rms (version 5.1-3.1)

Gls: 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).

For the print method, format of output is controlled by the user previously running options(prType="lang") where lang is "plain" (the default), "latex", or "html".


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

# S3 method for Gls print(x, digits=4, coefs=TRUE, title, …)



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, it must be specified in the form argument to the corStruct constructor. Defaults to NULL, corresponding to uncorrelated errors.


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 homoscesdatic errors.


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 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 (na.omit) results in deletion of observations having any of the variables of interest missing.


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


for Gls set to TRUE to store the design matrix in the fit object; otherwise the result of Gls


number of significant digits to print


specify coefs=FALSE to suppress printing the table of model coefficients, standard errors, etc. Specify coefs=n to print only the first n regression coefficients in the model.


a character string title to be passed to prModFit



an object of classes Gls, rms, and gls representing the linear model fit. Generic functions such as print, plot, ggplot, 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 (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). The \(g\)-index is also stored in the returned object under the name "g".


The na.delete function will not work with Gls due to some nuance in the model.frame.default function. This probably relates to na.delete storing extra information in the "na.action" attribute of the returned data frame.


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, GiniMd, prModFit, logLik.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
ggplot(Predict(f, tim, trt))
# v <- Variogram(f, form=~tim|id, data=d)
nlme:::summary.gls(f)$tTable   # print matrix of estimates etc.

# }