Theory and Application of the Linear Model

PREFACE 1. MATHEMATICAL CONCEPTS Introduction / Elementary Theorems on Linear and Matrix Algebra / Partitioned Matrices / Nonnegative Matrices / Generalized and Conditional Inverses / Solutions of Linear Equations / Idempotent Matrices / Trace of Matrices / Derivatives of Quadratic and Linear Forms; Expectation of a Matrix / Evaluation of an Integral 2. STATISTICAL CONCEPTS Introduction / Random Variables and Distribution Functions / Moment Generating Function / Independence of Random Vectors / Special Distributions and Some Important Formulas / Statistical Inference / Point Estimation / Hypothesis Testing / Confidence Intervals / Comments on Statistical Inference / Problems 3. THE MULTIDIMENSIONAL NORMAL DISTRIBUTION Introduction / The Univariate Normal Distribution / Multivariate Normal Distribution / Marginal Distributions / Independent and Uncorrelated Random Vectors / Conditional Distribution / Regression / Correlation / Examples / Problems 4. DISTRIBUTIONS OF QUADRATIC FORMS Introductions / Noncentral Chi-Square Distribution / Noncentral F and Noncentral t Distributions / Distribution of Quadratic Forms in Normal Variables / Independence of Linear Forms and Quadratic Forms / Expected Value of a Quadratic Form / Additional Theorems / Problems 5. MODELS Introduction / General Linear Model / Linear Regression Model / Design Models / Components-of-Variance Model 6. GENERAL LINEAR MODEL Introduction / Point Estimation standard deviation and Linear Functions of Beta [i]:Case 1 / Test of the Hypothesis Hb =h: Case 1 / Special Cases for Hypothesis Testing / Confidence Intervals Associated with the Test H[o]: Hb = h / Further discussion of Confidence Intervals Associated with the Test H[o]: Hb = h / Example / The General Linear Model, Case 1, and sum is not equal to the standard deviation x Y / Examination of Assumptions / Inference in the Linear Model: Case 2 / Further Discussion of the Test Hb =h 7. COMPUTING TECHNIQUES Introduction / Square root Method of Factoring a Positive Definite Matrix / Computing Point Estimates, Test Statistics, and Confidence Intervals / Analysis of Variance / The Normal / Equations Using Deviations from Means / Some Computing Procedures When cov[Y] = the standard deviation x V / Appendix / Problems 8. APPLICATIONS OF THE GENERAL LINEAR MODEL Introduction / Prediction Intervals / Tolerance Intervals / Other Tolerance and Associated Intervals / Determining x for a Given Value of Y (The Calibration Problem) / Parallel, Intersecting, and Identical Models / Polynomial Models / Trigonometric Models / Designing Investigations / Maximum or Minimum of a Quadratic Function / Point of Intersection of Two Lines / Problems 9. SAMPLING FROM THE MULTIVARIATE NORMAL DISTRIBUTION Introduction / Notation / Point Estimators of the population mean and the sum / Test of the Hypothesis H[o] :population mean = h[o] / Confidence Intervals on l' [I] population mean, for I = 1,2,?, q/ Computations / Additional Theorems about mu (hat) and sum (hat)/ Problems 10. MULTIPLE REGRESSION Introduction / Multiple Regression Model: Case I, Case II, and Point Estimation / Multiple Regression Model: Confidence Intervals and Test Hypothesis, Case I and Case II / Multiple Regression Model: Case III / Problems 11. CORRELATION Introduction, Simple Correlation, Partial Correlation, Multiple Correlation / Correlation for Non-normal p.d.f.'s / Correlation and Independence of Random Variables / Problems 12. SOME APPLICATIONS OF THE REGRESSION MODEL Introduction / Prediction / Selecting Variables for a Model / Growth Curves / Discrimination (Classification) / Problems 13. DESIGN MODELS Introduction / Point Estimation for the Design Model; Case I / Point Estimation for the Design Model; Case II / Confidence Intervals and Tests of Hypothesis for Case I of the Design Model / Computations / The One-Factor Design Model / Further Discussion of Tests and Confidence Intervals for the Design Models / Problems 14. TWO-FACTOR DESIGN MODEL Introduction / Two-factor Design Model, No Interaction, M > 1 Observations Per Cell / Two-factor Design Model, No Interaction, Unequal Numbers of Observation in Cells / Interaction in the Two-Factor Design Model / Two-Factor Design Model with Interaction and M > 1 Observations Per Cell / Two-Factor Design Model with Interaction and with M = 1 / Two-Factor Model with Interaction and Unequal Number of Observations in the Cells / Some Situations Described by Two-Factor Design Models / Balanced Incomplete Block Models / Test for Interaction / Problems 15. COMPONENTS-OF-VARIANCE MODELS Introduction / One-Factor Components-of-Variance Model; Point Estimation / A General Components-of-Variance Model / Two-Factor Components-of-Variance Model / Other Components-of-Variance Models / Additional Results on Components-of-Variance Models / Proof Theorem / Problems / TABLES / REFERENCES AND FURTHER READING / INDEX