LMM reduction following marginality taking out "item" before "subject:item" grouping factor

Maarten,

No, I would not agree that the Bates quote is referring to the principle
of marginality in the sense of e.g.:
https://en.wikipedia.org/wiki/Principle_of_marginality

Bates can chip in if he wants, but as I see it, the quote doesn't hint at
anything like this. It simply says that "variance components of
higher-order interactions should generally be taken out of the model before
lower-order terms nested under them" -- which I agree with. The reason this
is _generally_ true is because hierarchical ordering is _generally_ true.
But it looks like it's not true in your particular case.

can you think of a reason why they suggest to follow this principle other

than "higher-order interactions tend to explain less variance than
lower-order interations"?

No.

Jake

On Wed, Nov 28, 2018 at 12:53 PM Maarten Jung <
Maarten.Jung at mailbox.tu-dresden.de> wrote:

Hi Jake,

Thanks for your thoughts on this.

I thought that Bates et al. (2015; [1]) were referring to this principle
when they stated:
"[...] we can eliminate variance components from the LMM, following the
standard statistical principle with respect to interactions and main
effects: variance components of higher-order
interactions should generally be taken out of the model before
lower-order terms nested under them. Frequently, in the end, this leads
also to the elimination of variance
components of main effects." (p. 6)

Would you agree with me that this is referring to the principle of
marginality? And if so, can you think of a reason why they suggest to
follow this principle other than "higher-order interactions tend to explain
less variance than lower-order interations"?

Best regards,
Maarten

[1] https://arxiv.org/pdf/1506.04967v1.pdf

On Wed, Nov 28, 2018 at 7:24 PM Jake Westfall <jake.a.westfall at gmail.com>
wrote:

Maarten,

I think it's fine. I can't think of any reason to respect a principle of
marginality for the random variance components. I agree with the feeling
that it's better to remove higher-order interactions before lower-order
interactions and so on, but that's just because of hierarchical ordering
(higher-order interactions tend to explain less variance than lower-order
interations), not because of any consideration of marginality. If in your
data you find that hierarchical ordering is not quite true and instead the
highest-order interaction is important while a lower-order one is not, then
it makes sense to me to let your model reflect that finding.

Jake

On Wed, Nov 28, 2018 at 12:18 PM Maarten Jung <
Maarten.Jung at mailbox.tu-dresden.de> wrote:

Dear list,

In a 2 x 2 fully crossed design in which every participant responds to
every stimulus multiple times in each cell of the factorial design the
maximal linear mixed model justified by the design (using the lme4
syntax)
should be:
y ~ A * B + (1 + A * B | subject) + (1 + A * B | item)  + (1 + A * B |
subject:item)

Within a model reduction process, be it because the estimation algorithm
doesn't converge or the model is overparameterized or one wants to
balance
Type-1 error rate and power, I follow the principle of marginality
taking
out higher-order interactions before lower-order terms (i.e. lower-order
interactions and main effects) nested under them and random slopes
before
random intercepts.
However, it occurs that the variance components of the grouping factor
"item" are not significant while those of the grouping factor
"subject:item" are.

Does it make sense to remove the whole grouping factor "item" before
taking
out the variance components of the grouping factor "subejct:item"?

A reduced model would f.i. look like this:
y ~ A * B + (1 + A | subject) + (1 | subject:item)

I'm not sure whether this contradicts the principal of marginality and,
in
general, whether this is a sound approach.

Any help is highly appreciated.

Best regards,
Maarten

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