Linear Mixed Model

Kunio Takezawa
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Kunio Takezawa: National Agricultural and Food Research Organization

Chapter Chapter 6 in Learning Regression Analysis by Simulation, 2014, pp 269-294 from Springer

Abstract: Abstract 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).

Keywords: Generalized Additive Mixed Models (GAMM); 2nd Set; Generalized Linear Mixed Models (GLMM); Random Intercept Model; Nonrandom Part (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-4-431-54321-3_6

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