nlme (version 3.1-164)

lme: Linear Mixed-Effects Models


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, = TRUE)

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

resid, coef, fitted,

fixed.effects, and

random.effects can be used to extract some of its components.



an object inheriting from class lme, representing a fitted linear mixed-effects model.


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.


Changes to the fixed-effects formula -- see update.formula for details.


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.


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.


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.


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.


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.


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".


a function that indicates what should happen when the data contain NAs. The default action ( causes lme to print an error message and terminate if there are any incomplete observations.


a list of control values for the estimation algorithm to replace the default values returned by the function lmeControl. Defaults to an empty list.


an optional list. See the contrasts.arg of model.matrix.default.

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.


If TRUE evaluate the new call else return the call.


José Pinheiro and Douglas Bates


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").

See Also

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


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

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