  Regression models fitted to data can be assessed on their goodness of fit,
though models with many parameters should be disfavored to prevent
over-fitting. Statisticians' tools for this are little known to physical
scientists. These include the Akaike Information Criterion (AIC), a penalized
goodness-of-fit statistic, and the AICc, a variant including a small-sample
correction. They entered the physical sciences through being used by
astrophysicists to compare cosmological models; e.g., predictions of the
distance-redshift relation. The AICc is shown to have been misapplied, being
applicable only if error variances are unknown. If error bars accompany the
data, the AIC should be used instead. Erroneous applications of the AICc are
listed in an appendix. It is also shown how the variability of the AIC
difference between models with a known error variance can be estimated. This
yields a significance test that can potentially replace the use of `Akaike
weights' for deciding between such models. Additionally, the effects of model
misspecification are examined. For regression models fitted to data sets
without (rather than with) error bars, they are major: the AICc may be shifted
by an unknown amount. The extent of this in the fitting of physical models
remains to be studied.
