[R] negetative AIC values: How to compare models with negative AIC's

[Prof Brian Ripley]

AICs (like log-likelihoods) can be positive or negative.
However, you fitted a Gaussian and not a binomial glm (as lrm does if 
m.arson is binary).

For a discrete response with the usual dominating measure (counting 
measure) the log-likelihood is negative and hence the AIC is positive,
but not in general (and it is matter of convention even there).

In any case, Akaike only suggested comparing AIC for nested models, no one
suggests comparing continuous and discrete models.

On Fri, 15 Apr 2005, Jan Verbesselt wrote:

>
> Dear,
>
> When fitting the following model
> knots <- 5
> lrm.NDWI <- lrm(m.arson ~ rcs(NDWI,knots)
>
> I obtain the following result:
>
> Logistic Regression Model
>
> lrm(formula = m.arson ~ rcs(NDWI, knots))
>
>
> Frequencies of Responses
>  0   1
> 666  35
>
>       Obs  Max Deriv Model L.R.       d.f.          P          C        Dxy
> Gamma      Tau-a         R2      Brier
>       701      5e-07      34.49          4          0      0.777      0.553
> 0.563      0.053      0.147      0.045
>
>          Coef     S.E.    Wald Z P
> Intercept   -4.627   3.188 -1.45  0.1467
> NDWI         5.333  20.724  0.26  0.7969
> NDWI'        6.832  74.201  0.09  0.9266
> NDWI''      10.469 183.915  0.06  0.9546
> NDWI'''   -190.566 254.590 -0.75  0.4541
>
> When analysing the glm fit of the same model
>
> Call:  glm(formula = m.arson ~ rcs(NDWI, knots), x = T, y = T)
>
> Coefficients:
>            (Intercept)     rcs(NDWI, knots)NDWI    rcs(NDWI, knots)NDWI'
> rcs(NDWI, knots)NDWI''  rcs(NDWI, knots)NDWI'''
>                0.02067                  0.08441                 -0.54307
> 3.99550                -17.38573
>
> Degrees of Freedom: 700 Total (i.e. Null);  696 Residual
> Null Deviance:      33.25
> Residual Deviance: 31.76        AIC: -167.7
>
> A negative AIC occurs!
>
> How can the negative AIC from different models be compared with each other?
> Is this result logical? Is the lowest AIC still correct?

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
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