Generalized Linear Mixed-Effects Models

DESCRIPTION:

This generic function fits a generalized linear mixed-effects model in the formulation described in Breslow and Clayton (1993) but allowing for nested random effects. The within-group errors are allowed to be correlated and/or have additional heteroscedastic patterns, besides the one associated with the family argument.

USAGE:

glme(fixed, data, random, family, correlation, weights, dispersion,
     start, subset, method, na.action, control, verbose)

REQUIRED ARGUMENTS:

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, a glmList object, or a groupedData object. The method functions glme.glmList and glme.groupedData are documented separately.
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.
family
a family object - a list of functions and expressions for defining the link and variance functions, initialization and iterative weights. Families supported are gaussian, binomial, poisson, Gamma, inverse.gaussian and quasi. Functions like binomial produce a family object, but can be given without the parentheses. Family functions can take arguments, as in binomial(link=probit). Defaults to gaussian.
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 to be used in addition to the variance function corresponding to family. 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, in which case only the variance function corresponding to family is used.
dispersion
an optional numeric value for the dispersion parameter for the within-group errors. If positive, the dispersion is fixed at the specified value, otherwise, if less than or equal to zero, the dispersion is estimated with the remaining parameters. Default is 0.
start
an optional list with a single component, mu, with starting estimates for the fitted responses. Defaults to an empty list, in which case the starting estimates for the fitted responses are obtained using the initialization procedure in family.
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
an optional character string. If "(RE)PQL" the model is fit by maximizing the (restricted) penalized quasi-likelihood. If "(RE)MQL" the (restricted) marginal quasi-likelihood is maximized. Use AGQUAD for adaptive gaussian quadrature or LAPLACE for Laplacian approximation to the likelihood when fitting the model. The number of quadrature points can be specified through the control formal argument, see below. The default number of quadrature points is three. When using methods AGQUAD or LAPLACE, the family must be either binomial or poisson with their canonical links logit and log, respectively. Defaults to "REPQL". See the NOTES section below about using "REPQL" with the poisson or binomial families.
na.action
a function that indicates what should happen when the data contain NAs. 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. If method=AGQUAD is used, the number of quadrature points can be specified using Ngq=n, where n is a positive integer. 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 glme, also inheriting from class lme, representing the generalized linear mixed-effects model fit. Generic functions such as print, plot and summary have methods to show the results of the fit. See glmeObject for the components of the fit. The functions resid, coef , fitted, fixef , and ranef can be used to extract some of its components.

DETAILS:

The computational and estimation methods are described in Breslow and Clayton (1993) and Pinheiro and Chao (2004). The variance-covariance parameterizations and the different correlation structures available for the correlation argument are described in Pinheiro and Bates (1996, 2000) and Venables and Ripley (1999).

NOTES:

The PQL approximation of the likelihood for the binomial and poisson families is generally poor especially with small sample sizes. The LAPLACE approximation ( method="LAPLACE") may be a better in these situations.

REFERENCES:

Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized mixed models. Journal of the American Statistical Association, 88, 9-25.

Pinheiro, J. C. and Bates., D. M. (1996). Unconstrained parameterizations 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-Verlag, New York.

Pinheiro, J. C. and Chao, E. C. (2004). Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear mixed models, Journal of Computational and Graphics Statistics, (submitted).

Venables, W. N. and Ripley, B. D. (1999). Modern Applied Statistics with S-PLUS", 3rd Edition Springer-Verlag, New York.

SEE ALSO:

, , , , , , , , .

EXAMPLES:

fm1 <- glme(resp ~ trt, data=Clinic, random = ~1|clinic/trt,
            family=binomial)