AIC with weighted nonlinear regression (nls)

问题

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 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). 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 applicable in case of weighted fitting?

Cheers, Andrej

来源：https://stackoverflow.com/questions/12566905/aic-with-weighted-nonlinear-regression-nls

标签

regression

nls

model-comparison