Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

Experimental control over the shape of the response distribution provides opportunities to increase power. Some learning tasks, for example, may be designed to measure either errors in a fixed number of trials or errors to some specified criterion. The latter measure, being unbounded, might be questionable with young children or patients because a few might persevere with some inappropriate strategy, yielding extremely high error scores. In the cited example of easy multiple choice tests, skewness could be reduced by making the test of intermediate difficulty. Among the many vital functions of pilot work, the importance of distribution shape should not be overlooked.

Even after the data are in, their shape can still be changed by applying a statistical transformation (Section 12.4). A long-tailed distribution of response times, for example, becomes more normal by taking reciprocals, that is, by transforming time to speed. Even more useful may be the trimmed Anova of Section 12.1.

The Experimental Pyramid implies that the shape of the data depends primarily on empirical determinants. Task, procedure, and measurement constitute an empirical transformation that determines the shape of the data. Statistical theory, in contrast, concentrates on statistical transformations after the data have been obtained. Although statistical transformations can be helpful, they involve only the tip of the Pyramid. For empirical investigators, the main concern over the shape of the data should instead be focused at the lower levels of the Pyramid in planning the investigation.


NOTES
3.2.1a
The values of F* and t* are standardly called critical values, but critical over-states their function and meaning, as though they derived from some external standard of validity. The present term, criterial values, which recognizes that they function as somewhat arbitrary decision criteria, is suggested as being more suitable.
3.2.2a
A glance at Fisher's formula for the F distribution is worthwhile, not for illumination but just to see that it really exists:

In this equation, v1 and v2 are the df for numerator and denominator, respectively. The constant c is a complicated function of v1 and v2 that makes the area under this F distribution equal to 1.

The curve in Figure 3.1 labeled “H0 true” was obtained from this formula: Prob(F) is the vertical elevation of the curve, as a function of F on the horizontal axis. Different df give different curves of different shape.

-75-

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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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