R: Nonlinear Mixed-Effects Models

nlme {nlme} | R Documentation

## Nonlinear Mixed-Effects Models

### Description

This generic function fits a nonlinear mixed-effects model in the formulation described in Lindstrom and Bates (1990) but allowing for nested random effects. The within-group errors are allowed to be correlated and/or have unequal variances.

### Usage

    nlme(model, data, fixed, random, groups, start, correlation, weights,
         subset, method, na.action, naPattern, control, verbose)

### Arguments

`model` | a nonlinear model formula, with the response on the left of a `~` operator and an expression involving parameters and covariates on the right, or an `nlsList` object. If `data` is given, all names used in the formula should be defined as parameters or variables in the data frame. The method function `nlme.nlsList` is documented separately.
`data` | an optional data frame containing the variables named in `model`, `fixed`, `random`, `correlation`, `weights`, `subset`, and `naPattern`. By default the variables are taken from the environment from which `nlme` is called.
`fixed` | a two-sided linear formula of the form `f1+...+fn~x1+...+xm`, or a list of two-sided formulas of the form `f1~x1+...+xm`, with possibly different models for different parameters. The `f1,...,fn` are the names of parameters included on the right hand side of `model` and the `x1+...+xm` expressions define linear models for these parameters (when the left hand side of the formula contains several parameters, they all are assumed to follow the same linear model, described by the right hand side expression). A `1` on the right hand side of the formula(s) indicates a single fixed effects for the corresponding parameter(s).
`random` | optionally, any of the following: (i) a two-sided formula of the form `r1+...+rn~x1+...+xm | g1/.../gQ`, with `r1,...,rn` naming parameters included on the right hand side of `model`, `x1+...+xm` specifying the random-effects model for these parameters and `g1/.../gQ` the grouping structure (`Q` 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 two-sided formula of the form `r1+...+rn~x1+..+xm`, a list of two-sided formulas of the form `r1~x1+...+xm`, with possibly different random-effects models for different parameters, a `pdMat` object with a two-sided formula, or list of two-sided formulas (i.e. a non-`NULL` value for `formula(random)`), or a list of pdMat objects with two-sided formulas, or lists of two-sided formulas. In this case, the grouping structure formula will be given in `groups`, or derived from the data used to fit the nonlinear mixed-effects model, which should inherit from class `groupedData`,; (iii) a named list of formulas, lists of formulas, or `pdMat` objects as in (ii), 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; (iv) an `reStruct` object. See the documentation on `pdClasses` for a description of the available `pdMat` classes. Defaults to `fixed`, resulting in all fixed effects having also random effects.
`groups` | an optional one-sided formula of the form `~g1` (single level of nesting) or `~g1/.../gQ` (multiple levels of nesting), specifying the partitions of the data over which the random effects vary. `g1,...,gQ` must evaluate to factors in `data`. The order of nesting, when multiple levels are present, is taken from left to right (i.e. `g1` is the first level, `g2` the second, etc.).
`start` | an optional numeric vector, or list of initial estimates for the fixed effects and random effects. If declared as a numeric vector, it is converted internally to a list with a single component `fixed`, given by the vector. The `fixed` component is required, unless the model function inherits from class `selfStart`, in which case initial values will be derived from a call to `nlsList`. An optional `random` component is used to specify initial values for the random effects and should consist of a matrix, or a list of matrices with length equal to the number of grouping levels. Each matrix should have as many rows as the number of groups at the corresponding level and as many columns as the number of random effects in that level.
`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 `"ML"`.
`na.action` | a function that indicates what should happen when the data contain `NA`s. The default action (`na.fail`) causes `nlme` to print an error message and terminate if there are any incomplete observations.
`naPattern` | an expression or formula object, specifying which returned values are to be regarded as missing.
`control` | a list of control values for the estimation algorithm to replace the default values returned by the function `nlmeControl`. Defaults to an empty list.
`verbose` | an optional logical value. If `TRUE` information on the evolution of the iterative algorithm is printed. Default is `FALSE`.

### Value

an object of class `nlme` representing the nonlinear mixed-effects model fit. Generic functions such as `print`, `plot` and `summary` have methods to show the results of the fit. See `nlmeObject` 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.

### Note

The function does not do any scaling internally: the optimization will work best when the response is scaled so its variance is of the order of one.

### Author(s)

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

### References

The model formulation and computational methods are described in Lindstrom, M.J. and Bates, D.M. (1990). The variance-covariance parametrizations are described in Pinheiro, J.C. and Bates., D.M. (1996). The different correlation structures available for the `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. (2002). The use of variance functions for linear and nonlinear mixed effects models is presented in detail in Davidian, M. and Giltinan, D.M. (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.

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

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

Lindstrom, M.J. and Bates, D.M. (1990) "Nonlinear Mixed Effects Models for Repeated Measures Data", Biometrics, 46, 673-687.

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

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer.

Venables, W.N. and Ripley, B.D. (2002) "Modern Applied Statistics with S", 4th Edition, Springer-Verlag.

### See Also

`nlmeControl`, `nlme.nlsList`, `nlmeObject`, `nlsList`, `nlmeStruct`, `pdClasses`, `reStruct`, `varFunc`, `corClasses`, `varClasses`

### Examples

    fm1 <- nlme(height ~ SSasymp(age, Asym, R0, lrc),
                data = Loblolly,
                fixed = Asym + R0 + lrc ~ 1,
                random = Asym ~ 1,
                start = c(Asym = 103, R0 = -8.5, lrc = -3.3))
    summary(fm1)
    fm2 <- update(fm1, random = pdDiag(Asym + lrc ~ 1))
    summary(fm2)

[Package nlme version 3.1-118 Index]