anova() and the difference between (x | y) and (1 | y:x) in lme4

  Thanks Tierry.

  I would add:

  (1) for what it's worth, lme offers an intermediate model (compound
symmetry), which allows for homogeneous but _negative_ within-group
correlation ((1|var1:var2) only allows for non-negative within-group
correlation)

  (2) the 'unstructured' (var1|var2) and 'grouped/positive compound
symmetry' models (1|var1:var2) are in principle nested (all
off-diagonals equal to zero, all diagonals identical -> simpler model),
so you should be able to use a likelihood ratio test/ANOVA to test.

  (3) your max|grad| convergence warnings are probably false positives;
I would try scaling&centring your continuous predictors to see if that
makes the eigenvalue warnings go away.