[R-sig-ME] interpretation of range value from lme model with exponential correlation structure

[Kevin Wright]

I have an example in the agridat package you might find useful.  First
the code and then a bit of interpretation.

require(agridat)
dat <- harris.wateruse
dat <- subset(dat, day!=268)
# Rescale day for nicer output, and convergence issues, add quadratic term
dat <- transform(dat, ti=day/100)
dat <- transform(dat, ti2=ti*ti)
require(nlme)
m3l <- lme(water ~ 0 + age:species + age:species:ti + age:species:ti2,
          data=dat, na.action=na.omit,
          random = list(tree=pdDiag(~1+ti)),
          cor = corExp(form=~ day|tree),
          )
print(m3l)

Linear mixed-effects model fit by REML
  Data: dat
  Log-restricted-likelihood: -497.8969
  Fixed: water ~ 0 + age:species + age:species:ti + age:species:ti2

ageA1:speciesS1     ageA2:speciesS1     ageA1:speciesS2     ageA2:speciesS2
         -14.569534          -12.305968           -6.749603          -11.362943
 ageA1:speciesS1:ti  ageA2:speciesS1:ti  ageA1:speciesS2:ti  ageA2:speciesS2:ti
          14.408342           13.538021            7.920372           12.837023
ageA1:speciesS1:ti2 ageA2:speciesS1:ti2 ageA1:speciesS2:ti2 ageA2:speciesS2:ti2
          -2.980521           -2.856673           -1.844861           -2.940303

Random effects:
 Formula: ~1 + ti | tree
 Structure: Diagonal
        (Intercept)        ti  Residual
StdDev:   0.3935732 0.1668882 0.4000106

Correlation Structure: Exponential spatial correlation
 Formula: ~day | tree
 Parameter estimate(s):
   range
4.721752
Number of Observations: 953
Number of Groups: 40


This example comes from a book by Schabenberger & Pierce. See:
http://books.google.com/books?id=c2Rr_J7geGQC&pg=PA518#v=onepage&q&f=false

They say "since the measurement times are coded in days, this estimate
of phi=4.721752 implies that water usage exhibits temporal correlation
over 3*phi = 14.16 days."

They did not justify this...it looks like a rule-of-thumb.  Note that
exp(-14.16/4.7217) = .05.

Kevin


On Wed, Nov 19, 2014 at 10:53 PM, Janee Wilkinson
<janeemwilkinson at gmail.com> wrote:
> Hi there,
>
> I am fitting a model using the lme function with an exponential correlation
> structure, e.g.
> lme(y~fixed1+fixed2, random =~1|subject,
>         cor=corExp(form=~distance|subject,nugget=TRUE))
>
> How do I interpret the range value?
>
> In a paper I described the range as the distance at which observations are
> uncorrelated but a reviewer has said this is incorrect and that the range
> specifies the rate at which the correlation tends to zero.
>
> I thought that the usual interpretation of the range was that observations
> separated by less than the range are spatially autocorrelated, whereas
> observations further apart than the range are not.
>
> >From the help file for corExp, letting d denote the range and n denote the
> nugget effect, the correlation between two observations a distance r apart
> is (1-n)*exp(-r/d). So when the distance=range, the correlation will be
> (1-n)*exp(-1), which will not be zero.
>
> Any help would be much appreciated.
>
> Thanks,
>
> Janee.
>
>         [[alternative HTML version deleted]]
>
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-- 
Kevin Wright