General Linear Hypothesis and Analysis of Variance

Suppose that Y ₁,…,Y_n is a sequence of independent random variables with (15.1) E ( Y i ) = ∑ j = 1 p  β j x i j V ( Y i ) = σ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq3085.tif"/> for i = 1,…,n, where β₁,…,β _p , σ² are unknown constants (parameters) and x_ij, i = 1,…,n, j = 1,…,p are known constants. Since E(Y_i ) are linear functions of the parameters β₁,…,β _p , the equations given in (15.1) are called general linear hypothesis models. They are also called linear models for the expectations with independent covariance structure. Let Y = ( Y 1 ⋮ Y n ) ,        β = ( β 1 ⋮ β p ) ,         X = ( x 11 ⋯ x 1 p ⋮ ⋯ ⋮ x n 1 ⋯ x n p ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq3086.tif"/>