Mathematical Statistics

4Linear Models – General Theory

4.1 Linear Models with Fixed Effects

The theory of linear statistical models plays an important role in the applications. Mainly the standard methods of analysis of variance and regression analysis have become firmly established in evaluating biological and technological experiments.

In this chapter we introduce the general theory concerning methods of analysis of variance and regression analysis with fixed effects. In the following Ω ⊂ Rⁿ denotes a p‐dimensional linear subspace with p < n called parameter space, and θ ∈ Ω denotes a parameter vector with n coordinates θ_i(i = 1,  … , n).

Further, let Y be an n‐dimensional random variable (a random vector) with components y_i(i = 1,  … , n) and realisations Y from the n‐dimensional sample space Rⁿ. Finally, let e be an n‐dimensional random variable with E(e) = 0_n, var(e) = σ²V, where V is a symmetric and positive definite matrix of size (n, n) and rank n. For constructing tests and confidence intervals, we will later suppose that e (and hence also Y) are n‐dimensional normally distributed (satisfy n‐variate normal distributions).

Definition 4.1

The equation

(4.1)

including the constraints θ ∈ Ω, E(e) = 0_n, var(e) = σ²V is said to be a general linear model (with fixed effects). If ω ⊂ Ω is a linear subspace of Ω, then the hypothesis H₀ : θ ∈ ω is called linear.

The definition of a linear hypothesis ...