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NATBONAL BUREAU OF STANDARDS REPORT

6504

FRACTIONAL FACTORIAL DESIGNS FOR EXPERIMENTS

WITH FACTORS AT TWO AND THREE LEVELS

W, S. Connor and Shirley Young

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

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 amen»led b) Congress in Public I>aw 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 hack 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 peri-

odicals, available from the Go\ eminent Printing Office: The Journal of Research, which presents

complete papers repotting technii-al 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 N BS Circidar 460, Publications of

the National Bureau of Standarils ($1.25) and its Supplement ($0.75), available from the Superin-

tendent of Documents, Go\ eminent Printing Office, Washington 25, D. C,

Inquiries regarding the Bureau’s reports should be addressed to the Office of Technical Informa-

tion, National Bureau of Standards. Washington 2.5, D. C.

NATIONAL BUREAU OF STANDARDS REPORT NBS PROJECT

1103-40-5148 July 1959

NBS REPORT

6504

FRACTIONAL FACTORIAL DESIGNS FOR EXPERIMENTS

WITH FACTORS AT TWO AND THREE LEVELS

W, S. Connor and Shirley Young

NATIONAL BUREAU OF STAN^ Intended for use yvithin the Goi

to additional evaluation and revi

listing of this Report, either in v

the Office of the Director, Natioi

however, by the Government agt to reproduce additional copies fc

IMPORTANT NOTICE

Approved for public release by the

director of the National Institute of

Standards and Technology (NIST)

on Octobers, 2015

fress accounting documents

lly published it is subjected

reduction, or open-literature

is obtained in writing from

;h permission is not needed,

pared if that agency wishes

U. S. DEPARTMENT OF COMMERCE

uscohm-nbs-dcNational bureau of standards

PREFACE

The designs presented here are for experiments with

some factors at two levels and other factors at three levels

o

These designs were developed in the Statistical Engineering

Laboratory of the National Bureau of Standards under a pro-

gram sponsored by the Bureau of Ships_, Department of the

Navy. The work was performed under the direction of

W. S. Connor. Professor R. C. Bose served as consultant

and contributed to the development of related theory.

Shirley Young performed most of the work of constructing

the designs and working out the corresponding estimates,

Carroll Dannemiller devised an electronic computer program

which was used to check the normal equations. A program

previously developed by R, C. Burton was used to generate

n treatment combinations from 3' factorials. Also, Burton

participated during the summer of 1958 in certain aspects

of construction. Lola S, Doming supervised the preparation

of the manuscript in final form.

i

CONTENTS

Page

iPreface

1. Introduction 1

2. Construction of designs 2

3. Estimation of effects 5

4. Tests of significance and confidence intervals 15

5. An example 18

6. References 23

DESIGN

Number of Effects

Estimated

Number of Treatment Combinations

Fraction of Complete

Factorial

Page

2^*3^ 21 36 3/4 24

2^1 28 48 1/2 26

2®3^ 36 48 1/4 27

2*^3^ 45 96 1/4 29

2^3^ 55 96 1/8 31

2^3^ 66 128 1/12 34

2^3^ 27 36 1/2 37

2"^3^ 35 72 1/2 39

2V 44 72 1/4 41 2®32 54 96 1/6 43

2'^32 65 144 1/8 46

2®3^ 77 144 1/16 48

ii

DESIGN

Number of Effects

Estimated

Number of Treatment Combinations

Fraction of Complete

Factorial

Page

2^3^ 34 54 1/2 61

2^33 43 72 1/3 53

53 108 1/4 55

2®3^ 64 144 1/6 57

2®3^ 76 288 1/6 59

2^3^ 89 432 1/8 61

2^3 42 81 1/2 64

2^3^ 52 162 1/2 66

2^3^ 63 162 1/4 68

2^3^ 75 162 1/8 71

2V 88 216 1/12 74 263^ 102 324 1/16 77

2^3^ 62 162 1/3 81

2^3^ 74 162 1/6 83

2^3® 87 216 1/9 85

2^" 101 324 1/12 87

2S® 116 432 1/18 90

2^3® 86 243 1/6 92

2^3® 100 486 1/6 95

2®3® 115 486 1/12 98

2^^3^ 131 486 1/24 102

iii

DESIGN

Number of Effects

Estimated

Number of Treatment Combinations

Fraction of Complete Factorial

Page

114 243 1/18 106

130 486 1/18 109

2^3’ 147 486 1/36 112

146 243 1/54 116

2^38 164 486 1/54 120

2^3® 182 243 1/162 124

iv

1

1. Introduction

This catalogue is the sequel to [l] and [2], It

contains fractional factorial designs for use in experiments

which investigate m factors at two levels and n factors at

three levels. A design has been constructed for each of

the 39 pairs (ra, n) included from m + n = 5 through

m + n = 10, (m, n 0) . The design for (m, n) is designated

DESIGN .

It is believed that the method of construction des-

cribed in section 2 is new. Morrison [3] published several

designs which can be constructed by the present method, and

his paper was an inspiration to the authors in formulating

their method.

Section 3 contains a description of the mathematical

model, and of how to estimate the parameters contained in

the model. Section 4 contains a discussion of how to test

hypotheses and construct confidence intervals. A worked

example is presented in section 5.

-» 2 “

2. Construction of Designs

The designs are constructed by associating not neces-

R1sarily distinct fractions 82 ^ from the 2

complete factorial with not necessarily distinct fractions I r f

Sf, from the 3 complete factorial. The frac- f

tions and (i = 1 , 2 , t) are obtained by conven-

tional methods which have been described, for example, in

[4, 5 ]. The association is such that every treatment

combination in is adjoined to every treatment combination

in thus forming treatment combinations from the 2^‘^3^

complete factorial. The resulting fraction from the 2^3^

complete factorial may be denoted by

(2 . 1 )

3 2 To illustrate, consider the 2 3 complete factorial,

which contains 72 treatment combinations. The three factors

with two levels will be denoted by A^, and and the

3 two factors with three levels by and B

2 . The 2 complete

factorial may be fractionated into tv/o distinct sets and

S 2

by finding the treatment combinations (x^ Xg ^ 3 );

(x . = 0, 1; j = 1, 2, 3) having x’s which satisfy J

(2.2) x^ + Xg + = 0 and + Xg + x^ = 1 (mod 2),

respectively. These sets are as follows;

3

(2.3)

Sets of Treatment Combinations

from the 2^

tl

0 0 0

110 10 1 Oil

^1 ^2 ^3

111 10 0 0 10 0 0 1

2 The 3 complete factorial may be fractionated into

t » f

three distinct sets , ^2 ' ^3 finding the treat-

ment combinations (z^ Zg) , (zj^ =