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Design of Experiments The Design of Experiments By R. A. .Fisher, Sc.D., F.R.S. Formerly Fellow of Gonville and (Jams College, Cambridge Honorary Member, American Statistical Association and American Academy of Arts and Sciences Galton Professor, University of London Oliver and Boyd Edinburgh: Tweeddale Court London: 33 Paternoster Row, E.C. 1935 CONTENTS I. INTRODUCTION PAC.F 1. The Grounds on whidi Evidence is Disputed 1 2. The Mathematical Attitude towards Induction 3 3. The Rejection of Inverse Probability 6 >4- The Logic of the Laboratory 8 II. THE PRINCIPLES OF EXPERIMENTATION, ILLUSTRATED BY A PSYCIIO-PIIYSICAL EXPERIMENT 5. Statement of Experiment ....... 13 6. Interpretation and its Reasoned Basis ..... 14 -7.. The Test of Significance . 15 8. The Null Hypothesis ....... 18 9. Randomisation ; the Physical Basis of the Validity of the Test 20 10. The Effectiveness of Randomisation ..... 22 It. The Sensitiveness of an Experiment. Effects of Enlargement and Repetition ........ 24 12. Qualitative Methods of increasing Sensitiveness 26 III. A HISTORICAL EXPERIMENT ON GROWTH RATE. 13- ............................... 30 14. Darwin’s Discussion of the Data 31 15. Gajton’s Method of Interpretation 32 » 16. Pairing and Grouping 35 y ¡. “ Student’s ” t Test . 3« 18. Fallacious Use of Statistics 43 19. Manipulation of the Data . 44 20. Validity and Randomisation 46 21. Test of a Wider Hypothesis 50 vii * VIH CONTENTS IV. AN AGRICULTURAL EXPERIMENT IN RANDOMISED BLOCKS PAGE 22. Description of the Experiment ...... 55 23. Statistical Analysis of the Observations .... 57 24. Precision of the Comparisons ...... 64 25'. The Purposes of Replication ...... 66 26. Validity of the Estimation of Error ..... 68 27. Bias of Systematic Arrangements . t . ■ 71 28. Partial Elimination of Error ..... 72 29. Shape of Blocks and Plots . - 7 3 30. Practical Example . 75 V. THE LATÍN SQUARE 31. Randomisation subject to Double Restriction . 78 32. The Estimation of Error ....... 81 33. Faulty Treatment of Square Designs ..... 83 34. Systematic Squares ........ 85 35. Graeco-Latin and Higher Squares ..... 90 36. Practical Exercises ........ 93 VI. THE FACTORIAL DESIGN IN EXPERIMENTATION 37. The Single Factor ........ 96 38. A Simple Factorial Scheme . 98 39. The Basis of Inductive Inference . J06 40. Inclusion of Subsidiary Factors . 107 41. Experiments without Replication . .111 VII. CONFOUNDING 42. The Problem of Controlling Heterogeneity .... Í14 43. Example with 8 Treatments, Notation . .117 44. Design suited to Confounding the Triple Interaction . 119 45- Effect on Analysis of Variance . 120 46. Example with 27 Treatments .... .123 47. Partial Confounding . .131 VIII. SPECIAL CASES OF PARTIAL CONFOUNDING 48. ......................................................................................................138 49. Dummy Comparisons ....... 138 CONTENTS ix •■a g k 50. Interaction of Quantity and Quality . .140 51. Resolution of Three Comparisons among Four Materials 14.’ ^ 5-- An Early Example . 145 Interpretation of R e su lts ..................................................... 154 54. An Experiment with 81 Plots . 157 IX. THE INCREASE OF PRECISION BY CONCOMITANT MEASUREMENTS. STATISTICAL CONTROL X 55. Occasions suitable for Concomitant Measurements . 1O7 56. Arbitrary Corrections .... 174 .57. Calculation of the Adjustment 170 58. The Test of Significance . .181 59. Practical Example . .184 X. THE GENERALISATION OF NULL HYPOTHESES. FIDUCIAL PROBABILITY 60. Precision regarded as Amount of Information . .187 61. Multiplicity of Tests of the same Hypothesis . 190 62. Extension of the t Test ....... 195 63. The X" Test . .198 64. Wider Tests based on the Analysis of Variance . 201 65. Comparisons with Interactions . .211 XI. THE MEASUREMENT OF AMOUNT OF INFORMATION IN GENERAL 66. Estimation in General . 216 67. Frequencies of Two Alternatives . 218 68. Functional Relationships among Parameters . 221 69. The Frequency Ratio in Biological Assay .... 227 70. Linkage Values inferred from Frequency Ratios . 230 71. Linkage Values inferred from the Progeny of Self-fertilised or Intercrossed Heterozygotes ...... 234 72. Information*as to Linkage derived from Human Families . 240 73. The Information elicited by Different Methods of Estimation . 244 74. The Information lost in the Estimation of Error . .247 Index . • 251 PREFACE I n 1925 the author wrote a hook (Statistical Methods fo r Research Workers) with the object of supplying practical experimenters and, incidentally, teachers of mathematical statistics, with a connected account of Jthe applications in laboratory work of some of the more recent advances in statistical theory. Some of the new methods, such as the analysis of variance, were found to be so intimately related with problems of experimental design that a considerable part of the eighth Chapter was devoted to the technique of agricultural experimentation, and these Sections have been progressively enlarged with subsequent editions, in response to frequent requests for a fuller treatment of the subject. The design of experiments is, however, too large a subject, and of too great importance to the general body of scientific workers, for any incidental treatment to be adequate. A clear grasp of simple and standardised statistical procedures will, as the reader may satisfy himself, go far to elucidate the principles of experimentation ; but these procedures are themselves only the means to a more important end. Their part is to satisfy the requirements of sound and intelligible experimental design, and fo supply the machinery for unambiguous interpretation. To attain a clear grasp of these requirements we need to study designs which have been widely successful in many fields, and to examine their structure in relation to the requirements of valid inference. VI PREFACE The examples chosen in this book are aimed at illustrating the principles of successful experimenta­ tion ; first, in their simplest possible applications, and later, in regard to the more elaborate structures by which the different advantages sought may be combined. Statistical discussion has been reduced to a minimum, and all the processes required will be found more fully exemplified in the previous work. The reader is, however, advised that the detailed working of numerical examples is essential to a„ thorough grasp, not only ‘of the technique, but of the principles by which an experimental procedure may be judged to be satisfactory and effective. Galton Laboratory, July 1935- I AM very sorry, Pyrophilus, that to the many (elsewhere enumerated) difficulties which you may meet with, and must therefore surmount, in the serious and effectual prosecution of experimental philosophy I must add one discouragement more, which will perhaps as much surprise as dishearten you ; and it is, that besides that you will find (as we elsewhere mention) many of the experiments published by authors, or related to you by the persons you converse with, false and unsuccessful (besides this, I say), you will meet with several observations and experiments which, though communicated for true by candid authors or undistrusted eye-witnesses, or perhaps recommended by your own experience may, upon further trial, disappoint your expectation, either not at all succeeding constantly or at least varying much from what you expected. R o b e r t B o y l e , 16 7 3 , Concerning the Unsuccessfulness of Experiments. THE DESIGN OF EXPERIMENTS l INTRODUCTION 1. The Grounds on which Evidence is Disputed W h en any scientific conelusion is supposed to be proved on experimental evidence, critics who still refuse to accept the conclusion are accustomed to take one of two lines of attack. They may claim that the interpretation of the experiment is faulty, that the results reported are not in fact those which should have been expected had the conclusion drawn been justified, or that they might equally well have arisen had the conclusion drawn been false. Such criticisms of interpretation are usually treated as falling within the domain of statistics. They are often made by professed statisticians against the work of others whom they regard as ignorant of or incompetent in statistical technique ; and, since the interpretation of any considerable body of data is likely to involve computations, it is natural enough that questions involving the logical implications of the results cff the arithmetical processes employed, should be relegated to the statistician. At least I make no complaint of this convention. The statistician cannot evade the responsibility for understanding the processes he applies or recommends. My immediate 2 INTRODUCTION point is that the questions involved can be dis­ sociated from all that is strictly technical in the statistician’s craft, and, when so detached, are questions only of the right use of human reasoning powers, with which all intelligent people, who hope to be intelligible, are equally concerned, and on which the statistician, as such, speaks with no special authority. The statistician cannot excuse himself from the duty of getting his head clear on the principles of scientific inference, but equally no* other thinking man can avoid a like obligation. The other type of criticism to which experimental results are exposed is that the experiment itself was ill designed, or, of course, badly executed. If we suppose that the experirrfenter did what he intended to do, both of these points come down to the question of the design, or the logical structure of the experiment. This type of criticism is usually
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