[R] Discrepancies in weighted nonlinear least squares

[Andrej-Nikolai Spiess]

Dear all, 

 

I encounter some discrepancies when comparing the deviance of a weighted and
unweigthed model with the AIC values.

A general example (from 'nls'):

 

DNase1 <- subset(DNase, Run == 1)

fm1DNase1 <- nls(density ~ SSlogis(log(conc), Asym, xmid, scal), DNase1)

 

This is the unweighted fit, in the code of 'nls' one can see that 'nls'
generates a vector wts <- rep(1, n).

 

Now for a weighted fit:

 

fm2DNase1 <- nls(density ~ SSlogis(log(conc), Asym, xmid, scal), DNase1,
weights = rep(1:8, each = 2))

 

in which I assign dcreasing weights for each of the 8 concentrations with 2
replicates.

 

Now with 'deviance' I get:

 

> deviance(fm1DNase1)
[1] 0.004789569

 

> deviance(fm2DNase1)
[1] 0.0164259

 

Telling me that the weighted fit has a significantly higher deviance (is a
worse fit).

Now with AIC (or BIC) I get

 

> AIC(fm1DNase1)
[1] -76.41642

 

> AIC(fm2DNase1)
[1] -372.5437

 

Which tells me that the second fit is by orders of magnitude the better one.
Why so?

 

If I define AIC based on residual sum-of-squares as found in the textbooks

 

RSS <- function (object) 
{
    w <- object$weights
    r <- residuals(object)
    if (is.null(w)) 
        w <- rep(1, length(r))
    sum(w * residuals(object)^2)
}

 

AICrss <- function(object)

{

  n <- nobs(object)

  k <- length(coef(object))

  rss <- RSS(object)

  n * log((2 * pi)/n) + n + 2 + n * log(rss) + 2 * k

}

 

I get 

 

> AICrss(fm1DNase1)
[1] -76.41642

 

Which is the same value as the above AIC (stats:::AIC.logLik) based on
log-likelihood

 

but

 

> AICrss(fm2DNase1)
[1] -56.69772
 
which is higher and fits perfectly to the also higher deviance of the second
model.
 
Could anyone enlighten me? Is the standard AIC implementation for 'nls'
models not appliccable in case of weighted fitting?
 
Cheers,
Andrej
 

 

 

 

 

___________________________

 

Dr. rer. nat. Andrej-Nikolai Spiess

Department of Andrology

University Hospital Hamburg-Eppendorf

Martinistr. 52

20246 Hamburg

 

phone: +49-40-7410-51585

fax: +49-40-7410-51554

 

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