I have used the R lme function (nlme package) to construct linear mixed models, with a single random effect (as a random intercept) and a varIdent variance structure on a fixed effect (that is a factor).
The data is from a commercial fishery and I am looking to see which are the most important variables that need to be reported for this fishery. I have 3 biological variables (total length, weight, proportion of females) and two operational variables (effort and season) (unfortunately I can't post my data as it's confidential and I haven't managed to reproduce my problem with dummy data).
I have used the AICc to select the 'best' models from a list of a priori models and I am now calculating the R-squared values from each of the 'best' models (using the code from https://github.com/jslefche/rsquared.glmer/blob/master/rsquaredglmm.R from the blog post http://jonlefcheck.net/2013/03/13/r2-for-linear-mixed-effects-models/ and taken from Nakagawa and Schielzeth (2013) r2 for GLMMs). I have 6 models with a AICc < 10 and I've used that as my cut off as it gives me a balanced number of models to the calculate the relative importance of each variable of interest using the sum Akaike's weights (the purpose of the modelling).
However, the R-squared values are lower for my 'best' models (Akaike weights between 0.5-0.8; marginal $R^2$ = 0.48) than the model with all my fixed effects (Akaike weights 0.00; marginal $R2^$ = 0.89).
I've re-run the model selection using BIC and I get the same problem. My question is: Are most 'best' models the models that have a lower $R^2$ value (and lower AICc) or have higher R2 values but higher AICc values?