corRatio                package:nlme                R Documentation

Rational Quadratic Correlation Structure

Description:

     This function is a constructor for the 'corRatio' class,
     representing a rational quadratic spatial correlation structure.
     Letting d denote the range and n denote the nugget effect, the
     correlation between two observations a distance r apart is
     1/(1+(r/d)^2) when no nugget effect is present and
     (1-n)/(1+(r/d)^2) when a nugget effect is  assumed. Objects
     created using this constructor need to be later initialized using
     the appropriate 'Initialize' method.

Usage:

     corRatio(value, form, nugget, metric, fixed)

Arguments:

   value: an optional vector with the parameter values in constrained
          form. If 'nugget' is 'FALSE', 'value' can have only one
          element, corresponding to the "range" of the rational
          quadratic correlation structure, which must be greater than
          zero. If 'nugget' is 'TRUE', meaning that a nugget effect is
          present, 'value' can contain one or two elements, the first
          being the "range" and the second the "nugget effect" (one
          minus the correlation between two observations taken
          arbitrarily close together); the first must be greater than
          zero and the second must be between zero and one. Defaults to
          'numeric(0)', which results in a range of 90% of the minimum
          distance and a nugget effect of 0.1 being assigned to the
          parameters when 'object' is initialized.

    form: a one sided formula of the form '~ S1+...+Sp', or '~
          S1+...+Sp | g', specifying spatial covariates 'S1' through
          'Sp' and,  optionally, a grouping factor 'g'.  When a
          grouping factor is present in 'form', the correlation
          structure is assumed to apply only to observations within the
          same grouping level; observations with different grouping
          levels are assumed to be uncorrelated. Defaults to '~ 1',
          which corresponds to using the order of the observations in
          the data as a covariate, and no groups.

  nugget: an optional logical value indicating whether a nugget effect
          is present. Defaults to 'FALSE'.

  metric: an optional character string specifying the distance metric
          to be used. The currently available options are '"euclidean"'
          for the root sum-of-squares of distances; '"maximum"' for the
          maximum difference; and '"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 '"euclidean"'.

   fixed: an optional logical value indicating whether the coefficients
          should be allowed to vary in the optimization, or kept fixed
          at their initial value. Defaults to 'FALSE', in which case
          the coefficients are allowed to vary.

Value:

     an object of class 'corRatio', also inheriting from class
     'corSpatial', representing a rational quadratic spatial
     correlation structure.

Author(s):

     Jose Pinheiro Jose.Pinheiro@pharma.novartis.com and Douglas Bates
     bates@stat.wisc.edu

References:

     Cressie, N.A.C. (1993), "Statistics for Spatial Data", J. Wiley &
     Sons.

     Venables, W.N. and Ripley, B.D. (1997) "Modern Applied Statistics
     with S-plus", 2nd Edition, Springer-Verlag.

     Littel, Milliken, Stroup, and Wolfinger (1996) "SAS Systems for
     Mixed Models", SAS Institute.

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

See Also:

     'Initialize.corStruct', 'summary.corStruct', 'dist'

Examples:

     sp1 <- corRatio(form = ~ x + y + z)

     # example lme(..., corRatio ...)
     # Pinheiro and Bates, pp. 222-249
     fm1BW.lme <- lme(weight ~ Time * Diet, BodyWeight,
                        random = ~ Time)
     # p. 223
     fm2BW.lme <- update(fm1BW.lme, weights = varPower())
     # p 246 
     fm3BW.lme <- update(fm2BW.lme,
                correlation = corExp(form = ~ Time))
     # p. 249
     fm5BW.lme <- update(fm3BW.lme, correlation =
                        corRatio(form = ~ Time))

     # example gls(..., corRatio ...)
     # Pinheiro and Bates, pp. 261, 263
     fm1Wheat2 <- gls(yield ~ variety - 1, Wheat2)
     # p. 263 
     fm3Wheat2 <- update(fm1Wheat2, corr = 
         corRatio(c(12.5, 0.2),
            form = ~ latitude + longitude,
                  nugget = TRUE))

