\HeaderA{Variogram.lme}{Calculate Semi-variogram for Residuals from an lme Object}{Variogram.lme}
\keyword{models}{Variogram.lme}
\begin{Description}\relax
This method function calculates the semi-variogram for the
within-group residuals from an \code{lme} fit. The semi-variogram
values are calculated for pairs of residuals within the same group. If
\code{collapse} is different from \code{"none"}, the individual
semi-variogram values are collapsed using either a robust estimator
(\code{robust = TRUE}) defined in Cressie (1993), or the average of
the values within the same distance interval. The semi-variogram is
useful for modeling the error term correlation structure.
\end{Description}
\begin{Usage}
\begin{verbatim}
## S3 method for class 'lme':
Variogram(object, distance, form, resType, data,
       na.action, maxDist, length.out, collapse, nint, breaks,
       robust, metric, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{object}] an object inheriting from class \code{lme}, representing
a fitted linear mixed-effects model.
\item[\code{distance}] an optional numeric vector with the distances between
residual pairs. If a grouping variable is present, only the
distances between residual pairs within the same group should be
given. If missing, the distances are calculated based on the
values of the arguments \code{form}, \code{data}, and
\code{metric}, unless \code{object} includes a \code{corSpatial}
element, in which case the associated covariate (obtained with the
\code{getCovariate} method) is used.
\item[\code{form}] an optional one-sided formula specifying the covariate(s)
to be used for calculating the distances between residual pairs and,
optionally, a grouping factor for partitioning the residuals (which
must appear to the right of a \code{|} operator in
\code{form}). Default is \code{\textasciitilde{}1}, implying that the observation 
order within the groups is used to obtain the distances.
\item[\code{resType}] an optional character string specifying the type of
residuals to be used. If \code{"response"}, the "raw" residuals
(observed - fitted) are used; else, if \code{"pearson"}, the
standardized residuals (raw residuals divided by the corresponding
standard errors) are used; else, if \code{"normalized"}, the
normalized residuals (standardized residuals pre-multiplied by the
inverse square-root factor of the estimated error correlation
matrix) are used. Partial matching of arguments is used, so only the
first character needs to be provided. Defaults to \code{"pearson"}.
\item[\code{data}] an optional data frame in which to interpret the variables
in \code{form}. By default, the same data used to fit \code{object}
is used.
\item[\code{na.action}] a function that indicates what should happen when the
data contain \code{NA}s. The default action (\code{na.fail}) causes
an error message to be printed and the function to terminate, if there
are any incomplete observations.
\item[\code{maxDist}] an optional numeric value for the maximum distance used
for calculating the semi-variogram between two residuals. By default
all residual pairs are included.
\item[\code{length.out}] an optional integer value. When \code{object}
includes a \code{corSpatial} element, its semi-variogram values are
calculated and this argument is used as the \code{length.out}
argument to the corresponding \code{Variogram} method. Defaults to
\code{50}.
\item[\code{collapse}] an optional character string specifying the type of
collapsing to be applied to the individual semi-variogram values. If
equal to \code{"quantiles"}, the semi-variogram values are split
according to quantiles of the distance distribution, with equal
number of observations per group, with possibly varying distance
interval lengths. Else, if \code{"fixed"}, the semi-variogram values
are divided according to distance intervals of equal lengths, with
possibly different number of observations per interval. Else, if
\code{"none"}, no collapsing is used and the individual
semi-variogram values are returned. Defaults to \code{"quantiles"}.
\item[\code{nint}] an optional integer with the number of intervals to be
used when collapsing the semi-variogram values. Defaults to \code{20}.
\item[\code{robust}] an optional logical value specifying if a robust
semi-variogram estimator should be used when collapsing the
individual values. If \code{TRUE} the robust estimator is
used. Defaults to \code{FALSE}.
\item[\code{breaks}] an optional numeric vector with the breakpoints for the
distance intervals to be used in collapsing the semi-variogram
values. If not missing, the option specified in \code{collapse} is
ignored.
\item[\code{metric}] an optional character string specifying the distance
metric to be used. The currently available options are
\code{"euclidean"} for the root sum-of-squares of distances;
\code{"maximum"} for the maximum difference; and \code{"manhattan"}
for the sum of the absolute differences. Partial matching of
arguments is used, so only the first three characters need to be
provided. Defaults to \code{"euclidean"}.
\item[\code{...}] some methods for this generic require additional
arguments.  None are used in this method.
\end{ldescription}
\end{Arguments}
\begin{Value}
a data frame with columns \code{variog} and \code{dist} representing,
respectively, the semi-variogram values and the corresponding
distances. If the semi-variogram values are collapsed, an extra
column, \code{n.pairs}, with the number of residual pairs used in each
semi-variogram calculation, is included in the returned data frame. If
\code{object} includes a \code{corSpatial} element, a data frame with
its corresponding semi-variogram is included in the returned value, as
an attribute \code{"modelVariog"}. The returned value inherits from
class \code{Variogram}.
\end{Value}
\begin{Author}\relax
Jose Pinheiro \email{Jose.Pinheiro@pharma.novartis.com} and Douglas Bates \email{bates@stat.wisc.edu}
\end{Author}
\begin{References}\relax
Cressie, N.A.C. (1993), "Statistics for Spatial Data", J. Wiley \& Sons.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{lme}{lme}},
\code{\LinkA{Variogram}{Variogram}},
\code{\LinkA{Variogram.default}{Variogram.default}},
\code{\LinkA{Variogram.gls}{Variogram.gls}},
\code{\LinkA{plot.Variogram}{plot.Variogram}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
fm1 <- lme(weight ~ Time * Diet, data=BodyWeight, ~ Time | Rat)
Variogram(fm1, form = ~ Time | Rat, nint = 10, robust = TRUE)
\end{ExampleCode}
\end{Examples}

