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

[Douglas Bates]

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?

I'm not sure about this particular example but in general there is no 
problem with a negative AIC or a negative deviance just as there is no 
problem with a positive log-likelihood.  It is a common misconception 
that the log-likelihood must be negative.  If the likelihood is derived 
from a probability density it can quite reasonably exceed 1 which means 
that log-likelihood is positive, hence the deviance and the AIC are 
negative.

If you believe that comparing AICs is a good way to choose a model then 
it would still be the case that the (algebraically) lower AIC is preferred.