[R-meta] Prediction intervals for multilevel meta-analysis

Hi Paul,

In short, yes. The prediction interval incorporates two sources of
uncertainty: uncertainty from the estimate of the mean effect (the center
of the PI) and uncertainty from there being a distribution of effects about
the mean (measured by the sum of the random effects variance components).
In your case, my guess is that the change in the prediction intervals is
driven by the first source. When you add in additional levels, you are
acknowledging that there is additional dependence in the data structure,
and this dependence leads to more uncertainty about the average effect. You
should see this in how the standard error of the average effect increases
across the three models you described. Particularly if the data include
only a small number of top-level units, the increase in SE can lead to
fairly big changes in the width of the PI.

In the example you gave, it's interesting (at least to me
::nerdfaceemoji::) to see that the second source of uncertainty doesn't
actually change very much. We can see this by comparing the sums of
variance components (sigma-squared's rather than sigmas):
- Model 1 total RE variance: 0.41^2 = 0.1681
- Model 2 total RE variance: 0.279^2 + 0.317^2 = 0.1783 (square root =
0.4223)
- Model 3 total RE variance: 0.275^2 + 0.114^2+0.309^2 = 0.1841 (square
root = 0.4209)
The total variance increases slightly, but not enough to affect the width
of the PI by all that much. The differences between models indicate that
they're similar to variance decompositions. Model 1 is estimating the total
variance, Model 2 is telling how much of the total is at level 2 versus
level 1, Model 3 is further breaking out how much of the total is at level
3, level 2, and level 1.

James

--------------------------
James Pustejovsky
https://jepusto.com

Why do prediction intervals get so much wider when a multi-level approach
is used?

Prediction intervals are usually computed by +/- tau*1.96. Obtaining tau
is straightforward when doing a random-effects meta-analysis (e.g.,
function rma() with metafor).

When running a multilevel meta-analysis, things are a bit more
complicated. According to Wolfgang Viechtbauer, it is possible to take the
sum of the taus ? (or sigmas, as the taus are called in the output of the
rma.mv() function). However, this results in even wider prediction
intervals. For a random effects meta-analysis with over 300 effect sizes,
the width of the prediction interval is 1.60 (tau = 0.41). Command used:
rma(yi, vi, data = df)
When I run a multilevel meta-analysis with effect sizes nested in studies,
the width of the prediction interval is 2.34 (tau/sigma level 1 = .279,
level 2 = .317). Command used: rma.mv(yi, vi, random = list(~ 1 |
effectID, ~ 1 | StudyID), tdist = TRUE, data=df)
If I add yet another level, articles (i.e., effect sizes nested within
studies, studies nested within papers), the width of the prediction
interval gets even wider: 2.74 (tau/sigma level 1 = .275, level 2 = .114,
level 3 = .309). Command used: rma.mv(yi, vi, random = list(~ 1 |
effectID, ~ 1 | StudyID, ~ 1 | PaperID), tdist = TRUE, data=df)

Is it plausible that the prediction intervals get that much wider?

Thanks,
Paul



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