Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

The development of factorial design is principally due to pioneering work in the 1920s by R. A. Fisher (1960, pp. 93–94):

In expositions of the scientific use of experimentation it is frequent to find an excessive stress laid on the importance of varying the essential conditions only one at a time.… to study the effects of this single factor, is the essentially scientific approach to an experimental investigation. This ideal doctrine seems to be more nearly related to expositions of elementary physical theory than to laboratory practice in any branch of research.…

The modifications possible to any complicated apparatus, machine or industrial process must always be considered as potentially interacting with one another, and must be judged by the probable effects of such interactions. If they have to be tested one at a time this is not because to do so is an ideal scientific procedure, but because to test them simultaneously would sometimes be too troublesome, or too costly. In many instances, as will be shown in this chapter, the belief that this is so has little foundation. Indeed, in a wide class of cases an experimental investigation, at the same time as it is made more comprehensive (by including multiple determinants], may also be made more efficient … [so that] … more knowledge and a higher degree of precision are obtainable by the same number of observations.

Seventy years later, however, we read:

One-factor-at-a-time experiments and similar “common sense” design strategies continue to be prevalent in industrial experiments in spite of the strong emphasis in statistics courses that these design strategies should be avoided. In justifying the avoidance of such design strategies, technical criteria such as design efficiency and confounding … routinely are over-looked by experimenters in industry. (Gunst & McDonald, 1996, p. 44.)

Actually, factorial design may be considered an ideal form of varying one variable at a time. Each row in a two-factor, A × B design varies only B, for one fixed level of A. Each additional row is thus a systematic replication of the B effect, but at a different level of A. From this perspective, factorial design conjoins the rule of one-variable-at-a-time with the principle of replication.

Three distinct concepts are usually conglomerated under factorial design. One is the design itself, a rectangular matrix illustrated in Figure 5.2, which possesses advantages in estimation and generalization of main effects. Second is the Anova model of Equation 1a, which provides a conceptual model of multiple determination. Third is the analysis of variance, a flexible tool for detecting trends in the data.
Treating main effects as one-way designs yields correct values of SS A and SS B. The error term from these analyses, however, would be inflated because it includes SS AB. The two-way Anova excludes SS AB and so yields correct MSerror.
The term degrees of freedom comes from a geometric representation of the data in N-dimensional space. Each observation corresponds to one dimension in this space, and hence to one spatial degree of freedom. To some statisticians, this geometric representation is a helpful way of thinking. Fisher could visualize his theorems in this way, so he often did not bother to give a formal algebraic proof, distressing statisticians who lacked this geometric facility.


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Empirical Direction in Design and Analysis
Table of contents

Table of contents

  • Title Page iii
  • Dedication v
  • Foreword vi
  • Contents vii
  • Preface xvi
  • Chapter 1 - Scientific Inference 1
  • Preface 30
  • Chapter 2 - Statistical Inference 31
  • How to Do Exercises 54
  • Exercises for Chapter 2 54
  • Preface 58
  • Chapter 3 - Elements of Analysis of Variance I 59
  • Notes 75
  • Appendix: How to Randomize 77
  • Exercises for Chapter 3 84
  • Preface 90
  • Chapter 4 - Elements of Analysis of Variance II 91
  • Notes 111
  • Exercises for Chapter 4 113
  • Preface 118
  • Chapter 5 - Factorial Design 119
  • Notes 145
  • Appendix: Hand Calculation for Factorial Design 148
  • Exercises for Chapter 5 151
  • Preface 158
  • Chapter 6 - Repeated Measures Design 159
  • Notes 177
  • Exercises for Chapter 6 181
  • Preface 188
  • Chapter 7 - Understanding Interactions 189
  • Notes 209
  • Exercises for Chapter 7 214
  • Preface 218
  • Chapter 8 - Confounding 219
  • Notes 250
  • Preface 258
  • Chapter 9 - Regression and Correlation 259
  • Notes 280
  • Exercises for Chapter 9 282
  • Preface 286
  • Chapter 10 - Frequency Data and Chi-Square 287
  • Notes 300
  • Exercises for Chapter 10 302
  • Preface 306
  • Chapter 11 - Single Subject Design 307
  • Notes 338
  • Exercises for Chapter 11 345
  • Preface 350
  • Chapter 12 - Nonnormal Data and Unequal Variance 351
  • Notes 373
  • Exercises for Chapter 12 378
  • Preface 382
  • Chapter 13 - Analysis of Covariance 383
  • Notes 395
  • Exercises for Chapter 13 397
  • Preface 400
  • Chapter 14 - Design Topics I 401
  • Notes 431
  • Exercises for Chapter 14 437
  • Preface 442
  • Chapter 15 - Design Topics II 443
  • Notes 475
  • Exercises for Chapter 15 481
  • Preface 484
  • Chapter 16 - Multiple Regression 485
  • Notes 514
  • Exercises for Chapter 16 520
  • Preface 524
  • Chapter 17 - Multiple Comparisons 525
  • Notes 546
  • Exercises for Chapter 17 548
  • Preface 550
  • Chapter 18 - Sundry Topics 551
  • Notes 589
  • Exercises for Chapter 18 596
  • Preface 602
  • Chapter 19 - Foundations of Statistics 603
  • Notes 637
  • Preface 646
  • Chapter 20 - Mathematical Models for Process Analysis 647
  • Notes 677
  • Exercises for Chapter 20 681
  • Preface 688
  • Chapter 21 - Toward Unified Theory 689
  • Notes 729
  • Exercises for Chapter 21 742
  • Preface 750
  • Chapter 22 - Principles and Tactics of Writing Papers 751
  • Notes 761
  • Preface 764
  • Chapter 23 - Lifelong Learning 765
  • Notes 780
  • Preface 782
  • Chapter 0 - Basic Statistical Concepts 783
  • Notes 803
  • Exercises for Chapter 0 805
  • Statistical Tables 808
  • References 820
  • Author Index 847
  • Subject Index 854


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