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Linear mixed-effects models are extensions of linear regression models for data that are collected and summarized in groups. These models describe the relationship between a response variable and independent variables, with coefficients that can vary with respect to one or more grouping variables. A mixed-effects model consists of two parts, fixed effects and random effects. Fixed-effects terms are usually the conventional linear regression part, and the random effects are associated with individual experimental units drawn at random from a population. The random effects have prior distributions whereas fixed effects do not. Mixed-effects models can represent the covariance structure related to the grouping of data by associating the common random effects to observations that have the same level of a grouping variable. The standard form of a linear mixed-effects model is
$$y=\underset{fixed}{\underbrace{X\beta}}+\underset{random}{\underbrace{Zb}}+\underset{error}{\underbrace{\epsilon}},$$
where
y is the n-by-1 response vector, and n is the number of observations.
X is an n-by-p fixed-effects design matrix.
β is a p-by-1 fixed-effects vector.
Z is an n-by-q random-effects design matrix.
b is a q-by-1 random-effects vector.
ε is the n-by-1 observation error vector.
The assumptions for the linear mixed-effects model are:
Random-effects vector, b, and the error vector, ε, have the following prior distributions:
$$\begin{array}{l}b~N\left(0,{\sigma}^{2}D\left(\theta \right)\right),\\ \epsilon ~N\left(0,\sigma {}^{2}I\right),\end{array}$$
where D is a symmetric and positive semidefinite matrix, parameterized by a variance component vector θ, I is an n-by-n identity matrix, and σ^{2} is the error variance.
Random-effects vector, b, and the error vector, ε, are independent from each other.
Mixed-effects models are also called multilevel models or hierarchical models depending on the context. Mixed-effects models is a more general term than the latter two. Mixed-effects models might include factors that are not necessarily multilevel or hierarchical, for example crossed factors. That is why mixed-effects is the terminology preferred here. Sometimes mixed-effects models are expressed as multilevel regression models (first level and grouping level models) that are fit simultaneously. For example, a varying or random intercept model, with one continuous predictor variable x and one grouping variable with M levels, can be expressed as
$$\begin{array}{l}{y}_{im}={\beta}_{0m}+{\beta}_{1}{x}_{im}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{..},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M,\text{\hspace{1em}}{\epsilon}_{im}~N\left(0,{\sigma}^{2}\right),\\ {\beta}_{0m}={\beta}_{00}+{b}_{0m},\text{\hspace{1em}}{b}_{0m}~N\left(0,{\sigma}_{0}^{2}\right),\end{array}$$
where y_{im} corresponds to data for observation i and group m, n is the total number of observations, and b_{0m} and ε_{im} are independent of each other. After substituting the group-level parameters in the first-level model, the model for the response vector becomes
$${y}_{im}=\underset{fixed\text{\hspace{0.17em}}effects}{\underbrace{{\beta}_{00}+{\beta}_{1}{x}_{im}}}+\underset{random\text{\hspace{0.17em}}effects}{\underbrace{{b}_{0m}}}+{\epsilon}_{im}.$$
A random intercept and slope model with one continuous predictor variable x, where both the intercept and slope vary independently by a grouping variable with M levels is
$$\begin{array}{l}{y}_{im}={\beta}_{0m}+{\beta}_{1m}{x}_{im}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M,\text{\hspace{1em}}{\epsilon}_{im}~N\left(0,{\sigma}^{2}\right),\\ {\beta}_{0m}={\beta}_{00}+{b}_{0m},\text{\hspace{1em}}{b}_{0m}~N\left(0,{\sigma}_{0}^{2}\right),\text{\hspace{1em}}\\ {\beta}_{1m}={\beta}_{10}+{b}_{1m},\text{\hspace{1em}}{b}_{1m}~N\left(0,{\sigma}_{1}^{2}\right),\end{array}$$
or
$${b}_{m}=\left(\begin{array}{l}{b}_{0m}\\ {b}_{1m}\end{array}\right)~N\left(0,\left(\begin{array}{cc}{\sigma}_{0}^{2}& 0\\ 0& {\sigma}_{1}^{2}\end{array}\right)\right).$$
You might also have correlated random effects. In general, for a model with a random intercept and slope, the distribution of the random effects is
$${b}_{m}=\left(\begin{array}{l}{b}_{0m}\\ {b}_{1m}\end{array}\right)~N\left(0,\sigma {}^{2}D\left(\theta \right)\right),$$
where D is a 2-by-2 symmetric and positive semidefinite matrix, parameterized by a variance component vector θ.
After substituting the group-level parameters in the first-level model, the model for the response vector is
$${y}_{im}=\underset{fixed\text{\hspace{0.17em}}effects}{\underbrace{{\beta}_{00}+{\beta}_{10}{x}_{im}}}+\underset{random\text{\hspace{0.17em}}effects}{\underbrace{{b}_{0m}+{b}_{1m}{x}_{im}}}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M.$$
If you express the group-level variable, x_{im}, in the random-effects term by z_{im}, this model is
$${y}_{im}=\underset{fixed\text{\hspace{0.17em}}effects}{\underbrace{{\beta}_{00}+{\beta}_{10}{x}_{im}}}+\underset{random\text{\hspace{0.17em}}effects}{\underbrace{{b}_{0m}+{b}_{1m}{z}_{im}}}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M.$$
In this case, the same terms appear in both the fixed-effects design matrix and random-effects design matrix. Each z_{im} and x_{im} correspond to the level m of the grouping variable.
It is also possible to explain more of the group-level variations by adding more group-level predictor variables. A random-intercept and random-slope model with one continuous predictor variable x, where both the intercept and slope vary independently by a grouping variable with M levels, and one group-level predictor variable v_{m} is
$$\begin{array}{l}{y}_{im}={\beta}_{0im}+{\beta}_{1im}{x}_{im}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M,\text{\hspace{1em}}{\epsilon}_{im}~N\left(0,{\sigma}^{2}\right),\\ {\beta}_{0im}={\beta}_{00}+{\beta}_{01}{v}_{im}+{b}_{0m},\text{\hspace{1em}}{b}_{0m}~N\left(0,{\sigma}_{0}^{2}\right),\text{\hspace{1em}}\\ {\beta}_{1im}={\beta}_{10}+{\beta}_{11}{v}_{im}+{b}_{1m},\text{\hspace{1em}}{b}_{1m}~N\left(0,{\sigma}_{1}^{2}\right).\end{array}$$
This model results in main effects of the group-level predictor and an interaction term between the first-level and group-level predictor variables in the model for the response variable as
$$\begin{array}{l}{y}_{im}={\beta}_{00}+{\beta}_{01}{v}_{im}+{b}_{0m}+\left({\beta}_{10}+{\beta}_{11}{v}_{im}+{b}_{1m}\right){x}_{im}+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M,\\ \text{\hspace{1em}}\text{\hspace{1em}}=\underset{fixed\text{\hspace{0.17em}}effects}{\underbrace{{\beta}_{00}+{\beta}_{10}{x}_{im}+{\beta}_{01}{v}_{im}+{\beta}_{11}{v}_{im}{x}_{im}}}+\underset{random\text{\hspace{0.17em}}effects}{\underbrace{{b}_{0m}+{b}_{1m}{x}_{im}}}+{\epsilon}_{im}.\end{array}$$
The term β_{11}v_{m}x_{im} is often called a cross-level interaction in many textbooks on multilevel models. The model for the response variable y can be expressed as
$$\begin{array}{l}{y}_{im}=\left[\begin{array}{cccc}1& {x}_{1}{}_{im}& {v}_{im}& {v}_{im}{x}_{1im}\end{array}\right]\left[\begin{array}{c}{\beta}_{00}\\ {\beta}_{10}\\ {\beta}_{01}\\ {\beta}_{11}\end{array}\right]+\left[\begin{array}{cc}1& {x}_{1im}\end{array}\right]\left[\begin{array}{c}{b}_{0m}\\ {b}_{1m}\end{array}\right]+{\epsilon}_{im},\text{\hspace{1em}}i=1,2,\mathrm{...},n,\text{\hspace{1em}}m=1,2,\mathrm{...},M,\\ \text{\hspace{1em}}\text{\hspace{1em}}\end{array}$$
which corresponds to the standard form given earlier,
$$\text{\hspace{1em}}y=X\beta +Zb+\epsilon .$$
In general, if there are R grouping variables, and m(r,i) shows the level of grouping variable r, for observation i, then the model for the response variable for observation i is
$${y}_{i}={x}_{i}^{T}\beta +{\displaystyle \sum _{r=1}^{R}{z}_{ir}{b}_{m(r,i)}^{(r)}}+{\epsilon}_{i},\text{\hspace{1em}}i=1,2,\mathrm{...},n,$$
where β is a p-by-1 fixed-effects vector, b^{(r)}_{m(r,i)} is a q(r)-by-1 random-effects vector for the rth grouping variable and level m(r,i), and ε_{i} is a 1-by-1 error term for observation i.
[1] Pinherio, J. C., and D. M. Bates. Mixed-Effects Models in S and S-PLUS. Statistics and Computing Series, Springer, 2004.
[2] Hariharan, S. and J. H. Rogers. "Estimation Procedures for Hierarchical Linear Models." Multilevel Modeling of Educational Data (A. A. Connell and D. B. McCoach, eds.). Charlotte, NC: Information Age Publishing, Inc., 2008.
[3] Hox, J. Multilevel Analysis, Techniques and Applications. Lawrence Erlbaum Associates, Inc., 2002
[4] Snidjers, T. and R. Bosker. Multilevel Analysis. Thousand Oaks, CA: Sage Publications, 1999.
[5] Gelman, A. and J. Hill. Data Analysis Using Regression and Multilevel/Hierarchical Models. New York, NY: Cambridge University Press, 2007.
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