3 Linear Models
Sections 1 and 2 have been preparation. The analysis of variance models of Section 1 and the regression models of Section 4 are not two subjects: both are \(\underline {Y}=XB+\underline {\varepsilon }\), differing only in what the columns of \(X\) contain — indicators of group membership in one case, measured covariates in the other. This section treats them together.
The programme is: define the general linear model and its assumptions; obtain the least squares estimator and its distribution; establish that it is best among linear unbiased estimators; determine which parameters the data can speak about at all when \(X\) is rank deficient, as every analysis of variance design is; give one test that covers every linear hypothesis; and then apply the whole apparatus to designs with two and three factors.
3.2 Review of Linear Algebra
3.3 Estimation of Parameters in a Linear Model
3.4 Estimability and the Generalised Inverse
3.5 The General Linear Hypothesis
3.6 Practice Problems
3.7 Simultaneous Confidence Intervals and Multiple Comparison
One-dimensional case
Simultaneous Confidence Intervals
Using Bonferroni’s Inequality
Simultaneous Confidence Intervals by Scheffé’s Method
Tukey Method of Multiple Comparison
Studentised range Distribution
Multiple Comparison Confidence Intervals
3.8 Two-Factor Studies
Computation Formula
3.9 Three-Way Analysis Of Variance
3.10 Latin Square Design
3.11 Practice Problems
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