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

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

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

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