Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES
9.1.2a
For simplicity of exposition, this paragraph assumes that Y and X have symmetric distributions.
9.1.2b
The slope coefficient of ½ in Galton's equation comes from a more extensive follow-up by Pearson and Lee (1903), who obtained a stronger relation than Galton.
9.1.2c
Regression equations are misleading because they ignore the error of prediction. In Galton's father-son equation, sons of the same father usually differ considerably in height. The correlation is only.5; the scatterplot has a lot of scatter that almost obscures the relation. Freedman. et al. (1998, p. 172) comment, “It was a stroke of genius on Galton's part to see a straight line in the chaos.” The predictive power of GRE for success in graduate school is even smaller, although not less than that of an admissions committee.
9.1.3a
Formulas for the confidence band for a regression line, and for the confidence interval for any predicted value, are given in many texts (e.g., Myers & Well, 1991; Snedecor & Cochran, 1980). I have adopted the symbol, Xnew, from Myers and Well.
9.1.3b
Pure Error Through Within Cell Replication. The regression error term obtained from Equation 8b differs from the Anova error term of previous chapters because it includes systematic deviations from linearity as well as pure error. A pure error term can sometimes be obtained that requires no assumption about the form of the regression equation. Suppose X is controlled experimentally and that two or more independent cases are run at each value of X. Since these cases are treated alike, their variability is pure error. This variability may be pooled across all values of Xi to obtain a single pooled error term that is independent of the form of the regression model. In fact, the values of X may be considered the levels of a one-way Anova. If this error term is used, the linear regression is identical to the linear trend of Chapter 4.
9.1.4a
Assumptions for regression differ somewhat, depending on whether X is fixed, as when experimentally manipulated, or random, as with father's height in Galton's equation and with most observational data. In practice, these different assumptions lead to the same results. In either case, linearity and independence imply that b0 and b1, are unbiased estimates of the population parameters, β0 and β1. Confidence intervals and significance tests are also the same.
9.1.4b
Extreme cases can be extremely serious in regression analysis. Kahn and Udry (1986) reanalyzed data considered in a previous report that had claimed surprising conclusions about age changes in marital coital frequency, showing that these conclusions were largely due to eight extreme cases out of 2063 couples, four of which were thought to be keypunch errors in the archival data. When a few extreme cases make a big difference, including them will misrepresent the main population.
9.1.4c
The main generalization that can be hoped for from a predictive regression is that the predictor variables will retain some usefulness in other situations. The values of b0 and b1, would usually have to be calibrated anew for each new situation.

-280-

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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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