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A linear mixed-effects model is of the form

$$y=\underset{fixed}{\underbrace{X\beta}}+\underset{random}{\underbrace{Zb}}+\underset{error}{\underbrace{\epsilon}},$$

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 random-effects vector, *b*, and
the error vector, *ε*, are assumed to have
the following independent 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.

In this model, the parameters to estimate are the fixed-effects
coefficients *β*, and the variance components *θ* and *σ*^{2}.
The two most commonly used approaches to parameter estimation in linear
mixed-effects models are maximum likelihood and restricted maximum
likelihood methods.

The maximum likelihood estimation includes both regression coefficients and the variance components, that is, both fixed-effects and random-effects terms in the likelihood function.

For a linear mixed-effects model defined above, the conditional
response of the response variable *y* given *β*, *b*, *θ*,
and σ^{2} is

$$y|b,\beta ,\theta ,{\sigma}^{2}~N\left(X\beta +Zb,{\sigma}^{2}{I}_{n}\right).$$

The likelihood of *y* given *β*, *θ*,
and σ^{2} is

$$P\left(y|\beta ,\theta ,{\sigma}^{2}\right)={\displaystyle \int P\left(y|b,\beta ,\theta ,{\sigma}^{2}\right)}P\left(b|\theta ,{\sigma}^{2}\right)db,$$

where

$$\begin{array}{l}P\left(b|\theta ,{\sigma}^{2}\right)=\frac{1}{{\left(2\pi {\sigma}^{2}\right)}^{\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\frac{1}{{\left|D\left(\theta \right)\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\mathrm{exp}\left\{-\frac{1}{2{\sigma}^{2}}{b}^{T}{D}^{-1}b\right\}\text{\hspace{1em}}\text{and}\\ P\left(y|b,\beta ,\theta ,{\sigma}^{2}\right)=\frac{1}{{\left(2\pi {\sigma}^{2}\right)}^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\mathrm{exp}\left\{-\frac{1}{2{\sigma}^{2}}{\left(y-X\beta -Zb\right)}^{T}\left(y-X\beta -Zb\right)\right\}.\end{array}$$

Suppose Λ(*θ*) is the lower triangular
Cholesky factor of *D*(*θ*)
and Δ(*θ*) is the inverse of Λ(*θ*).
Then,

$$D{\left(\theta \right)}^{-1}=\Delta {\left(\theta \right)}^{T}\Delta \left(\theta \right).$$

Define

$${r}^{2}\left(\beta ,b,\theta \right)={b}^{T}\Delta {\left(\theta \right)}^{T}\Delta \left(\theta \right)b+{\left(y-X\beta -Zb\right)}^{T}\left(y-X\beta -Zb\right),$$

and suppose *b*^{*} is
the value of *b* that satisfies

$${\frac{\partial {r}^{2}\left(\beta ,b,\theta \right)}{\partial b}|}_{{b}^{*}}=0$$

for given *β* and *θ*.
Then, the likelihood function is

$$P\left(y|\beta ,\theta ,{\sigma}^{2}\right)={\left(2\pi {\sigma}^{2}\right)}^{-\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left|D\left(\theta \right)\right|}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{exp}\left\{-\frac{1}{2{\sigma}^{2}}{r}^{2}\left(\beta ,{b}^{*}\left(\beta \right),\theta \right)\right\}\frac{1}{{\left|{\Delta}^{T}\Delta +{Z}^{T}Z\right|}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}.$$

P(y|*β*,*θ*,σ^{2})
is first maximized with respect to *β* and
σ^{2} for a given *θ*.
Thus the optimized solutions $$\widehat{\beta}\left(\theta \right)$$ and $${\widehat{\sigma}}^{2}\left(\theta \right)$$are obtained as functions of *θ*.
Substituting these solutions into the likelihood function produces $$P\left(y|\widehat{\beta}\left(\theta \right)\text{,}\theta \text{,}{\widehat{\sigma}}^{2}\left(\theta \right)\right)$$. This expression is called a
profiled likelihood where *β* and σ^{2} have
been profiled out. $$P\left(y|\widehat{\beta}\left(\theta \right)\text{,}\theta \text{,}{\widehat{\sigma}}^{2}\left(\theta \right)\right)$$ is a function
of *θ*, and the algorithm then optimizes it
with respect to *θ*. Once it finds the optimal
estimate of *θ*, the estimates of *β* and
σ^{2} are given by $$\widehat{\beta}\left(\theta \right)$$ and $${\widehat{\sigma}}^{2}\left(\theta \right)$$.

The ML method treats *β* as fixed but
unknown quantities when the variance components are estimated, but
does not take into account the degrees of freedom lost by estimating
the fixed effects. This causes ML estimates to be biased with smaller
variances. However, one advantage of ML over REML is that it is possible
to compare two models in terms of their fixed- and random-effects
terms. On the other hand, if you use REML to estimate the parameters,
you can only compare two models, that are nested in their random-effects
terms, with the same fixed-effects design.

Restricted maximum likelihood estimation includes only the variance
components, that is, the parameters that parameterize the random-effects
terms in the linear mixed-effects model. *β* is
estimated in a second step. Assuming a uniform improper prior distribution
for *β* and integrating the likelihood P(*y*|*β*,*θ*,σ^{2})
with respect to *β* results in the restricted
likelihood P(*y*|*θ*,σ^{2}).
That is,

$$P\left(y|\theta ,{\sigma}^{2}\right)={\displaystyle \int P\left(y|\beta ,\theta ,{\sigma}^{2}\right)}P\left(\beta \right)d\beta ={\displaystyle \int P\left(y|\beta ,\theta ,{\sigma}^{2}\right)}d\beta .$$

The algorithm first profiles out $${\widehat{\sigma}}_{R}^{2}$$ and maximizes remaining objective
function with respect to *θ* to find $${\widehat{\theta}}_{R}$$. The restricted likelihood is
then maximized with respect to σ^{2} to
find $${\widehat{\sigma}}_{R}^{2}$$. Then, it estimates *β* by
finding its expected value with respect to the posterior distribution

$$P\left(\beta |y,{\widehat{\theta}}_{R},{\widehat{\sigma}}_{R}^{2}\right).$$

REML accounts for the degrees of freedom lost by estimating
the fixed effects, and makes a less biased estimation of random effects
variances. The estimates of *θ* and σ^{2} are
invariant to the value of *β* and less sensitive
to outliers in the data compared to ML estimates. However, if you
use REML to estimate the parameters, you can only compare two models
that have the identical fixed-effects design matrices and are nested
in their random-effects terms.

[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] Raudenbush, S. W. and A. S. Bryk. *Hierarchical
Linear Models: Applications and Data Analysis Methods*,
2nd ed. Thousand Oaks, CA: Sage Publications, 2002.

[4] Hox, J. *Multilevel Analysis, Techniques and
Applications*. Lawrence Erlbaum Associates, Inc, 2002.

[5] Snidjers, T. and R. Bosker. *Multilevel Analysis*.
Thousand Oaks, CA: Sage Publications, 1999.

[6] McCulloch, C.E., R. S. Shayle, and J. M. Neuhaus. *Generalized,
Linear, and Mixed Models*. Wiley, 2008.

`LinearMixedModel`

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