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.

This page describes the formula method;
the methods `lme.lmList`

and `lme.groupedData`

are documented separately.

```
lme(fixed, data, random, correlation, weights, subset, method,
na.action, control, contrasts = NULL, keep.data = TRUE)
```# S3 method for formula
lme(fixed, data, random, correlation, weights, subset, method,
na.action, control, contrasts = NULL, keep.data = TRUE)

# S3 method for lme
update(object, fixed., ..., evaluate = TRUE)

An object of class `"lme"`

representing the linear mixed-effects
model fit. Generic functions such as `print`

, `plot`

and

`summary`

have methods to show the results of the fit. See

`lmeObject`

for the components of the fit. The functions

`fixed.effects`

, and

`random.effects`

can be used to extract some of its components.

- object
an object inheriting from class

`lme`

, representing a fitted linear mixed-effects model.- fixed
a two-sided linear formula object describing the fixed-effects part of the model, with the response on the left of a

`~`

operator and the terms, separated by`+`

operators, on the right, an`"lmList"`

object, or a`"groupedData"`

object.There is limited support for formulae such as

`resp ~ 1`

and`resp ~ 0`

, and less prior to version`3.1-112`.- fixed.
Changes to the fixed-effects formula -- see

`update.formula`

for details.- data
an optional data frame containing the variables named in

`fixed`

,`random`

,`correlation`

,`weights`

, and`subset`

. By default the variables are taken from the environment from which`lme`

is called.- random
optionally, any of the following: (i) a one-sided formula of the form

`~ x1 + ... + xn | g1/.../gm`

, with`x1 + ... + xn`

specifying the model for the random effects and`g1/.../gm`

the grouping structure (`m`

may be equal to 1, in which case no`/`

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`~ 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`~ x1 + ... + xn`

, or a`pdMat`

object with a formula (i.e. a non-`NULL`

value for`formula(object)`

), or a list of such formulas or`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`"groupedData"`

; (iv) a named list of formulas or`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`reStruct`

object. See the documentation on`pdClasses`

for a description of the available`pdMat`

classes. Defaults to a formula consisting of the right hand side of`fixed`

.- correlation
an optional

`corStruct`

object describing the within-group correlation structure. See the documentation of`corClasses`

for a description of the available`corStruct`

classes. Defaults to`NULL`

, corresponding to no within-group correlations.- weights
an optional

`varFunc`

object or one-sided formula describing the within-group heteroscedasticity structure. If given as a formula, it is used as the argument to`varFixed`

, corresponding to fixed variance weights. See the documentation on`varClasses`

for a description of the available`varFunc`

classes. Defaults to`NULL`

, corresponding to homoscedastic within-group errors.- subset
an optional expression indicating the subset of the rows of

`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.- method
a character string. If

`"REML"`

the model is fit by maximizing the restricted log-likelihood. If`"ML"`

the log-likelihood is maximized. Defaults to`"REML"`

.- na.action
a function that indicates what should happen when the data contain

`NA`

s. The default action (`na.fail`

) causes`lme`

to print an error message and terminate if there are any incomplete observations.- control
a list of control values for the estimation algorithm to replace the default values returned by the function

`lmeControl`

. Defaults to an empty list.- contrasts
an optional list. See the

`contrasts.arg`

of`model.matrix.default`

.- keep.data
logical: should the

`data`

argument (if supplied and a data frame) be saved as part of the model object?- ...
some methods for this generic require additional arguments. None are used in this method.

- evaluate
If

`TRUE`

evaluate the new call else return the call.

José Pinheiro and Douglas Bates bates@stat.wisc.edu

`offset`

terms in `fixed`

are an error since 3.1-157
(2022-03): previously they were silently ignored.

The computational methods follow the general framework of Lindstrom
and Bates (1988). The model formulation is described in Laird and Ware
(1982). The variance-covariance parametrizations are described in
Pinheiro and Bates (1996). The different correlation structures
available for the `correlation`

argument are described in Box,
Jenkins and Reinsel (1994), Littell *et al* (1996), and Venables and
Ripley (2002). The use of variance functions for linear and nonlinear
mixed effects models is presented in detail in Davidian and Giltinan
(1995).

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.
tools:::Rd_expr_doi("10.1201/9780203745502").

Laird, N.M. and Ware, J.H. (1982).
Random-Effects Models for Longitudinal Data.
*Biometrics* **38**, 963--974.
tools:::Rd_expr_doi("10.2307/2529876").

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.
tools:::Rd_expr_doi("10.2307/2290128").

Littell, 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.
tools:::Rd_expr_doi("10.1007/BF00140873").

Pinheiro, J.C., and Bates, D.M. (2000).
*Mixed-Effects Models in S and S-PLUS*, Springer.
tools:::Rd_expr_doi("10.1007/b98882").

Venables, W.N. and Ripley, B.D. (2002).
*Modern Applied Statistics with S*, 4th Edition, Springer-Verlag.
tools:::Rd_expr_doi("10.1007/978-0-387-21706-2").

`corClasses`

,
`lme.lmList`

,
`lme.groupedData`

,
`lmeControl`

,
`lmeObject`

,
`lmeStruct`

,
`lmList`

,
`pdClasses`

,
`plot.lme`

,
`predict.lme`

,
`qqnorm.lme`

,
`residuals.lme`

,
`reStruct`

,
`simulate.lme`

,
`summary.lme`

,
`varClasses`

,
`varFunc`

```
fm1 <- lme(distance ~ age, data = Orthodont) # random is ~ age
fm2 <- lme(distance ~ age + Sex, data = Orthodont, random = ~ 1)
summary(fm1)
summary(fm2)
```

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