# [R-meta] variance explained by fixed & random effects

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Fri Mar 29 11:50:04 CET 2019

```Yes, this is a good reference.

1) What does it means if the pseudo R^2 statistic is negative?

R^2 type measures in meta-analysis are typically computed as proportional reductions in variance components (that reflect heterogeneity) when predictors/moderators are added to the model. However, for the models we typically use for a meta-analysis, there is no guarantee that the variance components always decrease (or remain unchanged) when we add predictors. So, it can happen that a variance component increases, in which case the R^2 value is technically negative. An example:

library(metafor)

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

res0 <- rma(yi, vi, data=dat)
res1 <- rma(yi, vi, mods = ~ alloc, data=dat)
res1

# proportional 'reduction' in tau^2 (how R^2 is computed here)
(res0\$tau2 - res1\$tau2) / res0\$tau2

If you look at 'res1', R^2 is given as 0%, since negative R^2 values are simply set to 0.

So what does this mean? I would say it simply means that there is no reduction in the estimate of tau^2, so 'alloc' does not account for any heterogeneity.

But also keep in mind that the value of R^2 is an estimate that can be quite inaccurate esp. if the number of studies is small (https://onlinelibrary.wiley.com/doi/abs/10.1111/bmsp.12002). One may need maybe 30 or even more studies to get a fairly decent estimate (but don't quote me on that number).

Another important point here is that we compute R^2 based on how much *heterogeneity* is accounted for, not how much of the *total variance* is accounted for. So, for example:

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

This suggests that ~76% of the heterogeneity is accounted for by 'ablat' (but again, this estimate could be way off with k=13). But the total variance in this example is tau^2 plus the amount of sampling variability in the estimates (the 'vi' values). Since each study has a different amount of sampling variability, it isn't super clear how we then should think of the total variance, but one commonly used approach is to compute a 'typical' sampling variance and use that to define the total variance. For example, Higgins and Thompson (2002) suggest to compute the 'typical' sampling variance with:

k <- res\$k
wi <- 1/dat\$vi
vt <- (k-1) / (sum(wi) - sum(wi^2)/sum(wi))
vt

Others (https://academic.oup.com/aje/article/150/2/206/55325) have suggested using the harmonic mean, which would be:

1/mean(wi)

Regardless, we can then use tau^2 + vt as the total variance.

2) How can we think of variance accounted for by random effects?

Another issue in this discussion is the question of how to think of

-----Original Message-----
From: Theresa Stratmann [mailto:theresa.stratmann using senckenberg.de]
Sent: Tuesday, 26 March, 2019 9:58
To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] variance explained by fixed & random effects

Hi all,

A paper from Nakagawa & Santos (2012) seems to answer my question (specifically the section "Heterogeneity - Quantifying Heterogeneity").

Nakagawa, S. & Santos, E.S.A. Evol Ecol (2012) 26: 1253. https://doi.org/10.1007/s10682-012-9555-5

Theresa

> On March 14, 2019 at 5:13 PM Theresa Stratmann <theresa.stratmann using senckenberg.de> wrote:
>
> Dear Wolfgang,
>
> Thank-you for the fast reply!
>
> So the code is easy enough, but I am struggling to understand some of the nuance/ how to properly think about the different numbers. I have added some questions, between the dashed lines, below your answers. If they are within the scope of the mailing list and not to cumbersome, I would be curious for some explanations to help clear up my confusion.
>
> Thanks!
>
> Theresa
>
> One way of estimating an R^2-type measure for the fixed effects is to fit the reduced model without the predictor and then compute the proportional reduction in the variance components:
>
> res0 <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat)
> pmax(0, (res0\$sigma2 - res\$sigma2) / res0\$sigma2) * 100
> max(0, (sum(res0\$sigma2) - sum(res\$sigma2)) / sum(res0\$sigma2)) * 100
>
> One can compute this per component or overall. There is no guarantee that the value is >= 0, so I use pmax()/max() to set negative values to 0. One could debate whether it even makes sense here to estimate how much of the school-level heterogeneity is accounted for by 'year'(since this is a district-level predictor), but that's another issue.
>
> -----------------------------------------------------------------------------------------------------------------------------------------
> ***
> So we have:
>
> sigma^2 for res0 .... heterogeneity in district & school:
> 0.06506194 0.03273652
> sigma^2 for res1 .... heterogeneity in district & school, accounting for year:
> 0.11482670 0.03208976
>
> (res0\$sigma2 - res\$sigma2) / res0\$sigma2
> -0.76488265  0.01975659
>
> So it looks like accounting for year increases the heterogeneity in district.
>
> Acknowledging and moving on from the fact that year is not the best fixed effect... What does it mean if the value is negative/zero?
> There is more variance in the random intercepts when you include the fixed effect?
> Adding this particular fixed effect does not improve the model?
> If I was writing this up, could I say: "Accounting for year does not explain any additional variation in the estimated variance of the random intercepts."
> More specifically:
> "The heterogeneity accounted for by year is 0% for district level variance, and 1.98% for school level variance."
> "The overall heterogeneity accounted for by year is 0%." ???
>
> Thinking about marginal and conditional R^2 for mixed-effects models, there is a separation of fixed and random effects: fixed effects account for X% of the variation
> and random effects accounting for Y% of the variation and Z% residual variance. But here the fixed effect is somehow included
> in the sigma^2 (estimated variance of the random intercepts)?  This just confuses me a bit/ I missed the thought process.
>
> ***
> -----------------------------------------------------------------------------------------------------------------------------------------
>
> I am not sure how to think of the question how much variance is explained by a random effect here. One could ask how much of the total heterogeneity
> is due to district- and school-level variance, which would be:
>
> res0\$sigma2 / sum(res0\$sigma2) * 100
>
> Or one could ask how much of the unaccounted for heterogeneity (so heterogeneity not accounted for by 'year') is due to district- and school-level variance, which would be:
>
> res\$sigma2 / sum(res\$sigma2) * 100
>
> -----------------------------------------------------------------------------------------------------------------------------------------
> ***
> So, to double check, the heterogeneity accounted for by year is, what you showed above:
>
> max(0, (sum(res0\$sigma2) - sum(res\$sigma2)) / sum(res0\$sigma2)) * 100
>
> We also know that:
>
> res0\$sigma2 / sum(res0\$sigma2) * 100  = 66.52655; 33.47345
>
> max(0, (sum(res0\$sigma2) - sum(res\$sigma2)) / sum(res0\$sigma2)) * 100 = 0
>
> Since these have the same denominator... is it true that:
> [max(0, (sum(res0\$sigma2) - sum(res\$sigma2)) / sum(res0\$sigma2)) * 100] + sum(res0\$sigma2 / sum(res0\$sigma2) * 100) = 100
> (I check on my data and the answer seems to be no...)
> ***
> -----------------------------------------------------------------------------------------------------------------------------------------
>
> Or one could ask how much of the total variance (so heterogeneity + sampling variance) is due to district- and school-level variance, which would be:
>
> k <- res\$k
> wi <- 1/dat\$vi
> s2 <- (k-1) * sum(wi) / (sum(wi)^2 - sum(wi^2))
>
> res0\$sigma2 / (sum(res0\$sigma2) + s2) * 100
>
> (here I am using the Higgins & Thompson, 2002, definition of a 'typical' sampling variance).
>
> Or one could ask how much of the unaccounted for variance (so unaccounted for heterogeneity [heterogeneity not accounted for by 'year'] + sampling variance)
> is due to district- and school-level variance, which would be:
>
> res1\$sigma2 / (sum(res1\$sigma2) + s2) * 100
>
> -----------------------------------------------------------------------------------------------------------------------------------------
> ***
> Can one ask here how much of the total variance (estimate of variance in true effects + sample variation) is accounted for by year? This is really what I want.
> ***
> -----------------------------------------------------------------------------------------------------------------------------------------
>
> > On March 13, 2019 at 6:00 PM "Viechtbauer, Wolfgang (SP)" <wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
> >
> >
> > Dear Theresa,
> >
> > You forgot some parts in your code, so to make this example reproducible, let's use this model:
> >
> > dat <- dat.konstantopoulos2011
> > res <- rma.mv(yi, vi, mods = ~ factor(year), random = ~ 1 | district/school, data=dat)
> >
> > And I would consider schools nested within districts here, but this isn't pertinent to the issue.
> >
> > One way of estimating an R^2-type measure for the fixed effects is to fit the reduced model without the predictor and then compute the proportional reduction in the variance components:
> >
> > res0 <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat)
> > pmax(0, (res0\$sigma2 - res\$sigma2) / res0\$sigma2) * 100
> > max(0, (sum(res0\$sigma2) - sum(res\$sigma2)) / sum(res0\$sigma2)) * 100
> >
> > One can compute this per component or overall. There is no guarantee that the value is >= 0, so I use pmax()/max() to set negative values to 0. One could debate whether it even makes sense here to estimate how much of the school-level heterogeneity is accounted for by 'year' (since this is a district-level predictor), but that's another issue.
> >
> > I am not sure how to think of the question how much variance is explained by a random effect here. One could ask how much of the total heterogeneity is due to district- and schol-level variance, which would be:
> >
> > res0\$sigma2 / sum(res0\$sigma2) * 100
> >
> > Or one could ask how much of the unaccounted for heterogeneity (so heterogeneity not accounted for by 'year') is due to district- and schol-level variance, which would be:
> >
> > res\$sigma2 / sum(res\$sigma2) * 100
> >
> > Or one could ask how much of the total variance (so heterogenity + sampling variance) is due to district- and schol-level variance, which would be:
> >
> > k <- res\$k
> > wi <- 1/dat\$vi
> > s2 <- (k-1) * sum(wi) / (sum(wi)^2 - sum(wi^2))
> > res0\$sigma2 / (sum(res0\$sigma2) + s2) * 100
> >
> > (here I am using the Higgins & Thompson, 2002, definition of a 'typical' sampling variance).
> >
> > Or one could ask how much of the unaccounted for variance (so unaccounted for heterogenity + sampling variance) is due to district- and schol-level variance, which would be:
> >
> > res1\$sigma2 / (sum(res1\$sigma2) + s2) * 100
> >
> > These are at least some possibilities.
> >
> > Best,
> > Wolfgang
> >
> > -----Original Message-----
> > From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Theresa Stratmann
> > Sent: Wednesday, 13 March, 2019 17:29
> > To: r-sig-meta-analysis using r-project.org
> > Subject: [R-meta] variance explained by fixed & random effects
> >
> > Dear Meta-Analysis Community,
> >
> > I am using the metafor package to help me summarize some data on habitat selection. I have a question whose answer I have not found yet on the useful metafor website, and was hoping that someone could help me (or point me to an online resource I missed).
> >
> > As part of the description of my meta-analysis I would like to explain how much of the variation in habitat selection is due to my fixed effects (season) versus my random effects (individual & year).
> >
> > For a normal mixed-effects model I know how to compute the marginal and conditional R^2 (Nakagawa & Schielzeth 2013), but I am wondering how to do this correctly for a meta-analysis, or maybe, better said, talk about variance explained by the fixed and random effects.
> >
> > First I looked into I^2 and the post on this for random effect models. But then I read more from Borenstein et al. (https://www.meta-analysis-workshops.com/download/common-mistakes1.pdf) and was not quiet sure anymore if this is what I want (I^2 = proportion of observed variance that reflects variance in the true effects and not the sampling error ... so this seems more of a summary for the entire model (?), which also needs to be treated with care and presented properly).
> >
> > Then I found this question on this mailing list: https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2017-September/000232.html  ...but cannot really get this code to produce anything sensible, because I guess I am not clear on the formula for res0 vs. res1.
> >
> > So I will give a different example, which closely approximates what I am trying to do (a model with fixed & random effects):
> >
> > dat <- dat.konstantopoulos2011
> > res <- rma.mv(yi, vi, mods = ~ factor(year), list( ~ 1 | district , ~ 1|school)  data=dat)
> > res
> >
> > How would you calculate / correctly describe variance explained by the fixed effects (year)? By the random effects (district, school)? I am not tied to a particular statistic, just want to make sure that whatever I do is an honest representation of my results and gives readers (& myself) the information they need to interpret the results.
> >
> > I am new to meta-analyses and this is just a small part of my project, so I have been able to do some reading, but not an in-depth dive. Therefore I would appreciate any help you can offer.
> >
> > Many thanks,
> >
> > Theresa Stratmann
> >
> > B.S. Ecology, The University of Georgia
> > M.S. Wildlife & Fisheries Biology, Clemson University
> >
> > PhD Student
> > Goethe University
> > and
> > Senckenberg Biodiversity and Climate Research Center
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