Nesting terminology in mixed models

Question

Suppose I have the following nested lmer structure:

lmer(Y ~ X1 + X2 + X1:X2 + (1 | A) + (1 | A:B), data=d)

which is the same as:

lmer(Y ~ X1 * X2 + (1 | A/B), data=d)

Now if I would write it out in a report, I'd say something like:

Y modelled by X1 and X2 including the interaction of X1 and X2 (as the fixed effects) and B nested in A (as the random effects).

Both + (1 | A/B) and + (1 | A) + (1 | A:B) symbolize nesting and are equivalent. But how about when I don't want the effect of A to be estimated:

lmer(Y ~ X1 + X2 + X1:X2 + (1 | A:B), data=d)

Can I still simply say:

1. B nested in A (even though the random effect for A is now omitted?)

Or should it be:

2. B nested in A without the effect of A

Or since the : in the fixed effects part denotes for an interaction, could I potentially say:

3. The random effects part of the model was given as the interaction of A and B.

Answer 1

I would describe the final model as simply having

a random intercept for every observed combination of A and B.

Although it looks like an interaction because you're just using R's interaction syntax, you're really just redefine a grouping variable in a slightly less structured way than in the previous specifications, ensuring that the random effect can now vary more freely than in the previous arrangements.

For expository purposes you may wish to give 'every observed combination of A and B' a name that makes sense to readers in terms of the study you are analyzing.