Linear Mixed Model

The simplest form of the linear mixed model is the random-effects model, which represents data using the regression equation: $$\displaystyle{ \mathbf{y}_{i} =\boldsymbol{\alpha } +\mathbf{b}_{i} +\boldsymbol{\epsilon } _{i} (1 \leq i \leq m), }$$ where \(\boldsymbol{\alpha }\), y i , b i , and \(\boldsymbol{\epsilon }_{i}\) are column matrices for which the lengths are n i and can be expressed in the form: $$\displaystyle\begin{array}{rcl} \boldsymbol{\alpha } = \left (\begin{array}{c} \alpha \\ \alpha \\ \alpha \\ \vdots\\ \alpha \end{array} \right ), \mathbf{y}_{i} = \left (\begin{array}{c} y_{1i} \\ y_{2i} \\ y_{3i}\\ \vdots \\ y_{n_{i}i} \end{array} \right ), \mathbf{b}_{i} = \left (\begin{array}{c} b_{i} \\ b_{i} \\ b_{i}\\ \vdots \\ b_{i} \end{array} \right ), \boldsymbol{\epsilon }_{i} = \left (\begin{array}{c} \epsilon _{1i}\\ \epsilon _{ 2i}\\ \epsilon _{3i}\\ \vdots \\ \epsilon _{n_{i}i} \end{array} \right ).& &{}\end{array}$$ Here, {y ji } (1 ≤ j ≤ n i ) are observations of the i-th treatment (1 ≤ i ≤ m); {b i } (1 ≤ i ≤ m) are realizations from N(0, d 2) (normal distribution with mean 0 and variance d 2).