# Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES
4.1.1a
The advantages of confidence intervals over standard t and F tests have been noted by many (e.g., Cochran & Cox, 1957, p. 5). In psychology, Grant (1962), Cohen (1988), Loftus (1993), and Reichardt and Gollob (1997), among many others, have been strong advocates of confidence intervals.
4.1.1b
With unequal n, the confidence interval for the difference between two independent means is

Note that the square root term is the standard deviation of the mean difference. You can check that this expression reduces to Expression 1a when n1 = n2.

4.1.3a
A seeming peculiarity of the significance test is that an observed difference between two means may be statsig but opposite to the true difference. Rejecting H0 only implies that the true means are not equal, not which is greater. The obvious solution is to decide direction by visual inspection. When the true difference is very close to zero, visual inspection will lead to an incorrect directional conclusion with probability very close to ½ α. This has been reified as “Type III” error to distinguish it from the Type I error (false alarm) of claiming a real effect when the true effect is exactly zero. There is no way to avoid this possibility when the real effect is small.

This issue has been discussed at length by Leventhal and Huynh (1996). They argue for a directional version of the null hypothesis, but this differs little from standard practice based on visual inspection. Their main finding is that standard power calculations yield slight overestimates in selected extreme cases.

In the range view of the null hypothesis, Type I error includes Type III error as a special case. If the true effect is substantial, on the other hand, a wrong directional conclusion is improbable.

4.1.6a
A coin toss helps clarify the difference between confidence and probability. Before we toss the coin, the probability of heads is ½. Now we toss the coin but do not look at the outcome. The outcome, however, is either heads or tails; there is no longer a chance element, which is essential to probability. In the frequentist view, our ignorance about the outcome is not probability.

But we may have 50% confidence in heads; even odds on the outcome is a fair bet. Confidence is thus a valid guide to action (see Confidence, Probability, and Belief, Section 19.1.2, page 607).

4.2.1a
Usually, the linear trend will be superior to the overall F. However, the numerical example in the text unduly favors the trend test because these data are perfectly linear. If the data depart substantially from linearity, the overall F might have more power, although it would be less informative (see further Sections 18.2 and 18.3).
4.2.1b
In a contrast, the weights must sum to zero. This can always be accomplished by subtracting the mean weight from each weight; the pattern remains unchanged. The weights for the linear trend example would thus be − 1.5, −.5, +.5, + 1.5, obtained by subtracting the mean of 2.5 from the listed values, 1, 2, 3, 4 (Section 18.2).

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#### Cited page

Empirical Direction in Design and Analysis

• 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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