Contents Preface xi 1 Overview of the General Linear Model 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The General Linear Model . . . . . . . . . . . . . . . . . . . . . . 1 1.3 The Restricted General Linear Model . . . . . . . . . . . . . . . . 3 1.4 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . 4 1.5 Elementary Properties of Normal Random Variables . . . . . . . . . 8 1.6 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Generating Multivariate Normal Data . . . . . . . . . . . . . . . . 10 1.8 Assessing Univariate Normality . . . . . . . . . . . . . . . . . . . 11 1.8.1 Normally and Nonnormally Distributed Data . . . . . . . . 12 1.8.2 Real Data Example . . . . . . . . . . . . . . . . . . . . . . 15 1.9 Assessing Multivariate Normality with Chi-square Plots . . . . . . . 15 1.9.1 Multivariate Normal Data . . . . . . . . . . . . . . . . . . 18 1.9.2 Real Data Example . . . . . . . . . . . . . . . . . . . . . . 19 1.10 Using SAS INSIGHT . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.10.1 Ramus Bone Data . . . . . . . . . . . . . . . . . . . . . . 19 1.10.2 Risk-taking Behavior Data . . . . . . . . . . . . . . . . . . 21 1.11 Three-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . 23 2 Unrestricted General Linear Models 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Linear Models without Restrictions . . . . . . . . . . . . . . . . . . 25 2.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Simultaneous Inference . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . 30 2.5.1 Classical and Normal Regression Models . . . . . . . . . . 31 2.5.2 Random Classical and Jointly Normal Regression Models . 42 2.6 Linear Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.7 One-Way Analysis of Variance . . . . . . . . . . . . . . . . . . . . 53 2.7.1 Unrestricted Full Rank One-way Design . . . . . . . . . . . 54 2.7.2 Simultaneous Inference for the One-Way Design . . . . . . 56 2.7.3 Multiple Testing . . . . . . . . . . . . . . . . . . . . . . . 58 2.8 Multiple Linear Regression:Calibration . . . . . . . . . . . . . . . 58 2.8.1 Multiple Linear Regression: Prediction . . . . . . . . . . . 68 2.9 Two-way Nested Designs . . . . . . . . . . . . . . . . . . . . . . . 70 2.10 Intraclass Covariance Models . . . . . . . . . . . . . . . . . . . . . 72 3 Restricted General Linear Models 77 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Estimation and Hypothesis Testing . . . . . . . . . . . . . . . . . . 77 3.3 Two-Way Factorial Design without Interaction . . . . . . . . . . . . 79 3.4 Latin Square Designs . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5 Repeated Measures Designs . . . . . . . . . . . . . . . . . . . . . 89 3.5.1 Univariate Mixed ANOVA Model, Full Rank Representation for a Split Plot Design . . . . . . . . . . . . . . . . . . . . 90 3.5.2 Univariate Mixed Linear Model, Less than Full Rank Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5.3 Test for Equal Covariance Matrices and for Circularity . . . 97 3.6 Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . 100 3.6.1 ANCOVA with One Covariate . . . . . . . . . . . . . . . . 102 3.6.2 ANCOVA with Two Covariates . . . . . . . . . . . . . . . 104 3.6.3 ANCOVA Nested Designs . . . . . . . . . . . . . . . . . . 106 4 Weighted General Linear Models 109 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Estimation and Hypothesis Testing . . . . . . . . . . . . . . . . . . 110 4.3 OLSE versus FGLS . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 General Linear Mixed Model Continued . . . . . . . . . . . . . . . 114 4.4.1 Example: Repeated Measures Design . . . . . . . . . . . . 117 4.4.2 Estimating the df for the F statistic in GLMMs . . . . . . . 118 4.5 Maximum Likelihood Estimation and Fisher’s Information Matrix . 119 4.6 WLSE for data Heteroscedasticity . . . . . . . . . . . . . . . . . . 121 4.7 WLSE for Correlated Errors . . . . . . . . . . . . . . . . . . . . . 124 4.8 FGLS for Categorical Data . . . . . . . . . . . . . . . . . . . . . . 127 4.8.1 Overview of the Categorical Data Model . . . . . . . . . . 127 4.8.2 Marginal Homogeneity . . . . . . . . . . . . . . . . . . . . 130 4.8.3 Homogeneity of Proportions . . . . . . . . . . . . . . . . . 132 4.8.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.8.5 Univariate Mixed Linear Model, Less than Full Rank Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5 Multivariate General Linear Models 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Developing the Model . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3 Estimation Theory and Hypothesis Testing . . . . . . . . . . . . . . 145 5.4 Multivariate Regression . . . . . . . . . . . . . . . . . . . . . . . . 152 5.5 Classical and Normal Multivariate Linear Regression Models . . . . 153 5.6 Jointly Multivariate Normal Regression Model . . . . . . . . . . . 163 5.7 Multivariate Mixed Models and the Analysis of Repeated Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.8 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 176 5.9 Multivariate Regression: Calibration and Prediction . . . . . . . . . 182 5.9.1 Fixed X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.9.2 Random X . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.9.3 Random X, Prediction . . . . . . . . . . . . . . . . . . . . 186 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
5.9.3 Random X, Prediction . . . . . . . . . . . . . . . . . . . . 186 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
X . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219 6 Doubly Multivariate Linear Model 223 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2 Classical Model Development . . . . . . . . . . . . . . . . . . . . 223 6.3 Responsewise Model Development . . . . . . . . . . . . . . . . . . 226 6.4 The Multivariate Mixed Model . . . . . . . . . . . . . . . . . . . . 227 6.5 Double Multivariate and Mixed Models . . . . . . . . . . . . . . . 231 7 The Restricted MGLM and the Growth Curve Model 243 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.2 The Restricted Multivariate General Linear Model . . . . . . . . . . 243 7.3 The GMANOVA Model . . . . . . . . . . . . . . . . . . . . . . . 247 7.4 Canonical Form of the GMANOVA Model . . . . . . . . . . . . . . 253 7.5 Restricted Nonorthogonal Three-Factor Factorial MANOVA . . . . 259 7.5.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 269 7.6 Restricted Intraclass Covariance Design . . . . . . . . . . . . . . . 269 7.6.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 275 7.7 Growth Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . . 279 7.7.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 283 7.8 Multiple Response Growth Curves . . . . . . . . . . . . . . . . . . 289 7.8.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 290 7.9 Single Growth Curve . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.9.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 294 8 The SUR Model and the Restricted GMANOVA model 297 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.2 The MANOVA-GMANOVA Model . . . . . . . . . . . . . . . . . 297 8.3 Tests of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.4 Sum of Profiles and CGMANOVA Models . . . . . . . . . . . . . . 305 8.5 The SUR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.6 The Restricted GMANOVA Model . . . . . . . . . . . . . . . . . . 314 8.7 GMANOVA-SUR: One Population . . . . . . . . . . . . . . . . . . 317 8.7.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 317 8.8 GMANOVA-SUR: Several Populations . . . . . . . . . . . . . . . 319 8.8.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 319 8.9 SUR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.9.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 323 8.10 Two-Period Crossover Design with Changing Covariates . . . . . . 328 8.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 329 8.11 Repeated Measurements with Changing Covariates . . . . . . . . . 334 8.11.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 335 8.12 MANOVA-GMANOVA Model . . . . . . . . . . . . . . . . . . . . 337 8.12.1 Results and interpretation . . . . . . . . . . . . . . . . . . 338 8.13 CGMANOVA Model . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.13.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 346 9 Simultaneous Inference Using Finite Intersection Tests 349 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 9.2 Finite Intersection Tests . . . . . . . . . . . . . . . . . . . . . . . . 349 9.3 Finite Intersection Tests of Univariate Means . . . . . . . . . . . . 350 9.4 Finite Intersection Tests for Linear Models . . . . . . . . . . . . . . 354 9.5 A Comparisons of Some Tests of Univariate Means . . . . . . . . . 355 9.5.1 Single-Step Methods . . . . . . . . . . . . . . . . . . . . . 355 9.5.2 Stepdown Methods . . . . . . . . . . . . . . . . . . . . . . 357 9.6 Analysis of Means Analysis . . . . . . . . . . . . . . . . . . . . . 358 9.7 Simultaneous Test Procedures for Mean Vectors . . . . . . . . . . . 360 9.8 Finite Intersection Test of Mean Vectors . . . . . . . . . . . . . . . 362 9.9 Finite Intersection Test of Mean Vectors with Covariates . . . . . . 366 9.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 9.11 Univariate: One-way ANOVA . . . . . . . . . . . . . . . . . . . . 369 9.12 Multivariate: One-way MANOVA . . . . . . . . . . . . . . . . . . 372 9.13 Multivariate: One-way MANCOVA . . . . . . . . . . . . . . . . . 379 10 Computing Power for Univariate and Multivariate GLM 381 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 10.2 Power for Univariate GLMs . . . . . . . . . . . . . . . . . . . . . 383 10.3 Estimating Power, Sample Size, and Effect Size for the GLM . . . . 384 10.3.1 Power and Sample Size . . . . . . . . . . . . . . . . . . . . 384 10.3.2 Effect Size . . . . . . . . . . . . . . . . . . . . . . . . . . 385 10.4 Power and Sample Size based upon Interval-Estimation . . . . . . . 388 10.5 Calculating Power and Sample Size for Some Mixed Models . . . . 390 10.5.1 Random One-Way ANOVA Design . . . . . . . . . . . . . 390 10.5.2 Two Factor Mixed Nested ANOVA Design . . . . . . . . . 396 10.6 Power for Multivariate GLMs . . . . . . . . . . . . . . . . . . . . 400 10.7 Power and Effect Size Analysis for Univariate GLMs . . . . . . . . 401 10.7.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . 401 10.7.2 Three-Way ANOVA . . . . . . . . . . . . . . . . . . . . . 403 10.7.3 One-Way ANCOVA Design with two covariates . . . . . . 405 10.8 Power and Sample Size based upon Interval-Estimation . . . . . . . 405 10.8.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . 407 10.9 Power Analysis for Multivariate GLMs . . . . . . . . . . . . . . . 409 10.9.1 Two Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.9.2 Repeated Measures Design . . . . . . . . . . . . . . . . . . 409 11 Two-level Hierarchical Linear Models 413 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 11.2 Two-level Hierarchical Linear Models . . . . . . . . . . . . . . . . 413 11.3 Random Coefficient Model: One Population . . . . . . . . . . . . . 424 11.4 Random Coefficient Model: Several Populations . . . . . . . . . . . 435 11.5 Mixed Model Repeated Measures . . . . . . . . . . . . . . . . . . 440 11.6 Mixed Model Repeated Measures with Changing Covariates . . . . 442 11.7 Two-Level Hierarchical Linear Models . . . . . . . . . . . . . . . . 443 12 Incomplete Repeated Measurement Data 455 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 12.2 Missing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 456 12.3 An FGLS Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 457 12.4 An ML Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 12.5 Imputations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 12.5.1 EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 462 12.5.2 Multiple Imputation . . . . . . . . . . . . . . . . . . . . . 463 12.6 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 464 12.7 Repeated Measures with Changing Covariates . . . . . . . . . . . . 464 12.8 Random Coefficient Model . . . . . . . . . . . . . . . . . . . . . . 467 12.9 Growth Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . . 471 13 Structural Equation Modeling 479 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 13.2 Model Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 13.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 13.4 Model Fit in Practice . . . . . . . . . . . . . . . . . . . . . . . . . 494 13.5 Model Modification . . . . . . . . . . . . . . . . . . . . . . . . . . 496 13.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 13.7 Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 13.8 Confirmatory Factor Analysis . . . . . . . . . . . . . . . . . . . . . 503 13.9 General SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 References 511
4.6 WLSE for data Heteroscedasticity . . . . . . . . . . . . . . . . . . 121 4.7 WLSE for Correlated Errors . . . . . . . . . . . . . . . . . . . . . 124 4.8 FGLS for Categorical Data . . . . . . . . . . . . . . . . . . . . . . 127 4.8.1 Overview of the Categorical Data Model . . . . . . . . . . 127 4.8.2 Marginal Homogeneity . . . . . . . . . . . . . . . . . . . . 130 4.8.3 Homogeneity of Proportions . . . . . . . . . . . . . . . . . 132 4.8.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.8.5 Univariate Mixed Linear Model, Less than Full Rank Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5 Multivariate General Linear Models 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Developing the Model . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3 Estimation Theory and Hypothesis Testing . . . . . . . . . . . . . . 145 5.4 Multivariate Regression . . . . . . . . . . . . . . . . . . . . . . . . 152 5.5 Classical and Normal Multivariate Linear Regression Models . . . . 153 5.6 Jointly Multivariate Normal Regression Model . . . . . . . . . . . 163 5.7 Multivariate Mixed Models and the Analysis of Repeated Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.8 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 176 5.9 Multivariate Regression: Calibration and Prediction . . . . . . . . . 182 5.9.1 Fixed X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.9.2 Random X . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.9.3 Random X, Prediction . . . . . . . . . . . . . . . . . . . . 186 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
5.9.3 Random X, Prediction . . . . . . . . . . . . . . . . . . . . 186 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
X . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.9.4 Overview - Candidate Model . . . . . . . . . . . . . . . . . 186 5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219
5.9.5 Prediction and Shrinkage . . . . . . . . . . . . . . . . . . . 187 5.10 Multivariate Regression: Influential Observations . . . . . . . . . . 189 5.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 191 5.11 Nonorthogonal MANOVA designs . . . . . . . . . . . . . . . . . . 192 5.11.1 Unweighted Analysis . . . . . . . . . . . . . . . . . . . . . 197 5.11.2 Weighted Analysis . . . . . . . . . . . . . . . . . . . . . . 198 5.12 MANCOVA Designs . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.1 Overall tests . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.12.2 Tests of Additional Information . . . . . . . . . . . . . . . 203 5.12.3 Results and Interpretation . . . . . . . . . . . . . . . . . . 204 5.13 Stepdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.14 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 208 5.14.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 209 5.15 Extended Linear Hypotheses . . . . . . . . . . . . . . . . . . . . . 216 5.15.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 219 6 Doubly Multivariate Linear Model 223 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.2 Classical Model Development . . . . . . . . . . . . . . . . . . . . 223 6.3 Responsewise Model Development . . . . . . . . . . . . . . . . . . 226 6.4 The Multivariate Mixed Model . . . . . . . . . . . . . . . . . . . . 227 6.5 Double Multivariate and Mixed Models . . . . . . . . . . . . . . . 231 7 The Restricted MGLM and the Growth Curve Model 243 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.2 The Restricted Multivariate General Linear Model . . . . . . . . . . 243 7.3 The GMANOVA Model . . . . . . . . . . . . . . . . . . . . . . . 247 7.4 Canonical Form of the GMANOVA Model . . . . . . . . . . . . . . 253 7.5 Restricted Nonorthogonal Three-Factor Factorial MANOVA . . . . 259 7.5.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 269 7.6 Restricted Intraclass Covariance Design . . . . . . . . . . . . . . . 269 7.6.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 275 7.7 Growth Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . . 279 7.7.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 283 7.8 Multiple Response Growth Curves . . . . . . . . . . . . . . . . . . 289 7.8.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 290 7.9 Single Growth Curve . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.9.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 294 8 The SUR Model and the Restricted GMANOVA model 297 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.2 The MANOVA-GMANOVA Model . . . . . . . . . . . . . . . . . 297 8.3 Tests of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.4 Sum of Profiles and CGMANOVA Models . . . . . . . . . . . . . . 305 8.5 The SUR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.6 The Restricted GMANOVA Model . . . . . . . . . . . . . . . . . . 314 8.7 GMANOVA-SUR: One Population . . . . . . . . . . . . . . . . . . 317 8.7.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 317 8.8 GMANOVA-SUR: Several Populations . . . . . . . . . . . . . . . 319 8.8.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 319 8.9 SUR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.9.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 323 8.10 Two-Period Crossover Design with Changing Covariates . . . . . . 328 8.10.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 329 8.11 Repeated Measurements with Changing Covariates . . . . . . . . . 334 8.11.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 335 8.12 MANOVA-GMANOVA Model . . . . . . . . . . . . . . . . . . . . 337 8.12.1 Results and interpretation . . . . . . . . . . . . . . . . . . 338 8.13 CGMANOVA Model . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.13.1 Results and Interpretation . . . . . . . . . . . . . . . . . . 346 9 Simultaneous Inference Using Finite Intersection Tests 349 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 9.2 Finite Intersection Tests . . . . . . . . . . . . . . . . . . . . . . . . 349 9.3 Finite Intersection Tests of Univariate Means . . . . . . . . . . . . 350 9.4 Finite Intersection Tests for Linear Models . . . . . . . . . . . . . . 354 9.5 A Comparisons of Some Tests of Univariate Means . . . . . . . . . 355 9.5.1 Single-Step Methods . . . . . . . . . . . . . . . . . . . . . 355 9.5.2 Stepdown Methods . . . . . . . . . . . . . . . . . . . . . . 357 9.6 Analysis of Means Analysis . . . . . . . . . . . . . . . . . . . . . 358 9.7 Simultaneous Test Procedures for Mean Vectors . . . . . . . . . . . 360 9.8 Finite Intersection Test of Mean Vectors . . . . . . . . . . . . . . . 362 9.9 Finite Intersection Test of Mean Vectors with Covariates . . . . . . 366 9.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 9.11 Univariate: One-way ANOVA . . . . . . . . . . . . . . . . . . . . 369 9.12 Multivariate: One-way MANOVA . . . . . . . . . . . . . . . . . . 372 9.13 Multivariate: One-way MANCOVA . . . . . . . . . . . . . . . . . 379 10 Computing Power for Univariate and Multivariate GLM 381 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 10.2 Power for Univariate GLMs . . . . . . . . . . . . . . . . . . . . . 383 10.3 Estimating Power, Sample Size, and Effect Size for the GLM . . . . 384 10.3.1 Power and Sample Size . . . . . . . . . . . . . . . . . . . . 384 10.3.2 Effect Size . . . . . . . . . . . . . . . . . . . . . . . . . . 385 10.4 Power and Sample Size based upon Interval-Estimation . . . . . . . 388 10.5 Calculating Power and Sample Size for Some Mixed Models . . . . 390 10.5.1 Random One-Way ANOVA Design . . . . . . . . . . . . . 390 10.5.2 Two Factor Mixed Nested ANOVA Design . . . . . . . . . 396 10.6 Power for Multivariate GLMs . . . . . . . . . . . . . . . . . . . . 400 10.7 Power and Effect Size Analysis for Univariate GLMs . . . . . . . . 401 10.7.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . 401 10.7.2 Three-Way ANOVA . . . . . . . . . . . . . . . . . . . . . 403 10.7.3 One-Way ANCOVA Design with two covariates . . . . . . 405 10.8 Power and Sample Size based upon Interval-Estimation . . . . . . . 405 10.8.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . 407 10.9 Power Analysis for Multivariate GLMs . . . . . . . . . . . . . . . 409 10.9.1 Two Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.9.2 Repeated Measures Design . . . . . . . . . . . . . . . . . . 409 11 Two-level Hierarchical Linear Models 413 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 11.2 Two-level Hierarchical Linear Models . . . . . . . . . . . . . . . . 413 11.3 Random Coefficient Model: One Population . . . . . . . . . . . . . 424 11.4 Random Coefficient Model: Several Populations . . . . . . . . . . . 435 11.5 Mixed Model Repeated Measures . . . . . . . . . . . . . . . . . . 440 11.6 Mixed Model Repeated Measures with Changing Covariates . . . . 442 11.7 Two-Level Hierarchical Linear Models . . . . . . . . . . . . . . . . 443 12 Incomplete Repeated Measurement Data 455 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 12.2 Missing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 456 12.3 An FGLS Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 457 12.4 An ML Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 12.5 Imputations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 12.5.1 EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 462 12.5.2 Multiple Imputation . . . . . . . . . . . . . . . . . . . . . 463 12.6 Repeated Measures Analysis . . . . . . . . . . . . . . . . . . . . . 464 12.7 Repeated Measures with Changing Covariates . . . . . . . . . . . . 464 12.8 Random Coefficient Model . . . . . . . . . . . . . . . . . . . . . . 467 12.9 Growth Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . . 471 13 Structural Equation Modeling 479 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 13.2 Model Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 13.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 13.4 Model Fit in Practice . . . . . . . . . . . . . . . . . . . . . . . . . 494 13.5 Model Modification . . . . . . . . . . . . . . . . . . . . . . . . . . 496 13.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 13.7 Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 13.8 Confirmatory Factor Analysis . . . . . . . . . . . . . . . . . . . . . 503 13.9 General SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 References 511 |