Fit Linear Model Using Generalized Least Squares

This function fits a linear model using generalized least squares. The errors are allowed to be correlated and/or have unequal variances. Gls is a slightly enhanced version of the Pinheiro and Bates Gls function in the nlme package to make it easy to use with the rms package and to implement cluster bootstrapping (primarily for nonparametric estimates of the variance-covariance matrix of the parameter estimates and for nonparametric confidence limits of correlation parameters).

Gls(model, data, correlation, weights, subset, method, na.action,
    control, verbose, B=0, dupCluster=FALSE, pr=FALSE,

## S3 method for class 'Gls': print(x, digits=4, \dots)

a two-sided linear formula object describing the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right.
an optional data frame containing the variables named in model, correlation, weights, and subset. By default the variables are taken from the environment from which gls is called.
an optional corStruct object describing the within-group correlation structure. See the documentation of corClasses for a description of the available corStruct classes. If a grouping variable is to be used,
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 do
an optional expression indicating which subset of the rows of data 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
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 gls 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 glsControl. Defaults to an empty list.
an optional logical value. If TRUE information on the evolution of the iterative algorithm is printed. Default is FALSE.
number of bootstrap resamples to fit and store, default is none
set to TRUE to have Gls when bootstrapping to consider multiply-sampled clusters as if they were one large cluster when fitting using the gls algorithm
set to TRUE to show progress of bootstrap resampling
specifies whether the optimize or the optim function is to be used for optimization
the result of Gls
number of significant digits to print

  • an object of classes Gls, rms, and gls representing the linear model fit. Generic functions such as print, plot, and summary have methods to show the results of the fit. See glsObject for the components of the fit. The functions resid, coef, and fitted can be used to extract some of its components. Gls returns the following components not returned by gls: Design, assign, formula, opmeth (see arguments), B (see arguments), bootCoef (matrix of B bootstrapped coefficients), boot.Corr (vector of bootstrapped correlation parameters), Nboot (vector of total sample size used in each bootstrap (may vary if have unbalanced clusters), and var (sample variance-covariance matrix of bootstrapped coefficients).


Pinheiro J, Bates D (2000): Mixed effects models in S and S-Plus. New York: Springer-Verlag.

See Also

gls glsControl, glsObject, varFunc, corClasses, varClasses

  • Gls
  • print.Gls
ns  <- 20  # no. subjects
nt  <- 10  # no. time points/subject
B   <- 10  # no. bootstrap resamples
           # usually do 100 for variances, 1000 for nonparametric CLs
rho <- .5  # AR(1) correlation parameter
V <- matrix(0, nrow=nt, ncol=nt)
V <- rho^abs(row(V)-col(V))   # per-subject correlation/covariance matrix

d <- expand.grid(tim=1:nt, id=1:ns)
d$trt <- factor(ifelse(d$id <= ns/2, 'a', 'b'))
true.beta <- c(Intercept=0,tim=.1,'tim^2'=0,'trt=b'=1)
d$ey  <- true.beta['Intercept'] + true.beta['tim']*d$tim +
  true.beta['tim^2']*(d$tim^2) +  true.beta['trt=b']*(d$trt=='b')
library(MASS)   # needed for mvrnorm
d$y <- d$ey + as.vector(t(mvrnorm(n=ns, mu=rep(0,nt), Sigma=V)))

dd <- datadist(d); options(datadist='dd')
f <- Gls(y ~ pol(tim,2) + trt, correlation=corCAR1(form= ~tim | id),
         data=d, B=B)
f$var      # bootstrap variances
f$varBeta  # original variances
plot(Predict(f, tim=., trt=.))
# v <- Variogram(f, form=~tim|id, data=d)
Documentation reproduced from package rms, version 2.0-2, License: GPL (>= 2)

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