory of symmetrioal factorial designs.Sankhya ~: 107-166.

Bose, R. C. 1950. Mathematics of faotorial designs. Proe. Inter­national Congress of Mathematioians 1950: 543-548.

Bose, R. C., and Kishen, K. 1940. On the problem of oonfounding inthe general symmetrical factorial design. Sankhya 2: 21-36.

Box, G. E. P. 1952. Multi-factor designs of first order.Biometrika ~: 49-57.

Box, G. E. P. 1954. The exploration and exploitation of responsesurfaoes: Some general oonsiderations and examples.Biometrics 10: 16-60.

Box, G. E. P., and Hunter, J. S. 1954. Multi-faotor experimentaldesigns. Inst. of Statistics Mimeo Series No. 92. NorthCarolina State College, Raleigh.

Box, G. E. P., and You1e, P. V. 1955. The Exploration and exploitationof response surfaces: An example of the link between the fittedsurface and the basic mechanism of the system. Biometrics 11:287-323.

108

Box, G. E. P., and Wilson, K. B. 1951. On the experimental attainmentof optimum conditions. Jour. Roy. Stat. Soc., Ser. B, 13: 1-45.

Brownlee, K. A., Kelly, B. K., and Loraine, P. K. 1948. Fractionalreplication arrangements for factorial experiments with factorsat two levels. Biometrika 35: 268-276.

Chinloy, T., Inne:s, R. F., and Finney, D. J. 1953. An example of frac­tional replication in an experiment on sugar cane manuring. Jour.Agr. Sci. 43: 1-11.

Cochran, W. G., and Cox, G. M. 1950. Experimental designs. John Wileyand Sons, New York.

Daniel, C. 1956. Fractional replication in industrial research. hoc.Third Berkeley Symposium on Math. Stat. and Prob. 2,:

Davies, O. L. (Ed.) 1954. Design and analysis of industrial experiments.Oliver and Boyd, Edinburgh and London.

Davies, O. L., and Hay, W. A. 1950. The construction and use of fractionaldesigns in industrial research. Biometrics 2: 233-249.

DeLury, D. B. 1950.up to n = 26.

Values and int egrals of the orthogonal polynomialsUniversity of Toronto Press.

Esscher, M. F. 1920. Uber die Sterblichkeit in Schweden. Medd.franLunds Astron. Obser. Ser. 2, No. ,23.

Esscher, M. F. 1930. On graduation according to the method of leastsqiiaresby means of c eTta.in polynomials. ForsakringsaktiebolagetSkandinavia, Festskrift. Stockholm.

Finney, D. J. 1945. The fractional replication of factorial experiments.Ann. Eugenics g:29l-301.

Finney, D. J. 1946. Fractional replication. Jour. Agr. Sci. ~: 184-191.

Fisher, R. A. 1921. Studies in crop variation I.the yield of dressed grain from Broadbalk.107-135.

An examination ofJour. Agr. Sci. 11:

Fisher,R. A. 1924.at Rothamst ed.

The influence of rainfall on the yield of wheatPhil.Trans. Roy. Soc. London B2l3: 89-142.

Fisher, R. A. 1926. The arrangement of field experiments. Jour.Ministry of Agr. 33: 503-513-

Fisher, R. A. 1935. The design of experiments. Oliver and Boyd,Edinburgh, First ed.

Fisher, R. A. 1942. The theory of confounding in factorial experimentsin relation to the theory of groups. Ann. Eugenics 11: 341-353.

109

Fisher, R. A. 1945. A system of confounding for factors with more thantwo alternatives, giving completely orthogonal cubes and higherpowers. Ann. Eugenics 12: 283-290.

Fisher, R. A., and Wishart, J. 1930. The arrangement of field experimentsand the statistical reduction of the results. Imperial Bureau ofSoil Science. Tech. Gomm. No. 10.

Fisher, R. A., and F. Yates. 1938. Statistical tables. Oliver andBoyd, Edinburgh. First Ed.

Gram, J. P. 1915.funktionen.

Uber partie11e Ausg1eichung mittelst Orthogona1­Mitt. Ver. schweiz. Versich. Math. 1£: 41-73.

Jordan, C. 1929. Sur la determination de 1atendence secu1airedesgrandeurs statistiques par 1a methode des moindres carres. Jour.de 1a Soc. Hongroise de Stat. I: 567-588.

Jordan, C. 1932. Approximation and graduation according to the principleof least squares by orthogonal polynomials. Ann. Math. Stat.1: 257-333.

Kempthorne, O. 1947. A simple approach to confounding and fractionalreplication in factorial experiments. Biometrika~: 255-272.

Kempthorne, O. 1952. The design and analysis of experiments. JohnWiley and Sons, New York.

Kitagawa, T., and Mitome, M. 1953. Tables for the design of factorialexperiments. Baifukan Co., Ltd., Tokyo.

Li, J. C.-R. 1844. Design and statistical analysis of so:meconfoundedfactorial experiments. Iowa State College Agr. Exp. Sta. Bull.333.

Lorenz, P. ,1931. Der Trend. Vierte1jahrshefte zur Konjunkturforschung.Sonderheft 21.

Nair, K. R. 1938. On a method of getting confounded arrangements inthe general symmetrical type of experiments. Sankhya 11: 121-138.

Nair, K. R. 1940.experiment s •

Balanced confounded arrangements forthe5n type ofSankhya ~: 57-70.

Nair, K. R., and Rao, C. R. 1948. Confounding in asymmetrical factorialexperiment s • Jour. Roy. Stat. Soc. Sar. B!2: 109-131.

Pearson, E. S., and Hartley, H. o. 1954. Biometrika tables forstatisticians. Vol. 1. Cambridge University Press.

P1ackett, R. L., and Burman, J. P. 1946. The design of optimum multi­factor experiments. Biometrika..12.: 305-325.

110

Poincare, H. L. 1896. Calcu1 des probabilitieSl. Hermann et Cie, Paris.

Tchebycheff, P. L. 1859. Sur l'interpo1ation par 1a methode des moindrescarres. Oeuvres Completes, Gauthier-Villars, Paris.

Thompson, H. R. 1952. Factorial design with small 'olocks. New ZealandJour. Sci. and Technol. B33: 319-344.

Thompson, H. R., and Dick,I. D. 1951. Factorial designs in small blocksderived from orthogonal latin squares. Jour. Roy. Stat. Soc.B13: 126-130.

Van der Reyden, D. 1943. Curve fitting by the orthogonal polynomials ofleast squares. Onderstepoort J. Vet. Sci. ,gQ: 355-404.

Yates, F. 1933. The principles of orthogonality and confounding inreplicated experiments. Jour. Agr. Sci. 23: 108-145.

Yates, F.2:

1935. Complex experiments.181-247.

Jour. Roy. Stat. Soc. Supple

Yates, F. 1937. The design and analysis of factorial experiments.Imperial Bureau of Soil Sci. Tech. Comm. No. 35.