[R] Deviance and AIC in weighted NLS

[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)

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 increasing 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
(lower AIC). 

However, 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 applicable in case of weighted fitting?

Do the weights have to be normalized somehow? I tried different stuff, but
to no avail.

Thanks in advance,

-ans

 

 

 

 

___________________________

 

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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