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Erschienen in:
2014 | OriginalPaper | Buchkapitel
6. Linear Mixed Model
verfasst von : Kunio Takezawa
Erschienen in: Learning Regression Analysis by Simulation
Verlag: Springer Japan
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Abstract
The simplest form of the linear mixed model is the random-effects model, which represents data using the regression equation: 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: 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).
$$\displaystyle{ \mathbf{y}_{i} =\boldsymbol{\alpha } +\mathbf{b}_{i} +\boldsymbol{\epsilon } _{i} (1 \leq i \leq m), }$$
$$\displaystyle\begin{array}{rcl} \boldsymbol{\alpha } = \left (\begin{array}{c} \alpha \\ \alpha \\ \alpha \\ \vdots\\ \alpha \end{array} \right ), \mathbf{y}_{i} = \left (\begin{array}{c} y_{1i} \\ y_{2i} \\ y_{3i}\\ \vdots \\ y_{n_{i}i} \end{array} \right ), \mathbf{b}_{i} = \left (\begin{array}{c} b_{i} \\ b_{i} \\ b_{i}\\ \vdots \\ b_{i} \end{array} \right ), \boldsymbol{\epsilon }_{i} = \left (\begin{array}{c} \epsilon _{1i}\\ \epsilon _{ 2i}\\ \epsilon _{3i}\\ \vdots \\ \epsilon _{n_{i}i} \end{array} \right ).& &{}\end{array}$$