[R-meta] I2 interpretation for Multilevel meta-analysis with moderators

[Viechtbauer, Wolfgang (SP)]

I can't answer your question (Is this correct?) at the moment. Instead, let me just explain again what I^2, R^2, and I^2_R (the latter denoting the I^2 from a model with moderators) represent.

Let's just go back to the case of a 'standard' random/mixed-effects models with no multilevel structure to keep things simple.

Draw a circle and then cut it up into three pieces of various sizes (i.e., draw a venn diagram). The three areas are:

(1) variability due to sampling variance
(2) heterogeneity accounted for by the moderators included in the model
(3) residual heterogeneity (i.e., heterogeneity that is not accounted for by the moderators)

One can think of tau^2 from the random-effects model (without any moderators) as the sum of (2) and (3) (i.e., the 'total' amount of heterogeneity). tau^2 from the mixed-effects meta-regression model (with the moderators) is (3). Therefore, tau^2 from the random-effects model minus tau^2 from the mixed-effects model is (2).

Then:

I^2   = 100 * ((2) + (3)) / ((1) + (2) + (3))
R^2   = 100 * (2) / ((2) + (3))
I^2_R = 100 * (3) / ((1) + (3))

So, I^2 estimates how much of the total amount of variability (the sum of (1), (2), and (3)) is due to heterogeneity.

R^2 estimates how much of the total amount of heterogeneity is accounted for by the moderators.

I^2_R estimates how much of the unaccounted for variability (the sum of (1) and (3)) is due to residual heterogeneity.

It is indeed the case that if (3) gets smaller, then R^2 increases and I^2_R decreases, but since the denominators of these statistic involve different quantities (beside (3)), they are only indirectly related to each other. And of all these, I still find I^2_R to be the least useful quantity to report.

And finally, an example to illustrate all of the above. Note that (1) is actually a bit of a tricky thing to quantify, since each study has a different sampling variance. So we compute an average/typical sampling variance for (1) and then use that for the computations. For the definitions above to be exactly true in the example below, I pretend that each study has the same sampling variance given by this somewhat odd-looking definition of the 'typical' sampling variance (vt).

dat <- dat.bcg
dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat)

k <- nrow(dat)
wi <- 1/dat$vi
vt <- (k-1) * sum(wi) / (sum(wi)^2 - sum(wi^2))
dat$vi <- vt

res1 <- rma(yi, vi, data=dat)
res1

res2 <- rma(yi, vi, mods = ~ ablat, data=dat)
res2

round(100 * res1$tau2 / (res1$tau2 + vt), 2)        # I^2
round(100 * (res1$tau2 - res2$tau2) / res1$tau2, 2) # R^2
round(100 * res2$tau2 / (vt + res2$tau2), 2)        # I^2_R

I will drop out of the discussion after this post. Everything is defined and illustrated at this point. Nothing wrong with reporting I^2_R, but I just don't find it very informative and I suspect most people will struggle to make sense of it. Heck, many people even misinterpret I^2 (as some kind measure of absolute heterogeneity), so chances that I^2_R gets correctly interpreted are quite slim.

Best,
Wolfgang

>-----Original Message-----
>From: Ivan Jukic [mailto:ivan.jukic using aut.ac.nz]
>Sent: Thursday, 14 October, 2021 7:51
>To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
>Subject: Re: I2 interpretation for Multilevel meta-analysis with moderators
>
>Hi Wolfgang,
>
>thank you for your prompt and helpful response.
>
>Indeed, I don't know how I missed this from the equation you provided. It clearly
>looks like an R2-type metric (Raudenbush, 2009). I guess I was too much focused
>on I2.
>
>Of course, R2 will always be more meaningful, but I'm still trying to make sense
>out of this while looking at both metrics at the same time (i.e., I2 and pseudo
>R2). Considering correct interpretations of both metrics, I would expect them to
>complement each other, well, at least in case of models with moderators. However,
>I then thought more about this and came up with the reasoning below.
>
>Let's consider my previous model with moderators having a total I2 of 68.35%
>(which can be decomposed to ~ 67.8, 0.51, and 31.7). From this, we can say that
>68.35% of the remaining variance (that is not accounted for already by the
>moderators) is due to the sum of study and effect size level heterogeneity (the
>rest is due to sampling variance). Then, taking the model's pseudo R2 into
>account, we can now conclude how 49% of the between-study heterogeneity (so 49%
>of the 67.8% from the above) and 92% of the within-study heterogeneity (so 92% of
>the 0.51% from the above) are accounted for by the moderators. Is this correct?
>If so, I would argue that we need both to have a complete picture, though, I'm
>afraid I might be wrong here.
>
>Cheers,
>Ivan
>
>From: Viechtbauer, Wolfgang (SP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
>Sent: Wednesday, 13 October 2021 10:32 PM
>To: Ivan Jukic <ivan.jukic using aut.ac.nz>; r-sig-meta-analysis using r-project.org <r-sig-
>meta-analysis using r-project.org>
>Subject: RE: I2 interpretation for Multilevel meta-analysis with moderators
>
>The 48.97833 and 92.06504 are pseudo R^2 statistics, so they tell you that 49% of
>the between-study and 92% of the within-study heterogeneity are accounted for by
>the moderators. To me, that's more informative than saying something about how
>much of the unaccounted for variance is due to (the sum of) between- and within-
>study heterogeneity.
>
>Best,
>Wolfgang
>
>>-----Original Message-----
>>From: Ivan Jukic [mailto:ivan.jukic using aut.ac.nz]
>>Sent: Wednesday, 13 October, 2021 4:32
>>To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
>>Subject: Re: I2 interpretation for Multilevel meta-analysis with moderators
>>
>>Dear Wolfgang,
>>
>>thank you for explaining this and providing an example - I really appreciate it.
>>
>>Indeed, in the posts I linked, you made it clear that I2 meaning is not very
>>intuitive (or even meaningful) in models with moderators. However, I would like
>>to use it because these moderators are the most important thing in my analysis.
>I
>>was thinking about just looking at I2 without moderators but these analyses are
>>less meaningful for the story that I'm trying to tell. I'm now considering
>>reporting both.
>>
>>I get (77.747826 3.745411 18.506763) for res0 (i.e., model without moderators),
>>and (67.8411969 0.5082705 31.6505327) for res1 (i.e., model with moderators).
>>
>>After reading your response I'm now unsure what the numbers obtained by 100 *
>>pmax(0, (res0$sigma2 - res1$sigma2) / res0$sigma2) actually mean? I get
>(48.97833
>>92.06504).
>>
>>In your response, you interpreted this as "how much of the between-study and
>>within-study heterogeneity is accounted for by the moderators". Based on these
>>numbers, it seems like a lot of between- and within-study heterogenity is
>>accounted for by the moderators. However, I can't interpret this in the same way
>>by just looking at I2 for res0 and res1 (81.49% and 68.35%, respectively, OR
>even
>>individual heterogeneity components from each model)? I guess I'm still missing
>>something here.
>>
>>Cheers,
>>Ivan