I use mgcv package in R to build a Generalized Additive Model. When looking at results of (e.g., summary()), there are two indices shown, and they are not the same, though not too different:
R-sq.(adj) = ... Deviance explained = ...%
In ?summary.gam, definitions of these two terms are:
r.sq
The adjusted r-squared for the model. Defined as the proportion of variance explained, where original variance and residual variance are both estimated using unbiased estimators. ... The proportion null deviance explained is probably more appropriate for non-normal errors. Note that r.sq does not include any offset in the one parameter model.
dev.expl
The proportion of the null deviance explained by the model. The null deviance is computed taking account of any offset, so dev.expl can be substantially lower than r.sq when an offset is present.
It seems that r.sq is about "variance", while dev.expl is about "deviance". However, I still don't know which one is better, r.sq or dev.expl?
Best Answer
In the most updated version of mgcv (1.8-37). Wood elaborated on the r.sq definition by stating "The proportion null deviance explained is probably more appropriate for non-normal errors." Therefore, deviance explained should be a more generalized measurement of goodness of fit especially for non-gaussian models.
More detailed explanation on deviance explained can be found at How I can interpret GAM results?