[R-sig-ME] A conceptual question regarding fixed effects

[Phillip Alday]

This differs somewhat depending on whether you're assuming an identity
link (as in linear mixed models) or a non-identity link (as in
generalized linear mixed models), see e.g. Dimitris Rizopoulos'
explanation of conditional vs. marginal effects on pdf-page 346 / slide
321 of his course notes http://drizopoulos.com/courses/EMC/CE08.pdf.

For LMM, once you've added a by-group intercept term, the biggest change
you'll generally see with adding addition by-group slopes is in the
standard errors of the fixed effects. The by-group intercept term
matters a lot because it begins to separate within vs. between/across
group effects and thus 'overcomes' Simpson's paradox. More directly,
introducing a by-group intercept allows the groups to have individual
lines instead of sharing one line, and thus you have a separation of
within vs between group effects. (Actually, this matters for any first
term, whether the intercept or not, but the first RE term is usually the
intercept.)

In Statistical Rethinking, Richard McElreath introduces random effects
as being a type of interaction, which is actually a fair intuition
(although there are substantial differences in estimation and formal
details). If you add in higher order effects, you also change the
precise interpretation of the lower-level effects, potentially along
with their estimates and standard errors. The same holds approximately
for adding in random effects.

Note that for the linked example, both the LM and LMM offer coefficients
with potentially meaningfully interpretations. Generally for a bigger
stimulus, you would expect a bigger response, which is a good prediction
if you don't know which subject a given observation came from. And thus
the LM tells you just that because it doesn't know which subject each
observation came from. But if you want to how a given subject will
respond to a larger stimulus, then the effect is paradoxically reversed.
And that's what the mixed model captures.

Or in yet other words, the LM assumes that there are no differences
between subjects and thus any differences are due to stimulus alone.
This isn't true, so it doesn't give a good estimate for different
subjects. Your choice of random effects is a statement about where you
assume differences to exist (and be measurable / distinguishable from
observation-level variance).

Note that there is one confound in the simulated data there: each
subject only saw stimuli within a relatively small range. If each
subject had seen stimuli across a wider range, then I suspect that each
subject's 2 very low response values would have had sufficient leverage
to flatten out the LM's slope estimate. (Such confounds of course exist
in reality in many practical contexts, but for a repeated-measures
design in biology/psychology/neuroscience, it would be great to have a
bit more control....)

Phillip

On 12/8/21 4:52 pm, Simon Harmel wrote:
> Dear Colleagues,
> 
> Can we say in mixed-effects models, a fixed-effect coef. is some kind of
> (weighted) average of its individual regression counterparts fit to each
> level of a grouping variable and that is why fixed-effect coefs in
> mixed-effects models can prevent things like a Simpson's Paradox case (
> https://stats.stackexchange.com/a/478580/140365) from happening?
> 
> If yes, then, would this also mean that if we fit models with the exact
> same fixed-effects specification but differing random-effect
> specifications, then the fixed coefs can be expected to be different in
> value but also meaning (i.e., what kind of [weighted] average they
> represent)?
> 
> For example, would the meaning of a fixed-effect coef. for variable X
> change if it has a corresponding random-effect in the model vs. when it
> doesn't, or if we allow X to vary across levels of 1 grouping variable (X |
> ID1) vs. those of 2 nested grouping variables (X | ID1/ID2)?
> 
> Many thanks for helping me understand this better,
> Simon
> 
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