Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES

A good short treatment of analysis of covariance is given by Snedecor and Cochran (1980, Chapter 18). The book by Huitema (1980) is well-informed, thorough, and reasonably elementary. Among psychological texts, Maxwell and Delaney (1990) give an extensive discussion with a positive outlook on Ancova. Reichardt (1979) gives an illuminating discussion of Ancova with nonrandom groups, some of which is summarized in Section 15.5 on quasi-experimental design. I am indebted to Anthony Greenwald and Charles Reichardt for helpful comments on this chapter.

13.1.1a
Strictly speaking, Ancova is not simply Anova on the deviations from the regression line (Maxwell, Delaney, & Mannheimer, 1985). Intuitively, however, it is helpful to visualize Ancova in this way. Computer programs use a least squares analysis of Equation 2, which yields correct results.
13.1.1b
The adjustment of the means in Equation 4 assumes the regression is linear. With random groups, this linearity assumption is usually reasonable, even when the true regression is substantially nonlinear, because the X .j will be relatively close together.
13.1.1c
A covariate may yield a worthwhile reduction in MSerror even though its true correlation with Y is only ρ =.4, too small to be dependably statsig at α =.05. Hence a covariate may be retained for future use even though it is not statsig in the present application. One determinant in this decision is prior belief in the evidence value of the covariate, to which the evidence of the present experiment is subsidiary. In this situation, of course, a “false alarm” usually has low statistical cost and an α of.20 would seem often appropriate.
13.1.2a
Using blocks and Ancova together may be most effective. The historically first application of Ancova (Fisher, 1932) was used to illustrate a blocks-and-Ancova analysis by Snedecor and Cochran (1980, p. 371). Y was the current yield on 16 plots of tea bushes in Ceylon; X was yield in the previous year, prior to applying the four experimental treatments. In one set of these data, blocking reduced MSerror from 136 to 48; Ancova produced a further reduction to 27. Good design and analysis, at almost no cost, thus increased precision five-fold. Even 30% increases, which are more to be expected, will often be worthwhile (see similarly Maxwell and Delaney, 1990, pp. 395 ff).
13.1.2b
Significance tests are not too meaningful for assessing differences on the covariate between randomized groups. The difference in any actual sample is real in that sample (barring measurement unreliability). Whether the group differences on X are statsig depends strongly on N, which is irrelevant to the issue.
13.1.2c
Ancova with a correlated response measure also suffers a statistical bias. Unreliability in X causes the sample regression coefficient b1 to underestimate the true coefficient β1. Causal interpretation requires a structural regression, whereas standard Ancova incorporates a predictive regression. When Ancova with the structural regression would equalize the adjusted Y means, the standard Ancova will not, and vice versa. The outcome is thus ambiguous regardless of whether Ancova leaves the differences statsig or nonstatsig. Standard Ancova is thus conceptually invalid, although correction for the unreliability may be possible (Note 13.2.2b).

-395-

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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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