## Theory and Application of the Linear Model (Duxbury Classic)

### Franklin A. Graybill

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In THEORY AND APPLICATION OF THE LINEAR MODEL, Franklin A. Graybill integrates the linear statistical model within the context of analysis of variance, correlation and regression, and design of experiments. With topics motivated by real situations, it is a time tested, authoritative resource for experimenters, statistical consultants, and students.

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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

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