[R-meta] Alternative view of fixed effects in meta-regression

[Farzad Keyhan]

Dear Lukasz,

The post you linked (had a fleeting look at it) is demonstrating my
logic for a 2-level model i.e. random effects of the form: ~1 | ID. In
that case, a single within and between is sufficient. But in the case
of Tim's question, we have a 3-level model i.e. random effects of the
form: ~ 1 | ID1/ID2 hence, additional betweens and withins.

To build some intuition, say you have a big bag with 5 marbles in it,
3 red ones, and 2 blue ones.

The mean of the entire bag gives you the bag-level mean
("X_btw_study"). The means of color-specific marbles gives you
color-level means ("X_btw_outcome") in the big bag.

Now, each marble --regardless of its color-- can differ from the mean
of the big bag ("X_wthn_study"). Also, each marble --given its color--
can differ from its own color-level mean ("X_wthn_outcome").

Finally, color-level means can differ from the big bag mean
("X_btw_outcome_study").

This is how my line of reasoning works. But certainly
correction/feedback/comments are more than welcome.

Best,
Fred

On Sat, Aug 28, 2021 at 2:17 PM Lukasz Stasielowicz
<lukasz.stasielowicz using uni-osnabrueck.de> wrote:
>
> Dear Fred,
>
> isn't it sufficient to include two variables rather than four variables
> when disentangling within-group-effects and between-group-effects?
> Some references:
>
> *Bell, A., Fairbrother, M. & Jones, K. Fixed and random effects models:
> Making an informed choice. Qual Quant 53, 1051–1074 (2019).
> https://doi.org/10.1007/s11135-018-0802-x
>
> *https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html#the-complex-random-effect-within-between-model-rewb
>
>
> According to the cited literature one could include two variables in
> multilevel models: X_within_group and X_btw_group
> X_btw_group refers to the group mean (e.g., mean age in the study j: x_j)
> X_within_group refers to the difference between each observation and its
> group mean (x_ij - x_j).
>
>
> Best,
>
> Lukasz
> --
> Lukasz Stasielowicz
> Osnabrück University
> Institute for Psychology
> Research methods, psychological assessment, and evaluation
> Seminarstraße 20
> 49074 Osnabrück (Germany)

> > Date: Fri, 27 Aug 2021 20:31:55 -0500
> > From: Farzad Keyhan <f.keyhaniha using gmail.com>
> > To: Timothy MacKenzie <fswfswt using gmail.com>
> > Cc: R meta <r-sig-meta-analysis using r-project.org>
> > Subject: Re: [R-meta] Alternative view of fixed effects in
> >       meta-regression
> > Message-ID:
> >       <CAEvy2r2yqKiR4UkL=9PNDC2n4c78sawxL+PXODzELCEoWnyfqQ using mail.gmail.com>
> > Content-Type: text/plain; charset="utf-8"
> >
> > Dear Tim,
> >
> > Unconditional 3-level models (i.e., models with no moderator) fit by
> > --rma.mv()-- assume: (A) normality of individual effects within
> > studies, (B) normality of level-specific effects, and that (C) the
> > relationship among the effects at each level is univariate linear.
> >
> > (If your model is a multivariate one, then those relationships are
> > assumed to be multivariate linear).
> >
> > Applying these assumptions to the model that you referred to (j cases
> > nested in k studies) would mean that the potential linear relationship
> > between case-specific effects can be estimated by adding a moderator
> > (e.g., --Age_jk--) that can vary at the case level.
> >
> > Now, if you add a moderator that varies among the cases, then, your
> > fixed-effect coefficient for --Age_jk-- would detonate the amount of
> > change in case-specific true effects (which are averages of individual
> > effect sizes for each case) relative to 1 year increase in --Age_jk--.
> >
> > Or equivalently: “the difference in average effect sizes between cases
> > that differ in age by one year”.
> >
> > So, you can add moderators at any level, and interpret the fixed
> > effects for those moderators as: the amount of change in
> > level-specific true effects relative to 1 unit increase in those
> > moderators.
> >
> > To (partially) answer your final question, for moderators that can
> > vary between more than one level, a single regression coefficient is a
> > mix of the moderators’ effects on more than one levels’ true effects.
> > Thus, it is a good idea to disentangle these effects. In the context
> > of multilevel meta-regression, I’m not sure if there is a suggested
> > procedure to do so. But *conceptually* something like what follows
> > *might* make sense:
> >
> > 1-    Create a variable called “X_btw_study”: Average X in each study.
> > 2-    Create a variable called “X_btw_outcome”: Average X in each
> > outcome in each study.
> > 3-    Create a variable called “X_btw_outcome_study”: Subtract (1) from (2).
> > 4-    Create a variable called “X_wthn_study”: Subtract (1) from each
> > X value in each study.
> > 5-    Create a variable called “X_wthn_outcome”: Subtract (2) from X
> > value of that outcome in each study.
> > 6-    Fit the following model: >> rma.mv(yi ~ X_btw_study +
> > X_btw_outcome + X_btw_outcome_study + X_wthn_study + X_wthn_outcome,
> > random=~1 | study/outcome) <<
> >
> > In my conceptual description above, I divided X into five parts
> > between two levels. But I leave it to other meta-regression experts to
> > comment on whether I've missed something or if they know of a
> > practical way of to deal with moderators that can vary across more
> > than one level
> >
> > Best,
> > Fred
> >
> > On Sat, Aug 21, 2021 at 9:09 AM Timothy MacKenzie <fswfswt using gmail.com> wrote:
> >>
> >> Dear Colleagues,
> >>
> >> I have some clarification questions.
> >>
> >> In multilevel models, what do the fixed-effect coefficients exactly
> >> predict? (change in the 'observed' effect yi for 1 unit of increase in
> >> moderator X OR change in some form of 'true effect' [depending on the
> >> random-part specification] for 1 unit of increase in moderator X)
> >>
> >> The reason I ask this is the bottom of p.26 of this paper (
> >> https://osf.io/4fe6u/). In this paper, Dr. Pustejovsky describes a 3-level
> >> model (j cases in k studies):
> >>
> >>                                         Rjk = Y0 + Y1(Age)jk + Vk + Ujk + ejk
> >>
> >> Then, he interprets Age's fixed effect coefficient as: *"the difference in
> >> average effect sizes between cases [level 2] that differ in age by one
> >> year"*.
> >>
> >> I wonder how this interpretation is possible and can be extended to other
> >> models (see below)?
> >>
> >> Say X is a continuous moderator that can vary between 'studies' and
> >> 'outcomes'. How can we apply Dr. Pustojuvsky's logic to the interpretation
> >> of 'X' fixed coefficient separately in:
> >>
> >> (A) 'rma.mv(yi ~ X, random=~1 | study)'
> >> vs.
> >> (B) 'rma.mv(yi ~ X, random=~1 | study/outcome)' differ?
> >>
> >> Thank you very much,
> >> Tim
> >>
> >>          [[alternative HTML version deleted]]
> >>
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