https://cc.tir.tw/wk/ 2026-05-19T10:29:57+00:00 https://cc.tir.tw/wk/ https://cc.tir.tw/wk/_media/wiki:logo.png text/html 2022-03-20T08:37:27+00:00 Anonymous (anonymous@undisclosed.example.com) https://cc.tir.tw/wk/notes:dan:contrast-matrix?rev=1647765447&do=diff Contrast Matrix Formulation The constraints on the full model can be written as $$\mathbf{L} 𝛃 = \mathbf{c} $$ Using Lagrange multiplier, we can get the estimate for null hypothesis $\widehat{𝛃}_0$ by minimizing $$\mathcal{L}\equiv\left(\mathbf{y}-\mathbf{X}𝛃\right)^{T}\left(\mathbf{y}-\mathbf{X}𝛃\right)+\left(\mathbf{L}𝛃-\mathbf{c}\right)^{T}𝛌,$$ where $𝛌$ is a column vector of multipliers. The minimization leads to $$ -2\mathbf{X}^{T}\left(\mathbf{y}-\mathbf{X}𝛃_{0}\right)+\mathbf{L}^{T}𝛌_{0}…