\Header{lme}{Linear Mixed-Effects Models}
\methalias{lme.formula}{lme}
\alias{update.lme}{lme}
\keyword{models}{lme}
\begin{Description}\relax
This generic function fits a linear mixed-effects model in the
formulation described in Laird and Ware (1982) but allowing for nested
random effects. The within-group errors are allowed to be correlated
and/or have unequal variances.
\end{Description}
\begin{Usage}
\begin{verbatim}
lme(fixed, data, random, correlation, weights, subset, method,
    na.action, control, contrasts = NULL)
## S3 method for class 'lme':
update(object, fixed, data, random, correlation, weights,
      subset, method, na.action, control, contrasts, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{object}] an object inheriting from class \code{lme}, representing
a fitted linear mixed-effects model.
\item[\code{fixed}] a two-sided linear formula object describing the
fixed-effects part of the model, with the response on the left of a
\code{\textasciitilde{}} operator and the terms, separated by \code{+} operators, on
the right, an \code{lmList} object, or a \code{groupedData}
object. The method functions \code{lme.lmList} and
\code{lme.groupedData} are documented separately.
\item[\code{data}] an optional data frame containing the variables named in
\code{fixed}, \code{random}, \code{correlation}, \code{weights}, and
\code{subset}.  By default the variables are taken from the
environment from which \code{lme} is called.
\item[\code{random}] optionally, any of the following: (i) a one-sided formula
of the form \code{\textasciitilde{}x1+...+xn | g1/.../gm}, with \code{x1+...+xn}
specifying the model for the random effects and \code{g1/.../gm} the
grouping structure (\code{m} may be equal to 1, in which case no
\code{/} is required). The random effects formula will be repeated
for all levels of grouping, in the case of multiple levels of
grouping; (ii) a list of one-sided formulas of the form
\code{\textasciitilde{}x1+...+xn | g}, with possibly different random effects models
for each grouping level. The order of nesting will be assumed the
same as the order of the elements in the list; (iii) a one-sided
formula of the form \code{\textasciitilde{}x1+...+xn}, or a \code{pdMat} object with
a formula (i.e. a non-\code{NULL} value for \code{formula(object)}),
or a list of such formulas or \code{pdMat} objects. In this case, the
grouping structure formula will be derived from the data used to
fit the linear mixed-effects model, which should inherit from class
\code{groupedData}; (iv) a named list of formulas or \code{pdMat}
objects as in (iii), with the grouping factors as names. The order of
nesting will be assumed the same as the order of the order of the
elements in the list; (v) an \code{reStruct} object. See the
documentation on \code{pdClasses} for a description of the available
\code{pdMat} classes. Defaults to a formula consisting of the right
hand side of \code{fixed}.
\item[\code{correlation}] an optional \code{corStruct} object describing the
within-group correlation structure. See the documentation of
\code{corClasses} for a description of the available \code{corStruct}
classes. Defaults to \code{NULL},
corresponding to no within-group correlations.
\item[\code{weights}] an optional \code{varFunc} object or one-sided formula
describing the within-group heteroscedasticity structure. If given as
a formula, it is used as the argument to \code{varFixed},
corresponding to fixed variance weights. See the documentation on
\code{varClasses} for a description of the available \code{varFunc}
classes. Defaults to \code{NULL}, corresponding to homocesdatic
within-group errors.
\item[\code{subset}] an optional expression indicating the subset of the rows of
\code{data} that should be used in the fit. This can be a logical
vector, or a numeric vector indicating which observation numbers are
to be included, or a  character  vector of the row names to be
included.  All observations are included by default.
\item[\code{method}] a character string.  If \code{"REML"} the model is fit by
maximizing the restricted log-likelihood.  If \code{"ML"} the
log-likelihood is maximized.  Defaults to \code{"REML"}.
\item[\code{na.action}] a function that indicates what should happen when the
data contain \code{NA}s.  The default action (\code{na.fail}) causes
\code{lme} to print an error message and terminate if there are any
incomplete observations.
\item[\code{control}] a list of control values for the estimation algorithm to
replace the default values returned by the function \code{lmeControl}.
Defaults to an empty list.
\item[\code{contrasts}] an optional list. See the \code{contrasts.arg}
of \code{model.matrix.default}.
\item[\code{...}] some methods for this generic require additional
arguments.  None are used in this method.
\end{ldescription}
\end{Arguments}
\begin{Value}
an object of class \code{lme} representing the linear mixed-effects
model fit. Generic functions such as \code{print}, \code{plot} and
\code{summary} have methods to show the results of the fit. See
\code{lmeObject} for the components of the fit. The functions
\code{resid}, \code{coef}, \code{fitted}, \code{fixed.effects}, and
\code{random.effects}  can be used to extract some of its components.
\end{Value}
\begin{Author}\relax
Jose Pinheiro \email{jose.pinheiro@pharma.novartis.com} and
Douglas Bates \email{bates@stat.wisc.edu}
\end{Author}
\begin{References}\relax
The computational methods are described in Bates, D.M. and Pinheiro
(1998) and follow on the general framework of Lindstrom, M.J. and Bates,
D.M. (1988). The model formulation is described in Laird, N.M. and Ware,
J.H. (1982).  The variance-covariance parametrizations are described in
<Pinheiro, J.C. and Bates., D.M.  (1996).   The different correlation
structures available for the \code{correlation} argument are described
in Box, G.E.P., Jenkins, G.M., and Reinsel G.C. (1994), Littel, R.C.,
Milliken, G.A., Stroup, W.W., and Wolfinger, R.D. (1996), and Venables,
W.N. and Ripley, B.D. (1997). The use of variance functions for linear
and nonlinear mixed effects models is presented in detail in Davidian,
M. and Giltinan, D.M. (1995). 

Bates, D.M. and Pinheiro, J.C. (1998) "Computational methods for
multilevel models" available in PostScript or PDF formats at
http://franz.stat.wisc.edu/pub/NLME/

Box, G.E.P., Jenkins, G.M., and Reinsel G.C. (1994) "Time Series
Analysis: Forecasting and Control", 3rd Edition, Holden-Day. 

Davidian, M. and Giltinan, D.M. (1995) "Nonlinear Mixed Effects Models
for Repeated Measurement Data", Chapman and Hall.

Laird, N.M. and Ware, J.H. (1982) "Random-Effects Models for
Longitudinal Data", Biometrics, 38, 963-974.  

Lindstrom, M.J. and Bates, D.M. (1988) "Newton-Raphson and EM
Algorithms for Linear Mixed-Effects Models for Repeated-Measures
Data", Journal of the American Statistical Association, 83,
1014-1022. 

Littel, R.C., Milliken, G.A., Stroup, W.W., and Wolfinger, R.D. (1996)
"SAS Systems for Mixed Models", SAS Institute.

Pinheiro, J.C. and Bates., D.M.  (1996) "Unconstrained
Parametrizations for Variance-Covariance Matrices", Statistics and
Computing, 6, 289-296.

Venables, W.N. and Ripley, B.D. (1997) "Modern Applied Statistics with
S-plus", 2nd Edition, Springer-Verlag.
\end{References}
\begin{SeeAlso}\relax
\code{\Link{lmeControl}}, \code{\Link{lme.lmList}},
\code{\Link{lme.groupedData}}, \code{\Link{lmeObject}},
\code{\Link{lmList}}, \code{\Link{reStruct}}, \code{\Link{reStruct}},
\code{\Link{varFunc}}, \code{\Link{pdClasses}},
\code{\Link{corClasses}}, \code{\Link{varClasses}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(Orthodont)
fm1 <- lme(distance ~ age, data = Orthodont) # random is ~ age
fm2 <- lme(distance ~ age + Sex, data = Orthodont, random = ~ 1)
summary(fm1)
summary(fm2)
\end{ExampleCode}
\end{Examples}

