[R-sig-ME] A conceptual question regarding fixed effects
sree datta
@reedt@8 @end|ng |rom gm@||@com
Fri Aug 13 05:55:20 CEST 2021
Thanks for sharing this Phillip, a fabulous and a very helpful document!
Sree
On Thu, Aug 12, 2021 at 9:19 PM Phillip Alday <me using phillipalday.com> wrote:
> 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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> > >
> >
>
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