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

[Simon Harmel]

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> 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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> >
>

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