R: Rational Quadratic Correlation Structure

corRatio {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))

[Package nlme version 3.1-89 Index]