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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

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    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^ =