Given a model $M$, $n$ is the number of predictors, $m$ is the number of columns of $M$'s model matrix.

Define two sets, $\mathcal{C} = \{x \in \mathbb{N}\, |\, 1 \leq x \leq m\}$, the index of columns and $\mathcal{P} = \{x \in \mathbb{N}\, |\, 1 \leq x \leq n\}$, the index of predictors.

A map $id_X: \mathcal{C} \mapsto \mathcal{P}$ maps the index of columns into the corresponding predictor sequentially, i.e.,

\[\begin{aligned} \forall i \in \mathcal{C}, id_X(i) = k &\implies i\text{th column} \text{ is a component of } k\text{th predictor}\\\\ \forall i, j \in \mathcal{C}, i \lt j &\implies id_X(i) \leq id_X(j) \end{aligned}\]

We can define a vector of index set for each predictors,

\[\mathbf{I} = (I_1, ..., I_n)\]

where $\forall i \in I_k, id_X(i) = k$.

The degrees of freedom (dof) is

\[\mathbf{df} = (n(I_1), ..., n(I_n))\]

where $n(I)$ is the size of $I$.

F-value is a vector

\[\mathbf{F} \sim \mathcal{F}_{\mathbf{df}, \mathbf{df_r}}\]

where $\mathbf{df_r}$ is estimated by between-within method.

F-value is computed directly by the variance-covariance matrix ($\boldsymbol \Sigma$) and the coefficients ($\boldsymbol \beta$) of the model.

Type I

Calculate F-value by the the upper factor of Cholesky factorization of $\boldsymbol \Sigma^{-1}$ and multiplying with $\boldsymbol \beta$

\[\begin{aligned} \boldsymbol{\Sigma}^{-1} &= \mathbf{LU}\\\\ \boldsymbol{\eta} &= \mathbf{U}\boldsymbol{\beta}\\\\ F_j &= \frac{\sum_{k \in I_j}{\eta_k^2}}{df_j} \end{aligned}\]

Type II

Define two vectors of index sets $\mathbf J$ and $\mathbf K$ where

\[\begin{aligned} J_j &= \{i \in \mathcal{C}\, |\, id_X(i) \text{ is an interaction term of }j\text{th predictor and other terms}\}\\\\ K_j &= J_j \cup I_j \end{aligned}\]

And F-value is

\[F_j = \frac{\boldsymbol{\beta}_{K_j}^T \boldsymbol{\Sigma}_{K_j; K_j}^{-1} \boldsymbol{\beta}_{K_j} - \boldsymbol{\beta}_{J_j}^T \boldsymbol{\Sigma}_{J_j; J_j}^{-1} \boldsymbol{\beta}_{J_j}}{df_j}\]

Type III

\[F_j = \frac{\boldsymbol{\beta}_{I_j}^T \boldsymbol{\Sigma}_{I_j; I_j}^{-1} \boldsymbol{\beta}_{I_j}}{df_j}\]


Given a vector of models

\[\mathbf{M} = (M_1, ..., M_n)\]

The $\mathcal{D}$ is $-2loglikelihood(\mathbf{M})$ for linear mixed-effect models or ordinary linear models; unit deviance for generalized linear mixed-effect model or generalized linear models.

The likelihood ratio is a vector

\[\begin{aligned} \mathbf{L} &= \mathcal{D}_{[1, n - 1]} - \mathcal{D}_{[2, n]}\\\\ \mathbf{L} &\sim \chi^2_{\mathbf{df}} \end{aligned}\]

where $df_i = dof(M_i) - dof(M_{i+1})$