Section 11.3

Oo not remove

NATIONAL BUREAU OF STANDARDS REPORT

4412

FRACTIONAL FACTORIAL DESIGNS FOR THE 1/2 S x 2 n SERIES

FOR n = 12(1)16 AND s - 6,7,8.

by

R. C. Burton,

F. L. Miller, Jr.,

and H. M. Pettigrew

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

1103-40-5118

17 November 1955

NBS REPORT

4412

FRACTIONAL FACTORIAL DESIGNS FOR THE 1/2

S

x 2 n SERIES

FOR n = 12(1)16 AND s = 6,7,8.

t>y

R. C. Burton,

F. L. Miller, Jr.,

and H. M. Pettigrew

Statistical Engineering Laboratory

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Approved for public release by the

The publicatic r

. director of the National Institute of

unless permiss ^

25,0,0. sue Standards and Technology (NIST)

cal ly preparer On October 9, 2015

le or in part, is prohibited

of Standards, Washington

i a report has been specifilar report for its own use,

FOREWARD

This report brings together work on the construction of two-level fractional

factorial designs begun during the summer of 195# , and is an extension of the

work of Clatworthy, Connor, and Zelen [l] . The experimental designs catalogued

here may be useful in reducing the amount of experimentation for those experimental situations where the joint effects of many factors are to be evaluated.

This work is part of a continuing program of research on mathematical

statistics and its applications carried out at the Statistical Engineering

Laboratory, National Bureau of Standards for the Chemical Corps, U. S. Department of the Army (NBS Project Number 11030-1j0-5>118/5>2-1).

C. Eisenhart

Chief, Statistical Engineering Laboratory

E. W* Cannon

Chief, Applied Mathematics Division

National Bureau of Standards
:

:

'
TABLE OF CONTENTS

Page

Introduction 1

References 13

Classification of factorial designs according to fractional replication.

number of factors, and number of experimental units per block.

Plan Number of measurements

required

6*4.12.8 . « & © 0 0 0 • • 0 6*4 • 0 0 € 0 1*4 6it.12.l6 . 0 0 - 0 6*4 0 0 • e • 15

61t.12.32 . • 0 © 0 © 0 © 0 e 6*4 0 0 0 9 O 16

6*4.13.8 . © 0 © © 0 0 e 0 0 128 0 0 0 © 0 17

6*4.13.16 . • 0 0 0 0 0 O 0 0 128 0 © 0 © 0 19

6it.13.32 . 0 © 0 © 0 © 0 0 0 128 • 0 0 © 20

6i4.i3.6i4 . © O © © • 0 0 0 0 128 • 0 0 0 O 21

6i4.li4.8 . 256 © 0 0 O O 22

6i4.li4.16 « • e 0 0 0 0 © 0 © 256 0 0 © © © 2*4 6i4.li4.32 . e © © © © © 0 0 0 256 0 • © © O 25

6i4.li4.6i4 . • © 0 <* © © © © 0 256 © e 0 O O 26

6i4.li4.128. 0 0 0 0 0 e 0 0 0 256 0 0 0 © © 27

128.12.8 . 0 © © © © 0 9 0 © 32 0 0 • O © 28

128.12.16. © © • O © O © 0 0 32 0 © © © O 29

128.13.8 . © u 0 O O © 0 e © 0 6*4 0 0 • O © 30

128.13.16. • © O 0 © © 0 © 0 6*4 0 © 0 © e 31

128.13.32. 0 © 0 O © © 0 0 © 6*4 © 0 a • © 32

128.1*4.8 . 0 © 0 O © 0 0 0 © 128 0 © © O 0 33

128.1*4.16. • 0 O © © © © 0 © 128 O © © © 0 35

128.1*4. 32. 0 © O © © © 0 © © 128 a © © O 0 36

128.1*4. 6*4. 0 O © © e • 0 0 0 128 0 0 0 O 0 37

128.15.8 . © © 0 0 © © © © 0 256 0 0 © O 0 38

128.15.16. • 0 O © • © 0 © © 256 0 © 0 © 0 *40 128.15.32. © © 0 O 0 0 0 © © 256 0 0 0 © 0 *a

128.15. 6*4. 0 0 0 0 • 0 0 © 0 256 0 © 0 © © *42 128.15.128 © © © 0 0 0 © © 0 256 0 © © • 0 h3

256.13.8 0 © O 0 © © © © 0 0 32 • 0 © © 0 *4*4 256.13.16. 0 O © 0 © © © © © 32 • • 0 © 0 *46 256.1*4.8 . • 0 © 0 © 0 © • 6*4 © 0 0 O © *47 256.1*4. 16. • • 0 0 © © 0 © © 6*4 © © 0 © 0 *49 256.lU.32. © • 0 © 0 0 0 © © 6*4 0 © 0 O 0 50

256.15.8 . 0 0 © 0 © © 0 0 c 128 • © 0 © 0 51

256.15.16. 0 © © © O 0 0 0 0 128 0 © © © 0 5*4 256.15.32. © 0 © © 0 0 0 © © 128 © 0 © 0 © 56

256.15.6U. • 0 © © 0 O 0 0 0 128 0 © 0 © © 57

256.16.8 . © © © © 0 0 © © • 256 © 0 © 0 © 58

256.16.16. © © © © 0 © © 0 0 256 0 © 0 © 0 61

256.16.32. • 0 • © © 0 © © • 256 0 0 0 © © 62

256.16.6*4. 0 © © © © • 0 0 © 256 0 © © • 0 63

256.16.128 • © O O © © 0 0 © 256 © © © O © 6*4

ii

1

2

s

Fractional Factorial Designs for the

x 2n

Series for n = 12(1)16 and s = 6, 7 , 8

fry

R.C. Burton, F.L. Miller, and H.M. Pettigrew

Introduction . This report is an extension of the catalogue of two-level

fractional factorial replicate designs compiled by W.H. Clatworthy, W.S. Connor,

and M. Zelen CD- The designs are of the 2nseries, where n, the number of

factors, ranges from 12 to 16 and each factor is at two levels. This report

includes designs which are l/6i|, 1/128, and 1/256 of a full replication.

Each design has a designation r. n. k., where r is the replication, n

the number of factors, and k the number of experimental units per block. For

example, plan 61uliu8 refers to a 1/6U replication of llj factors in (32) blocks

of 8 units each.

A main effect or interaction is said to be measurable if it is confounded

only with higher-order interactions. In all designs, the main effects of each

of the factors are confounded with interactions involving three or more factors.

The information as to which two-factor interactions are not measurable is given

for each design. Those two-factor interactions lost in choosing the fraction

of treatment combinations are indicated under the heading "Without blocks."

Additional two-factor interactions lost as a result of blocking are given under

the heading "With blocks."

In all designs, capital letters are used to refer to factors, their main

effects, and interactions. The lower case letters are used to denote the various

treatment combinations and indicate the levels of each factor appliedj the
,
absence of a letter indicates application of the lower level of the given factor,

and the presence of a letter indicates the application of the upper level of that

factor* For example, in a design of five factors A, B, Cf D; E$ the treatment combination acd denotes the higher level of A, G and D, and the lower level of

B and E* The symbol "(1)" denotes the treatment combination consisting of the

low levels of all the factors. These are the usual conventions, cf. Cochran and

Cox ^2 J, Davies [

3], and Kempthorne M •

With each design is given the fundamental identity (denoted by I) used to

choose a subset of the set of all possible treatment combinations. Also given

are the block confounding relationships used to subdivide this subset of treatment combinations into experimental blocks* The equal signs are read, "is confounded with." Additional details concerning the construction and statistical

analysis of the designs may be found in the references cited above.

In each design, all the treatment combinations of the first block are given

explicitly. One treatment combination for every other block is given. This

enables the remaining treatment combinations for any other block to be found by

multiplying the first treatment of the block by all the treatment combinations

in the first block. Multiplication, which is commutative, is defined such that

a*b = ab, a*a = 1, and l*a = a. For example, the treatment combinations in

block 2 of plan 6U.1U.8 are found by multiplying the corresponding treatments

in block 1 by abfhkls

2

abfhkl

abcdfmn

bceghln

bdegkm

aghjmo

acdgjklno

cefhjkmno

defjlo
'
- 3 -

These designs represent a considerable effort on the part of the authors to

retain as many measurable two-factor interactions as possible while at the same

time allowing main effects to be only confounded with three-factor or higher order

interactions. The designs have been carefully checked but there is always a

possibility of error. We would appreciate hearing from anyone who improves the

designs ' or finds errors in them.

The authors wish to express their thanks to Mr. Marvin Zelen who supervised

this project, and to Mrs. Mary E. McKinley and Miss Caroline Yick for their

painstaking efforts in typing the manuscript for reproduction.

Conversion of Two-level Designs to Four Levels .

It is possible to adapt the two-level designs in this catalogue to

experiments in which the number of levels of the factors is a power of

two, or where there is a mixture of levels each a power of two, such

as experiments having some factors at two levels and some at four. To

illustrate the conversion procedure, a 1/6U replication of lU factors at

two levels will be converted to a completely randomized four-level design

- a

-

involving seven factors* The fundamental identity in plan 6U.ll4.a8 (and its

resulting treatment combinations) will be used without regard to the

grouping of treatments into blocks*

Let the seven new factors be denoted byo< ,/3 ,V , <£, © , and^ , and

the four levels of each new factor be called the Oth (lowest) level, the

1st level, the 2nd level, and the 3rd (highest) level* To show the conventions used in writing the four-level treatment combinations, the following

table has been made, (with the factors listed down the side and the levels

across the topi):

The application of the 1st level of x is denoted by x^, or simply x,

and the applications of the 2nd and 3rd levels of x are denoted by

x2 and x3 , respectively*

The treatment combination (1) represents that treatment where all

seven factors are at their lowest, or Oth, level* For all other treatment

combinations, the absence of a factor from the written treatment combination

indicates that that factor is present in its lowest level, and the (1) is

not written in the treatment , As an example, the treatment combination

o<. y £ ft implies O(

2/9

0

, and represents the combination

of the Oth levels of /3 , @ , and ^ , the 1st levels of d and , the 2nd

level of o( , and the 3rd level of T* 0 (It should be noted here that in

the two-level designs capital letters were used to refer to the factors

and their main effects and interactions, and lower case letters to treatment combinations, but in the four-level designs a given letter can refer

to any of these)*

Now if the fourteen factors of the two-level design are thought of as

“pseudofactors" of the proposed four-level plan, a correspondence can be

set up by grouping the pseudofactors in pairs, one pair with each fourlevel factor. One way to group them would be|A, Bsocj, (read "A and B with

D.Tjj.jE, F:^, etc. As will be seen later, however, this is not

the best grouping in this case, and instead the following groupings have

been used;

A, J % Of E,Ls 9

B, C;/9 K,N; A

D,G;r

H,0s d

Now a correspondence must be set up between the four-level treatment combinations and the treatment combinations of the pseudofactors. This is

done as follows. The application of the Oth level of factor 0( may be

thought of. as the application of the lower levels of both of the pseudofactors A and J, the application of the 1st level of <X as the application

of the upper level of pseudofactor A, the 2nd level of o( as the upper level
’
of J, and the 3rd and highest level as the upper levels of both A and J.

In other words, c<

0 or (1) in the four-level design will correspond to afj

0

or (1) in the two-level design, designating the lower level of both pseudofactors A and J,« will correspond to a , (X to j , and ©f to aj. This is

done in a similar manner for the other factors, as has been done in the

following tables

Four-level Treatments in Terms of Pseudofactors

Levels

Oth 1st 2nd 3rd

Ok (1) o< = a C* 2 = j 0<3 = aj

CO ft (1) /3 = b = c /3 3 = be

ShO Tr (1) r = d r® = g rf = dg

•po <S (1) $ = h S = o 6

3 = ho

ctf (1) B = e e\ = i ^

~ n e

3 = el

(1) $ = k $

3 ~ kn

r (1) * = f = m — fm

Using this table we can "express" any four-level treatment combination in

pseudofactors having two levels5 and vice-versa. For example (1), representing the lower level of all fourteen pseudofactors, corresponds to the (1)

representing the Oth level of the seven four-level factors; Q< r

3

,

representing the Oth level of factors/3 ,

6

, and ^9the 1st level of factors

CX and Jr , the 2nd level of and the 3rd level of T , corresponds to

adgof , or adfgo. The two-level treatment combination abdefghl corresponds

to Of /3 ?~3 & B

3

fh . This last translation can be done in either of two ways:

* -either by taking each two-level letter in order and writing the corresponding four-level letter. Offt rBjh r

2

<f s\ and multiplying algebraically
' ' .

.

‘
= 7 -

o * Q giving (X/Sr , or by noting the pairs that occur in the two-level

treatment , in this case dg and el^ , and immediately writing the highest

3 3

levels of the corresponding four-level factors,, Y ©

Thus there is a one to one correspondence between the two-level and

the four-level treatment combinations. Therefore the 256 two-level treatment combinations, when "translated” into four-level combinations, will

give the proper subset for the 1/6U replicate of a factorial design.

One may either obtain all 256 treatments in terms of pseudofactors

and then translate them one at a time into four-level terms, or translate

only the treatments of the initial block and the block multipliers. If

the latter method is used, multiplication by the block multipliers must

follow these rules: X° * X = X

* X

1

* X

2

X° * ?6

3 = X x y = %

3

% x X

3

= X * X

3 - X

X x X - x* x X

s - X

3 * X

3 s X° ~

(1)

Given below are the eight treatments in the initial block obtained by

translating the treatments, (of the two-level design), and the second

group of eight treatments found by multiplying by abfhkl - <X/3 & Q $ ' ID

/fi-yS <9

8 fi

c fir

«7-V03 P

«vs

8 >*f3 fir

<x/3f eV ?

ex/3 3rf2 f

3

(3*vztf

3

ot 3

fi*y3

Of^i'e3 r
.

'

.

.
The block groupings here are used only as a convenient means of generating

all 2^6 treatment combinations 0 The treatments are of course not grouped

into blocks in this completely randomized design,.

In the four-level designs, as in the two-level designs, the fundamental

identity is used to determine what information is lost through confounding.

In the two-level designs each main effect and each two-factor interaction,

having only one degree of freedom, was either completely measurable or

completely lost. In the four-level design, each of the seven main effects

has three degrees of freedom and each of the twenty-one two-factor interactions has nine degrees of freedom. The three degrees of freedom for

main effects can be broken up into three individual degrees of freedom,

given by the ’’main effects" of the two pseudofactors and their "interaction",

e,g, for 0{ by A,J, and AJ, Similarly the nine degrees of freedom for

each two-factor interaction can be broken up into nine individual degrees

of freedom, given by multiplying the individual degrees of freedom of the

main effects of the two factors, e,g,, for©(/3 by (A,J,AJ)x(B,C,BC) or

AB, AC, ABC, BJ , CJ , BCJ, ABJ, ACJ, and ABCJ,

Each individual degree of freedom may or may not be measurable, so

it is possible to have partial information on a main effect or two-factor

interaction. To determine which are measurable, the aliases of each are

found from the fundamental identity. For example, by checking AJ

through, the - first few terms of the identity?

Ad = BCDJO = BEFJLNO = ACDEFJLN = 9 0 0
'
- 9 -

it is seen that it is confounded with BCDJO (fromo^T'cf ) , BEFJLNO

(from«'/3$ and ACDEFJLN (from oifi'ySft'jfi = If AJ is checked

through each remaining term of the identity., it will be found that in

all cases its aliases are composed of three-factor or higher interactions* Therefore AJ is termed measurable* Similarly, the other

twenty individual degrees of freedom for main effects are measurable.

However, when the degrees of freedom for the two-factor interactions are

checked, it is found that some are confounded with each other and hence

are not measurable* The following table shows what information is

measurable, and how lost information is confoundeds

Summary of Confounding in 1/61+ x

Main measurable from 2-factor

effects degrees of freedom Lost d e f o confounded with • interaction i ©C A, J, AJ none

/3 B, G, BC ?> y D, G, DG n s H, 0, HO n 9 E, L, EL it K, N, KN ti Tb F, M, FM n

2-Factor

interactions

o(/3

AB, CJ , ACJ, ABGJ AG EFm ©^

ABC DO rtf

BJ DEGL re

BCJ EH 6e

ABJ FHM, KL Sjh,6fl

10

2-Factor

interactions

measurable

degrees of freedom Lost d 0 f« confounded with;

from 2-factor

interaction:

CX 7" AG, DJ, agj, adgj AD BCO /3S

ADG EK e?

GJ HKO

DGJ BEL f3 0

ADJ EHO 6 0

cx i. AHs AHO, JO, AJO AO BCD fay

HJ BCE /3 @

HJO GK r $

AHJ BFM

AHJO DE

©C 0 AXj , AEL , JL , AEJL AE CFM, DGK /3jb9

EJ BCH /3 S

EJL BDG /3 y

AEJ DHO T'S

AJL BK /3/

Q< $ AN, AKN, JN, JKN, AK DEG y ©

AJN, AJKN JK GHO y€

AJK BL /aa

Of

5^ AF,AM,FJ,JM,FJM AFM CE /38

AFJ,AJM AFJM BH fU

BD, BG, CD, CG, BCG

,

BDG EJL « a

BCDG BCD AO

CDG HL„ FKM <fSi tit

BO, BHO, CO, CHO BH AFJM

CH DGL re

BCH EJ ae

BCO AD a y

BCHO FKN

/BO BE, CEL , BCL

,

BL AJK cx BCEL BEL DGJ cx y

CE AFM exit'

CL DGH r<T

BCE HJ 6

31 -

2-Factor

interactions

measurable

degrees of freedom Lost d e f o confounded with!

from 2-factor

interaction

!

BN, BKN, CN, CKN, BK AJL CX Q

BCK, BCN CK DFGM y It

BCKN FHO

f3 p* ' BF, BM, CF, CM, BFM AHJ CK &

BCM, BCFM GFM DGK, AE

BCF HKNO

'r£ DH, GH, GO, DO ABC CK ft

DGO, DGHO DHO AEJ CK 9

GHO JK «* *

DGH CL /3 6

DL, DEL, EG, GL DE AHJO CX&

EGL KMN

DEG AK ay

DGL CH /3£

DEGL BJ

$ DK, DN, DKN, GN, GK HJO o< 8

DGN, DGKN GKN ELM Qf.

DGK AE, CFM

'YJh DF, DM, DFM, FG, GM EKLN ©

FGM, DFG, DGM DFGM CK /3 &

$ e EHL , EO , LO

,

EH BCJ ot ELO , HLO , EHLO HL CDG, FKM ar, ty*

EHO ADJ o<~y

HN, HKN, KD, NO, HK ELM eft

KNO, HNO HKO GJ y

HKNO BCF /3fr

§Th FH, HM, FO, MO, FHM KL, ABJ 6<J> ? cx/3

FMD, HMO, FHMO FHO BCKN Gj>
>

.

i m
-12

2-Factor

interactions

measurable

degrees of freedom Lost dof o confounded with*.

from 2-factor

interaction;

9<f EN, EKN, LN, KLN,

EKL, ELN

EK

KL

EKLN

ADG

ABJ, FHM

GM

cx y

©</3, f ^

yp*

ey EF, EM, FL, LM, EFM AC a/3

EFL7* EFLM FLM HK $ &

ELM GKN y&

$ FK, KM, FN, MN, FKM HL, CDG 8 B s fhv

FMN, FKMN FKN BCHO /Qg

KMN EGL re

In general it is possible to adapt any two-level design of the form

2^m into a four-level design of form Um° The two-level factors should

be paired off so as to furnish, if possible, all information on main

effects, and then to provide as much information as possible on twofactor interactions^ as determined by the fundamental identity,, (To

illustrate the importance of proper choice, consider the term ABe If

we had grouped our letters [AjBsocj, etc., AB would have been part of

the main effect of factor OS , and the first term of the identity shows

AB to be confounded with CDO, part of a two-factor interaction,, Thus

at least one degree of freedom for a main effect would not have been

measurable.) Finally after pairing the two-level factors, the treatment

combinations of the two-level design are translated into terms of the

four-level factors

- 13 -

REFERENCES

Cl] Clatworthy, W.H., Connor, W.S., and Zelen, M», Some Fractional Factorial

Arrangements for Factors at Two Levels , National Bureau of

Standards Report, (195k)*

DO Cochran, W.G. and Cox, G.M., Experimental Designs, John Wiley and Sons,

Inc., New York, 195>0.

C3] Davies, O.L. (editor). The Design and Analysis of Industrial Experiments ,

Chapter 10, Hafner Publishing Co», New York, 195U*

m Kempthorne, 0., The Design and Analysis of Experiments , John Wiley and

Sons, Inc., New York, 19!?2.
.
- 11+ -

Plan 6Iul2.8 . 1/61; replication of 12 factors in 8 blocks of 8 units each.

Factors; A,B,C,D,E,F,G,H,J.,K,L,M.

I = ABCD = ABEFL = CDEFL = ABGHL = CDGHL = EFGH = ABCDEFGH = ABJKL = GDJKL

= EFJK = ABGDEFJK = GHJK * ABCDGHJK = ABEFGHJKL = CDEFGHJKL = ACEGJL

= BDEGJL = BCFGJ = ADFGJ * BCEHJ = ADEHJ = ACFHJL = BDFHJL = BCEGK

= ADEGK = ACFGKL « BDFGKL = ACEHKL = BDEHKL = BCFHK = ADFHK = ADLM = BCLM

= BDEFM = ACEFM = BDGHM = ACGHM = ADEFGHLM = BCEFGHLM = BDJKM = ACJKM

= ADEFJKLM = BCEFJKLM = ADGHJKLM = BCGHJKLM = BDEFGHJKM - ACEFGHJKM

= CDEGJM = ABEGJM = ABCDFGJLM = PGJLM - ABGDEHJLM = EHJLM = CDFHJM

= ABFHJM = ABCDEGKLM = EGKLM * CDFGKM - ABFGKM = GDEHKM = ABEHKM

= ABCDFHKLM = FHKLM.

Block confounding; AB, AC, BC, CL, ABCL, AL, BL0

Without blocks; All main effects, and all two-factor interactions

except the following are measurable

s

ab, AC, AD, AL, AM,

bc. BD, BL, BM, CD,

cl. CM, DL, DM, EF,

EG, EH, EJ, EK, FG,

FH, FJ, FK, GH, GJ,

GK, HJ, HK, JK, LM„

With blocks; Same as above*

Blocks;

1 2 3 h

(1)

efgh

giyk

efjk

abcdehklm

abcdf gklm

abcdegjlm

abcdfhjlm

abed cdfhkl abfhkl

6 7 8

acghl bdghl ad%k bcfgk

- 15 -

Plan 61;. 12.16. 1/61; replication of 12 factors in U blocks of 16 units each.

Factors: A5 B5 G,D,E,F,G5 H5 J,K,L5 M.

I: Same as plan 61ul2.8.

Bloek confounding? AB, AC, BC.

Without blocks? All main effects, and all two-factor interactions except

the following are measurable:

AB, AC , AD , AL , AM, BC , BD, BL

,

BM,CD,CL,CM,DL,DM,EF,EG,

EH, EJ , EK, FG, FH, FJ , FK,

GH,GJ,GK,HJ,HK,JK,LM«

With blocks: Same as above.

Blocks:

1 2

(D cdfhkl

efgh

ghjk 3

efjk acghl

abcdehklm

abcdfgklm h

abcdegjlm adTgk

abcdfhjlm

abed

ab cdefgh

abcdghjk

abedefjk

ehklm

fgklm

egjlm

fhjlm
. ' ; >: ,
- 16 -

Plan 61|.,12,32 , I/6I4. replication of 12 factors in 2 blocks of 32 units each.

Factors; A,B,C,D,E,F,G,H,J,K,L,M,

I; Same as plan 61|.„12,8,

Block confounding: AB,

Without blocks: All main effects, and all two-factor interactions except

the following are measurable:

AB,AC,AD,AL,AM,BC,BD,BL,

EM, CD, CL, CM, DL, DM, EF, EG,

EH,EJ,EK,FG,FH,FJ,FK,

GH,GJ,GK,HJ,HK,JK,LM« ,

With blocks: Same as above.

Blocks

I X _

(1) acghl

I ghjk

efjk

abcdehklm

abcdfgklm

abcdegjlm

abcdfhjlm

abed

abcdefgh

abcdghjk

abedefjk

ehklm

fgklm

egjlm

fhjlm

cdfhkl

cdegkl

cdfgjl

cdehjl

abefm

abghm

abefghjkm

abjkm

abfhkl

abegkl

abfgjl

abehjl

cdejfm

cdghm

cdej'ghj km

cdjkm

- 17 -

Plan 6i| 0 13o8 o 1/61; replication of 13 factors in 16 blocks of 8 units each0

Factors; A^B^C^D^E^F^G^H^J^K^L^M^No

I = ABCD = ABEFLN = CDEFLN = ABGHL = CDGHL = EFGHN = ABCDEFGHN = ABJKL

= CDJKL = EFJKN = ABCDEFJKN = GHJK = ABCDGHJK = ABEFGHJKLN = GDEFGHJKLN

= ACEGJL = BDEGJL = BCPGJN = ADFGJN = BCEHJ = ADEBJ = ACFHJLN = BDFKJLN

= BGEGK = ADEGK = ACFGKLN = BDPGKLN = ACEHKL = BDEHKL = BCFHKN = ADFHKN

= ADLM = BCLMN = BDEFM = ACEFM = BDGHMN = ACGHMN = ADEFGHLM = BGEFGHLM

= BDJKMN = ACJKMN = ADEFJKLM = BCEFJKLM = ADGHJKLMN = BCGHJKLMN = BDEFGBJKM

= ACEFGHJKM = CDEGJMN = ABEGJMN = ABGDFGJLM = FGJLM = ABCDEHJLM = EHJLMN

= CDFHJM = ABFBJM = ABCDEGKLMN = EGKLMN - CDPGKM = ABFGKM = CDEHKMN

= ABEHKM = ABCDFHKLM = FHKLMo

Block confounding; AB5 AC,BC,ABCFN5 CFN5 BFN,AFN,ABCDEFGJKLM,CDEFGJKLM,

BDEPGJKLM, ADER3JKLMN, DEGJKLM,ABDEGJKLM, ACDEGJKLM, BCDEGJKLM,

Without blocks; All main effects, and all two-factor interactions except

the following are measurable;

AB, AG , AD, BC, BD, CD,

GH,GJ,GK,HJ,HK,JK0

With blocks; Same as above, except the two-factor interactions FJ,GN,

and HM also are not measurable <,

1

Blocks

2 3 k

(1) ghjk ef jk efgh

ehklm

abcdfhjlm

abcdefjk

efgjjln

fghjkran

abcdeghmn

abcdgkln

5 6 7 8

cdfgjl cdfhkl cdegkl cdehjl
: :
- 18 -

9

acghl

13

adfhj

10 11 12

acjkl acefghjkl acefl

Ik l£ 16

adfgk adehk adegj
'

. v

'

.

'

• • ,

- •
- 20 -

Plan 6U<.13»32 <> 1/61; replication of 13 factors in 1* blocks at 32 units each.

Factors;

I; Same as plan 6Uol3«8 0

Block confounding: AB 5 AC,BC,

Without blocks; All main effects^ and all two~factor interactions except

the following are measurable?

ABj AC 5 AD,BC,,BD<,CDj GH5 GJ 5 GK,HJ 5 HK5 JK e

With blocks: Same as above,,

Blocks:

1

(1) efjk

ehklm fhjlm

abcdfhjlm abcdehklm

abcdefjk abed

efgjln gkln

fghjkmn eghmn

abcdeghmn abcdfghjkmn

abcdgkln abcdefgjln

ghjk efgh

egjlm fgklm

abcdfgklm abcdegjlm

abcdefgh abcdghjk

efhkln hjln

fmn ejkmn

abodejkmn abcdfmn

abcdhyln abcdefhkln

2

cdfgjl

3

acghl

k

adfhj

- 21 -

Plan 614.0 13 o6I|.o l/61j. replication of 13 factors in 2 blocks of 61± units each*

Factors; A$B, C,D5 E,F,G,H,JjKjLs M,N«

Is Same as plan 6Iul3o8 0

« Block Confounding; AB*

Without blocks; All main effect s ? and all two “factor interactions except

the following are measurable!

AB5 AC 3 AD5 BC } BD5 CDS GH5 GJ a GK^HJ 5 HK5 JKo

With blocks; Same as above*

Blocks;

1

(1) efjk cdfgjl cdegkl

ehklm fhjlm cdefghjkm cdghm

abcdfhjlm abcdehklm abghm abefghjkm

abcdefjk abed abegkl abfgjl

efgjln gkln eden cdfjkn

fghjkmn eghmn cdhklmn cdefhjlmn

abcdeghmn abcdfghjkmn abefhjlmn abhklran

abcdgkln abcdefgjln abfjkn aben

ghjk efgh cdfhkl cdehjl

egjlm fgklm edefm cdjkm

abcdfgklm abcdegjlm abjkm abefm

abcdefgh abcdghjk abehjl abfhkl

efhkln hjln cdeghjkn cdfghn

fmn ejkmn cdgjlmn cdefgklmn

abcdejkmn abcdfmn abefgklmn abgjlmn

abcdhjln abcdefhkln abfghn abeghjkn

2

acghl

22

Plan 61|.«>llio8o I/6I4. replication of lU factors in 32 blocks of 8 units each.

Factors; A5 B 5 C,D5 E5 F^G5 H<,J,K,L,M,N,0 8

I = ABCDO = ABEFLNO = CDEFLN = ABGHLO = CDGHL = EPGHN = ABCDEPGHNO = ABJKL

= CDJKLO = EFJKNO = ABCDEFJKN = GHJKO = ABCDGHJK = ABEFGHJKLN = CDEFGHJKLNQ

= ACEGJLO = BDEGJL = BCFGJN = ADFGJNO « BCEHJ = ADEHJO = ACFHJLNO = BDFHJLN

= BGEGKO « ADEGK « ACFGKLN = BDFGKLNO = AGEHKL = BDEHKLO = BCFHKNO = ADFHKN

= ADLMN = BCLMNO = BDEFMO = ACEFM * BDGHMNO = ACGHMN = ADEFGHLM = BCEFGHLMO

= BDJKMN = AGJKMNO = ADEFJKLMO = BGEFJKLM = ADGKJKLMNO = BCGHJKLMN = BDEFGHJKM .

« ACEFGHJKMO - ODEGJMNO = ABEGJM = ABCDFGJLM = FGJLMO = ABCDEIiJLMN = EHJLMNO

= CDFHJMO = ABFHJM = ABCDEGKLMNO = EGKLMN = CDFGKM - ABPGKMO = CDEHKMN

= ABEHKMO = ABCDFHKLMO = FHKLM.

Block confounding; ABE05 BGK5 ACEK05 ABCM5 CEMO, AKM5 BEKMO, ACL05 BCEL5 ABKL05 EKLs BLMO^ aelm5 gklmo, ABCEKLM* AE^ B05 ABCEK, CKO^ BCEM, ACMO, EKM,

ABKM05 CELO, ABGL5 BEKL05 AKL5 ABELMO^ LM5 AGEKLMO, BCKLMo

Without blocks; All main effects and all two-factor interactions are measurable

„

With blocks; Same as above 5 except that the following two-factor interactions are not measurable s AE, BJ 5 BO, CN, DH, FG, LM, JO,

Blocks

2 3 k

(I)

cdhklmn

acefgkn

adefghlm

bfgjklmo

bcdfghjno

abcejlmno

abdehjko

abfhkl cdjkm abcdfhjlm

5 6 7 8

fmn abhklmn cd£|kn abcdhjln

9 10 11 12

ghjk abfgjl cdghm abcdfgklm

13 lU 15 16

fgEJkran abgjlmn ccLfghn abcdgkln

- 23 -

Blocks (Continued)

17

abcdeghmn

18

cdefgklmn

19

abeghjkn

20

efgjln

21

abcdefgh

22

cdegkl

23

abefghjkm

2k

egjlm

25

abodejkmn

26

cdefhjlran

27

aben

28

efhkln

29

abcdefjk

30

edeKjl

31

abefm

32

eEELm

- 2h -

Plan 6l4-.lU.l6. 1/6U replication of llj. factors in 16 blocks of 16 units each.

Factors: A,B,C,D,E,F,G,H,J,K,L,M,N,0.

I: Same as plan 6U.li4.08,

Block confounding: ABEO, BCK, ACEKO, ABCM, CEMO, AKM, BEKMO, ACLO, BCEL,

ABKLO, EKL, BLMO, AELM, CKLMO, ABCEKLM,

Without blocks: All main effects and all two-factor interactions are measurable.

With blocks: Same as above, except the following two -factor interactions

are not measurable:

ON, JO.

Blocks:

1 2

(1) abfhkl cdjkm

cdhklran abcdfmn

acefgkn bceghln

.adefghlm bdegkm

bfgjklmo aghjmo

bcdfghjno acdgjklno

abcejlmno cefhjkmno

abdehjko defjlo

3 U 5 6

fmn cdfjkn ghj'k cdghm

7 8 9 10

fghjkmn cdfghn abcdeghmn abeghjkn

11 12 13 lk

abcdefgh abefghjkm abcdejkran aben

15 16

abcdefjk abefm

- 25 -

Plan 6U«lU«32. 1/61; replication of ll; factors in 8 blocks of 32 units each*

Factors: A,B_,C_,D,E,F,G,H, J,K,L*M,N,0.

I: Same as plan 61ull4-<*8,

Block confounding: ABEO, BCK, ACEKO} ABCM, CEMO, AKM, BEKMO*

Without blocks: All main effects and two-factor interactions are measurable.

With blocks: Same as above*

Blocks:

1 2

(1) cdjkm fmn

cdhklmn hjln

acefgkn adefgjmn 3

adefghlm acefghjkl g^Jk

bfgjklmo bcdfglo

bcdfghjno bfghkmno h

abcejlmno abdeklno fghjknn

abdehjko abcehmo

abfhkl abodfhjlm 5

abcdfmn abfjkn abcdeghmn

bceghln bdeghjklmn

bdegkm bcegj 6

aghjmo acdghko abcdefgh

acdgjklno aglmno

cefhjkmno defhno 7

defjlo cefklmo abcdejkmn

8

abcdefjk
I
- 26 -

Plan 6U, 114.0614. 0 I/6I4 replication of 14 factors in b blocks of 6I4. units each.

Factors; k9 B9 C3 D5 E,F,G5 H^ J 5 K,L_,M,N,0.

I; Same as plan 6l4.oli4.080

Block confounding; ABE05 BCK 5 ACEK0 o

Without blocks; All main

measurable.

effects and all two=factor interactions

With blocks; Same as above c

Blocks

1 2 (1) acegkm gbjk

cdhklmn adeghln

acefgkn bgjklno 3

adefgblm bcdghjmo abedeghmn

bfgjklmo abcefjlo

bcdfghjno abdefhjkmno b

abcejlmno abhklmn abcdejkmn

abdehjko abed

abfhkl bcefghlm

abcdfmn bdefgkn

bceghln afghjno

bdegkm acdfgjklmo

aghjmo cehjko

acdgjklno dejlrano

cefhjkrano cdfjkn

defjlo fhjlm

cdjkm adegj

hjln aceghj klmn

adefgjmn bcdglmno

acefghjkl bghko

bcdfglo abdefklmo

bfghkmno abcefhno

abdeklno abcdhjln

abcehmo abjkm

abcdfhjlm bdefghjkl

abfjkn bcefgjmn

bdeghjklmn acdfghkrano

bcegj afglo

acdghko dehmo

aglmno

defhno

cefklmo

fran

cdfhkl

ceklno

- 2 ? -

Plan 6I4

. s ll|., 128o I/6I4

. replication of II4. factors in two blocks of 128 units each

Factors; A,B,C, E3 F5 G5 H5 J 9 M,N5 0,

Is Same as plan 61ullu8,

Block confounding; ABEO,

Without blocks; All main effects and two -fact or interactions are measurable,

With blocks; Same as above,

1

Blocks

2

(1)

cdhklmn abed gkln

abedeghmn

abghm

acefgkn bcefghlm adefhkmn bdefl

adefghlm bdefgkn acefl bcehkmn

bfgjklmo afghjno bcdfhjklo aedfjmno

bcdfghjno acdfgjklmo bfjmno afhjklo

abcejlmno cehjko abdeghjlno degjkmo

abdehjko dejlmno abcegjkmo ceghjlno

abfhkl cdfjkn abcdfgklm

abcdfmn fhjlm abfghn

bceghln adegj bdelmn

bdegkm aceghj klmn bcehk

aghjmo bcdglmno acdjo

acdgjklno bghko ahjklmno

cefhj krnno abdefklmo defgjkno

defj lo abcefhno cefghj Imo

cdjkm abcdhjln fghjkmn

hjln abjkm cdfgjl

adefgjmn bdefghjkl acehjm

acefghjkl bcefgjmn adejkln

bcdfglo acdfghkmno bhlno

bfghkmno afglo bcdkmo

abdeklno dehmo abcefghklo

abcehmo ceklno abdefgmno

abcdfhjlm ghjk abgjlmn

abfjkn cdgjlran abcdghjk

bdeghjklmn acefhjn beefjklm

bcegj adefjklm bdefhjn

acdghko bfhlmo afkno

aglrano bcdfkno acdfhlmo

defhno abceghklmno cego

cefklmo abdego deghklmno

fmn abfgjl cdfghn

cdfhkl abcdfghjkmn fgklm

acegkra bcejkln adehk

adeghln bdehjm acelmn

bgjklno akmo bcdhjklmno

bcdghjmo acdhlno bjo

abcefjlo cefgmno ab defghj Imo

abdefhjkmno defghklo abcefgjkno

abhklmn cdghm abcdgkln

- 28 -

Plan 128,1208 0 1/128 replication of 12 factors in 4 blocks of 8 units eache

Factors; A^B^C^DjE^F^G^H^J

I = ABEG = ACEF = BCFG = DEFG = ABDF = ACDG = BGDE = ADHK = BDEGHK = CDEFHK

= ABGDFGHK = AEFGHK « BFHK = GGHK = ABCEHK = BCHJ = ACEGHJ = ABEFRJ = FGHJ

= BCDEFGHJ = ACDFHJ = ABDGEJ = DEHJ = ABCDJK * CDEGJK = BDEFJK = ADFGJK

= ABCEFGJK « CFJK = BGJK = AEJK = ABKL = EGKL = BGEFKL = AGFGKL = ABDEFGKL

= DFKL = BGDGKL = ACDEKL = BDKL - ADEGHL = ABGDEFHL - CDFGHL = BEFGHL

= AFHL = ABCGHL - CEHL = ACEJKL - BGEGHJKL -EFEJKL - ABPGHJKL = AGDEFGHJKL

» BCDFKJKL = DGHJKL = ABDEHJKL - CDJL * ABCDEGJL = ADEFJL - BDFGJL = CEPGJL

= ABGFJL = AGJL - BEJL = GDGHJM = ABCDEHJM - ADEFGEJM = BDFHJM - CEFHJM

= ABCFGKJM = ABJM - BEGHJM = ACGJKM = BCEJKM - EFGJKM = ABFJKM = ACDEFJKM

= BCDFGJKM = DJKM - ABDEGJKM = BDGM = ADEM - ABCDEFGM = CDFM = BEFM = AFGM

= ABCM = GEGM - ABGHKM = EHKM - BCEFGHKM = ACFHKM = ABDEFHKM « DFGHKM

= BGDHKM = ACDEGHKM = ABCDGHJKLM « CDEHJKLM = BDEFGEJKLM - ADFHJKLM

« ABCEFBJKLM - CFGSJKLM « BHJKLM = AEGHJKLM *5 . BCGJLM = ACEJLM = ABEFGJLM

** FJLM = BGDEFJLM = ACDFGJLM = ABDJLM = DEGJLM = ADGKLM = BDEKLM = CDEPGKLM

= ABGDFKLM = AEFKLM - BFGKLM - CKLM = ABCEGKLM = GHLM = ABEHLM = ACEFGHLM = BCfmM

= DEFHLM = ABDFGHLM = ACDHLM = BCDE&iLM*

Block confounding; DG* DH5 GH0

Without blocks; All main effect s s but no two-factor interactions., are measurable.

With blocks; Same as above.

Blocks

1 2 3 4

(1) hjkl cegjm ceghklm

abcklm

bdghlm

acdghk

acefjk

befjlm

abcdefghjklm

defghj

- 29

flan 128. 12.16. 1/128 replication of 12 factors in 2 blocks of 16 units each.

Factors: A,B, G,D5 E,F,G,H,J,K,L,M.

I: Same as plan 128.12.8.

Block confounding: DG.

Without blocks: All main effects,, but no two-factor interactions, are

meaerrn*aifber?r •

With blocks: Same as above.

Blocks

1 2

(1) cegjm

abcklm

bdghlm

acdghk

acefjk

bef^lm

abcdefghjklm

defghj

hjkl

abchjm

bdgjkm

acdgjl

acefhl

befhkm

abcdefgra

defgkl

- 30 -

Plan 128 , 1308 , 1/128 replication of 13 factors in 8 blocks of 8 units

each*

~

Factors? - A,B,C,D,E,F,G,H,J,K,L,M,N.

I “ ABCD = ABEFL = CDEFL = ABGHL = CDGHL = EFGH = ABCDEFGH = ABJKL = CDJKL = EFJK = ABCDEFJK = GHJK = ABCDGHJK = ABEFGHJKL = CDEFGHJKL = ACEGJL = BDEGJL = BCFGJ = ADFGJ = BCEHJ = ADEHJ = ACFHJL = BDFHJL = BCEGK = ADEGK = ACFGKL = BDFGKL = ACEHKL = BDEHKL = BCFHK = ADFHK = ADLM = BCLM = BDEFM = ACEFM = BDGHM = ACGHM = ADEFGHLM = BCEFGHLM = BDJKM = ACJKM = ADEFJKLM = BCEFJKLM = ADGHJKLM = BCGHJKLM = BDEFGHJKM = ACEFGHJKM = CDEGJM = ABEGJM - ABCDFGJLM - FGJLM - ABGDEHJLM = EHJLM = CDFHJM = ABFHJM - ABCDEGKLM - EGKLM = CDFGKM = ABFGKM - CDEHKM = ABEHKH - ABCDFHKLM - FHKLM - EFGHJKLMN - ABCDEFGHJKLMN = ABGHJKM

= CDGHJKMN - ABEFJKMN " CDEFJKMN TM JKLMN — ABCDJKLM ~ ABEFGHM

= CDEFGHMN ~ GHLMM = ABCDGHLMN “ EFLMN - ABCDEFLM - ABM = GDM

= ACFHKMN = BDFHKM - BCEHKLMN - ADEHKLMN - BCFGKLM - ADFGKLM = ACEGKM

= BDEGKMN - BCFHJLM - ADFHJLM = ACEHJM - BDEHJM ~ ACFGJM - BDFGJM

= BGEGJLMN = ADEGJLM - ADEFGHJKN ~ BCEFGHJKN = BDGHJKLN = ACGHJKLN = BDEFJKLN ACEFJKLN. ADJK1M = BCJKN - BDEFGHLN =• ACEFGHLN = ADGHN = BGGHN = ADEFN = BCEFN = EDLN~= ACLN - GDFHKLN = ABFHKLN - ABCDEHKN = EHKN — ABCDFGKN = FGKN = CDEGKLN - ABEGKLN = ABCDFHJN - FHJTJ = CDEHJLN = ABEHJLN - CDFGJLN = ABFGJLN = ABCDEGJN “ EGJN.

Block confounding? AB, AC,* BC, CL, ABCL, AL, BL„

Without blocks? All main effects and the following two-factor interactions

are measurable.

AE, AF, AG, AH,. AJ, AK, BE, BF, BG,

BH, BJ, BK, CE, CF, CG, CH, CJ, CK,

DE, DF, -DG, DH, DJ, DK, EL,. EM, FL,

FM, GL, GM,. HL, HM, JL, JM, KL,. KM.

With blocks t Same as above

,

Blocks?

1 2 3 k

(1)

efgh

ghjk

efjk

abcdehklmn

abcdfgklmn

abodegjlmn

abcdfhjlmn

abed cdfhkl abfhkl

£

6*. 7 8

bdegkm acegkm bcefghim adefghlm
.
31

Plan 128.13 »16. 1/128 replication of 13 factors in U blocks of 16 units

each.

Factors?: A, B,C,D,E,F,G,H, J,K,L,M,N.

I?. Same as plan 12 8.13 o 8.

Block confounding; AB, AC, BC.

Without blocks; All main effects and the following two-factor interactions

are measurable.

AE, AF, AG, AH, AJ, AK, BE, BF, BG,

BH,. BJS; BK, CE, CP, CG, CH, CJ, CK.

DE,. DF,. DG, DH, DJ, DK, EL, EM, FL < FMJ;, GL, GM, HL, HM, JL, JM, KL, m

With blocks t Same as above.

Blocks;

1 2 3

(1 ) cdfhkl bdegkm

efgh

ghjk

efjk

abcdehklmn

abcdfgklmn

abcdegjlmn

abcdfhjlmn

abed

abcdefgh

abcdghjk

abcdefjk

ehklmn

fgklmn

egjlmn

fhjlmn

k

bcefghlm

- 32 -

Plan 128JL3o32o 1/128 replication of 13 factors in 2 blocks of 32 units

each*

Factors ? A,B,C,D,E,F,G,H,J,K,L,M,N<>

I? Same as plan 128„13o8*

* Block confounding?: AB.

Without blocks? All main effects and the following two-factor interactions

are measurable,, *

AE, AF, AG, AH, AJ, AK, BE, BF, BG,

BH, BJ, BKS CE, CF, CG, CH, CJ, CK,

DE,, df5

, DG,. DH, DJ, DK, EL, EM, FL,

fm. GM, HL, HM, JL, JM, KL, KM*

With blocks? Same as above*

Blocks

?

1 2. (1) bdegkm

efgh cdefmn

ghjk cdghmn

efjk cdefghjkmn

abcdehklmn

abedfgklmn

abcdegjlmn

abcdfhjlmn

cdjkmn

abed

abcdefgh

abcdghjk

abedefjk

ehklmn

fgklmn

egjlmn

fhjlmn

cdfhkl

cdegkl

cdfgjl

cdehjl

abefmn

abghmn

abefghjkmn

abjkmn

abfhkl

abegkl

abfgjl

abehjl

•

- 33 -

Plan 128 e ll4-o8 9 1/128 replication of ll; factors in 16 blocks of 8 units each,

factors? A,B,C,D,E,F,G5 H, J,K,L,M,N,0.

I = ABGD - ABEFL = CDEF1 = ABGHL = CDGHL = EFGH = ABCDEFGH = ABJKL = GDJKL

= EFJK = ABCDEFJK = GHJK = ABGDGHJK = ABEFGHJKL = CDEPGHJKL = ACEGJL

= BDEGJL = BCFGJ = ADFGJ - BCEHJ * ADEHJ = ACFHJL = BDFEJL = BGEGK = ADEGK

= ACFGKL = BDFGKL = ACEHKL = BDEHKL = BCFHK = ADFHK = ADLM = BCLM = BDEFM

= ACEFM = BDGHM = ACGHM = ADEFGHLM = BCEFGHLM = BDJKM = ACJKM = ADEFJKLM

= BCEFJKLM = ADGHJKLM = BGGHJKLM = BDEFGHJKM = ACEPGHJKM = CDEGJM = ABEGJM

= ABCDFGJLM = FGJLM = ABCDEHJLM - EHJLM = CDFKJM - ABFHJM = ABCDEGKLM

= EGKLM = CDFGKM = ABFGKM = GDEHKM = ABEHKM - ABCDFHKLM = FHKLM = EFGBJKLMNO

= ABCDEPGBJKLMNO = ABGHJKMNO = GDGHJKF^O = ABEFJKMNO * CDEFJKMNO = JKLMNO

= ABCDJKLMNO = ABEPGHMNO - CpEFGHMO - GHLMNO = ABCDGRLMO = EFLMNO

= ABCDEFLMNO = ABMNO = GDMO = ACFHKMO = BDFHKMNO = ElbEHKLMO = ADEHKLMO

= BCFGKLMNO * ADFGKLMNO = AGEGKMNO « BDEGKMO = ADFHJLMNO

= ACEHJMNO = BDEBJMNO = ACFGJMNO = BDFGJMNO = BCEGJLMNO = ADEGJLMNO

= ADEPGHJKNO = BCEBGHJKNO - BDGHJKLNO = AGGHJKLNO = BDEFJKLNO = ACEFJKLNO

= ADkJKNO = BCJKNO = BDEFGHLNO = ACEPGHLNO = ADGHNO = BCGHNO = ADEFNO

= BCEFNO = BDLNO = AGLNO - CDFHKLNO = ABFHKLNO = ABCDEHKNO = EHKNO

= ABGDUjKNO = FGKNO = CDEGKLNO - ABEGKLNO = ABCDFHJNO = FHJNO = CDEHJLND

= ABEHJLNO = CDF3JLNO = ABPGJLNO = ABCDEGJNO = EGJNO,

Block confounding; AB,AC.,BC, CL,ABCL,AL,BL,GO,ABGO,ACGO,BCGO, CGLO,ABCGLO,

AGLOjBGLO,

Without blocks; All main effects, and all two-factor interactions except

following are measurable

s

ab,ac,ad,al,am,bc,

BD,3L,3M,CD,CL,CM,

DL, ]jJM„EF, EG, EH, EJ , EK,PG,FH,FJ,FK,GH,

GJ ,GK, HJ , HK, JK, LMo

- 3k -

With blocks: Same as above, except that the two-factor interactions GO

and HN also are not measurable?

Blocks:

1 2 3 k

(1)

efjk

abcdehklmn

abcdfhjlmn

efghno

ghjkno

abcdfgklmo

abcdegjlmo

no abed abedno

£ 6 7 8

cdfhkl edfhklno abfhkl abfhklno

9 10 11 12

bdegkmno bdtegka aeeg^mno acegkm

13 lk 15 16

bcefghlmno bcefghlm adefghlmno adefghlm
.

.
Plan 128.lU.l6 . 1/128 replication of lU factors in 8 blocks of 16 units each.

Factors; A,B,C,D,E,F,G,H,J,K,L,M,N,0.

I; Same as plan 128.lUc8<>

Block confounding; AB, AC,BC, CL,ABCL, AL,BL.

Without blocks: All main effects, and all two factor interactions except the

following are measurable;

AB,AC,AD,AL,AM,BC,

BD,BL,BM, CD, CL, CM,

DL,DM,EF,EG,EH,EJ,

EK,FC,FH,FJ,FK,GH,

GJ,GK,HJ,HK,JK,LM0

With blocks: Same as above.

Blocks :

1 2 3 U

(l) abed cdfhkl abfhkl

efjk

abcdehklmn

abcdfhjlmn

efghno

ghjkno

abcdfgklmo

abcdegjlmo

no

efjkno

abcdehklmo

abcdfhjlmo

efgh

ghjk

abcdfgklmn

abcdegjlmn

5

bdegkmno

6

acegkmno

7

bcefghlmno

8

adefghlmno
-
- 36 -

Plan 128oli4.«32« 1/128 replication of lk factors in I4 blocks of 32 units each.

Factors: A,B,C,D,£,F,G,H,J,K,L,M,N,0,

I: Same as plan 128.114,. 8,

Block confounding: AB>AC,BC.

Without blocks: All main effects, and all two-factor interactions except

the following are measurable;

AB,AC,AD,AL,AM,BC,

BD, BL, BM, CD, CL, CM,

DL, DM, EF, EG, EH, EJ

,

EK,PG,FH,FJ,FK,GH,

GJ,GK,HJ,HK,JK,LM0

With blocks: Same as above.

Blocks:

1 2

(1) abed cdfhkl

efjk abedefjk

abcdehklmn ehklmn 3

abcdfhjlmn fhjlmn bdegkmno

efghno abedefghno

ghjkno abcdghjkno k

abcdfgklmo fgklmo bcefghlmno

abcdegjlmo egjlmo

no abedno

efjkno abedefjkno

abcdehklmo ehklmo

abcdfhjlmo fhjlmo

efgh abcdefgh

ghjk abcdghjk

abcdfgklmn fgklmn

abcdegjlmn egj Iran

- 37 ~

Plan 128 oliu 614- q 1/128 replication of 11; factors in 2 blocks of 61; units each e

Factors: A,B,C,D,E,F,G,H,J,K,L,M,N,0 o

I: Same as plan 128.lli.o8o

Block confounding: AB.

Without blocks: All main effects, and all two-factor interactions except the

following are measurable:

AB,AC,AD,AL,AM,BC,

BD,BL,BM,CD,CL,CM,

DL , DM, EF, EG , EH, EJ ,,

ek,fg,fh,fj 3 fk,gh,

GJ,GK,HJ,HK,JK,LM.

With blocks: Same as above.

Blocks:

1

(1) abed cdfhkl abfhkl

efjk abedefjk cdehjl abehjl

abcdehklmn ehklmn abefmn cdefmn

abcdfhjlmn fhjlmn abjkmn cdjkmn

efghno abedefghno cdegklno ab egkino

ghjkno abcdghjkno cdfgjlno abfgjlno

abcdfgklmo fgklmo abghmo cdghmo

abcdegjlmo egjlmo abefghjkmo cdefghjkmo

no abedno cdfhklno abfhklno

efjkno abedefjkno cdehjlno abehjlno

abcdehklmo ehklmo abefmo cdefmo

abcdfhjlmo fhjlmo abjkmo cdjkmo

efgh abcdefgh cdegkl ab egkl

abcdghjk cdfgjl abfgjl

abcdfgklmn fgklmn abghmn cdghmn

abcdegjlmn egjlmn abefghjkmn cdefghjkm 1

2

bdegkrano

- 38 -

Plan 128,15,8 , 1/128 replication of 15 factors in 32 blocks of 8 units each.

Factors; A^B^C^D^E^F^G^H^J^K^L^M^N^O^Po

I = ABEGN = ACEFNP » BCfGP - DEFGO = ABDFWO = ACDGNOP = BCDEOP = ADHKO

= BDEGHKNO * CDEFHKNOP = ABGDFGHKOP = AEFGHK = BFHKN = CGHKNP = ABCEHKP

= BCBJNOP = ACEGHJOP = ABEFHJO = FGHJNO » BGDEFGHJNP = ACDFHJ P = ABDGHJ

= DEHJN = ABCDJKNP - GDEGJKP = BDEFJK = ADFGJKN = ABGEFGJKNOP = CFJKOP

» BGJKO - AEJKNO = ABKLOP - EGKLNOP = BCEFKLNO = ACFGKLO = ABDEFGKLP

» DFKLNP = BCDGKLN = ACDEKL = BDHLP = ADEGHLNP. = ABGDEFHLN = CDFBHL

= BEPGHLOP « AFHLNOP = ABCGHLNO = CEHLO = ACHJKLN = BCEGHJKL = EFHJKLP

». ABfGHJKXiNP - AGDEFGBJKLNO = BGDFHJKLO » DGHJKLOP - ABDEHJKLNOP

s= CDJLNO = ABCDEGJ LO = .ADEFJLOP «= BDFGJLNOP = CEFGJLM = ABGFJL = AGJ LP

= BEJLNP = CDGHJMO = ABGDEHJMO » ADEFGHJMNOP = BDFHJMOP = GEFHJM

= ABCFGBJMN = AHJMNP = BEGBJMP = AGGJKM = BCEJKMN = EFGJKMNP = ABFJKMP

= ACDEFJKMO = BCDF3JKMO = DJKMOP » ABDEGJKMOP » BDGMP = ADEMP

= ABGDEFGM = CDFMN = BEFMNOP - AFGMOP - AJBCMO = CEGMO = ABGHKMNOP

= EHKMOP = BCEFGHKMO « ACFHKMNQ = ABDEFHKMNP - DFGHKMP - BCDKKM = ACDEGHKMN

* ABGDGHJKLMP = GDEHJKLMNP = BDEFGHJKLMN = ADFHJKLM = ABGEFHJKLMOP

= CFGHJKLMNOP = BHJKLMNO = AEGHJKLMD - BCGJLMOP - ACEJLMNOP = ABEFGJLMNO

« FJLMD = BGDEFJLMP » AGDFGJ LMNP = ABBJLMN = DEGJLM = ADGKLMNO = BDEKLMD

= CDEFGKLMOP = ABCDFKLMNOP = AEFKLMN = BFGKLM = CKLMP = ABCEGKLMNP

< = GHLM = ABEHLM = ACEFGHLMP = BGFHLMNP = DEFHLMNO = ABDFGHLMO = ACDHLMDP

= BGDEGHLMNOP,

Block confounding: i£BD,ACF,BCDF,ABCE,CDE,BEF,ADEF,FJ,ABDFJ,ACJ,BCD J,

ABGEFJ 5 CDEFJ ^BEJ 5 ADEJ , EH5 ABDEH,ACEFH, BCDEFH, ABCH, CDH,BFH, ADFH, EFHJ

,

ABDEFKJ , ACEHJ , BGDEHJ s ABCFHJ , CDFHJ , BHJ , ADHJ

Without blocks: All main effects and all two=factor interactions are measurable.
'
39 -

With Blocks: Same as above 5 except the following two-factor interactions

are not measurable.

AP5 BG , CM, DO, EH, EL* FJ , HL* KN.

Blocks:

1 2 3 k

(1)

cefhjklmn

adefhjlop

acdkinnop

bcdfgjmo

bdeghklno

abceghlrap

abfgjknp

acdghk acdgjl hjkl

£

6 7 8

abcklm bdghlm bdgjkm abchjm

9 10 U 12

ceghklm adelm adehjkm cegjm

13 Ik 15 16

abegh bcdek bcdehjl abegjkl

17 18 19 20

dgknp achnp acjklnp dghjlnp

21 22 23 2h

abcdglmnp bhklmnp bjmnp abcdghjkmnp

25 26 27 28

cdeHjrmp aegklranp aeghjmnp cdejkmnp

29 30 31 32

abdehknp bcegnp bceghjklnp abdejlnp

- Uo -

Plan 128.15 .16 . 1/128 replication of 15 factors in 16 blocks of 16 units each.

Factors: A,B,C,D,E,F,G,H,J,K,L,M,N,0,P.

Is Same as plan 128.15.8.

Block confounding: ABD, ACF, BCDF, ABCE, CDE, BEF, ADEF, FJ, ABDFJ, ACJ,

BCDJ, ABCEFJ, CDEFJ , BEJ, ADEJ.

Without blocks: All main effects and all two-factor interactions are measurable.

With blocks: Same as above, except the two--factor interactions EL and

FJ are not measurable.

Blocks:

1 2 3 k

(1) acdgjl abcklm bdgjkm

cefhjklmn

adefhjlop 5 6 7

acdkmnop

bcdfgjmo ceghklm adehj km abegh

bdeghklno 8 9 10

abceghlmp

abfgiknp bcdehjl dgknp acjklnp

acdghk

adefgjlmn 11 12 13

cefgjklop

ghmnop

abcdglmnp bjmnp cdehlmnp

abfhjkmo 16 Ik 15

abcelno

bdeklmp

bcdfhjnp

aeghjmnp abdehknp bceghjklnp
'
- Ill -

Plan 128.1^.32. 1/128 replication of 1$ factors in 8 blocks of 32 units each.

Factors? A5 B5 C,D,E,F',G5 H,Js K,L,M,Ns O,Po

I: Same as plan 128.15. 8.

Slock confounding! ABD, ACF, BCDF, ABCE? CDE, BEF5 ADEF.

Without blocks! All main effects and all two-factor interactions are measurable.

With blocks! Same as above.

Slocks

1

(1) abcklm

2

ceghklm

2k

abegh

cefhjklmn

adefhjlop £6 7

acdkmnop

bedfgjmb

bdeghklno

abceghlmp

dgknp abcdglmnp

8

cdehlmnp

abfgjknp

acdghk

adefgjlmn

cefgjklop

ghmnop

abfhjkmo

abcelno

abdehknp

bdeklmp

bcdfhjnp

acdgjl

adefghkmn

cefghop

gjklmnop

abflmo

abcehjkno

bdehgmp

bcdfklnp

hjkl

cefmn

adefkop

acdhjlmnop

bcdfghklmo

bdegjno

abcegjkmp

abfghlnp

- 1*2 -

Plan 128„l5o61* . 1/128 replication of 15 factors in 1* blocks of 61* units each.

Factors s AjBjC^DjEjFjGjHjJ^KjLyMjN^OjP.

Is Same as plan 128 .15. 8 „

Block confounding; ABDS ACF, BCDF.

Without blocks; All main effects and all two-factor interactions are measurable.

With blocks; Same as above.

Blocks;

(1) acdgjl

1

abcklm bdgjkm

cefhjklmn adefghkmn abefhjn bcdefghln

adefhjlop cefghop bcdefhj kmop abefghklmop

acdkmnop gjkLmnop bdlnop abcgjnop

bcdfgjmo abflmo adfgjklo cfko

bdeghklno abcehjkno acdeghmno ehjlmno

abceghlmp bdehjmp eghkp aedehj kip

abfgjknp bcdfklnp cfgjlmnp adfmnp

acdghk hjkl bdghlm abchjm

adefgjlmn cefmn bcdefgjkn abefkln

cefgjklop adefkop abefgjmop bcdeflmop

ghmnop acdhjlmnop abcghklnop bdhj knop

abfhjkmo bcdfghklmo cfhjlo adfgho

abcelno bdegjno ekmno acdegjklmno

bdeklmp abcegjkmp acdep egjlp

bcdfhjnp abfghlnp adfhjklmnp cfghkmnp

2 3 b

ceghklm dgknp cdehlmnp

- h3 -

Plan 128 e l5ol28o 1/12Q replication of 15> factors in 2 blocks of 128 units

each*

Factors? As B3>C,D ? E3 F,G,H,J 5 K,L,M,N? 0 3,P o

I s Same as plan 128ol5<,8<>

Block confounding? ABD»

Without blocks? All main effects and all two-factor interactions are measurable

With blocks? Same as above*

( 1 ) cefhjklmn

adefhjlop

aedkmnop

bcdfgjmo

bdeghklno

abceghlmp

abfgjknp

acdghk

adefgjlmn

cefgjklop

ghranop

abfhjkmo

abcelno

bdeklmp

bedfhjnp

ceghklm

fgjn

acdfgjkmop

adeghlnop

bdefhjklo

bcdmno

abkp

abcefhjlmnp

adelm

acdfhjkn

fhjmop

ceklnop

abcefgjlo

abghkmno

bcdghp

bdefgjklmnp

acdgjl

adefghkmn

cefghop

gjklmnop

abflmo

abeehjkno

bdehjmp

bcdfklnp

hjkl

cefrnn

adefkop

aedhjlmnop

bcdfghklmo

bdegjno

abcegjkmp

abfghlnp

adehjkm

acdfln

fklmop

cehjnop

abeefghko

abgjlmno

bcdgjklp

bdefghmnp

cegjm

fghkln

acdfghlmop

adegjknop

bdefo

bcdhjklmno

abhjlp

abcefkmnp

abcklm

abefhjn

bcdefhjkmop

bdlnop

adfgjklo

acdeghmno

eghkp

cfgjlmnp

bdghlm

bcdefgjkn

abefgjmop

abcghklnop

cfhjlo

ekmno

acdep

adf'hjklmnp

abegh

abcfgjklmn

bdfgjlop

bcdeghkmnop

acdefhjmo

adklno

clmp

efhjknp

bcdek

bdfhjlmn

abcfhjklop

abemnop

efgjkmo

cghlno

adghklmp

acdefgjnp

bdgjkm

bcdefghln

abefghklmop

abcgjnop

cfko

ehjlmno

acdehjklp

adfmnp

abchjm

abefkln

bcdeflmop

bdhjknop

adfgho

acdegj klrnno

egjlp

cfghkmnp

bcdehjl

bdfkmn

abcfop

abehjklmnop

efghlmo

cgjkno

adgjmp

acdefghklnp

abegjkl

abcfghmn

bdfghkop

bcdegjlmnop

acdefklmo

adhj no chjkmp

eflnp

Blocks

1

2

dgknp
'
- uu -

Flan 256eX3»8«» 1/2$6 replication of 13 factors in h blocks of 8 units

each®

Factors? A,B,C.,D,E,F,,G,H, J,K,L,M,No

I = ABCEGHJKMN = ADFGJN = BCDEFHKM - AEFKLN = BCFGHJLM = DEGJKL = ABC-DHLMN - ACFHJKMN = BEFG = CDGHKM - ABDEJN = CEHJLM = ABGKLN = ACDEFGHLMN - BDFJKL = ABCDFGHKMN = DEFJ - BCHJKM - AEGN - BCDEGHLM = ADJKLN = ABCEFHJLMN = FGKL - BDGJ = ACDEHKMN = ABFN = CEFGHJKM = ABDEFGJKLN = CDFHLM - BEKL = ACGHJLMN = ADEGHJLN = BCDKLM = EFHL = ABCFGJKLMN = DFGHJK - ABCDEFMN = AHKN =• BCEGJM - CDEFGKLM - ABDFHJLN = ACEJKLMN = BGHL = ACDGMN = BDEHJK = CFJM - ABEFGHKN = BCEFJKLM = AFGHLN = ABCDEGKLMN - DHJL = ABCJM - EGHK = BCDFGM = ADEFHJKN = ABEHLN - CGJKLM = BDEFGHJL - ACDFKLMN = BFHK - ACEFGJMN = ABDGHJKN = CDEM “ AEFGHJKLMN = BCFL = DEHKLM - ABCDGJLN - GHJM = ABCEKN = ADFHMN = BCDEFGJK - CBGL ~ ABHJKLMN = ACDEFJLN = BDFGHKLM = ACFGKN = BEFHJM = CDJK = ABDEGHMN - BCDEJL = ADGHKLM - ABCEFGLN = FHJKLM - ABCDFJKN = DEFGHH = BCGK = AEHJMN = ABDEFHKLMN - CDFGJL = BEGHJKLM - ACLN = BDHM = ACDEGJKN = ABFGHJMN — CEFK - DFKM - ABCDEFGHJN - AGJKMN = BCEH = ADELMN = BCDGHJKL = EFGJLM - ABCFHKLN = ACDHJN = BDEGKM = CFGH = ABEFJKMN = CDEFHJKL * ABDFGLMN = ACEGHKLN - BJLM “ ABCGHN = EJKM = BCDFHJ = ADEFGKHN - BCEFGHKL = AFJLMN = ABCDEHJKLN = DGLM = BFGJKM = ACEFHN = ABDKMN = CDEGHJ = ABEGJLMN - CHKL = BDEFLM = ACDFGHJKLN = AFHJ = BCEFGKMN = DGHN = ABCDEJKM = EHJKLN = ABCGLM = ADEFGHKL = BCDFJLMN = CKMN - ABEGHJ ~ ACDFGJKM - BDEFHN = ACEFLM - BFGHJKLN = CDEGJLMN = ABDHKL = BCDGJKMW * ADEH - ABCFKM - EFGHJN - ABCDEFGJLM - DFHKLN = BGELMN ~ AGHJKL = ABDFGH - CDEFJKMN = BHJN - ACEGKM = BDEGHKLN = ACDJLM = ABEFHJKL = CFGLMN = DEFGLN - ABCDFHJKLM = AEJL - BCGHKLMN = ADGK = BCDEHJMN = FJKN = ABCEFGHM = ACDEGHJKLM = BDLN = CEFHKLMN = ABFGJL

=' CDFGHJMN - ABDEFK = ACHM - BEGJKN - ABCEHKLM - GJLN = BCDEFGHJKLMN - ADEL = BCFHMN = AEFGJK = ABCDGHJM = DEKN - BEFJLN - ACFGHKLM = ABDEGL = CDHJKLMN = ABJK = CEGHMN - BDFGKN = ACDEFHJM - EGKLMN = ABCHJL = ADEFJKLM - BCDFGHLN - AFGM - BCEFHJKN - DJMN = ABCDEGHK - ACEFGHJL = BFKLMN = CDEHLN - ABDGJKLM = CGHJKN - ABEM ~ ACDFHK = BDEFGJMft

* ABCDEFHL = DFGJKLMN - BCEGHJLN = AKLM - BCDHKN = ABEGJM - ABCFGHJK - EFM = BDEJKLMH = ACDGHL ~ ABEFGKLM = CFHJLN - ABDFJM = CDEFGHKN - BGMN ~ ACEHJK = ADHJKM - BCDEGN - FGHKMN - ABCEFJ - DEFHJLMN = ABCDFGKL = AEGHLM - BCJKLN - CDFN - ABDEFGHJKM = ACGJ “ BEHKMN - ACDEKL - BDGHJLMN = GEFGJKLN - ABFHLM =- BCFGJN - AEFHKM = ABCD = DEGHJKMN = ABCEGJKL ~ HLMN = BCDEFKLN - ADFGHJLM - ABGHKM - CEJN = BDFHJKMN = ACDEFG - BEFGHLM - AGFJKL - ABDEHJLM - CDGKLW,

Block confounding? FGS GH* FHo

Without blocks? All main effeets s but no two-factor interactions, are measurable

With blocks? Same as above

- 1*5 -

Plan 256q13o 8 0 (Continued

)

0

Blocks £

1 2 3

(1) fgjkln eghjkm

abcdefghj klran

adjmn

bcefghkl

cdfghn

abejklm

acfghjm

bdekln

h

efhlmn

- h6 -

Flan 2$6ol3 <) l6o 1/256 replication of 13 factors in 2 blocks of 16 units

eacho

Factors? A,B? C,D,,E,F ,G,H, J,K,L? M,No

I? Same as plan 2£6ol3o8 c

Block confounding? FG<>

Without blocks? All main effects., but no two-factor interactions, are measurable*

With blocks? Same as above*

Blocks ?

1 2

(1) eghjkm

abedefghjklmn

adjmn

bcefghkl

cdfghn

abejklm

acfghjm

bdekln

fgjkLn

abcdehm

adfgklm

bcehjn

cdhjkl

abefgmn

achklmn

bdefgj

hi -

Plan 256oll4.o-8 0 1/256 replication of lit factors in 8 blocks of 8 units

each o

Factors s A, B, C, D,. E, F, G, H, J, K, L, M, N, 0®

1 = ABCEGHJKMNO = ADFGJNO = BCDEFHKM = AEFKLNO = BCFGHJLM = DEGJKL = ABCDHLHNO = ACFHJKMNO = BEFG - CDGHKM = ABDEJNQ = CEHJLM = ABGKLNO = ACDEFGHLMNO = BDFJKL - ABCDFGHKMNO = DEFJ - BGHJKM = AEGNO = BCDEGHLM = ADJKLNO = ABCEFHJLMNO = FGKL = BDGJ = ACDEHKMNO = ABFNO - CEFGHJKM - ABDEFGJKENO - GDFHLM - BEKL — ACGHJLMNO = ADEGHJLNO - BCDKLM = EFHL = ABCFGJKLMNO = DFGHJK = ABCDEFMNO = AHKNO - BCEGJM - CDEFGKLM = ABDFHJLNO = ACEJKLMNO = BGHL = ACDGMNO = BDEHJK = CFJM = ABEFGHKNO = BCEFJKLM = AFGHLNO - ABCDEGKLMNO = DHJL - ABCJMNO = EGHK = BCDFGM - ADEFHJKNO = ABEHLNO = CGJKLM - BDEFGHJL - ACDFKLMNO = BFHK - ACEFGJMNO = ABDGHJKNO - CDEM - AEFGHJKLMNO = BCFL - DEHKLM - ABCDC-JLNO = GHJM - ABCEKNO - ADFHMNO = BCDEFGJK - CEGL - ABHJKLMNO = ACDEFJLNO = BDFGHKLM = ACFGKNO - BEFHJM = CDJK = ABDEGHMNO - BCDEJL = ADGHKLMNO = ABGEFGLNO - FHJKLM - ABCDFjfcNQ = DEFGHM = BCGK = AEHJMNO - ABDEFHKLMNO = CDFGJL - BEGHJKLM = ACLNO = BDHM - ACDEGJKNO = ABFGHJMNO = CEFK = DFKM - ABCDEFGHJNO = AGJKMNO = BCEH - ADELMNO = BCDGHJKL = EFGJLM = ABCFHKLNO = ACDHJNO = BDEGKM = CFGH = ABEFJKMNO = CDEFHJKL = ABDFGLMNO = ACEGHKLNO = BJLM = ABCGHNO = EJKM = BCDFHJ = ADEFGKMNO = BCEFGHKL = AFJLMNO = ABCDEHJKLNO = DGLM = BFGJKM = ACEFHNO = ABDKMNO - CDEGHJ = ABEGJLMNO - CHKL = BDEFLM = ACDFGHJKLNO = AFHJ = BCEFGKMNO = DGHNO = ABCDEJKM = EHJKLNO = ABCGLM = ADEFGHKL = BCDFJLMNO = CKMNO = ABEGHJ = ACDFGJKM = BDEFHNO = ACEFLM = BFGHJKL.NO - CDEGJLMNO =- ABDHKL - BCDGJKMNO = ADEH = ABCFKM = EFGHJNO - - ABCDEFGJLM =• DFHKLNO - BCELMNO - AGHJKL = ABDFGH -- CDEFJKMNO = BHJNO - ACEGKM - BDEGHKLNO = ACDJLM - ABEFHJKL = CFGLMNO = DEFGLNO = ABCDFHJKLM = AEJL = BCGHKLMNO = ADGK = BCDEHJMNO = FJKNO = ABCEFGHM = ACDBGHJKLM = BDLNO = CEFHKLMNO = ABFGJL = CDFGHJMNO = ABDEFK = ACHM = BEGJKNO = ABCEHKLM = GJLNO - BCDEFGHJKLMNO = ADFL = BCFHMNO = AEFGJK = ABCDGHJM - DEKNO = BEFJLNO - ACFGHKLM = ABDEGL = CDHJKLMNO = ABJK = CEGHMNO = BDFGKNO = ACDEFHJM = EGKLMNG = ABCHJL - ADEFJKLM = BCDFGHLNO = AFGM = BGEFHJKNO = DJMNO - ABCDEGHK = ACEFGHJL = BFKLMNO = CDEHLNO = ABDGJKLM = CGHJKNO = ABEM - ACDFHK = BDEFGJMNO = ABCDEFHL = DFGJKLMNO = BCEGHJLNO = AKLM = BCDHKNO = ADEGJM = ABCFGHJK = EFMNO = BDEJKLMNO = A.CDGHL = ABEFGKLM = CFHJLNO = ABDFJM - CDEFGHKNO = BGMNQ = ACEHJK - ADHJKM - BCBEGNO = FGHKMNO = ABCEFJ - DEFHJLMNO = ABCDFGKL = AEGHLM - BCJKLNO = CDFNO - ABDEFGHJKM = ACGJ - BEHKMNO = ACDEKL - BDGHJLMNO - CEFGJKLNO = ABFHLM = BCFGJNO ~ AEFHKM = ABCD = DEGHJKMNQ = ABCEGJKL - HLMNO = BCDEFKLNO - ADFGHJLM = ABGHKM = CEJNO = BDFHJKMNO = ACDEFG = BEFGHLMNO = ACFJKL = AEDEHJLM = CDGKLNO 0

HLock confounding? EG, GH, FH, DJ, DFGJ, DGHJ, DFHJo

Without blocks? All main effects and the following two-factor interactions

are measurable s

AN, AO, BN, BO, CN,

CO, DN, DO, EN, EO,

FN, FO, GN, GO, HN,

HO, JN, JO, KN, KO,

LN, LO, MN, MO, NO®

With blocks? Same as above

Blocks?

1 1 3

(!) edfghn cdhjkl

abcdefghjklmn

adjinn

beefghkl

noabcdefghj klmo

adjmo

bcefghklno

5 6 x

cdefjkmn eghj Ion efhlmn

k

fgj kin

/

” h9 *»

Plan 256,1110160 1/256 replication of lij. factors in h blocks of 16 units

each*

Factors s A, B, C, D, E, F, G, H, J, K9 L, M, N,. 0 *

I s Same as plan 256*lU*8 *

Block confounding s FG9 GH, FH*

Without blocks?; All main effects and the following two—factor interactions

are measurable?

AN, AO, BN, BO, CN,

CO, DN, DO, EN, BO,

FN, fo. ON, GO, HN,.

HO, JN, JO, KN,. KD,.

LN, 10, MN,. M), NO*

With blocks?, Same as above*

Blocks

1 2 3 k

(1) cdhjkl cdefjkmn efhlmn

abcdefghj klmn

adjmn

bcefghkl

noabcdefghjklmo

adjmo

bcefghklno

cdfghn

abejklm

acfghjm

bdekln

cdfgho

abej klmno

acfghjmno

bdeklo

- £0 -

Plan 256*llu32o 1/256 replication of lii factors in 2 blocks of 32 units

eacEo’”'””

Factors? A, B, C, D, E, F, G, H, J, K, L, M, N, 0 o

1 ? Same as plan 256®Hu8<>

Block confounding? FG®

Without blocks? All main effects and the following two-factor interactions

are meastables

AN, AO, BN, BO, CN,

CO, DN, DO, EN, EO,

FN, FO, ON, GO, HN,

HO, JN, JO, KN, KO,

LN, LO, MM, MD,. NO®

With blocks? Same as above®

Blocks t

1 2

(1 ) cdefj kmn

abcdefghj klmn bcehjn

adj ran fgjkLo

bcefghki abcdehmno

no adfgklmno

abedefghjklmo

adjmo

bcefghklno

cdfghn

abej klm

acfghjm

bdekln

bcehjo

cdfgho

abej klmno

acfghjmno

bdeklo

edhjkl

abefgmn

achklmn

bdefgj

edhjklno

abefgmo

achklmo

bdefgj no fgjkln

abedehm

adfgklm

- 51 -

Plan 256ol5«8. 1/2^6 replication of V~> factors in 16 blocks of 8 units each.

Factors: A,B,C,D,E,F,G,H,J,K,,L,M,N,0,P*

I = ABCEGHJKMNO = ADPGJNP = BCDEFHKMOP = AEFKLNP = BCPGHJLMOP = DEGJKL

= ABGDHLMNO = ACFHJKMNO = BEPG = CDGHKMOP = ABDEJNP = CEHJLMDP

= ABGKLNP = ACDEFGHLMNO = BDFJKL = ABCDFGHKMP = DEFJOP = BCHJKM = AEGNO

= BGD0GHLM = ADJKLNO = ABCEFHJ LMNP = FGKLOP = BDGJOP = ACDEHKMP = ABFNO

= GEFGHJKM = ABDEFGJKLNO = CDFHLM = BEK10P = ACGEJLMNP = ADEGKJ LNOP

« BCDKLMP = EFHLO = ABCPGJKLMN = DPG&JKO = ABCDEFMN = AHKNOP = BGEGJMP

= CDEPGKLMP = ABDFHJ LNOP = ACEJKLMN = BGHLO = ACDGMN * BDEHJKO = GFJMP

= ABEFGHKNOP = BCEFJKLMO = AiGHLN - ABCDEGKLMNOP = DliJLP * ABGJMNOP

= EGHKP = BCDFGMO = ADEFHJKN = ABEHLN = GGJKLMO = BDEFGHJ LP = ACDFKLMNOP

= BFHKP = ACEPGJMNOP = ABDGHJKN = CDEMO = AEFGiiJKLMNP - BGFLOP = DEHKLM

= ABCDGJ LNO = GHJM = ABGEKNO = ADFHMNP = BCDEPGJKOP = CEGLOP = ABHJKLMNP

= AGDEFJ LNO = BDPGHKLM = AGFGKNO = BEFHJM = C&JKOP = ABDEGHMNP = BGDEJL

= ADGHKLMNO = ABCEFGLNP = FHJKLMOP = ABGDFJKNP = DEFGHMOP » BCGK

= AEHJMNO o ABDEFHKLMNO = CDFGJL = BBGHJKLMOP = ACLNP = BDHMOP = ACDEGJKNP

= ABFGHJMNO = CEFK = DFKMO = ABCDEFGHJN = AGJKMOP = BGEHP = ADELMNOP

= BCDGHJKLP = EFGJLMO = ABCFHKLN = ACDHJN = BDEGKMD = CPGHP = ABEFJKMNOP

= CDEFHJKLP = ABDPGLMOP = ACEGHKLN = BJLMO = ABCGHNOP = EJKMP = BCDFHJO

= ADEFGKM = BCEFGHKLO = AFJLMN = ABCDEHJKLNOP = DGLMP = BPGJKMP

= AGEFHNOP = ABDKM = CDEGHJO = ABEGJLM = CHKLO = BDEFLMP = ACDFGHJKLDJOP

= AFHJOP = BCEFGKMNP = DGHNO = ABGDEJKM = EHJKLNO = ABCGLM = ADEFGHKLOP

= BCDFJLMNP = CKMP = ABEGHJOP = ACDFGJKM = BDEFHNO = ACEFLM = BFGHJKLNO

= CDEGJLMNP = ABDHKLOP = BCDGJKMNO = ADEH = ABGFKMOP = EFGBJNP

= ABCDEPGJLMDP = DFHKLNP =BCELMNO = AGHJKL = ABDFtlH = CDEFJKMNO = BHJNP

= ACEGKMQP = BDEGHKLNP = AGDJLMOP = ABEFHJKL = CFGLMNO = DEFGLN
*
I (Continued) = ABCDFHJKLMO = AEJLP = BGGHKLMNOP - ADGKP = BCDEHJMNOP = FJKN

= ABCEFGHMO = ACDEGHJKLMO a BDLN a CEFHKLMOP = ABF3JLP = CDPSHJMNOP

= ABDEFKP = ACHMO « BEGJKN = ABCEHKLMP - GJLNOP = BCDEFGHJKLM = ADFLO

= BCFHMN = AEFGJKO = ABCDGHJMP = DEKNOP = BEFJLNOP * ACFGHKLMP = ABDEGLO

= CDHJKLMN = ABJKO = CEGHMN = BDFGKNOP - ACDEFHJMP = EGKLMNO = ABCHJL

a ADEFJKLMOP = BCDFGHLNP = AFGMOP a BCEFHJKNP = BJMNO - ABCDEGHK

= ACEFGHJL = BFKLMO - CDEHLNP » ABDGJKLMDP = CGHJKNP - ABEMOP - ACDFHK

= BDEFGJMO = ABCDEFHLOP = DFGJKLMNP a BCEGHJLNO - AKLM = BCDHKNO

- ADEGJM = ABCFGHJKOP a EFMNP - BDEJKLMNP «= ACBGHLOP - ABEFGKLM

a CFHJLNO a ABDFJM a CDEFGHKNO a BGMNP a ACEHJKOP a ADHJKMP a BGDEGNOP

FGHKMN = ABCEFJO « DEFHJLMN a ABCDFGKLO « AEGHLMP a BCJKLNOP a CDFNOP

= ABDEFGHJKMP = ACGJO = BEHKMf = ACDEKLO - BDGHJLM =CEFGJKLNOP

« ABFHLMP » BCFGJN a AEFHKMD - ABCDP - DBGHJKMNOP - ABCEGJKLP - HLMNOP

= BCDEFKLN = ADFGHJLMO = ABGHKMO a CEJN a BDFHJKMMOP - ACDEFGP

a BEFGHLMNOP = ACFJKLP = ABDEHJLMO a CDGKLN*

Block confounding? AK5 EK,AE5 ACEK5 CE5 AC3 CK^BC? ABCK? BCEKj) ABCE,ABEK5 BE5 ABJ BK0

Without blocks? All main effects and all two-factor interactions except

the following are measurable?

ad*ae5 ah5 ak5 al5 am5

BC s BD5 BEj BF5 BG^ BKs BL 3 BN5 CEj CF5 CG 5 CJ 9

ck5 cn5 de,dh5 dl5 dn,

EF5 BG 5 EH5 EJ s EK5 EN5

FG5 FJ 5 FK5 FN*GH5 GJ s GK5 GM5 HJ 5 HMs JK5 JM5 JN*KL*KM,KN*LM,LN0

With blocks: Same as above s except that the following two-factor interactions also are not measurable?

- 53 -

Blockss

1 2 3 k

(1)

. bdgho bcfjlm cdfghjlmo

abcdefghjkljflji

dhlmp

abcefgjknp

abcdefghko

jlmno

ab cefgklmop

dhjnop

£ 6 7 8

bcehjn cdegjno efhlmn bdefglmno

9 10 U 12

abcdfln acfghlno adjmn abghjmno

13 ik 15 16

adefbjl abefgjlo abcdehra acegmo

- 5U -

Plan 256.15 .16 , 1/256 replication of 15 factors in 8 blocks of 16 units each.

Factors; A5 B,C,D,E,F,G,H,J,K5 L,M,N3 0 5 P„

Is Same as plan 256.15*8.

Block confounding; AK^EK^AEjACEK^GE^AC^CK.

Without blocks: All main effects^ and all two“factor interactions except the

following are measurable:

AD5 AE,AH,AK5 AL5 AM5 BG jBD5 BE5 BF5 BG5 BK,

bl5 bn5 ge5 gf5 gg5 cj 5

ck5 cn3 de5 dh5 dl5 dn3 ef2 bg5 eh5 kj 5 ek5 en5

fg 5 fj 5 fk5 fn5 gh5 gj,

gk,gm,hj,hm,jk,jm,

jn5 kl5 km,kn5 lm„ln.

With blocks; Same as above except that the following two“fact or interactions

also are not measurable:

ag5 af„

Blocks:

1 2 3 h

(1)

abcdefghj klmn

dhlmp

bcfjlm bcehjn efhlran

abcefgjknp

abcdefghko

jlmno

abcefgklmop

dhjnop

bdgho

acefjklmho

bglmop

acdefhjknop

acefk

bdghjlran

acdefhklmp

bgjnp

5

abcdfln 6 1 8

adjmn adefhjl abcdehm

Plan 256.15.32. 1/256 replication of 15 factors In I4. blocks of 32 units each,

Factors: A,B,C,D,E,F,G,H,J ,K,L,M,N,0,Po

I: Same as plan 256.15*8.

Block confounding: AK,EK,AE.

Without blocks: All main effects, and all two~factor interactions except the

following are measurable:

AD,AE,AH,AK,AL,AM,

BC,BD,BE,BF,BG,BK,

BL,BN,CE,CF,CG,CJ,

CK,CN,DE,DH,DL5 PN,

EF,EG,EH,EJ,EK,EN,

FG,FJ,FK,FN,GH,GJ,

GK,GM,HJ,HM,JK,JM,

JN,KL,KM,KN,LM,LN.

With blocks: Same as above.

Blocks :

1 2 3 k

(1) bcehjn abcdfln adefhjl

abcdefghjklmn adeghkn

dhlmp bcdfhjp

abcefgjknp aegklmnp

abcdefghko adeghjklmo

jlmno bcfno

abcefgklmop aegjkop

dig nop bcdfhlmnop

bdgho cdfghjlmo

acefjklmno abekno

bglmop cf ''iop

acdefhj knop r\'y diklranop

acefk abe.-jklm

bdghjlmn cdf >hn

acdefhklmp abdohj 1. p bgjnp cfglmnp

Plan 256.l5.6Iu 1/256 replication of 15 factors in 2 blocks of 6U units each,

Factors: A,B,C,D,E,F,G,H,J,K,L,M,N,0,P.

I: Same as plan 256. 15.8.

Block confounding : AK.

Without blocks: All main effects, and all two-factor interactions except the

following are measurable:

AD,AE,AH,AK,AL,AM,

BC,BD,BE,BF,BG,BK,

BL, BN, CE, CF, CG, CJ ,

CK, GN, DE, DH, DL, DN,

EF,EG,EH,EJ,EK,EN,

FG,FJ,FK,FN,GH,GJ,

GK,GM,HJ,HM,JK,JM,

JN,KL,KM,KN,LM,LN.

With blocks: Same as above.

Blocks:

1

(1) bcfjlm bcehjn eflilmn

abcdefghjklmn adeghkn adfgklm abcdgjk

dhlmp bcdfhjp bcdejlmrp defnp

abcefgjknp aegklmnp afghkp abcghj klmp

abcdefghko adeghj klmo adfgjkno abcdgklmno

jlmno bcfno bcehlmo efh.j o abcefgklmop aegjkop afghjklmnop abcghknop

dig nop bcdfhlmnop bcdeop defjlmop

bdgho cdfghjlmo cdegjno bdefglmno

acefjklmno abekno * abfhklmo achj ko

bglmop cfgjop ceghjlranop befghnop

acdefhjknop abdehklmnop abdfkop -cdjklmop

acefk abej klm abfhjkn v.chklmn

bdghjlmn cdfghn cdeglm biefgj

acdefhklmp abdehj k

abdfjklmnp acdknp

bgjnp cfglmnp ceghp befghjlmp

2

abcdfln
.
- 58 -

Plan 256, 1608 . 1/256 replication of 16 factors in 32 blocks of 8 units each„

Factors? AJ B,C,D5 E,F5 G,H,JjK,L,M.> N,,0 5 Pi,S o

I - ABCEGEJKMOS = ADFGJNP = BCDEFHKMOPS = AEFKLNPS = BCFGEJLMOP = DEGJKLS

= ABCDHLMNO = ACFEJKMO = BEFGS = CDGHKMOP - ABDEJNPS - CEHJLMOPS = ABGKLNP

= ACDEFGHLMNOS = BDFJKL = ABCDFGHKMNP = DEFJ0P3 « BCHJKM o AEGNOS = BGDEGHLMS

= ADJKLNO = ABCEFHJ LMNPS = FGKLOP = BDGJOP = ACDEHKMNPS = ABFNO « CEFGEJKMS

* ABDEFGJKLNOS = GDFHLM = BEKLOPS - AGGEJLMNP - ADEGHJLNOP = BGDKLMPS - EFHLO

= ABGFGJKLHNS = DF3HJK0S = ABCDEFMN - AHKNOPS - BCEJSJMP - CDEFGKLMP

= ABDFHJLNOPS = ACEJKLMN - BGHLOS - AGDGMNS - BDEHJKO - CFJHPS - ABEFGHKNOP

= BCEFJKLMO = AFGHLNS = ABCDEGKLMNOP DHJLPS - ABCJMOPS = EGHKP

= BCDFGMOS = ADEFEJKN = ABEHLN = CGJKLMOS - BDEFGEJLP - ACDFKLMNOPS

= BFHKPS = ACEFGJMNOP = ABDGHJKNS « CDEMO * AEFGHJKLMP = BCFLOPS - DEHKLM

= ABCDGJ LNOS = GEJMS = ABCEKNO = ADFHMNPS - BGDEFGJKOP - CEGLOP

= ABEJKLMNPS = AGDEFJLNO = BDFGHKLMS - AGFGKNOS = BEFHJM = GDJKOPS

= ABDEGHMNP = BCDEJL = ADGHKLMOS * ABCEFGLNP = FHJKLMOPS = ABCDFJKNPS

= DEFGHMOP « BGGKS = AEHJMNO - ABDEFHKLMNO « CDF3JLS - BEGHJKLMOP = ACLNPS

= BDHMOPS = ACDEGJKNP - ABFGHJMOS » CEFK = DFKMO - ABCDEEGHJNS = AGJKMNOP

= BCEHPS = ADELMOPS = BGDGHJKLP = EFGJLMQS = ABCFHKLN « AGDHJN = BDEGKMGS

= CFGHP = ABEFJKMOPS = GDEFHJKLPS - ABDFGLMOP = ACEGHKLNS - BJLMO

= ABCGHNOP = EJKMPS = BCDFHJO = ADEFGKMS - BCEFGHKLOS = AFJLMN

= ABGDEHJKLNOPS = DGLMP = BFGJKMP = ACEFHNOPS = ABDKMN = CDEGHJOS

= ABEGJLMNS = CHKLO = BDEFLMPS = ACDFGHJKLNOP = AFHJOP * BCEFGKMNPS = DGHNO

= ABCDEJKMS = EHJKLNOS = ABCGLM = ADEPGHKLOPS = BCDFJLMNP = CKMNP = ABEGEJOPS

= AGDFGJKM = BDEFHNOS = ACEFLMS = BFGEJKLNO * CDEGJ LMNPS = ABDHKLOP

= BCDGJKMNO = ADEHS = ABCFKMOP = EFGHJNPS = ABCDEPGJ LMOPS = EFHKLNP

* BCELMNOS = AGEJKL = ABDFGH = CDEFJKMNOS = BEJ.NP = ACEGKMOPS » BDEGHKLNPS

= AGDJLMOP = ABEFEJKLS = CFGLMNO = DEFGLN = ABCDFEJKLMDS = AEJLP
*
- 59 ~

I (Continued) s BCGHKLMNOPS = ADGKPS - BCDEHJMNOP = FJKNS = .ABCEFGHMO

= ACDEGHJKLMO = BDLNS = CEFHKLMNOP » ABEGJLPS - CDFGHJMNOPS = ABDEFKP

= ACHMOS = BEGJKN = ABCEHKLMP = GJLNOPS - BCDEFGHJKLMN = ADFLOS = BCFHMNS

= AEFGJKO = ABCDGHJMPS = DEKNOP = BEFJLNOP » AGFGHKLMPS = ABDEGLO

= CDHJKLMNS = ABJKOS = CEGHMN = BDFGKNOPS = ACDEEHJMP = EGKLMNO = ABCBJLS

= ADEFJKLMOP = BCDFGHLNPS = AFGMOPS = BCEFHJKNP = DJMOS = ABCDEGHK

= ACEFGHJ L = BFKLMNOS * CDEHLNP = ABDGJKLMOPS » GGHJKNPS - ABEMOP = ACDFHKS

= BDEFGJMNO = ABCDEFHLOP ---- DFGJKLMNPS - BGEGBJLNO - AKLMS BCDHKNOS

= ADEGJM = ABCFGHJKOPS = EFMNP -BDEJKLMNP - ACDGHLOPS - ABEFGKLM = CFBJLNOS

= ABDFJMS = GDEFGHKNO = BGMNPS = ACEHJKOP * ADHJKMP = BCDEGNOPS - FGHKMN

= ABCEFJOS = DEFHJLMNS = ABGDFGKLO - AEGHLMPS « BCJKLNOP - CDFNOP

= ABDEFGHJKMPS = AGGJO = BEHKMNS - ACDEKLQS = BDGHJLM - CEFGJKLNOPS

= ABFHLMP = BCFGJN = AEFHKMOS * ABCDP = DEGHJKMOPS = ABCEGJKLPS - HLMNOP

= BCDEFKLNS = ADFGHJLMO = ABGHKMO = CEJNS = BDFHJKMNOP = AGDEFGPS

= BEFGHLMNOPS = ACFJKLP = ABDEHJ LMOS - CDGKLN,

Block confounding : CE, CF, EF, DFG, CDEFG, CDG,BEG , OS , CEOS , GFOS, EFOS, DFGOS,

CDEFGOS, CDGOS , DEGOS, LM, CELM, CFLM, EFLM, DFGLM, CDEFGLM, GDGLM, DEGLM, LMOS, GELMOS,

CFLMOS , EFLMOS , DFGLMOS , CDEFGLMOS, CDGLMOS , DEGLMOS

Without blocks: All main effects, and all two-factor interactions except

the following are measurable:

CE,CF,CK,EF,EK,FK c

With blocks: Same as above, except that the following two -factor interactions

also are not measurable:

BN, CP,DH, EP,FP,GJ ,KP,LM,0S o
-
- 60 -

Blocks;

1 2

(1) bcefghkl

bdghjlmn

cefgjklmops

bcdefhknops

acdefhklmp

abcefgjknp

adghjos

ablmnos

5 6

ab c defghj klmn adjim

9

cdeglm

10

bdfhkm

13

abfhjkn

ik

acegjln

17

cfglranp

18

behkmnp

21

abdehjkp

22

acdfgjlp

25

defnp

26

bcdghklnp

29

abcghjklmp aefjmp

3

jlSio

k

bcefghgkrano

1

abcdefghko

8

adlo

11

cdegjno

12

bdfhjklno

15

abfhklmo

16

acegrao

cig,]

M

op

20

bebjklop

23

ab dehklranop

ik

acdfgmnop

n

defjlraop

28

bcdghjkmop

31

abcghknop

32

aeflnop

- 61 -

Plan 256. 16, 16 . 1/256 replication of 16 factors in 16 'blocks of 16 maits each.

Factors % J^KjLjM^N^O^PjS,,

Is Same as plan 256.16,8,

Block confounding! CE, CF^EF^DFG^CDEFG^ CDG^ DEGS OS s CEOS s CFOS ^EFOS^DFGOg^

CDEFGOS , CDGOS , DEGOS

Without blocks: All main effects 5 and all two=factor interactions except the

following are measurable 5

CE S CF^fH^EF^EK^FK,

With blocks s Same as above5 except the two^faotor interaction OS als© is

not measurable.

Blocks g

1

(T)

bdghjlmn

cefgjklmops

bcdefhknops

acdefhklmp

abeefgjknp

adghjos

ablnmos

bcefghkl

cdef jlaisi

bhjmops

dglnops

abdgmp

ahjlnp

abcdefjklos

acefghkmnos

2 1 k

jlmno ab cdefghjklmn abcdeTghko

s 6 . 7

odeglm cdegjno abf'Ejkn

8 9 10

abfhklmo cfgTamp ofgjop

11 12 13

abdeKjkp abdel2cirnnop deTnjp

1h a IS

defjlmop abcghjklmp abcghkuop
'
- 62 »

Plan 256,16,32., 1/256 replication of 16 factors in 8 blocks of 32 units each.

Factors g A,B,€,D,E,F,G,HJ JjKjL^MjN^O^P^S,

I? Same as plan 256,16,8,

Block confounding? CE,'OF,EF,DFG, CDEFG, CDG,DEG,

Without blooks ? All main effects, and all two«=factor interactions except

the following are fflaagUJ’ables

CE,CF,CK,EF,EK,FK,

With blocks g Same as above.

Blocks g

1

(l)

bdghjlmn

oefgjklraops

bodefhlmops

acdefhklmp

abeefgjknp

adghjos

ablmnos

bcefghkl

®lef jkmn

bhjmops

dglnops

abdgmp

ahjlnp

abcdefjklos

acefghlonnos

jlm©

bdgh©

cefgkaps

bcdefhjklmps

&ed©fh$kn©p

abcefgklmop

adghlims

abjs

bcefghjkaoio

cdefkl©

bhlnpg

dgjmps

abdgjlnop

ataaop

abcdefkmns

aoefghjkls

2

abcdefghjklmn

3

©deglm

h

abfhjkn

cfgTmnp A

6

abdeEjkp

7 8

deFnp a,b ogJijklmp

- 63 -

Plan 256,16.6k. l/2$6 replication of 16 factors in 1* blocks of SU units each.

Factors s A,B i) C,D,E> F,G,H, J^K^L^M^N^O^PpS,,

Is Sam© as plan 256. 16. 8.

Block confounding s CE,CF? EF.

Without blocks? All main effects s and all two=factor interactions except

the following are measurable?

CE5 CF^ CK^EF^EK^FK,,

With blocks s Same as above.

Blocks g 1

(X) jlmno ab©defghjklm abodefghko

bdghjlmn bdgho aoefk ac@fjklmo

cefgjklmops ©efgknps abdhaops abdhjlmpg

bcdefhknops bcdefhjklmps agjlmops agaps

acdefhklmp aedefhjkaop bgjnp bglmop

abcefgjknp abcefgklmop dklmp dhjnop

adghjos adghlmns boefklmaos beef jks

ablmnos abjs ©defghjkos cdefghk Inns

bcefghkl beefghjbmo adjam adlo

cdef jkmn cdefklo abghl abghjnmo

bhjmops bhlnps aedefgklnops acdefgjkmps

dglnops dgjmps abcefhjkmops aboefhklnps

abdgmp abdgjlnop cefhjklnp cefhkmop

ahjlnp ahmop bcd@i^Joup bcdeffejklnop

abQdefjklos abed©flams ghmios ghjls

acefghkmnos acefghjkls bdjlos bdmns

2 3 h

edeglm efgTmp deUkp

- 6b -

Plan 256»16»128 » 1/256 replication of 16 factors in 2 blocks of 123 units each*

Factors : A, B, C, D, E,F, G,H, J^K,LS M,N, 0 , P, S

Is Same as plan 256*16,8*

HLock confounding: CE,

Without blocks: All main effects, and all two-factor interactions except the

following are measurable:

CE,CF,CK,EF,EK,FK0

With blocks: Same as above*

Blocks:

(1) abcdefghj klmn cdeglm abfhjkn

bdghjlmn acefk bcehjn adfgklm

cefg.jklmops abdhnops dfj kops abceghlmnops

bcdefhknops agjlmops bfghklmnops aedej ops

acdefhklmp bgjnp afghkp bcdejlmnp

abcefgjknp dhlmp abdfjklmnp ceghp

adghjos bcefklmnos acehjlmos bdfgknos

ablmnos cdefghj kos abedegnos fhj klmos

bcefghkl adjmn bdfhkm aeegjin

cdefjlatm abghl fgjkln abedehm

bhjmops acdefgklnops bedsghjlops afkmnops

dglnops abcefhj kmops cemnops abdfghjklops

abdgmp cefhj klnp abeelp dfghjkmnp

ahjlnp bcdefgkmp acdeghjmnp bfklp

abcdefjklos ghmnos abfgjkmos cdehlnos

acefghkmnos bdjlos adfhklnos bcegjmos

jlmno abcdefghko edegjno abfhklmo

bdgho acefj klmno bcehlmo adfgjkno

cefgknps abdhjlmps dfklmnps abceghjps

bcdefhjklmps agnps bfghjkps acdelmnps

acdefhjknop bglmop afghjklmnop bedeop

abcefgklmop dhjnop abdfkop ceghjlmnop

adghlmns beefj ks acehns bdfgj klms

abjs cdefghlclmns abodegj1ms fhkns

bcefghjkmno adlo bdfhjkLno acegmo

cdefklo abghjmno fgkmo abedehjino

bhlnps acdefgjkmps bcdeghmnps afj kips

dgjmps abcefhklnps cejlps abdfghkmnps

abdgjlnop cefhkmop abcejmnop dfghklop

ahmop bcdefgjklnop aedeghlop bfjkmnop

abcdefkmns ghjls abfgklns cdehjms

acefghjkls bdmns adfhjkms bceglns

2

cfglmnp
'

.
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.25) and its Supplement ($0.75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.
f.