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

[Phillip Alday]

Here's a  second try with the link:

http://www.drizopoulos.com/courses/EMC/CE08.pdf

On 12/8/21 6:17 pm, Simon Harmel wrote:
> Dear Phillip,
> 
> Thank you very much. Unfortunately, I couldn't open the link you shared
> (I get: This site can’t be reached). So, I mainly want to focus on the
> second paragraph of your answer. My focus is only on LMMs. 
> 
> To be clear, I gather that you believe it is correct to think that 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 this issue is helpful in preventing Simpson's paradox-type conclusions?
> 
> Now, suppose X is a continuous predictor. It can vary across levels of
> ID1 and ID2; where ID2 is nested in ID1.
> 
> I fit three models with X:
> 
> 1) y ~ X + (X | ID1)  
> 2) y ~ X + (X | ID1 / ID2)
> 3) y ~ X
> 
> Can I interpret X in (1) as: Change in y for 1 unit of change in X
> averaged across levels of ID1 disregarding combination of ID1-ID2 levels?
> Can I interpret X in (2) as: Change in y for 1 unit of change in X
> averaged across levels of ID1 and combination of ID1-ID2 levels?
> Can I interpret X in (3) as: Change in y for 1 unit of change in X
> disregarding levels of ID1 and combination of ID1-ID2 levels?
> 
> Thanks,
> Simon
> 
> On Thu, Aug 12, 2021 at 5:46 PM Phillip Alday <me using phillipalday.com
> <mailto:me using phillipalday.com>> wrote:
> 
>     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
>     >
>     >       [[alternative HTML version deleted]]
>     >
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