If I understand your question correctly, I think there is one exception.
Whenever you are dealing with balanced 2 x 2 x 2 x ... design all main
effects and interactions have equal statistical power. Whether or not main
effects are more important than interactions depends only on the coding of
the factors, that is you can recode any interaction as a main effect (e.g.,
congruency effects are often interactions). Of course, in a typical
research program main effects usually represent large known effects, but
this is a conceptual reason, not a mathematical implication. Therefore, in
my opinion, if a large effect maps onto a higher-order interaction, it may
well be associated with larger effects (and variance components) than main
effects.  The first paper I know that discusses this issue is:
Shaffer, J.P. (1977). Reorganization of variables in analysis of variance
and multidimensional contingency tables. Psychological Bulletin, 84,
220-228. I stumbled upon this when I realized that a significant 2 x 2 x 2
interaction mapped onto a simple main effect in a psycholinguistic
experimental design.

I am not sure whether this case generalizes to factorial designs with
factors with more than 2 levels, but in LMMs we usually test
single-degree-of-freedom contrasts. So, perhaps as long as the contrasts
are orthogonal it may hold here as well. (Shravan Vasishth recently
expressed some doubts about this, but we have not sorted this out
completely--I think.)

When you are dealing with continuous covariates (non-uniformly, usually
uni-modally distributed variables), interactions terms have definitely less
power. My classic reference for this is McClelland, G.H., & Judd, C.M.
(1993). Statistical difficulties of detecting interactions and moderator
effects. Psychological Bulletin, 114, 376-390. So in this case, I think the
low statistical power will lead to small effects and little chance to
detect significant variance components for interactions involving such
variables.

Reinhold






On Fri, Nov 8, 2013 at 7:15 PM, Jake Westfall <jake987722@hotmail.com>wrote:

> Hi all,
> I wanted to follow up with a tentative answer to the question I posted to
> the list a few weeks ago concerning lower-order random effects tending to
> have greater variance than higher-order random effects. There were not a
> lot of takers to my question either here or on stats.stackexchange.com,
> which is not too surprising really since it was kind of a long, involved,
> and ambiguous question.
>
> I have discovered that the regularity I described in my question has in
> fact been written about by several authors in the literature on Design of
> Experiments (DoE). It has been called the "hierarchical ordering principle"
> and also sometimes the "sparsity-of-effects principle."
>
> In the chapter on fractional factorial designs in Montgomery (2013, p.
> 290), he writes:
> >The successful use of fractional factorial designs is based on three key
> ideas:
> >
> >1. The sparsity of effects principle. When there are several variables,
> the system or process is likely to be driven primarily by some of the main
> effects and low-order interactions.
> >
> >...
>
>
> Wu & Hamada (2000, p. 143) instead call this the "hierarchical ordering
> principle", and use the phrase "sparsity of effects" to refer to a related
> but distinct observation:
> >Three fundamental principles for factorial effects:
> >
> >Hierarchical ordering principle: (i) Lower order effects are more likely
> to be important than higher order effects, (ii) effects of the same order
> are likely to be equally important.
> >
> >Effect sparsity principle: The number of relatively important effects in
> a factorial experiment is small.
> >
> >...
>
>
>
>  - Montgomery, D. C. (2013). Design and analysis of experiments (Vol. 8).
> New York: Wiley.
>  - Wu, C. J., & Hamada, M. S. (2000). Experiments: planning, analysis, and
> optimization (Vol. 552). John Wiley & Sons.
>
> Jake
> From: jake987722@hotmail.com
> To: r-sig-mixed-models@r-project.org
> Subject: RE: Variances of higher-order vs. lower-order random terms
> Date: Tue, 15 Oct 2013 02:25:29 -0600
>
>
>
>
> Follow-up: I cross-posted this question here:
>
> http://stats.stackexchange.com/q/72819/5829
>
> Jake
>
> From: jake987722@hotmail.com
> To: r-sig-mixed-models@r-project.org
> Subject: Variances of higher-order vs. lower-order random terms
> Date: Tue, 15 Oct 2013 02:08:58 -0600
>
>
>
>
> Hi everyone,
>
> **TL, DR summary:**
> Is there any theoretical or empirical basis to support the following
> statement being true as a general rule of thumb?
> "When estimating a mixed model, typically the estimated variances/standard
> deviations of random effects associated with 'higher-order' terms (e.g.,
> random effects of two-way, three-way, and beyond interaction terms) turn
> out to be *smaller* than the estimated variances/standard deviations of
> random effects associated with 'lower-order' terms (e.g., the residual
> variance, variances associated with simple effects of grouping factors)."
>
> The source of this claim is me. ;)
>
> ****
>
> Okay, now for the longer version...
>
> Typically when I sit down to start analyzing a new dataset which I know
> will call for a mixed model, one of the first models that I fit (after the
> statistical foreplay of looking through the observations in the dataset,
> plotting various things, cross-tabulating different factors, etc.) is one
> that is pretty close to the "maximal" random effects specification, where
> every random effect that is in-principle possible to estimate from the
> data, is estimated.
>
> Naturally, it is not uncommon that this nearly-maximal model will have
> some computational problems (convergence errors, or wacky
> variance/covariance estimates, or etc.) and that I have to trim back this
> model to find one that my data can more easily support. Fine.
>
> In these situations, the method I have come to prefer for trimming random
> terms is not to rely on significance tests or likelihood ratios, but rather
> to just identify the random effects that seem to have the smallest standard
> deviations (which can admittedly be a little tricky when predictors are on
> very different scales, but I try to take account of this in my appraisal)
> and remove these terms first, sequentially in an iterative process. The
> idea being that I want to alter the predictions of the model as little as
> possible while still reducing the complexity of the model.
>
> One pattern that I seem to have noticed after a pretty good amount of time
> spent doing this is that following this method very often leads me to trim
> random effects associated with higher-order terms (as defined above) of the
> model first. This is not always true, and occasionally some of the
> higher-order terms explain a lot of variance, but this doesn't seem to be
> the general pattern. In sharp contrast, I usually find that lower-order
> random terms -- particularly those associated with simple effects of the
> grouping factors -- explain a pretty good amount of variance and are fairly
> essential to the model. At the extreme, the residual term commonly accounts
> for close to the most variance, although of course removing this term
> wouldn't be sensible.
>
> This entirely informal observation leads me to form the hypothesis that I
> stated at the beginning of this email.
>
> If it is true, then it constitutes a useful piece of advice that might be
> passed down to people who are less experience with this kind of model
> selection process. But before I begin doing so, I want to check with other,
> more experienced users about their reactions to this observation. Does it
> seem more or less true to you? Is it roughly consistent with your
> experience fitting many different mixed models to many different datasets?
> Do you know of any sensible, theoretical reasons why we might actually
> *expect* this to be true in a lot of cases? Or does it just seem like
> bullshit?
>
> One possible answer here is that it is not true even in my own case, and I
> have simply deceived myself. Certainly a possibility that I am open to.
>
> Another possibility is that it might be true in my own case, but that this
> could simply be a kind of coincidence having to do with the kinds of
> datasets that I tend to work with routinely (which, FYI, are datasets in
> psychological / social sciences, a slight majority being experimental in
> origin, but also a fair proportion of non-experimental stuff). If this is
> the case then there is probably no good reason for expecting my
> observations to hold in general in other fields that handle very different
> kinds of data. Still, if there is a coherent non-coincidental reason for
> why this might be expected to be true, even if only for these particular
> kinds of datasets, I would love to hear it.
>
> And of course another possibility is that others *have* noticed similar
> patterns in their own data, and that it represents some kind of general
> rule of thumb that people find useful to keep in mind as they fit mixed
> models to various different data. If this is the case then it seems like
> there must be some compelling statistical-theoretical reason for why this
> pattern arises. But I really don't know what that reason would look like.
>
> I welcome anyone's thoughts and opinions about this. Totally legitimate
> responses to this might be as simple as comments like "Yeah I have noticed
> something similar in the data I've worked with, but I have no idea why it
> should be true" or conversely "I have noticed nothing like this in the data
> I've worked with." Of course I also welcome longer and more involved
> discussions...
>
> FULL DISCLOSURE: I think I am probably also going to post this question to
> stats.stackexchange.com. If/when I do, I will send along the link to that
> question thread.
>
> Jake
>
>
>
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