Analysis of Messy Data, Volume III: Analysis of Covariance

George A. Milliken, Dallas E. Johnson158488083X, 9781584880837

Analysis of covariance is a very useful but often misunderstood methodology for analyzing data where important characteristics of the experimental units are measured but not included as factors in the design. Analysis of Messy Data, Volume 3: Analysis of Covariance takes the unique approach of treating the analysis of covariance problem by looking at a set of regression models, one for each of the treatments or treatment combinations. Using this strategy, analysts can use their knowledge of regression analysis and analysis of variance to help attack the problem.

The authors describe the strategy for one- and two-way treatment structures with one and multiple covariates in a completely randomized design structure. They present new methods for comparing models and sets of parameters, including beta-hat models. They carefully investigate the effect of blocking, explore mixed models, and present a new methodology for using covariates to analyze data from nonreplicated experiments.

Analysis of covariance provides an invaluable set of strategies for analyzing data. With its careful balance of theory and examples, Analysis of Messy Data: Volume 3 provides a unique and outstanding guide to the strategy’s techniques, theory, and application.

---

Table of contents :
Analysis of Messy Data VOLUME III: ANALYSIS OF COVARIANCE……Page 1
Table of Contents……Page 3
Preface……Page 10
Contents……Page 0
1.2 THE COVARIATE ADJUSTMENT PROCESS……Page 13
1.3 A GENERAL AOC MODEL AND THE BASIC PHILOSOPHY……Page 19
REFERENCES……Page 22
2.1 THE MODEL……Page 23
2.2 ESTIMATION……Page 24
2.3 STRATEGY FOR DETERMINING THE FORM OF THE MODEL……Page 26
2.4 COMPARING THE TREATMENTS OR THE REGRESSION LINES……Page 29
2.4.1 EQUAL SLOPES MODEL……Page 30
2.4.2 UNEQUAL SLOPES MODEL-COVARIATE TREATMENT INTERACTION……Page 33
2.6 SUMMARY OF STRATEGIES……Page 37
2.7.1 USING PROC GLM PROC MIXED……Page 38
2.7.2 USING JMP……Page 43
2.8 CONCLUSIONS……Page 50
EXERCISE……Page 51
3.2 CHOCOLATE CANDY Û EQUAL SLOPES……Page 52
3.2.1 A NALYSIS U SING PROC GLM……Page 53
3.2.2 A NALYSIS U SING PROC MIXED……Page 58
3.2.3 A NALYSIS U SING JMP ®……Page 61
3.3 EXERCISE PROGRAMS AND INITIAL RESTING HEART RATE — UNEQUAL SLOPES……Page 65
3.4 EFFECT OF DIET ON CHOLESTEROL LEVEL: AN EXCEPTION TO THE BASIC ANALYSIS OF COVARIANCE STRATEGY……Page 77
3.5 CHANGE FROM BASE LINE ANALYSIS USING EFFECT OF DIET ON CHOLESTEROL LEVEL DATA……Page 81
3.6 SHOE TREAD DESIGN DATA FOR EXCEPTION TO THE BASIC STRATEGY……Page 85
3.7 EQUAL SLOPES WITHIN GROUPS OF TREATMENTS AND UNEQUAL SLOPES BETWEEN GROUPS……Page 89
3.8 UNEQUAL SLOPES AND EQUAL INTERCEPTS — PART 1……Page 94
3.9 UNEQUAL SLOPES AND EQUAL INTERCEPTS — PART 2……Page 96
EXERCISES……Page 101
4.2 THE MODEL……Page 103
4.4 EXAMPLE: DRIVING A GOLF BALL WITH DIFFERENT SHAFTS……Page 105
4.5 EXAMPLE: EFFECT OF HERBICIDES ON THE YIELD OF SOYBEANS — THREE COVARIATES……Page 109
4.6 EXAMPLE: MODELS THAT ARE QUADRATIC FUNCTIONS OF THE COVARIATE……Page 115
4.7 EXAMPLE: COMPARING RESPONSE SURFACE MODELS……Page 122
EXERCISES……Page 131
5.2 THE MODEL……Page 133
5.3 USING THE SAS SYSTEM……Page 137
5.3.1 USING PROC GLM PROC MIXED……Page 138
5.3.2 USING JMP……Page 139
5.4 EXAMPLE: AVERAGE DAILY GAINS AND BIRTH WEIGHT Û COMMON SLOPE……Page 140
5.5 EXAMPLE: ENERGY FROM WOOD OF DIFFERENT TYPES OF TREES – SOME UNEQUAL SLOPES……Page 146
5.6 MISSING TREATMENT COMBINATIONS……Page 154
5.7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH MISSING CELLS……Page 157
5.8 EXTENSIONS……Page 168
EXERCISES……Page 170
6.2 THE BETA-HAT MODEL AND ANALYSIS……Page 172
6.3 TESTING EQUALITY OF PARAMETERS……Page 174
6.4 COMPLEX TREATMENT STRUCTURES……Page 175
6.5 EXAMPLE: ONE-WAY TREATMENT STRUCTURE……Page 176
6.6 EXAMPLE: TWO-WAY TREATMENT STRUCTURE……Page 180
EXERCISES……Page 183
7.2 PROCEDURE FOR EQUAL SLOPES……Page 184
7.3 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH EQUAL SLOPES MODEL……Page 186
7.4 SOME THEORY……Page 193
7.5 WHEN SLOPES ARE POSSIBLY UNEQUAL……Page 194
EXERCISES……Page 195
8.1 INTRODUCTION……Page 197
8.2 TESTING EQUALITY OF MODELS FOR A ONE-WAY TREATMENT STRUCTURE……Page 198
8.3 COMPARING MODELS FOR A TWO-WAY TREATMENT STRUCTURE……Page 199
8.4 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH ONE COVARIATE……Page 201
8.5 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH THREE COVARIATES……Page 203
8.6 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH ONE COVARIATE……Page 205
8.7 DISCUSSION……Page 208
EXERCISES……Page 209
9.2 COMPLETE BLOCK DESIGNS……Page 210
9.3 WITHIN BLOCK ANALYSIS……Page 211
9.4 BETWEEN BLOCK ANALYSIS……Page 213
9.5 COMBINING WITHIN BLOCK AND BETWEEN BLOCK INFORMATION……Page 214
9.6 DETERMINING THE FORM OF THE MODEL……Page 216
9.7 COMMON SLOPE MODEL……Page 218
9.8 COMPARING THE TREATMENTS……Page 221
9.9 CONFIDENCE INTERVALS ABOUT DIFFERENCES OF TWO REGRESSION LINES……Page 222
9.9.2 Combined Within Block and Between Block Analysis……Page 223
9.10 COMPUTATIONS FOR MODEL 9.1 USING THE SAS® SYSTEM……Page 224
9.11 EXAMPLE: EFFECT OF DRUGS ON HEART RATE……Page 228
9.12 SUMMARY……Page 233
EXERCISES……Page 238
10.2 RCB DESIGN STRUCTURE — WITHIN AND BETWEEN BLOCK INFORMATION……Page 240
10.3 INCOMPLETE BLOCK DESIGN STRUCTURE — WITHIN AND BETWEEN BLOCK INFORMATION……Page 241
10.4 COMBINING BETWEEN BLOCK AND WITHIN BLOCK INFORMATION……Page 243
10.5 EXAMPLE: FIVE TREATMENTS IN RCB DESIGN STRUCTURE……Page 247
10.6 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS……Page 254
10.7 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS USING JMP®……Page 258
10.8 SUMMARY……Page 261
EXERCISES……Page 263
11.1 INTRODUCTION……Page 265
11.2 THE WITHIN BLOCK MODEL……Page 266
11.4 COMBINING WITHIN BLOCK AND BETWEEN BLOCK INFORMATION……Page 267
11.5 COMMON SLOPE MODEL……Page 269
11.6.2 Non-Parallel Lines Model……Page 270
11.7 EXAMPLE: TWO TREATMENTS……Page 271
11.8 EXAMPLE: FOUR TREATMENTS IN RCB……Page 275
11.9 EXAMPLE: FOUR TREATMENTS IN BIB……Page 283
11.10 SUMMARY……Page 288
EXERCISES……Page 290
12.2 THE MODEL……Page 292
12.3 ESTIMATION OF THE VARIANCE COMPONENTS……Page 297
12.4 CHANGING LOCATION OF THE COVARIATE CHANGES THE ESTIMATES OF THE VARIANCE COMPONENTS……Page 302
12.5 EXAMPLE: BALANCED ONE-WAY TREATMENT STRUCTURE……Page 304
12.6 EXAMPLE: UNBALANCED ONE-WAY TREATMENT STRUCTURE……Page 309
12.7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE……Page 314
12.8 SUMMARY……Page 320
REFERENCES……Page 325
EXERCISES……Page 326
13.2 THE MATRIX FORM OF THE MIXED MODEL……Page 330
13.4 ESTIMATION OF FIXED EFFECTS AND SOME SMALL SAMPLE SIZE APPROXIMATIONS……Page 334
13.5 FIXED TREATMENTS AND LOCATIONS RANDOM……Page 336
13.6 EXAMPLE: TWO-WAY MIXED EFFECTS TREATMENT STRUCTURE IN A CRD……Page 337
13.7 EXAMPLE: TREATMENTS ARE FIXED AND LOCATIONS ARE RANDOM WITH A RCB AT EACH LOCATION……Page 342
REFERENCES……Page 355
EXERCISES……Page 356
14.2 THE UNEQUAL VARIANCE MODEL……Page 358
14.3.1 Levene’s Test for Equal Variances……Page 359
14.3.3 Bartlett’s Test for Equal Variances……Page 360
14.4.1 Least Squares Estimation……Page 361
14.5 DETERMINING THE FORM OF THE MODEL……Page 362
14.6.1 Comparing the Nonparallel Lines Models……Page 364
14.6.2 Comparing the Parallel Lines Models……Page 366
14.8 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES……Page 367
14.9 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES……Page 374
14.10 EXAMPLE: TREATMENTS IN MULTI-LOCATION TRIAL……Page 386
EXERCISES……Page 394
15.1 INTRODUCTION……Page 396
15.3 COVARIATE MEASURED ON THE WHOLE PLOT OR LARGE SIZE OF EXPERIMENTAL UNIT……Page 397
15.4 COVARIATE IS MEASURED ON THE SMALL SIZE OF EXPERIMENTAL UNIT……Page 400
15.5 COVARIATE IS MEASURED ON THE LARGE SIZE OF EXPERIMENTAL UNIT AND A COVARIATE IS MEASURED ON THE SMALL SIZE OF EXPERIMENTAL UNIT……Page 403
15.6 GENERAL REPRESENTATION OF THE COVARIATE PART OF THE MODEL……Page 404
15.6.1 Covariate Measured on Large Size of Experimental Unit……Page 406
15.6.2 Covariate Measured on the Small Size of Experimental Units……Page 408
15.6.3 Summary of General Representation……Page 410
15.7 EXAMPLE: FLOUR MILLING EXPERIMENT — COVARIATE MEASURED ON THE WHOLE PLOT……Page 411
15.8 EXAMPLE: COOKIE BAKING……Page 419
15.9 EXAMPLE: TEACHING METHODS WITH ONE COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT AND ONE COVARIATE MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT……Page 431
15.10 EXAMPLE: COMFORT STUDY IN A STRIP-PLOT DESIGN WITH THREE SIZES OF EXPERIMENTAL UNITS AND THREE COVARIATES……Page 437
15.11 CONCLUSIONS……Page 449
EXERCISES……Page 451
16.1 INTRODUCTION……Page 456
16.2 THE COVARIANCE PART OF THE MODEL — SELECTING R……Page 458
16.3 COVARIANCE STRUCTURE OF THE DATA……Page 461
16.4 SPECIFYING THE RANDOM AND REPEATED STATEMENTS FOR PROC MIXED OF THE SAS® SYSTEM……Page 462
16.5 SELECTING AN ADEQUATE COVARIANCE STRUCTURE……Page 463
16.6 EXAMPLE: SYSTOLIC BLOOD PRESSURE STUDY WITH COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT……Page 464
16.7 EXAMPLE: OXIDE LAYER DEVELOPMENT EXPERIMENT WITH THREE SIZES OF EXPERIMENTAL UNITS WHERE THE REPEATED MEASURE IS AT THE MIDDLE SIZE OF EXPERIMENTAL UNIT AND THE COVARIATE IS MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT……Page 475
16.8 CONCLUSIONS……Page 484
EXERCISES……Page 492
17.1 INTRODUCTION……Page 497
17.2 EXPERIMENTS WITH A SINGLE COVARIATE……Page 499
17.3 EXPERIMENTS WITH MULTIPLE COVARIATES……Page 503
17.4 SELECTING NON-NULL AND NULL PARTITIONS……Page 505
17.5 ESTIMATING THE PARAMETERS……Page 506
17.6 EXAMPLE: MILLING FLOUR USING THREE FACTORS EACH AT TWO LEVELS……Page 507
17.7 EXAMPLE: BAKING BREAD USING FOUR FACTORS EACH AT TWO LEVELS……Page 512
17.8 EXAMPLE: HAMBURGER PATTIES WITH FOUR FACTORS EACH AT TWO LEVELS……Page 515
17.9 EXAMPLE: STRENGTH OF COMPOSITE MATERIAL COUPONS WITH TWO COVARIATES……Page 516
17.10 EXAMPLE: EFFECTIVENESS OF PAINT ON BRICKS WITH UNEQUAL SLOPES……Page 524
17.11 SUMMARY……Page 531
REFERENCES……Page 533
EXERCISES……Page 534
18.2 BLOCKING AND ANALYSIS OF COVARIANCE……Page 536
18.3 TREATMENTS HAVE DIFFERENT RANGES OF THE COVARIATE……Page 546
18.4.1 Heart Rate Data from Exercise Programs……Page 555
18.4.2 Average Daily Gain Data from a Two-way Treatment Structure……Page 558
18.5 CROSSOVER DESIGN WITH COVARIATES……Page 562
18.6 NONLINEAR ANALYSIS OF COVARIANCE……Page 567
18.7 EFFECT OF OUTLIERS……Page 575
EXERCISES……Page 593