# Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES

Wickens (1989) gives a good treatment of multiway contingency tables. His basic chapters require study, but seem essential for anyone who wishes to do serious work with frequency data. Usefully different perspectives appear in Fleiss (1981) and in Agresti (1996), who give many illustrative sets of data as well as helpful exercises.

10.1.1a
The polio cases in Table 10.1 include both paralytic and nonparalytic cases listed in Meier's (1989) Table 1. The frequencies of paralytic cases were 33 and 115 in the Vaccine and Placebo groups, respectively.
10.1.4a
Homogeneity and Independence. Contingency tables are mainly of two kinds. In one kind, one variable (row or column) represents different groups of subjects, as with the polio vaccine experiment, and the other variable represents the subject classification. The null hypothesis says that the proportion of each class of response is the same for each group—that the groups are homogeneous.

In the other kind of contingency table, there is only a single group. Row and column variables represent two characteristics of the same subjects, as with the height-happiness example of Table 10.3. The null hypothesis says that the two subject characteristics are uncorrelated, or independent.

For two-way contingency tables, fortunately, the X2 test is exactly the same for both homogeneity and independence. With more than two variables, however, homogeneity and independence may yield different E values and hence different values of X2 for the same numerical data. This is one complication that arises with more than two variables.

10.1.4b
The chi-square distribution was introduced in 1900 by Karl Pearson, the major figure in the early development of statistics. Pearson, however, thought that the df for a contingency table equaled rows × columns − 1. In Pearson's view, a 2 × 2 table would thus have 3 df, not 1. When Fisher solved this statistical problem, obtaining Equation 2, Pearson gave him a hostile reception, quoted by Agresti (1996, pp. 251 f):

I hold such a view [Fisher's] is entirely erroneous.… pardon me for comparing him with Don Quixote tilting at the windmill; he must either destroy himself, or the whole theory of probable errors.

To which Fisher replied (rather mildly for Fisher; see Fisher, 1956, pp. 2–3):

If peevish intolerance of free opinion in others is a sign of senility, it is one which he [Pearson] has developed at an early age.

Pearson's substantial contributions to the early development of statistics would shine more brightly today had he been more tolerant, for Fisher was mercilessly correct. Agresti concludes his historical survey of chi-square (p. 265) by saying:

And so, it is fitting that we end this brief survey by giving yet further credit to R. A. Fisher for his influence on the practice of modern statistical science.

10.1.5a
The theoretical chi-square distribution (χ2) is almost as important as the normal distribution in statistical theory. In fact, χ2 is distributed as the square of a unit normal variable. The χ2 distribution thus appears in Anova: Under the null hypothesis, numerator and denominator of the F ratio have (independent) χ2 distributions.

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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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