[R-lang] Comparing effect sizes in lmer

[Ariel M. Goldberg]

Dear R-lang,

I am interested in comparing the effect sizes of two predictors, A and B.  When I standardize them and enter them both into my lmer model, the coefficient of one is twice the size of the other (-9 vs -4.5).  Is there a principled way to determine if A has a "significantly" greater effect?  I was thinking that one way might be to compare the increase in loglikelihood over a baseline model but I'm not sure how to do it.  I'm envisioning something along the lines of creating 3 models: baseline, which contains neither of the two predictors, and baseline+A and baseline+B (which contain the baseline model and A and B, respectively), and then comparing how much each of the latter two models increases the loglikelihood over the baseline model.

Any thoughts would be much appreciated!
AG