LMM reduction following marginality taking out "item" before "subject:item" grouping factor
Maarten, So, regarding this issue, there is no difference between taking out
variance components for main effects before interactions within the same grouping factor, e.g. reducing (1 + A*B | subject) to (1 + A:B | subject), and taking out the whole grouping factor "item" (i.e. all variance components of it) before "subject:item"?
I think that if you have strong evidence that this is the appropriate random effects structure, then it makes sense to modify your model accordingly, yes. Do all variances of the random slopes (for interactions and main effects)
of a single grouping factor contribute to the standard errors of the fixed main effects and interactions in the same way?
No -- in general, with unbalanced datasets and continuous predictors, it's hard to say much for sure other than "no." But it can be informative to think of simpler, approximately balanced ANOVA-like designs where it's much easier to say much more about which variance components enter which standard errors and how. I have a Shiny power analysis app, PANGEA (power analysis for general anova designs) <http://jakewestfall.org/pangea/>, which as a side feature you can also use to compute the expected mean square equations for arbitrary balanced designs w/ categorical predictors. Near the bottom of "step 1" there is a checkbox for "show expected mean square equations." So you can specify your design, check the box, then hit the "submit design" button to view a table representing the equations, with rows = mean squares and columns = variance components. (A little while ago Shiny changed how it renders tables and now the row labels no longer appear, which is really annoying, but they are given in the reverse order of the column labels, so that the diagonal from bottom-left to top-right is where the mean squares and variance components correspond.) The standard error for a particular fixed effect is proportional to the (square root of the) corresponding mean square divided by the total sample size, that is, by the product of all the factor sample sizes. So examining the mean square for an effect will tell you which variance components enter its standard error and which sample sizes they are divided by in the expression. I find this useful for getting a sense of how the variance components affect the standard errors, even though the results from this app are only simplified approximations to those from more realistic and complicated designs. Jake On Wed, Nov 28, 2018 at 2:33 PM Maarten Jung <
Maarten.Jung at mailbox.tu-dresden.de> wrote:
Jake, thanks for this insight. So, regarding this issue, there is no difference between taking out variance components for main effects before interactions within the same grouping factor, e.g. reducing (1 + A*B | subject) to (1 + A:B | subject), and taking out the whole grouping factor "item" (i.e. all variance components of it) before "subject:item"? And, I would be glad if you could answer this related question: Do all variances of the random slopes (for interactions and main effects) of a single grouping factor contribute to the standard errors of the fixed main effects and interactions in the same way? Regards, Maarten On Wed, Nov 28, 2018, 20:03 Jake Westfall <jake.a.westfall at gmail.com wrote:
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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