Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

reality to some extent if the measuring scale is nonlinear. Confidence intervals are useful, however, because the semilinearity of most empirical scales of measurement means they are usually approximately correct.

To improve scales of measurement is a continuing goal of empirical science. Among the many aspects of this endeavor, closer approach to linearity is an important one. Although the problem of linearity is basically substantive and extrastatistical, Anova-regression can provide unique assistance as shown in the discussion of psychological measurement theory in Chapter 21.


NOTES
19.1.1a
Few statistical issues have generated as much controversy as Fisher's concept of fiducial probability. In their concluding discussion comparing the three concepts of fiducial interval, confidence interval, and Bayesian belief interval, Kendall and Stuart (1979, p. 165) state

There has been so much controversy … that, at this point, we shall have to leave our customary objective standpoint and descend into the arena ourselves. The remainder of this chapter is an expression of our personal views. We think that it is the correct viewpoint; and it represents the result of many years' silent reflexion on the issues involved, a serious attempt to understand what the protagonists say, and an even more serious attempt to divine what they mean.

19.1.2a
Usefulness of confidence as personal, extrastatistical belief seems to be ignored by Bayesians, who contrast the statistical concept of confidence with the Bayesian concept of personal, statistical belief.

The qualifications noted in the second sentence of the quotation from Mosteller and Tukey deserve comment. “Typicality, ” I take it, rules out samples considered to be biased. “Absence of selection” refers to the diverse dangers of sample selection that capitalize on chance outcomes listed in the later subsection, Bayesian Theory and Randomization. Similar qualifications apply to Bayesian belief intervals.

19.1.2b
Lehmann (1993) considers that the Fisher and Neyman–Pearson theories are complementary, not contradictory (aside from fiducial probability). The main difference is whether tests of composite hypotheses should be conditional, as Fisher later maintained, or unconditional, as Neyman and Pearson claimed. This issue, he says, cannot be decided by abstract principles, but depends on the empirical context of application.
19.1.2c
. Examples in which the confidence interval may be known to be incorrect after the data have been collected are given by Jaynes (1976, p. 198) and by Mosteller and Tukey (1968, p. 181). These examples are not realistic, but they show that the concept of confidence interval is no panacea.

-637-

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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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