Linear Statistical Models Developed and refined over a period of twenty years, the material in this book offers an especially lucid presentation of linear statistical models. These models lead to what is usually called "multiple regression" or "analysis of variance" methodology, which, in turn, opens up a wide range of applications to the physical, biological, and social sciences, as well as to business, agriculture, and engineering. Unlike similar books on this topic, Linear Statistical Models emphasizes the geometry of vector spaces because of the intuitive insights this approach brings to an understanding of the theory. While the focus is on theory, examples of applications, using the SAS and S-Plus packages, are included. Prerequisites include some familiarity with linear algebra, and probability and statistics at the postcalculus level. Major topics covered include: * Methods of study of random vectors, including the multivariate normal, chi-square, t and F distributions, central and noncentral * The linear model and the basic theory of regression analysis and the analysis of variance * Multiple regression methods, including transformations, analysis of residuals, and asymptotic theory for regression analysis. Separate sections are devoted to robust methods and to the bootstrap. * Simultaneous confidence intervals: Bonferroni, Scheffe, Tukey, and Bechhofer * Analysis of variance, with two- and three-way analysis of variance * Random component models, nested designs, and balanced incomplete block designs * Analysis of frequency data through log-linear models, with emphasis on vector space viewpoint. This chapter alone is sufficient for a course on the analysis of frequency data. Linear Statistical Models......Page 5 Contents......Page 9 Preface......Page 13 1.1. Introduction......Page 17 1.2. Vectors, Inner Products, Lengths......Page 19 1.3. Subspaces, Projections......Page 23 1.4. Examples......Page 34 1.5. Some History......Page 41 1.6. Projection Operators......Page 45 1.7. Eigenvalues and Eigenvectors......Page 53 2.1. Covariance Matrices......Page 61 2.2. Expected Values of Quadratic Forms......Page 66 2.3. Projections of Random Variables......Page 69 2.4. The Multivariate Normal Distribution......Page 74 2.5. The x2, F, and t Distributions......Page 78 3.1. The Linear Hypothesis......Page 91 3.2. Confidence Intervals and Test on n = c1β1 + ··· ckβk......Page 99 3.3. The Gauss-Markov Theorem......Page 103 3.4. The Gauss-Markov Theorem for the General Case......Page 108 3.5. Interpretation of Regression Coefficients......Page 111 3.6. The Multiple Correlation Coefficient......Page 113 3.7. The Partial Correlation Coefficient......Page 116 3.8. Testing H0:θεΕV0 [omitted] V......Page 121 3.9. Further Decomposition of Subspaces......Page 133 3.10. Power of the F-Test......Page 137 3.11. Confidence and Prediction Intervals......Page 139 3.12. An Example from SAS......Page 143 3.13. Another Example: Salary Data......Page 154 4.1. Linearizing Transformations......Page 161 4.2. Specification Error......Page 168 4.3. “Generalized” Least Squares......Page 179 4.4. Effects of Additional or Fewer Observations......Page 183 4.5. Finding the “Best” Set of Regressors......Page 190 4.6. Examination of Residuals......Page 198 4.7. Collinearity......Page 203 4.8. Asymptotic Normality......Page 213 4.9. Spline Functions......Page 219 4.10. Nonlinear Least Squares......Page 224 4.11. Robust Regression......Page 229 4.12. Bootstrapping in Regression......Page 237 5. Simultaneous Confidence Intervals......Page 246 5.1. Bonferroni Confidence Intervals......Page 247 5.2. Scheffé Simultaneous Confidence Intervals......Page 248 5.3. Tukey Simultaneous Confidence Intervals......Page 251 5.4. Comparison of Lengths......Page 255 5.5. Rechhofer’s Method......Page 257 6.1. Two-way Analysis of Variance......Page 261 6.2. Unequal Numbers of Observations per Cell......Page 275 6.3. Two-way Analysis of Variance, One Observation per Cell......Page 279 6.4. Design of Experiments......Page 281 6.5. Three-Way Analysis of Variance......Page 282 6.6. The Analysis of Covariance......Page 292 7.1. The Random Effects Model......Page 299 7.2. Nesting......Page 304 7.3. Split Plot Designs......Page 308 7.4. Balanced Incomplete Block Designs......Page 311 8. Analysis of Frequency Data......Page 320 8.1. Examples......Page 321 8.2. Distribution Theory......Page 323 8.3. Confidence Intervals on Poisson and Binomial Parameters......Page 340 8.4. Log-Linear Models......Page 352 8.5. Estimation for the Log-Linear Model......Page 364 8.6. The Chi-square Statistics......Page 382 8.7. The Asymptotic Distributions of β, μ, and m......Page 389 8.8. Logistic Regression......Page 407 References......Page 417 Appendix......Page 424 Answers......Page 447 Author Index......Page 459 Subject Index......Page 461
Linear Statistical Models Developed and refined over a period of twenty years, the material in this book offers an especially lucid presentation of linear statistical models. These models lead to what is usually called "multiple regression" or "analysis of variance" methodology, which, in turn, opens up a wide range of applications to the physical, biological, and social sciences, as well as to business, agriculture, and engineering. Unlike similar books on this topic, Linear Statistical Models emphasizes the geometry of vector spaces because of the intuitive insights this approach brings to an understanding of the theory. While the focus is on theory, examples of applications, using the SAS and S-Plus packages, are included. Prerequisites include some familiarity with linear algebra, and probability and statistics at the postcalculus level. Major topics covered include:
* Methods of study of random vectors, including the multivariate normal, chi-square, t and F distributions, central and noncentral
* The linear model and the basic theory of regression analysis and the analysis of variance
* Multiple regression methods, including transformations, analysis of residuals, and asymptotic theory for regression analysis. Separate sections are devoted to robust methods and to the bootstrap.
* Simultaneous confidence intervals: Bonferroni, Scheffe, Tukey, and Bechhofer
* Analysis of variance, with two- and three-way analysis of variance
* Random component models, nested designs, and balanced incomplete block designs
* Analysis of frequency data through log-linear models, with emphasis on vector space viewpoint. This chapter alone is sufficient for a course on the analysis of frequency data.
Linear Algebra, Projections -- Random Vectors -- Linear Model -- Fitting Of Regression Models -- Simultaneous Confidence Intervals -- Two-way And Three-way Analyses Of Variance -- Miscellaneous Other Models -- Analysis Of Frequency Data. James H. Stapleton. A Wiley-interscience Publication. Includes Bibliographical References (p. 401-407) And Indexes. This graduate-level text concentrates on linear statistical models: mathematical models which are linear in the unknown parameters and which include a random error term. This error term is what makes the models "statistical". The author focuses on the geometry of these vector spaces.
