SJ=rJ^ FiL

NATIONAL BUREAU OF STANDARDS REPORT

3481

SOME FRACTIONAL FACTORIAL ARRANGEMENTS

FOR FACTORS AT TV/O LEVELS

by

W. H. Clatworthy

W. S. Connor

M. Zelen

<NBS>

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

cover of this report.

Electricity. Resistance and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines

and Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid

State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy. Nuclear Physics. Radioactivity. X-Ray. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Radiation Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry.

Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass

and Scale. Capacity, Density, and Fluid Meters. Combustion Control.

Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery . Glass. Refractories. Enameled Metals. Concreting

Materials. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation. Process Technology.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Frequency Utilization Research. Tropospheric Propagation Research. High Frequency

Standards. Microwave Standards.

^Office of Basic Instrumentation ^Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-40-5118/^2-1 28 July 1954 3481

SOME FRACTIONAL FACTORIAL ARRANGEMENTS

FOR FACTORS AT TWO LEVELS

by

W. H. Cl atworthy

W. S. Connor

M. Zelen

Statistical Engineering Laboratory

<5gs>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, rep Approved for public release by the

unless permission is Director of the National Institute of

25 ,d.c. Suchperr Standards and Technology (NIST)

cal ly prepared If th on October 9, 2015

n part, is prohibited

indards, Washington

iort has been specifiport for its own use,
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FOREWORD

This is a technical report on Research in Applications

of Mathematical Statistics to Problems of the Chemical

Corps, for the Biological Laboratories, Chemical Corps,

U. S, Department of the Army carried out in the Statistical

Engineering Laboratory ( NBS Project Number 1103-40-5118/52-1).

o « F .L . A1

1

Acting Chief,

National Applied Mathematics

Laboratories

A.V. Astin

Director,

National Bureau of Standards

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TABLE OF CONTEM'S

Introduction

Bibliography

Classification of factorial arrangements according to

number of factors, fractional replication, and

number of experimental units per block

Plan 5 » 2 .Il: measurements required 16 • • 9 9 • 9

11 5. 2.8 It 16 • * * 9 • 9

ti 6 . 2.4 It 32

it 6 . 2.8 11 32

it 6.2.16 " It 32 * • 9 9 • 9

it 7.2.4 It 64 % « 1 9 * »

tt 7.2.8 It 64 * * -» 9 9 *

it 7.2.16 " It 64 • 9 9 9 »

tt 7 . 2.32 "

It 6k t ¥ 9 9 * •

tt 8 . 2.8 "

It 128 6 * • <9 9

tt 8.2.16 " tt 126 9 8- ¥ a

it 8 . 2.32 "

It 128 • » • 9 9 *

it 8.2.64 " tt 128 • • • 9 9 •

tt 9.2.8 tt 256 • % • 9 • 9

tt 9.2.16 " tt 256 • • 9 • •

tt 9.2.32 " It 256 9 • 9 9 9 •

tt 6.4.2 " tt 16 9 • 9 9 9 •

it 6.4.4 It 16 -«* * 9 9 9 a

it 6 . 4.8 It 16 * 9 9 9 9

it 7.4.4 II 32 4 * ¥ 9 % it 7.4.8 II 32 ¥ V » 6 0 9

it 7.4.16 tt 32 • 9 • 9 » 9

it 8.4.4 " II 64

tt 8.4.8 II 64 • • • 9 • •

it 8.[|..16 " It 64

it 8.4.32 " II 64 9 * 9 9 9

it 9.4.8 II 128 % * 9 # 9

it 9.4.16 II 128 9 • # 9 't «

tt 9.4.32 It 128 9 a # 9 9 9

11 10.4.8 " It 256 •t # & 9 9 9

11 10.4,16 " It 256 « * 9 9 9 9

11 10.4.32 " tt 256 * a 9 9 9

page

1

6

7

8

9

10

11

12

13

14

15

16

18

20 • 21

22

24

25

26

27

28

29

30

31

32

33

34

35

36

38

39

40

41

42

ii

Plan 8.8.4, measurements

" 8 . 8.8 " " 8 . 8.16 "

" 9 . 8 . 1; " 9 . 8.8 " " 9 . 8.16 "

" 9.8.32 " "10.8.3 "

" 10 , 8.16 " " 10,8.32 " " 11 . 8.8 "

" 11 . 8.16 " " 11.8.32 " " 7.16.1;

" 7.16.8 "

" 8 . 16 . 1; "

" 8 . 16.8 "

" 8 . 16.16

" 9 . 16 . 1; " 9 . 16,8

" 9 . 16 .16

" 10 . 16.8

" 10 . 16.16

" 11 . 16.8

" 11 . 16.16

" 12 . 16.8

" 12 . 16.16

" 10.32.U

" 10.32.8

" 10 , 32,16

" 11.32.8

" 11 . 32.16

" 11.32.32

" 12.32.8

" 12 . 32.16

" 12.32.32

required.

m

32 9 9 • 9 9 c 43

32 • • % 9 9 9 44

32 • • « 9 • 9 43

64 46

64 48

64 • • • • • « 49

64 • • 9 # « 50

128 % 9 9 ar 51

128 9 9 9 # 9 33

128 •t 9 « -» * 54

236 « 9 m « 9 « 33

256 * 9 • 9 • 37

256 • • m * * • 38

8 39

8 * 9 9 9 60

16 • # * * 9 9 61

16 • • 9 9 * « 62

16 63

32 64

32 9 • m 9 9 9 63

32 * * 9 e 9 9 66

64 9 9 9 « 67

6!l 9 9 9 9 68

128 • 9 9 » 9 9 69

128 • 9 9 9 9 71

236 9 9 • • 9 • 72

256 9 9 9 9 9 74

32 9 9 m • • 9 73

32 9 9 9 9 • 9 76

32 •m m 9 9 • « 77

6If. t « • 9 9 # 78

64 m 9 9 9 9 9 80

6k m 9 9 • # 9 81

128 9 9 9 9 • 9, 82

128 * « 9 9 9 • 84

128 * 83

111

Some Fractions! factorial Arrangements for

Factors at Two Levels

by

W.H. Clatworthy, W.S. Connor, and M. Zelen

1, Introduct ion « This report contains 68 experimental

arrangements or plans for fractional replication of factorial

designs of the 2

n

series, where n, the number of factors, ranges

from 5 bo 12 and each factor is at two levels. The plans include

arrangements which are 1/2, l/k, l/8, l/l6, and 1/32 of a full

replication and which involve 8, 16, 32 , 61j.,,128, and 2.56 measurements. With but two exceptions the experimental units have

been assigned to blocks*

A bibliography which is not exhaustive is given at the

end of this introduction* Several of the designs given in this

report may be found in these references. However this report

represents an independent effort.

A brief description of the plans follows. Each plan has

a designation of the form

Plan n.r.k

which refers to the l/r replication of the 2

n

factorial design

with 2

n/rk incomplete blocks of k experimental units each.

For example. Plan 7*2.k refers to the 1/2 replicate of the 2^

factorial design with 16 blocks of k experimental units each.
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2

In all the designs the n factors, each at two levels, are

designated by capital letters A, B, C, etc* ' Sra&ll * letters refer to assignment of the factors to the experimental

units, the presence of small letter x meaning that the high

level of factor X is applied and the absence of small letter x

meaning that the low level of factor X is applied. For example,

in Plan 5*2*4 the five factors are A, B, C, D, and E. In block

1 of this design one of the experimental units receives the

treatment combination de , i.e», low levels of factors A, B,

and C and high levels of factors D and E are applied to the

experimental unit. The symbol "(l) 11 is also found in block 1,

indicating that low levels of all factors of the design are

applied to one experimental unit of block 1„

A main effect or interaction is said to be estimable if it

is confounded only with higher order interactions. In any plan

in which a main effect is confounded with two-factor interactions,

it is explicitly stated which two-factor interactions are involved.

For each design there is given a fundamental confounding

relationship composed of groups of capital letters (representing

main effects and interactions)- connected with equal signs.

The equal signs are read "is confounded with". The fundamental

confounding relationship is useful in determining how the main

effects and interactions are confounded with each other as a
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result of the design being a fractional part of a complete

replication. For example, in Plan 6 .Ip .Ip the fundamental confounding relationship is

I = ABCE = ABDF = CDEF .

To determine how the main effects and interactions are confounded

by the 1/Ip replication of the full 2 factorial, each term of

the fundamental confounding relationship is multiplied by the

main effect or interaction in quest ion. with the understanding

2

that X = 1 where X = A, B, C, etc, and I(X.,,Z) = X ... Z.

For example, the confounding of main effect A is given by

A = BCE = BDF = ACDEF

Note that multiplication by BCE, BDF, and ACDEF give exactly the

same result, except for order of the groups. The other confounding

relationships are:

B = ACE = ADF = BCDEF

C = ABE = ABCDF = DEF

D = ABCDE = ABF = CEF

E = ABC = ABDEF = CDF

F = ABCEF = ABD = CDE

AB = CE = DF = ABCDEF
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AC = BE = BCDP = ADEP

AD = BCDE = BP = ACEF

AE = BC = BDEF * ACDF

AP = BCEP = BD = ACDE

CD = ABDE = ABCF = EP

CP = ABEP = ABCD = DE

ACD = BDE = BCF = AEP

ACF = BEP = BCD = ADE

The treatment combinations were assigned to the blocks so

as to confound interactions AB, BC, and AC with the blocks. If

b is the number of blocks, then certain b-1 of the confounding

relationships designate which effects cannot be estimated because

of block confounding. These are listed for each design as

"Block confounding" wherein each group of letters designates

a confounding relationship*

Examination of the totality of confounding relationships

for Plan reveals that the main effects A, B, C, D, E, and

F are estimable, that the two-factor interactions which are not

already confounded with blocks are confounded with other twofactor interactions and hence are not estimable, and that no

three-factor interaction is estimable. Thus in this design

it is impossible to estimate any interaction unless one is prepared

to assume that some other interact ion( s) of the same order is(are)

negl igible.
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In some of the plans where the number of treatment combinations is large, the blocks are not written out in full*

For such designs only the initial block is recorded in detail

and one treatment combination is given for each of the remaining

blocks. The other blocks are obtained by successively multiplying

the treatment combinations in the initial block by each of the

2 given treatment combinations with the proviso that x = 1,

(x = a,b,c, etc.). For example, Plan 10.ip. 8 is such a design.

The initial block consists of the followihg eight treatment

combinations

(1), abed, efhj, abcdefhj, eghk, abedeghk,

fgjk, abcdfgjk.

To obtain the treatment combinations for block 2, say, one

multiplies these eight groups by ab (the treatment combination

listed for block 2), obtaining

ab, cd, abefhj, cdefhj, abeghk, edeghk,

abfgjk, cdfgjk.

Although the plans in this report, have been carefully

checked, there is always the possibility of errors. The authors

would appreciate hearing from anyone discovering errors.

The authors wish to express their thanks to Mrs. Lola

Deming for checking the plans and to Mrs. Yvette Cocozzella who

did the typing*
.

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BIBLIOGRAPHY

[1] Brownlee, K.A., Kelly, B.K., and Loraine, P.K., ’’Fractional

replication arrangements for factorial experiments

with factors at two ' levels” , Biometrika , Vol. XXXV

( 194$ ) , pp. 268-282.

[2] Cochran, W.G. and Cox, G.M., Experimental Designs . John

Wiley and Sons, Inc., New Y0rk", 1950. !

[3] Finney, D.J., ” The fractional replication of factorial

arrangements". Annals of Eugenics . Vol. 12(1945),

pp. 291-301.

[41 Finney, D.J., 11 Recent developments in the design of field

experiments. III. Fractional replication". Journal ' of Agricultural Science . Vol. 38 (1948), pp. 184-191*

[5] Kempthorne, 0., "A simple approach to confounding and

and fractional replication in - factorial experiments",

Biometrika, Vol. 34 (1947) pp. 255-272.

[6] Kempthorne, 0., The Design and Analysis of Experiments . John Wiley and Sons, Inc., New York, 1952.

[7] Kitagawa, Tosio and Mltome, Michiwo, Tables for the Design

of Factorial Experiments, Baifukan Co., Ltd., Tokyo,

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Plan g,2,lj. , factors, l/2 replication, 1| blocks of 1| units

each.

Factors: A,B,C,D,E

I = ABODE

Block confounding: AB, AC, BC.

All main effects and all two-factor interactions except AB,

AC, and BC are estimable.

Blocks

I 2 1 k

(1) ab ac ad

abed cd bd be

de abde aede ae

abce ce be bede

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Plan 5.2.8 . 5 factors, 1/2 replication, 2 blocks of 8 units

each.

Factors: A,B,C,D,E

I = ABODE

Block confounding: AB

All main effects and all two-factor interactions except AB are

estimable

.

Blocks

1 2

(1) ac

abed bd

de aede

abce be

ab be

cd ad

abde bede

ce ae
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Plan 6.2.lu 6 factors, 1/2 replication, 8 blocks of I|. units

each.

Factors: A,B,C,D,E,F

I = ABCDEF

Block confounding: ABF, ACF, BC, ABE, EF, BCEF, ACE.

All main effects, all two-factor interactions except AD, BC,

and EF are estimable. Twelve of the three-factor interactions

are confounded in pairs as follows;

ABC = DEF ADE = BCF

ABD = CEF ADF = BCE

ACD = BEF AEF = BCD , while

with

the remaining eight three

blocks*

-factor interactions are confounded

Blocks

1 2 i it £ 6 1

8

(i) ab ac be ae be ce abce

abed cd bd ad bede aede abde de

beef acef abef ef abef cf bf af

adef bdef edef abodef df abdf aedf bedf

- 10

Plan 6,2,8 , 6 factors, 1/2 replication, I4

. blocks of 8 units

each*

Factors: A,B,C,D,E,F

I = ABCDEF

Block confounding: ABF, ACF, BC

All main effects and all two-factor interactions except BC are

estimable* Sixteen of the twenty three-factor interactions

are confounded in pairs as follows:

ABC = DEF ACE = BDF

ABD = CEF ADE = BCF

ABE = CDF ADF = BCE

ACD = BEF AEF = BCD , while the remaining four three-factor interactions are confounded

with blocks*

1

(1)

abed

beef

adef

abce

de

af

Blocks

2 1 k

ab ac be c cd bd ad

acef abef ef

bdef edef abedef

ce be ae

abde aede bede

bf cf abef

bedf aedf abdf df
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Plan 6*2*16 * 6 factors, 1/2 replication, 2 blocks of 16 units

each*

Factors: A,B,C,D,E,F

I = ABCDEF

Block confounding: ABF

All main effects and all two-factor interactions are estimable.

The three-factor interactions ABF and CDE are confounded with

blocks while all others are confounded in pairs as follows:

ABC = DEF ACD = BEF ADS = BCF

ABD = CEF ACE = BDF ADF = BCE

ABE = CDF ACF = BDE AEF = BCD .

Blocks

1

(1) ab

abed cd

beef acef

adef bdef

abce ce

de abde

af bf

bedf aedf

2

ac be

bd ad

abef ef

edef abedef

be ae

aede bede

cf abef

abdf df
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Plan 7 factors, 1/2 replication, 16 blocks of Ij. units

each.

Factors: A,B,C,D,E,F,G

I = ABCDEFG

Block confounding: ABFG, ACFG, BC, ABEG, EF, BCEF, ACEG, ABEF,

EG, BCEG, ACEF, FG, AB, AC, BCFG

All main effects, 15 of the 21 two-factor interactions, and 26

of the 35 three-factor interactions are estimable. The twofactor interactions

AB, AC, BC, EF, EG, FG

and the three-factor interactions

ADE, ADF, ADG, BDE, BDF, BDG, CDE, CDF, CDG

are confounded with blocks.

Blocks

1 2 i k l 6

(i) ab ac be ae be

abed cd bd ad bede aede

defg abdefg aedefg bedefg ad-fg bdfg

ab c efg cef£ bef£ aefg befg acfg

i 8 9 10 11

ce abce af bf cf

abde de bedf aedf abdf

cdfg abedfg adeg bdeg edeg

abfg fg bceg aceg abeg

12 n Ik il 16

abef ef* abef acef beef

df abedef edef bdef adef

abedeg dg abdg aedg bedg

eg abeg eg *>g ag
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Plan 7*2,8 , 7 factors, 1/2 replication, 8 blocks of 8 units

each*

Factors: A,B,C,D,E,F,G

I = ABCDEFG

Block confounding: ABFG-, ACF, BCG, BCEF, ACEG, ABE, EFG

All main effects, all two-factor interactions, and all threefactor interactions except

ABE, ACF, ADG, BCG, BDF, CDE, and EFG

are estimable.

Blocks

1 2 3 k

(1) ab ac be

abed cd bd ad

beef acef abef ef

adef bdef edef abedef

cdfg abedfg adfg bdfg

abfg fg befg acfg

bdeg adeg abedeg edeg

aceg bceg eg abeg

1 ' 1 i 8

ae ce be abce

bede abde aede de

abef bf cf af

df aedf abdf bedf

aedefg defg bedefg abdefg

befg abcefg aefg cefg

abdg bedg dg aedg

eg ag abeg bg

Plan 7*2*16 , each.

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7 factors, 1/2 replication, I|. blocks of 16 units

Factors: A,B,C ,D,E,F,G

I = ABCDEFG

Block confounding: ABFG, aCF, BCG

All main effects, all two-factor interactions, and all threefactor interactions except ACF, BCG, and CDE are estimable.

Blocks

1 2

(1) abce ab ce

abed de cd abde

beef af acef bf

adef bedf bdef aedf

cdfg abdefg abedfg defg

abfg cefg fg abcefg

bdeg aedg adeg bedg

aceg bg bceg ag

3 h

ac be be ae

bd aede ad bede

abef cf ef abef

edef abdf abedef df

adfg bedefg bdfg aedefg

befg aefg aofg befg

abedeg dg edeg abdg

eg abeg abeg eg
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Flan 7 >2,32 , 7 factors, 1/2 replication, 2 blocks of 32 units

each.

Factors: A,B,C,D,E,F,Gr

I = ABCDEFG

Block confounding: ABFGAll main effects, all two-factor interactions and all threefactor interactions except CDE are estimable*

Blocks

1

m abce ab ce

abed de cd abde

beef af acef bf

adef bedf bdef aedf

cdfg abdefg abedfg defg

abfg cefg fg abcefg

bdeg aedg adeg bedg

aceg bg bceg ag

2

ac be be ae

bd aede ad bede

abef cf ef abef

edef abdf abedef df

adfg bedefg bdfg aedefg

befg aefg acfg befg

abedeg dg edeg abdg

eg abeg abeg eg

- 16 -

Plan 8,2,8 , 8 factors, 1/2 replication, 16 blocks of 8 units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCDEFGH

Block confounding: ABCD, ABCE, DE, ABDEG, CEG, CDG, ABG, BCH,

ADH, AEH, BCDEH, ACDEGH, BEGH, BDGH, ACGH.

All main effects, all two-factor interactions except BF and DE,

and all three-factor interactions except ABG, ADH, AEH, BCH,

CDG, and CEC are estimable. Hone of the four- factor interactions

is estimable.

Blocks

1 2 2 k £ 6

(1) ab ac be ad bd

abfh fh befh acfh bdfh adfh

bcfg acfg abfg fg abedfg cdfg

acgh bcgh gh abgh edgh abedgh

cdeh abcdeh adeh bdeh aceh bceh

abcdef cdef bdef adef beef acef

bdefgh adefgh abcdefgh cdefgh abefgh efgh

adeg bdeg edeg abedeg eg abeg

2 8 9 10 11

cd abed ae be ce

abcdfh cdfh befh aefh abcefh

bdfg adfg abcefg cefg befg

adgh bdgh cegh abcegh aegh

oh abeh aedh bedh dh

abef of bedf aedf abdf

bcefgh acefgh abdfgh dfgh be dfgh

aceg bceg dg abdg aedg
.

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Plan 8+2.8. (Continued).

Blocks

12 13 Ik 1

£

16

abce de abde acde bode

cefh abdefh defh bcdefh aedefh

aefg bcdefg acdefg abdefg defg

begh acdegh bcdegh degh abdegh

abdh ch abch ah bh

df abcf cf bf af

acdfgh bfgh afgh abcfgh cfgh

bcdg ag bg eg abeg
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Plan 8,2,16 , 8 factors, 1/2 replication, 8 blocks of 16 units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCDEFGH

Block confounding: ABCD, ABEF, CDEF, BCEG, ADEG, ACFG, BDFG*

All main effects, all two-factor interactions, and all threefactor interactions are estimable* Fourteen of the 70 fourfactor interactions are confounded with blocks while the remaining £6 are confounded in pairs.

Blocks

1 2 2 k

(1) ab ac be

abed cd bd ad

abef ef beef acef

edef abedef adef bdef

bceg aceg abeg eg

adeg bdeg edeg abedeg

acfg befg fg abfg

bdfg adfg abedfg cdfg

efgh abefgh acefgh bcefgh

abcdefgh cdefgh bdefgh adefgh

abgh gh begh aegh

edgh abedgh adgh bdgh

befh acfh abfh fh

adfh bdfh cdfh abedfh

aceh be eh eh abeh

bdeh adeh abedeh edeh
•' ' ' ... -V , " • • ; .
Plan 8

- 19 -

2,16, (Continued)

Blocks

6 i 8

ae be ce abce

bcde acde abde de

bf af abef cf

acdf bcdf df abdf

abcg eg bg ag

dg abdg aedg bedg

cefg abcefg aefg befg

abdefg defg bedefg aedefg

afgh bfgh cfgh abcfgh

bcdfgh acdfgh abdfgh dfgh

begh aegh abcegh cegh

acdegh bedegh degh abdegh

abcefh cefh befh aefh

defh abdefh aedefh be defh

ch abch. ah bh

abdh dh bedh aedh
. ' V . ' to • • ' . . ' .
- 20

Plan 8.2,32, 8 factors, 1/2 replication, 4 blocks of 32 units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCDEFGH

Block confounding: ABCD, ABEF, CDEF

All main effects, all two-factor interactions, and all threefactor interactions are estimable. Six of the 70 four«factor

interactions are confounded with blocks and the other 64 are

confounded in pairs.

Blocks

Combine blocks in Plan 8.2.16 as follows:

1 and 2

3 and 4

5 and 6

7 and 8 •
-
21

Plan 8*2*61i « 8 factors, l/2 replication, 2 blocks of 6 I

4

. units

each.

Factors: A,B ,C,D,E,F,(x,H

I = ABCDEFGH

Block confounding: ABCD

All main effects, all two-factor and all three-factor interactions are estimable* Two of the 70 four-factor interactions

are confounded with blocks while the remaining 68 are confounded

in pairs.

Blocks

Combine blocks in Plan 8. 2.16 as follows

1 , 2 , 3, and

5 ,

• 6, 7, and 8 .

22

Plan 9 .2«8 » 9 factors, l/2 replication, 32 blocks of 8 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCDEFGHJ

Block confounding: ABFGJ, ACFG, BCJ, ADG, BDFJ, CDF, ABCDGJ,

FGHJ, ABH, ACHJ, BCFGH, ADFHJ, BDGH, CDGHJ,

ABCDFH, ACE, BCEFGJT, EFG, ABEJ, CDEG, ABCDEFJ,

ADEF, BDEG J, ACEFGHJ, BCEH, EHJ, ABEFGH

,

CDEFHJ, ABCDEGH, ADEGHJ, BDEFH.

The main effects, 33 of the 38 two-factor interactions, 71 of

the 8 I

4

. three-factor interactions and 111 of the 126 four-factor

interactions are estimable. The following interactions are

confounded with blocks:

BD,

CDF,

ACFH

BDGH

FJ, GH, ABG, ABH, ACE, ADG, ADH, BCF, BCJ,

CDJ, EFG, EFH, EGJ, EHJ, ABEF, ABEJ, ACFG,

,,ACGJ, ACHJ, ADEF, ADEJ, BCEG, BCEH, BDFJ,

, CDEG, CDEH, FGHJ.
. ..•&

. .. , .

" 3

- :

:

... :

-

. P

. r

'

- ' •

*

’•

'

' • ' ‘ C'

: ' -

.

.

•

. . .

'

. ,

-

'

' J •
- 23 -

Plan 9,2*8 , (Continued).

Blocks

1. (Initial Block) Block Multipliers

(1) 2 3 k 2 6 2

abed

aegh

ab ac be ae be ce

be aegh

cef j

8 9 10 11 12 12

abdef j

acfghj

abce af bf cf abef ef

bdfghj

Ik 11 16 12 18 12

abef acef beef ag t>g eg

20 21 22 23 kk 22

abeg eg abeg aceg bceg fg

26 21 28 29 30 21

abfg acfg befg aefg befg oefg

32

abcef

g
'

;

’ '

'

. ; .

'

.

' v

•

• . . . r.
Plan 9*2.16 » 9 factors, 1/2 replication, 16 blocks of 16 units

each*

Factors: A,B,C,D,E,F,G,H,

J

I = ABCDEFGHJ

Block confounding: ABODE, ABCF, DEF, BDEFGH, ACFGH, ACDEGH,

BGH, BCEH, ADH, AEFH, BCDFH, CDFG, ABEFG, ABDG, CEG.

All main effects, all two-factor interactions, all three-factor

interactions except

AC J, ADH, BFJ, BGH, CEG, and DEF

and all four-factor interactions except

ABCF, ABDG, AEFH, AEGJ, BCEH, BDEJ , CDFG, CDHJ, FGHJ

istimable . Blocks

1 2 3 ii. 5_ _ 6

(i) befg ab ac be ad bd

acde abcdfg

adfj

cef J

abdeg j

beg j

8 9 10 11

abhj aefghj cd abed af bf cf

bcdehj cdfghj

bdfh

abcefh

degh

aegh 12 13- it 15 16

abef df abdf aedf bedf

- 25

Plan 9*2.32 . 9 factors, 1/2 replication, 8 blocks of 32 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCDEFGHJ

Block confounding: ABCD, CDEF, ABEF, ACEG, BDEG, ADFG, BCFG

The main effects, all two-factor interactions, all three-factor

interactions, and all four-factor interactions except those

mentioned above are estimable.

~

Blocks

1. (Initial Block) Block Multipliers

(1) hj 2. ab

abed abedh j 3. ac

abef abefhj 4. be

edef cdefhj 5. ae

bdeg bdeghj 6. be

aceg acep’bl 7. ce

adfg adfghj 8. abce

befg bcx&iij

bceh bee j

adeh ade j

acfh acf j

bdfh bdf J

edgh edg j

abgh abgj

abcdefgh abedefg

j

efgh efgj
v - - ... i

-•

.. ; . , - " • : ‘ • ' no • It t i : ' • :C .. c . *
26

Plan '6.1u2 . 6 factors, 1/lj. replication, 8 blocks of 2 units

each*

Factors: A,B,C,D,E,F

I = ABCE = ABDF = CDEF

Block confounding: AB, BC, AC, EF, ABEF, BCEF, ACEF

All main effects are estimable. All two-factor interactions

have two-and four-factor interactions as aliases. Threefactor interactions are aliases of main effects or of other

three-factor interactions:

ACD = BDE = BCF = AEF

ACF = BEF = BCD = ADE .

Blocks

1 2 i k

(1) ab ce abce

abcdef cdef abdf df

i 6 I 8

acd bed ade bde

bef aef bef aef
' ' V

) .

. :• .y f. y X

-

’ " l:

-

' ' ••••* •

•

. • •• ' ...

:
27

Plan 6»lulu 6 factors, l/lj. replication, Ij. blocks of Ij. units

each.

Factors: A,B,C,D,E,F

I = ABCE = ABDF = CDEF

Block confounding: AB, BG, AC

The main effects are estimable, but all two factor-interactions

have two- and four-factor aliases while all three-factor interactions are aliases of either main effects or three-factor

int er actions

.

Blocks

1 • 2 1 Jl

(1) ab acd bed

abcdef cdef bef aef

abce ce bde ade

df abdf acf bef

28

Plan 6«lu8 « 6 factors, l/Ij. replication, 2 blocks of 8 units

each.

Factors: A,B,C,D,E,F

I = ABCE = ABDF = CDEF

Block confounding: AB

The main effects are estimable. All two-factor interactions

have two- and four-factor aliases while all three-factor interactions are aliases of either main effects or three-factor

interactions

Blocks

1 2

(1) acd

abcdef bef

abce bde

df acf

ab bed

cdef aef

ce ade

abdf bef
- -I

• •

' • ‘

-

. .

' - .
29

Plan 7 factors, 1/4 replication, 8 blocks of 4 units

each.

Factors: A,B,C,D,E,F,G

I = ABCE = ABDFG = CDEFG

Block confounding: ACD, BEF, ABCDEF, BC, ABD, CEF, ADEF

The main effects, 12 of the 21 two-factor interactions, and

5 of the higher order interactions are estimable. The following

two-factor interactions are estimable:

AD, AF, AG, BD, BF, BG, CD, CF, CG, DE, EF, EG,

Blocks

1 2 1 k 1 ..

_ 6 i 8

(1) ab df abdf dg abdg fg abfg

abce ce abcdef edef abedeg edeg abcefg cefg

adefg bdefg aeg beg aef bef ade bde

bcdfg acdfg beg aeg bef acf bed acd
; • ' - .

• 4 - —

•

. r; :

. ,

.

=

.

,• ' l [

! ,

• ••
- 30 -

Plan 7A«8 « 7 factors, l/[|. replication, I4

. blocks of 8 units

each.

Factors: A,B,C,D,E,F,G

I = ABCEG = ABDF = CDEFG

Block confounding: AB, AD, BD

The main effects and all of the two-factor interactions except

(for those already confounded in pairs by the fundamental

identity)

AB, AD, AF, BD, BF, DF

are estirrable. Six of the three-factor interactions have as

aliases six other three-factor interactions*

Blocks

1 2 3

( 1 ) ab bed acd

ce abce bde ade

eg abeg bdg adg

eg abeg bedeg aedeg

abdf df acf bef

abedef edef aef bef

abedfg cdfg afg bfg

abdefg defg aeefg bcefg
. , . .

•

'

V

rEH

:

•

j, •

.

‘‘ ''

.

.

•
- 31 -

Plan 7«^ul6 » 7 factors, I/I4. replication, 2 blocks of 16 units

each*

Factors* A,B,C,D,E,F,G

I = ABCEG = ABDF = CDEFG

Block confounding: AB

All main effects and all two-factor interactions except

AB, AD, AF, BD, BF, DF

are estimable*

Blocks

Combine blocks in Plan 7*4-*8 as follows*

1 and 2

3 and Ip .

- 32 -

Plan 8 , lulu 8 factors, l/[|. replication, 16 blocks of I4. units

each*

Factors: A,B,C,D,E,F,G,H

I = ABCEG = ABDFH = CDEFGH

Block confounding: ACD, BEF, ABCDEF, BC, ABD, CEF, ADEF, DE,

ACE, BDF, ABCF, BCDE, ABE, CDF, AF

The main effects, all two-factor interactions except

AF, AH, BC, BG, CG, DE, FH , and 19 of the higher order interactions are estimable.

Blocks

1 2 3 k 1 6

(1) ab acd bed aef bef

adefh bdefh cefh abcefh dh abdh

bcdeg acdeg abeg eg abedfg cdfg

abcfgh cfgh bdfgh adfgh bcegh acegh

2 8 9 10 11

cdef abcdef agh bgh cdgh

ach bch defg abdefg acefg

bfg afg abcdeh edeh beh

abdegh degh bcf acf abdf

12 13 Ik 11 16

abcdgh efgh abefgh acdefgh bcdefgh

bcefg adg bdg eg abeg

aeh bcdfh acdfh abfh fh

df abce ce bde ade
'

.
- 33 -

Plan 8.4»

8

, 8 factors, 1/4 replication, 8 blocks of 8 units

each*

Factors: A,B,C,D,E,F,G,H

I = ABCEG- = ABDFH = CDEFGH

Block confounding: ACD, BEF, ABCDEF, BC, ABD, CEF, ADEF

The main effects, all two-factor interactions except BC and FH,

and 22 higher order interactions are estimable.

Blocks

Combine blocks in Plan 8*4.14. as follows:

1 and 13 5 and 9

2 and ik 6 and 10

3 and 15 7 and 11

4 and 16 8 and 12

- 34 -

Plan 8.4*16 , 8 factors, I/I4

. replication, 4 blocks of 16 units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCEG = ABDFH = CDEFGH

Block confounding: ACD, BEF, ABCDEF

The main effects ? all two-factor interactions, and 24 higher order

interactions are estimable.

Blocks

Combine blocks in Plan 8 •4*4 as follows:

1, 7 , 11, and 13

2, 8, 12, and 14

3, 5, 9, and 15

t 4, 6 , 10 , and 16 ,
IID - '

II ! . . . HI I

Li . . ' . • '
- 35 -

Plan 8. Ip. 32. 8 factors, l/lp replication, 2 blocks of 32 units

each.

Factors: A,B ,C,D,E,F,G,H

I = ABCEG = ABDFH = CDEFGH

Block confounding: ACD

The main effects, all two-factor interactions, and 26 higher

order interactions are estimable.

Blocks

Combine blocks in Plan 8 .Ip . Ip as follows:

1, Ip, 6, 7 , 10, 11, 13, and 16

2, 3, 5, 8, 9, 12, lip, and 1

5

.

- 36 -

Plan 9 factors, 1/il|. replication, 16 blocks of 8 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCEGJT = ABDFHJ == CDEFGH

Block confounding: ACDJ, BEFJ, ABCDEF, BCJ, ABD, CEF, ADEFJ,

DEJ, ACE, BDF, ABCFJ, BCDE, ABEJ, CDFJ, AF.

The main effects, all two-factor interactions except AF and CG,

and 69 of the higher order interactions are estTTmabTe.

Blocks

1 2 3 k i 6

(1) ab ce abce eg abeg

bcdeg aedeg bdg ado- bde ade

adefh bdefh aedfh bedfh acdefgh bedefgh

abcfgh cfgh abefgh efgh abfh fh

bdhj adh3 bcdeh3 acdeh3 bcdgh.3 acdghj

ceghj abcegh3 g^3 abgh3 eh3 abeh3

abef j ®f 3 abef 3 cf3 abcefg3 cefg3

acdfgj bcdfg3 adefg3 bdefg3 adf 3 bdf 3

2 8 9 10 11

eg abeg df abdf edef

bed acd bcefg acefg bfg

adfgh bdfgh aeh beh ach

abcefh cefh abedgh edgh abdegh

bdeghj adegh3 bfh3 afh3 bcefh3

chj abch3 cdefgh3 abcdefgh3 dfgh3

abfgj fgj abde3 de3 abcd3

aedef

j

bedef

3

acg3 bcg3 aeg3
I J : v

•j i

. .

.

.

r.

•

.

.

"

.

.
- 37

Plan 9.1i«8 . (Continued)

Blocks

12 12 lit 1^ 16

abcdef cdfg abcdfg defg abdefg

afg bef aef bcf acf

bch acegh bcegh agh bgh

degh r^dh dh abcdeh cdeh

acefhj bcfghj acfghj befghj aefghj

abdfghj defhj ab defhj cdfhj ab cdfhj

cdj abcdeg

j

cdegj abdg j dgj

beg j aj bj ace j bee j

- 38 -

Plan 9.1ul6 . 9 factors, l/I|. replication, 8 blocks of 16 units

each.

Factors: A,B,C,D,E,F,Gr,H, J

I = ABCEGJ = ABDFHJ = CDEFGH

Block confounding: ACDJ, BEFJ, ABCDEF, BCJ, ABD, CEF, ADEFJ.

The main effects, all two-factor interactions, and 75 higher

order interactions are estimable.

Blocks

Combine blocks in Plan 9 as follows:

and k 9 and 12

and 3 10 and 11

and 8 13 and 16

and 7 11+ and 15
• . I

*;

- : . * <

'

.

' V
- 39 -

Plan 9»l4-»32 > 9 factors, 1/1+ replication, l\. blocks of 32 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCEGJ = ABDFHJ = CDEFGH

Block confounding: ACETJ, BEFJ, ABCDEF

All main effects, all two-factor interactions , and 79 higher

order interactions are estimable.

Blocks

C omb ine blocks in Pl,an 9 tp* 3 :as' .f oil ow s

:

1, 4, O

i

—

1

and 11

2, 3, 9, and 12

5, 8, and 15

6, 7, 13, and 16
1

. .. ...

' ..

, :• • ""

.

• ' -

.
Plan 10,Iu8 « 10 factors , l/l|_ replication, 32 blocks of 8 units

each.

Factors: A,B , C ,D,E,F,G,H, J,K

I = ABCDEFG = ABCDHJK = EFGHJK

Block confounding: ABEFHJ, CDEFHJ, ABCD, ACEH, BCFJ, ADFJ,~

BDEH, ACGK, BCEFGHJK, ADEFGHJK, BDGK, EGHK, ABFGJK, CI'FGJK*/

ABCDEGHK, ACFJ, BCEH, ADEK, BDFJ, EFHJ, AB , CD, ABCDEFHJ,

FGJK, ABEGHK, CDEGHK, ABCDFGJK, ACEFGHJK, BCGK, ADGK, BDEFGHJK

All main effects, 36 of the l±5 two-factor interact ions , and 178

higher order interactions estimable. The following twofactor interactions are confounded with blocks:

AB, AC, AD, BC, BD, CD, EH, FJ, and GK.

Blocks

1 2 3 k £ 6 2

(1) ab ac b c ef abef acef

abed

efhj

abedefh

j

9 10 11 12 u Ik

eghk

abedeghk

fgjk

eg abeg aceg bceg fg abfg

abedfg jk 20

ii 16 1? 18 12

acfg befg aeh beh oeh abceh

21 22 23 2k 25 26

afh bfh cfh abefh agh bgE

21 28 29 30 31 22

cgh abegh aefgh befgh cefgh ab cefgh
. - • 'si . U. . ! . « . • . , ’ • , . . '

•• •
- l4l

Plan 10.1+.16 . 10 factors, l/li replication, 16 blocks of 16

units each.

Factors: A,B,C,D,E,F,G,E, J,K

I = ABCDEFG = ABCDKJK = EFGH<IK

Block confounding: ABEFHJ, CDEFHJ, ABCD, ACEH, BCFJ, ADFJ,

BDEH, ACGK, BCEFGEJK, ADEFGHJK, BDGK, EGHK, ABFGJK, CDFGJK,

ABCDEGHK.

All main effects, 1+2 of the two-factor inter act ions , and 188

higher order interactions are estimable. Interactions AD, BC,

and FJ are confounded with blocks.

Blocks

Combine blocks in Plan 10.1+.8 as follows

1 and 12

17

and 28

2

and 11

18

and 27

3

and 1

0

19

and 26

4 and 9

20

and

5 and 16

21

and 32

6 and 1 $

22

and 31

7 and lip

8 and 13

23

and 30

21+ and 29
... . -• 1

•

• II I

1

.

.

- - -

. a ••
- 42 -

Plan 10.1}-. 32 . 10 factors, l/lj. replication, 8 blocks of 32 units

each.

Factors: A,B ,C,D,E,F,G,H, J,K

I = ABtJDEFG = ABCDHJK = EFGHJK

Block confounding: ABEFHJ, CDEFHJ, ABCD, ACEH, BCFJ, ADFJ, BDEH.

All main effects, all two-factor interactions, and 193 higher

order interactions are estimable*

Blocks

Combine blocks in Plan 10. 1(-e8 as follows:

1, 6, 12, and 1

5

2, 5, 11, and 16

3, 8, 10, and 13

k, 7, 9 ;, and ll±

17 , 22, 28, and 31

18 , 21, 27 , and 32

19 , 21+

,

26, and 29

20, 23, 23 , and 30

Plan 8 ,8,1-1- . 8 factors, 1/8 replication, 8 blocks of ij. units

each.

t Factors: A,B,C,D,E,F,G,H

I = ABEGH = ACFG = BCEFH = ABCD = CDEGH = BDFG = ADEFH

Block confounding: EGH, FG, EFH, BEH, BG, BEFGH, BF

All main effects and the following two-factor interactions

are estimable:

AE, AH, BE, BH, CE, CH de. DH, EF,

EG, EH, FE, GH

Blocks

1 2 2 1 6 2 8

(1) abed acefRh bdefgh cdf abf adegh bcegh

abcdefg efg bdh ach abeg edeg befh adfh

eh abedeh acfg ace edefh abefh adg beg

abcdfgh fgh bde bdfg abgh edgh beef adef

- 44 -

Plan 8.8.8 . 8 factors, 1/8 replication, blocks of 8 units each.

Factors: A,B ,C,D,E,F,G,H

I = ABEGH = ACFG = BCEFH = ABCD = CDEGH = BDFG = ADEFH

Block confounding: EGH, FG, EFH

All main effects and the following two-factor interactions are

estimable

:

AE, AH, BE, BH, CE, CH, DE, DH, EF, EG, EH, FH, GH . The following two-factor interactions are confounded with one

another:

AF = CG, AG = CF, BF = DG, BG = DF .

Blocks

1 2 i k

(1) acefgh cdf adegh

abcdefg bdh abeg befh

eh acfg edefh adg

abcdfgh bde abgh beef

abed bdefgh abf bcegh

efg ach edeg adfh

abedeh ace abefh beg

fgh bdfg edgh adef
•4 -

11
Plan 8 >8,16 , 8 factors, 1/8 replication, 2 blocks of 16- units

each.

Factors: A,B,C,D,E,F,G,H

I = ABEGH = ACFG = BCEFH = ABCD = CDEGH = BDFG = ADEFH

Block confounding: EGH

All main effects are estimable and the following two-factor

interactions are estimable:

AE, AH , BE, BH, CE, CH, DE, DH, EF, EG, EH, FH, GH . The following two-factor interactions are confounded with one

another:

AF=CG, AG=CF, BF=DG , BG=DF , AD=BC .

Blocks

1 2

(1) cdf

abcdefg abeg

eh edefh

abcdfgh abgh

abed abf

efg edeg

abedeh abefh

fgh edgh

acefgh adegh

bdh befh

acfg adg

bde beef

bdefgh bcegh

ach adfh

ace beg

bdfg adef

-k.6 -

Plan 9,S»Iu 9 factors, 1/8 replication, 16 blocks of If. uhits

each*

.Factors: A,B , C,D,E,F,G,H,

J

I = ABEGHJ = ACFGJ = BCEFH = ABCD = CDEGHJ = BDFG'J = ADEFH

Block confounding:

EGHJ, FGJ, EFH, AFHJ, AEFG, AGH, AEJ, ABEG,

ABHJ, ABFGH, ABEFJ, BEFGHJ, BF, BEH, BGJ

All main effects and all two-factor interactions are estimable

with the exception of:

AB, AC, AD, AF, BC, BD, BF, CD, CF,

DF, EG, HJ . Blocks

1 2 i k

(1) fgh abcdfgh abed

abcdefg abcdeh eh efg

eghj efj abode f j abedeghj

abcdfh

j

abcdg

j

gj fhj

5 6 7 8

bdefgh bde ace bdegj

ach acfg bdfg acefgh

bdf j bdghj acghj bdh

acegj acefhj bdefh

j

acf j

Plan 9«8«Iu (Continued)

Blocks

9 10

abf abgh

cdeg cdefh

abefgh

j

abe j

cdhj cdfgj

u ik

adegh adef

bcfh beg

adj adfghj

bcefg

j

bceh j

11 12

edgh cdf

abefh abeg^

ede j cdefghj

abfgj abhj

n 16

beef bcegh

adg^' adfh

bcfghj be j

adehj adefg

j

- 48 -

Plan 9 * 8 * 8 * 9 factors 1/8 replication, 8 blocks of 8 units

each*

Factors: A,B,C,D,E,F,G,H,

J

I = ABEGHJ = ACFGJ = BCEFH = ABCD = CDEGHJ = BDFGJ = ADEFH

Block confounding: EGHJ, FC-J, EFH, AFHJ, AEFG, AGH, AEJ

All main effects and all two-factor interactions are estimable

with the except ion of:

AB , AC, AD , BC, BD

Blocks

1 2 3 it

(i) ab c cl bde ace

abcdefg efg acfg bdfg

fgh abcdfgh bdefgh acefgh

abcdeh eh ach bdh

ef j abcdef

j

bdf j acf j

abcdgj gj aceg j bdeg j

eghj abcdegh

j

bdghj acghj

abcdfh

j

fhj acefhj bdefhj

CD

5 6 2 8

abf cdf ad'ef beef

cdeg abeg beg adg

abgh cdgh adegh bcegh

cdefh abefh befh adfh

abe j cde j ad j he j

cdfg j abfg j bcefg

j

adefg

j

abefghj cdefghj adfghj bcfghj

cdhj abhj bcehj adehj

- Ii-9

Plan 9.8.16 . 9 factors, 1/8 replication, if blocks of 16 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABEGHJ = ACFGJ = BCEFH = ABCD = CDEGHJ = BDFGJ = ADEFH

Block confounding: EGHJ, FGJ, EFH

All main effects and all two-factor interactions are estimable

with the exception of,*

AB, AC, AD, BC, BD, CD .

Blocks

Combine blocks in Plan 9.8*8 as follows:

1 and 2

3 and if 5 and 6

7 and 8 •

50

Plan 9^8^32 ^ 9 factors, 1/8 replication, 2 blocks of 32 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABEGHJ = ACFGJ = BCEFH = ABCD = CDEGHJ = BDFGJ = ADEFH

Block confounding: EGHJ

All main effects and all two-factor interactions are estimable

with the except ion of:

AB , AC , AD , 30 , 3D , CD •

Blocks

Combine blocks in Plan 9«8«8 as follows:

1, 2, 3, and ij. 5, 6, 7, and 8 .
; . • ' I l , ' .
- 51 -

Plan 10 #8 #8 # 10 factors, l/8 replication, 16 blocks of 8 units

each#

Factors: A,B,C,D,E,F,G,H, J,K

I = ABEGHJ = ACFGJK = BCEFHK = ABCDK = CDEGHJK = BDFGJ = ADEFH

Block confounding: GHJ, EHK, EGJK, FHJK, FGK, EFJ, EFGH, EG,

EHJ, GHK, JK, EFGHJK, EFK, FGJ, FH

All main effects and all two-factor interactions are estimable

with the except ion of:

BD , EG, FH, JK

Blocks

1 2 i k

(i) eghj dfk defghjk

abed abedeghj abefk abcefgh jk

aeg jk ahk \ adefgj adfh

be deg jk bedhk bcefg

j

befh

acefgh acf.j aedeghk acd jk

bdefgh bdf j beghk t>jk

cfhjk cefgk cdhj edeg

abdfhjk abdefgk abhj abeg

£ 6 2 8

dhjk degk fhj efg

abchjk abcegk abcdfhj abedefg

adegh adj aefghk af jk

bcegh bej bcdefghk bedf jk

acdefgjk acdfhk acegj ach

befg jk bfhk bdeg j bdh

cdf cdefghj ck ceghjk

abf abefghj iabdk abdeghjk

f

- 52 -

Plan 10,8,8« (Continued).

Blocks

9 10 11 12

abe j abgh abdef jk abdfghk

cde j cdgh cef jk cfghk

bgk behjk bdfg: bdefhj

acdgk acdeh jk acfg acefhj

bcfghj beef bedgh jk bedek

adfghj adef aghjk aek

abcefhk abefg jk abedeh abedg

j

defhk dfg jk eh si • 13 Ik 15 16

abdehk abdgjk abefh abfg j

cehk eg jk edefh cdfg j

bdghj bde bfghjk befk

acghj ace acdfgh jk aedefk

bcdfgk bedefh jk beg b c eh j

afgk aefh jk adg adehj

abcdef

j

abcdfgh abce jk abeghk

efj fgh de jk dghk

- 53 -

Plan 10.8.16 . 10 factors, 1/8 replication, 8 blocks of 16

units each.

Factors: A,B,C,D,E,F,G,H, J,K

I = ABEGHJ = ACFGJK = BCEFHK = ABCDK = CDEGHJK = BDFGJ = ADEFH

Block confounding: GHJ, EHK, EGJK, FHJK, FGK, EFJ, EFGH

All main effects and two-factor interactions are estimable.

Blocks

Combine blocks in Plan 10.8.8 as follows:

1 and 2 9 and 10

3 and k 11 and 12

5 and 6 13 and 14

7 and 8 15 and 16

- 54 -

Plan 10,8,32 , 10 factors, 1/8 replication, l|_ blocks of 32

units each.

Factors: A,B,C ,D,E,F,G,H, J,K

I = ABEGHJ = ACFGJK = BCEFKK = ABCDK = CDEGHJK = BDFGJ = ADEFH

Block confounding: GHJ, EHK, EGJK

All ma,in effects and two factor interactions are estimable.

Blocks

Combine blocks in Plan 10.8.8 as follows:

1, 2, 3, and 4

5, 6, 7j and 8

9, 10, 11, and 12

13 , 14, 13, and 16 .

- 55 -

Plan 11,8,8 , 11 factors, l/Q replication, 32 blocks of 8 units

each.

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

I = ABEGHJ = ACFGJK = BCEFEK = A3CDKL. = CDEGHJKL = BDFGJE

= ADEFHL

Block confounding: C-HJL, EHKL , EGJK, FHJK, FGKL , EFJL , EFGH,

EGL" , EHJ, GHK, JEL , EFGHJEL , EFK, FGJ,

FHL , AEG, AEHJL , AGHKL , AJK, AEFGHJK,

AEFKL, AFGJL , AFH, AL , AGHJ, AEHK, AEGJKL

,

AFHJEL, AFGK, AEFJ, AEFGHL

All main effects and all two-factor interactions are estimable

with the exception of:

AL , C «T , DG «

Blocks

I 2 2 k £ 62.8

(1) eghj dfk defghjk dhjk degk fhj efg

bcdeg jk

bdefgh

cfhjk

adfgkl

abcef jl

abehkl

acdghjl

Plan II .8 > 8» (Continued)

- 56 -

Blocks

9 10 11 12

abe j abgh abdef jk abdfghk

13 Ul ii 16

abdehk abdg jk abefh abfg j

11 18 ±i 20

abdfl abdefghjl abkl abegh jkL

21 22 23 lit

abfh jkl abefgkl ab dh j 1 abdegl

22 26 21 28

def jl dfghl e jkl ghkl

29 3£ 32

efhkl fg jkl dehl dgjl

- 57 -

Plan 11.8.16 . 11 factors, 1/8 replication, 16 blocks of 16

units each.

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

I = ABEGHJ = ACFGJK = BCEFHK - ABCDKL = C DEGHJKL = BDFGJE

= ADEFHL

Block confounding: GHJL, EHKL , EGJK, FHJK, FGKL , EFJL , EFGH,

EGL, EHJ, GHK, JKL , EFGHJKL , EFK, FGJ, FHL

All main effects and two-factor interactions are estimable.

Blocks

1 2 i it

(1) eghj dfk defghjk

bcdeg jk i

*7 bdefgh o 1

cfhjk dh jk degk fhj

adfgkl

abcdf jl 8 9 10

ab ehkl efg abe j abgh

acdgh jl

def jl 11 12 11

bcfgkl

bghjl

abdef jk abdfghk abdehk

cdehkl

aeg jk lit 16

abed abdg jk abefh abfgj

abdfhjk

acefgh
• •'

•• * -

:•

. 1:

-

* ;

'

'

.

.

.

... .

.

-
- 58

Plan 11.8.32 . 11 factors, 1/8 replication, 8 blocks of 32 units

each!.

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

I = A3FGHJ " ACFCtJK- ~ BCF.HiK = AB0D1L ” CTEGHJ35& = BDFG JL = ADFFHL

Block confounding: GHJL , EHKL , EGJK, FHJK, FGKD , EFJL , EFGH

All main effects and two-factor interactions are estimable.

Blocks

A 1

(1) abfgj

bcdegjk aedefk

bdefgh adehj

cfhjk abeghk

adfgkl bdjkl

abcef jl cegl

abehkl efghjkl

acdghjl bcdfhl

def jl abdegl

bcfgkl ac jkl

bghjl afhl

cdehkl abcdefgh jkl

aeg jk befk

abed cdfg j

abdfh jk dghk

acefgh bcehj

2 3 it £

eghj dfk defghjk dhjk

6 7 . 8

. degk fhj efg
• ' v'. • ' ••• : \
59

Plan 7«l6+i4.« 7 factors, 1/16 replication, 2 blocks of 4 units

each*

Factors* A,B,C,D,E,F,G

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFG

= EFG = CDG = ABG = ADF = BCF = BDE = ACE

Block confounding: A

All main effects except A are estimable. They are identified

with two-factor interactions as follows:

B = AG - CF = DE E = FG = BD = AC

C = DG = BF = AE F = EG = AD = BC

D = CG. = AF = BE G = EF = CD = AB

Blocks

1 2

(1) acfg

cdef adeg

bdfg abed

bceg abef

60

Plan 7»l6»8 « 7 factors, 1/16 replication, 1 block of 8 units

each#

Factors* A,B,C,D,E,F,G

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFG

= EFG = CDG = ABG = ADF = BCF = BDE = ACE

All main effects are estimable. They are identified with twofactor interactions as follows:

A sr BG sr DF = CE E = FG - BD = AC

B = AG - CF - DE F = EG = AD =r BC

C = DG - BF = AE G a EF = CD =r AB

D = CG = AF = BE

Block

(i) abed

acfg

adeg

bdfg

bceg

edef abef
, • . - : ' . ; t .... ' . • . : • . ... : . . ; . - . , .
- 61

Plan 8«l6«iu 8 factors, 1/16 replication, blocks of I4. units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFGH

= EFGH = CDGH = ABGH = ADFH = BCFH = BDEH = ACEH

Block confounding: AB, AC, BC

All main effects are estimable.

Blocks

1 2 2 k

(1) acfg edef ad eg

abed bdfg abef bceg

abcdefgh bdeh abgh befh

efgh aceh edgh adfh

62

Plan 8*16,8 * 8 factors, l/l6 replication, 2 blocks of 8 units

each*

Factors: A,B,C,D,E,F,G,H

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABGDEFGH

- EFGH = CDGH = ABGH = ADFH = BCFH = BDEH = ACEH

Block confounding: AB

All main effects are estimable*

Blocks

1 2

(1) acfg

cdef adeg

abed bdfg

abef bceg

abcdefgh bdeh

abgh befh

efgh aceh

edgh adfh

- 63 -

Plan 8.16.16 . 8 factors 1/16 replication, 1 block of 16 units

each.

Factors: A,B,C,D,E,F,G,H

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABGDEFGH

= EFGH = CDGH = ABGH = ADFH = BCFH = BDEH = ACEH

All main effects are estimable.

Block

(1) abcdefgh

acfg bdeh

adeg bcfh

cdef

abed

abgh

efgh

bdfg

bceg

aceh

adfh

abef edgh

- 64 -

Plan 9«l6.1]. . 9 factors, 1/16 replication, 8 blocks of units

each.

Factors: A,B,C,D,E,F,G,H, J

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFGHJ

= EFGHJ = CDGHJ = ABGHJ = ADFHJ = BCFHJ = BDEHJ = ACEHJ

Block confounding: AB, AC, BC, AE, BE, CE, ABCE

All main effects and the following two-factor interactions are

estimable: AH, AJ, BH, BJ, CH, CJ, DH, DJ, EH, EJ, FH, FJ,

GH,GJ,HJ.

Blocks

(1) aceh

2

edgh

Ik

adeg

abcdefgh bdfg abef befh

abcdefg

j

bdfghj abefh j bef j

hj ace j edgj adeghj

5 6 i 8

abed bdeh abgh bceg

efgh acfg edef adfh

efgj acfghj cdefhj adf j

abedh j bde j abgj bceghj
.. ' . ' ' • ; . t • . v ' ; : ; - ' * ;

•= •" - o " , .. . : : . • : ' - ' . - . 1
65

Plan 9«l6»8 » 9 factors, l/l6 replication. Ip. blocks of 8 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFGHJ = EFGHJ = CDGHJ = ABGHJ = ADFHJ = BCFHJ = BDEHJ = ACEHJ

Blork confounding: AB, AC, BC

All main effects and the following two-factor interactions

are estimable: AH, AJ, BH, BJ, CH, CJ, DH, DJ, EH, EJ, FH,

FJ, GH, GJ, HJ.

Blocks

(1)

1

aceh

2 i it

cdgh adeg

abed bdeh abgh be eg

abcdefgh bdfg abef befh

efgh acfg edef adfh

efgj acfghj cdefhj adfj

abcdefgj bdfghj abefhj befj

abedhj bdej abgj bceghj

ace j edgj adeghj
'

'

,

i
- 66

Plan 9«l6«l6 « 9 factors, 1/16 replication, 2 blocks of 16 units

each.

Factors: A,B,C,D,E,F,G,H,

J

I = ABCD = ABEF = CDEF = BCEG = ADEG = ACFG = BDFG = ABCDEFGHJ = EFGHJ = CDGHJ = ABGHJ = ADFHJ = BCFHJ = BDEHJ = ACEHJ

Block confounding: AB

All main effects and the following two-factor interactions are

estimable: AH, AJ, BH, BJ, CH, CJ, DH, DJ, EH, EJ, FH, FJ,

GH, GJ, HJ.

Blocks

1 2 m edgh aceh adeg

abed abgh bdeh bceg

abcdefgh abef bdfg befh

efgh edef acfg adfh

efgj cdefhj acfghj adfj

abedefg

j

abefhj bdfghj bcf j

abedh

j

abg j bde j bceghj

hi edgj ace j adeghj
' -

' ' "

- •

' ;

... ,

'

• ,

' ;

•'

" 1

I I

, i

; * ''

' ’

.

. *

'

‘ ''

.

. . .

.

‘

. -
- 67 -

Plan 10.16.8 . 10 factors, l/l6 replication, 8 blocks of 8 units

each#

Factors: A,B,C,D,E,F,G,H, J,K

I = ABCDJK = ABEFJ = CDEFK = BCEGJK = ADEG , = ACFGK = BDFGJ

= ABCDEFGK = EFGHJK = CDGHJ = ABGHK = ADFHJK = BCFH

= BDEHK = ACEHJ

Blobk confounding: AD* AE, DE, BC, ABCD, ABCE, BCDE

All main effects are estimable. All two-factor interactions are

estimable, except AD=EG, AE=DG, AG=DE , BC=FH, BF=CH,BH=CF, and JK.

1

Blocks ” 2

(1) aek

adeg dgii!

bcfh abcefhk

abcdefgh bcdfghk

be jk abce j

abedeg jk bedg j

fhjk aefhj

adefgh jk dfghj

i 6

de j abeh jk

agj bdghjk

bcdefhj acef jk

abcfghj cdfg jk

bedek aceh

abegk edgh

defhk abef

afghk bdfg

1 k

abed bhj

bceg abdegh

j

adfh of j

~ efgh aedefg

j

ad jk chk

eg jk aedeghk

abcdfhjk bfk

be efgh jk abdefgk

2 8

aedhj bdeh

ceghj abgh

abdf j edef

befgj acfg

abdhk edehjk

beghk aeghjk

aedfk bdef jk

cefgk abfgjk

- 68

Plan 10*16*16 * 10 factors, 1/16 replication, l\. blocks of 16

units each*

Factors: A,B,C,D,E,F,G,H, J,K

I = ABCDJK = ABEFJ = CDEFK = BCEGJK = ADEG = ACFGK = BDFGJ

= ABCDEFGH = EFGHJK = CDGHJ = ABGHK = ADFHJK = BCFH = BDEHK = ACEHJ

Block confounding: AD, AE, DE

All main effects are estimable* All two-factor interactions are

estimable, except AD=EG, AE=DG, AG=DE , BC=FH, BF=CH , and BH=CF

Blocks

1 2 i k

(1) aek aedh j dej

adeg dgk ceghj agj

bcfh abcefhk abdfj bedefh

j

abcdefgh bcdfghk befg j abcfghj

be jk ab c e j abdhk bedek

abedeg jk beds; j beghk abegk

fh ,1k aefh j aedfk defhk

adefgh jk dfghj cefgk afghk

bhj abeh jk abed bdeh

abdegh

j

bdghjk bceg abgh

cf j acef jk adfh edef

aedefg

j

cdfg jk efgh acfg

chk aceh ad jk edeh jk

aedeghk edgh eg jk aeghjk

bfk abef abedfh jk bdef jk

abdefgk bdfg bcefghjk abfg jk
• * ' s * - , . . ' • > ' : . ' ' • - - ' • ? * . , ; “ ; - •: • •' ' ' \ ' ' - ' . ' ' • • ' ' ; ' ! J
- 69 -

Flan 11 *16*8 * 11 factors, l/l6 replication, 16 blocks of 8

units each*

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

I = ABCDJK = ABEFJL = CDEFKL = BCEGJKL = ADEGL = ACFGK = BDFGJ

= ABCDEFGH = EFGHJK = CDGHJL = ABGHKL = ADFHJFL = BCFKL = BDEHK

= ACEHJ

Block confounding: DEFG, BCFG, BCDE, AGEF, ACDG, ABEG, ABDF,

ABCDEFGHJKL, ABCHJKL , ADEHJKL , AFGHJKL , BDGHJKL , BEFHJKL,

CDFHJKL, CEGHJKL,

All main effects are estimable. All two-factor interactions

except the following are estimable:

EF, DG, AC, BH .

Blocks

1 2 3 k

(1) abdf j abed cfj

abcdefgh ceghj efgh abdegh

j

defgjl abegl abcefg jl cdegl

abchjl cdfhl dhjl abfhl

acef jk bcdek bdef jk aek

bdghjk afghk aeghjk bcdi'ghk

acdgkl bcfgjkl bgkl adfgjkl

befhkl adeh jkl acdefhkl bcehjkl

- 70

Plan 11 < 16 ,.8 » (Continued) »

Blocks

5 6 i

bceg bhj abef

adfh acdefg

j

cdgh

bcdf jl bdefghl abdg jl

aeghjl acl eefhjl

abfg jk abcefhk be jk

cdeh jk dgk adefgh jk

abdekl abcdgh jkl bcdefgkl

cfghkl ef jkl aiikl

2 10 11

adeg acdh j edef

bcfh befg j abgh

af jl acefghl cgjl

bcdeghjl bdl abdefh jl

cdfg jk defhk ad jk

abeh jk abcgk bcefgh jk

cekl gh jkl aefgkl

abdfghkl abcdef jkl bcdhkl

13 Ik 15

acfg hcdg j bdfg

bdeh aefhj aceh

acde jl bcefl he jl

bfghjl adghl acdfgh jl

eg jk abdefgk abedeg jk

abcdfhjk chk fhjk

dfkl ab jkl abcfkl

abceghkl cdefgh jkl deghkl

8

de j

ab c fah j

*L o

ab cdbhl

acdfk

b'eghk

acegflkl

bdfhjkl

12

abce j

dfghj

abcdfgl

ehl

bfk

acdeghk

bdeg jkl

acfh jkl

16

as i bcdefh

j

adefl

bcghl

cefgk

abdhk

cd jkl

abefgh jkl
.

;

’

.

•

' ;

...

‘

.

.

*

. ; ^ <f ” .

.

.

. u •? C .

..

'

.

=

. ;

. , . . .. .

. 1

*

.
71

Plan 11 .16 .16 . 11 factors, l/l6 replication, 8 blocks of 16

units each.

I = ABCDJK = ABEFJL = CDEFKL = BCEG JKL = ADEGL = ACPGK = BDFGJ

= ABCDEFGH = EFGHJK = CDGHJL = ABGHKL = ADFHJKE = BCFHD

= BDEHK = ACEHJ

Block confounding: DEFG, BCFG, BCDE, ACEF, ACDG, ABEG,ABDF

All main effects and two-factor interactions are estimable.

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

Blocks

Combine blocks in Plan 11.16.8 as follows:

1 and 2 9 and 10

3 and Ij.

5 and 6

7 and 8

13 and lk

15 and 16 •

11 and 12

72

Plan 12,16,8 , 12 factors, 1/16 replication, 32 blocks of 8

units each.

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

I = ABCU’JK = ABEFJL = CDEFKL = BCEGJKLM = ADEGLM = ACFGKM

= BDFGJM = ABCDEFGH = EFGHJK = CDGHJL = ABGHKL = ADFHJKLM

= BCFHLM = BDEHKM = ACEHJM

Block confounding: DEFG, BCFG, BCDE, ACEF, ACDG, ABEG,

ABDF, ABCDEFGHJKL , ABCHJKL , ADEHJKL , AFGHJKL , BDGHJKL , BEFHJKL , CDFHJKL , CEGHJKL, LM, DEFGLM, BCFGLM, BODED M, ACEFLM, ACDGLM,

ABEGLM, ABDFLM, ABCDEFGHJKM, ABGHJKM, ADEHJKM, AFGHJKM,

BDGHJKM, BEFHJKM, CDFHJKM, CEGHJKM

All main effects are estimable. All two factor interactions

except the following are estimable;

AF, GH, CE, BD- .

Blocks

1 Z

(1) defgjl

abcdefgh

acef jk

bdghjk

bcdejlm

afghjlm

abdfklm

c eghklm

1 6

bceg bedf jl

9 10

abdf j abegl

i it

abed bgkl

i

2 8

abef ahkl

11 12

ad eg afjl

- 73

Plan 12.16.8. (Continued).

Blocks

13 Ik 12 16

cdef Cgjl efj cdegl

12 18 12 20

acfg dfkl bhj acl

21 22 22 3k

dej fgl bdfg be J1

22. 26 SI 28

acdh j bdl bfk ehl

22 30 21 32

chk bcefl agj adefl
< ,
74

Plan 12.l6.l6 . 12 factors, l/l6 replication, 16 blocks of 16

units each.

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

I = ABCDJK = ABEFJL = CDEFKL = BCEGJKLM = ADEGLM = ACFGKM

= BDFGJM = ABCDEFGH = EFGHJK = CDGHJL = ABGHKL = ADFHJKLM

=- BCFHLM = BDEHKM = ACEHJM

Block confounding: DEFG, BCFG, BCDE, ACEF, ACDG, ABEG, ABDF,

ABCDEFGHJKL, ABCHJKL, ADEHJKL , AFGHJKL , BDGHJKL, BEFHJKL , CDFHJKL , CEGHJKL

All main effects and two-factor interactions are estimable.

Blocks

Combine blocks in Plan 12.16.8 as follows:

1 and 2 17 and 18

3 and k 19 and 20

5 and 6 21 and 22

7 and 8 23 and 2k

9 and 10 23 and 26

11 and 12 27 and 28

13 and 14 29 and 30

15 and 16 31 and 32
.

J
75

Plan 10.32.lu 10 factors, 1/32 replication, 8 blocks of i|

units each*

Factors: A,B , C,D,E,F,G ,H, J,K

I = ABCD = ABEF = 'CDEF = ABGH = CDGH = EFGH = ABCDEFGH = ABJK

= CDJK = EFJK = ABCDEFJK = GHJK = ABCDGHJK = ABEFGHJK = CDEFGHJK

= ACEGJ = BDEGJ = BCFGJ = ADFGJ = BCEHJ = ADEHJ = ACFHJ = BDFHJ

= BCEGK =•' ADEGK = ACFGK = BDFGK = ACEHK = BDEHK = BCFHK = ADFHK

Block confounding: CD, CE, DE, ACEF, ADEF, AF, ACDF

All main effects are estimable, but no two-factor interactions

are estimable.

Blocks

1 2 3 k

(1) cdef abed abef

abcdefgh abgh efgh edgh

ghjk cdefgh jk abedgh jk abefghjk

abcdef jk ab jk ef jk cd jk

£ 6 7 8

bceg j bdfg j adeg j acfg j

adfhj aceh j bcfhj bdehj

bcehk bdfhk adehk acfhk

adfgk acegk bcfgk bdegk
,

•

;

:

- .

.
- 76 -

Plan 10.32.8 . 10 factors, 1/32 replication, Ij. blocks of 8

units each.

Factors: A,B ,C,D,E,F,G,H, J,K

I: Same as Plan 10.32.Ij.

Block confounding: CD, CE, DE

All main effects are estimable, but no two-factor interactions

are estimable.

Blocks

Combine blocks in Plan 10.32.Lj. as follows:

1 and 2

3 and Ij.

5> and 6

7 and 8 •
..

. :

' . ;

;

..

• C. ' ;A

’

: : : v

•••'•* • '

• t,

;
- 77 -

Plan 10. 32.16 . 10 factors, 1/32 replication, 2 blocks of 16

units each.

Factors: A,B,C,D,E,F,G,H, J,K

I: Same as Plan 10. 32. Ij. Block confounding: CD

All main effects are estimable, but no two-factor interactions

are estimable.

Blocks

Combine blocks in Plan 10. 32. ip as follows:

1, 2, 3, and 4

5, 6, 7, and 8 .
, s

;:. ;i

"

!

.

' ' '

•

.

'

'

.

.

,

• •

. .......

•

’ 'v-
78

Plan 11.32*8 • 11 factors, 1/32 replication, 8 blocks of 8 units

each.

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

I = ABCDL = ABEFL = CDEF = ABGH = CDGHL = EFGHL = ABGDEFGH

= ABJK = CDJKL = EFJKL = ABCDEFJK = GHJK = ABCDGHJKL

= ABEFGHJKL = CDEFGHJK = ACEGJL = BDEGJ = BGFGJ = ADFGJL

= BCEHJL = ADEHJ = ACFHJ = BDFHJL = BCEGKL = ADEGK = ACFGK = BDFGKL = ACEJ.IKL = BDEHK : = BCFIIK»= ADFHKL

Block confounding: CD, GE, DE, ACEF, ADEF, AF, ACDF

All fli&in effects are estimable; the following two-factor interactions

are estimable:

AL, BC, BD, BE, BF, BL, CG, CH, CJ, CK, CL

DG, DH, DJ, DK, DL, EG, EH, EJ, EK, EL

,

FG

FH, FJ, FK, EL, GL, HL, JL

,

KL

Blocks

1 2 i it

(i)

abcdefgh

ghjk

abcdef jk

bhjl

acdefg jl

bgkl

acdefhkl

cdef abed abef

abgh efgh edgh

cdefghjk abcdghjk abefghjk

abjk ef jk cdjk

bcdefhjl acdhjl aefhjl

ag jl befgjl bcdgjl

bcdefgkl acdgkl aefgkl

ahkl b e fhkl b c dhkl

- 79

Plan 11»32«8» (Continued),

Blocks

5 6 I 8

bcegj bdfgj adeg j acfg j

adfhj aceh j bcfhj bdeh j

bcehk bdfhk adehk acfhk

adfgk acegk bcfgk bdegk

ceghl dfghl abdeghl abcfgbl

abdfl abcel cfl del

ce jkl df jkl abde jkl abcf jkl

abdfgh jkl abceghjkl cfghjkl degh jkl
.. ’ • . ' ' ' . . 0
- 80 -

Plan 11,32,16 , 11 factors, 1/32 replication, Ij. blocks of 16

units each.

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

I: Same as Plan 11 ,32.8

Block confounding: CD, CE, DE

All main effects and the following two-factor interactions are

estimable:

AC, AD, AE, AF, AL , BC, BD, BE, BF, BL, CG

CH, CJ, CK, CL, DG, DH, DJ, DK, DL, EG, EH

EJ, EK, EL f FG, FH, FJ, FK, GL, HL, JL

KL

Blocks

Combine blocks in Plan 11. 32,8 as follows:

1 and 2

3 and l\. 5 and 6

7 and 8 .
. - > » • - - • • - .*< * V

- .

' , .

• "

.

.

\
Flan 11 ,32,32 , 11 factors, 1/32 replication, 2 blocks of 32

units each.

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

I: Same as Plan 11.32.8

Block confounding: CD

All main effects and the following two-factor interactions are

estimable:

AC, AD, AE, AF, AL, BC, BD, BE, BF, BL, CG

CH, CJ, CK, CL, DG, DH, DJ, DK, DL, EG, EH

EJ, EK, EL

,

FG, FH, FJ, FK, FL. GL

.

HL

,

JL

KL

Blocks

in Plan 11.32.8 as follows:

1 , 2 , 3 , and 4

5 , 6 , 7 , and 8 .

Combine blocks

aa

Plan 12.32.8 . 12 factors, 1/32 replication, 16 blocks of 8

units each.

Factors: A,B,C,D,E,F,G,H,

I = ABCDLM = ABEFL = CDEFM = ABGHM = CDGHL = EFGHLM = ABCDEFGH

= ABJKM = CDJKL = EFJKLM = ABCDEFJK = GHJK = ABCDGHJKLM

= ABEFGHJKL = vDEFGHJKM = ACEGJL = BDEGJM = BCFGJ = ADFGJLM

= BCEHJLM = ADEHJ = ACFHJM = BDFHJL - BCEGKLM = ADEGK = ACFGKM

= BDFGKL = ACEHKL = BDEBKM = BCFHEC = ADFHKLM

Block confounding: CD, CE, DE, ACEF, ADEF, AF, ACDF, KLM,

CDKEiM, CEKLM, DEKLM, ACEFKLM, ADEFKLM,

AFKLM, ACDFKEK

All main effects and two-factor interactions are estimable

except for the following:

AF, BG, CD, CE, CH, DE, DH, EH, GH, GJ, HJ,

JK, JM . Blocks

I 2 2 k

(1) bcdem abed aem

abcdefgh

af jkm

bcdeghjkm

bgkl

acdefhkl

abfer,ilm

cdehjlm
r i i

• '

.

-

- :

•

-

’ ~ 3A. **

' .

' - -

»"v • '.Vi

'

•

:

.

:

' - ”

•5
83

Plan 12»32»8 » (Continued)

Blocks

I 6 I 8

cdef bfm abef acdfm

9 10 11 12

del bclm abcel adlm

JU ik ii 16

on bdeflm abdn aceflm

- 8k -

Plan 12. 32.16 * 12 factors, 1/32 replication, 8 blocks of 16

units each*

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

I: Same as Plan 12.32*8

Block confounding: CD, CE, DE, ACEF, ADEF, AF, ACDF

All main effects and two-factor interactions are estimable except

for the following:

AF, CD, CE, DE, GH, GJ, HJ, JK

Blocks

Combine blocks in Plan 12.32.8 as follows:

1 and 2 9 and 10

3 and k 11 and 12

5 and 6 13 and llj7 and 8 13 and 16
I

'

.

'

. *

’ -

• '

.... -V .

- .

•

•

. .

V

' (9 . .

'

.

• :

5

..

hi

'

-

.
- 85 -

Plan 12,32.32 . 12 factors, 1/32 replication, [j. blocks of 32

units each.

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

I: Same as Plan 12.32.8

Block confounding: CD, ACEF, ADEF

All main effects and two factor interactions are estimable

except for CD.

Blocks

Combine blocks in Plan 12.32.8 as follows:

1, 2, 3, and 1+

5, 6, 7, and 8

9, 10, 11, and 12

13, 1 1+, 15* and 16 •

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.