In jaromilfrossard/lme4permuco: Permutation and randomization tests for mixed models

Description of resampling methods in mixed models

The general form of the model is written:

$$y = X\beta + Zb + \epsilon,$$

where $Z\gamma = \sum_j^J Z_jb_j$ and $b_j\sim (0, \Sigma_{j})$ and $\epsilon\sim (0,I\sigma_\epsilon^2)$. The optimization give us the estimates of the parameters $\theta$ which are the elements of the matrices $\Sigma_j$. We will focusing on case where $\Sigma_j = I\sigma_j$.

Using ML, REML, or assuming all $b$ as fixed effects, the estimation gives us the equation:

$$y = X \hat{\beta} + Z \hat{b} + r.$$

The permutation procedure implemented in this package are of the general form:

$$y^\ast = X \hat{\beta} + \rho_j\sum_j^J Z_j C_jT_jC_j^-\hat{b}_j + T_r r,$$

where $T_j$ and $T_r$ are transformation matrices, like bootstrap, flip or permutation, $C_j$ contrasts matrices and $\rho_j$ is a scaling coefficient.

Transformation

Assuming $T_j$ is a flipping matrix.

We have the average variance:

$$\frac{1}{n_\mathcal{T}}\sum_{\mathcal{T}}\frac{1}{n_j-1}\hat{b}j'T_j'R\textbf{1}T_j\hat{b}_j= \frac{1}{n_j-1}\hat{b}'_j\hat{b}_j$$

jaromilfrossard/lme4permuco documentation built on May 26, 2019, 4:42 p.m.