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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

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: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

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

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

or

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

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

If you express the group-level variable, *x*_{im},
in the random-effects term by *z*_{im},
this model is

In this case, the same terms appear in both the fixed-effects
design matrix and random-effects design matrix. Each *z _{im}* and

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

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

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

which corresponds to the standard form given earlier,

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

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* r*th 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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