[R-meta] Dealing with effect size dependance with a small number of studies

-----Original Message-----
From: Danka Puric [mailto:djaguard at gmail.com]
Sent: Tuesday, 05 January, 2021 10:36
To: Viechtbauer, Wolfgang (SP)
Cc: James Pustejovsky; R meta
Subject: Re: [R-meta] Dealing with effect size dependance with a small
number of studies

Dear James, Wolfgang,

Thanks a lot for the quick and informative responses!

1. I guess I made things unnecessarily complicated :) The thing is, I know
that all these models are essentially the same:

notationa <- rma.mv(ES_corrected, SV, random = ~ factor(IDeffect) | IDstudy,
data=MA_dat_raw)
notationb <- rma.mv(ES_corrected, SV, random = ~ 1 | IDstudy/IDeffect,
data=MA_dat_raw)
notationc <- rma.mv(ES_corrected, SV, random = ~ IDeffect | IDstudy,
data=MA_dat_raw)

but I read in the Konstantopoulos (2011) example that they only deal with
the dependence arising from effect sizes coming from the same studies, but
NOT with dependence arising from multiple ES coming from the same group of
participants. I then erroneously concluded that in order to deal with this
type of dependence I would need to use struct = "UN", but I understand now
that's not the case.

Also, indeed, IDeffect does not refer to the type of outcome in a study.
Actually, we do have an outcome variable DV which could be used instead of
IDeffect, but sometimes it has the same value for several ESs in the same
group of participants, so it didn't seem appropriate to use it in this case.
I did realize the model with IDeffect was not structured like Berkey at al.
but thought it would be a better option, as IDeffect variables have unique
values across IDstudy.

As for sigma1.1 and sigma2.1. it's quite possible I just got something mixed
up when I compared different notations (I may have plotted the?wrong model),
but anyway, this model is definitely wrong for the data, so I'll just leave
it at that.

2. So, just to be sure I got this right, the following model

model <- rma.mv(ES_corrected, SV, random = ?~ 1 | IDstudy / IDsubsample/
IDeffect, data=MA_dat_raw)

in combination with clubSandwich robust estimates will yield adequate effect
size estimates for the situation where the same group of participants
provided more than one ES? That's actually the model I fit first, but then
thought wasn't appropriate after all.

I will also now look into inputting the covariance matrices and see if it's
possible to implement with the data we have. Thanks for suggesting this,
James.

3. Great, it does make the most sense to include all three levels of random
effects, I'm glad the small amount of variance is not an issue.

4. @James, unfortunately, our moderator is categorical and I'm not sure if
it theoretically makes sense to center it in any way... And by "taking care"
of this problem, I mostly meant that we're not making a huge mistake for not
explicitly modeling this if we use robust estimates. So, I think we'll
probably just leave it as it is.

Thanks again, this is really immensely helpful.

Best,
Danka

On Mon, Jan 4, 2021 at 11:35 PM Viechtbauer, Wolfgang (SP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
A few additional comments from my side below as well.

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-

project.org]

On Behalf Of James Pustejovsky
Sent: Monday, 04 January, 2021 22:25
To: Danka Puric
Cc: R meta
Subject: Re: [R-meta] Dealing with effect size dependance with a small
number of studies

Hi Danka,

Responses inline below.

Kind Regards,
James

On Mon, Jan 4, 2021 at 5:41 AM Danka Puric <djaguard at gmail.com> wrote:

Hi everyone,

Apologies for the long post and lots of questions.

We are doing a meta-analysis where a single study sometimes included more
than one subsample and also the same subsample (same group of

participants)

sometimes yielded more than one effect size.

1. Following the Berkey et al. (1998) example in metafor, we tried

fitting

the following ?basic? model:
nested_UN <-rma.mv(ES_corrected, SV, random = ~ IDeffect | IDstudy,

struct

= "UN", data=MA_dat_raw)
where individual effect sizes are nested within studies. This model,
however, produces profile likelihood plots which have flat parts (both

for

sigma2.1 and sigma2.2), which (if I?m not mistaken) indicates model
overparametrization. We believe this is most likely due to a small number
of effect sizes (k = 69, from 53 subsamples, from 20 studies).

This model should not have any variance components called "sigma2.1" or
"sigma2.2". When using the "random = ~ IDeffect | IDstudy" notation, you
should get "tau2" and "rho" values.

However, this model doesn't make much sense. I assume that different values
of "IDeffect" are just used to differentiate multiple effects within the
same study, but the levels are not meaningful in themselves (as opposed to
the Berkey example, where the two levels of the 'inner' factor differentiate
the two different outcomes). It would make more sense to use 'random = ~
IDeffect | IDstudy, struct = "CS"' which is in essence the same as 'random =
~ 1 | IDstudy / IDeffect'. See:

http://www.metafor-project.org/doku.php/analyses:konstantopoulos2011

We tried a similar model with random = ~ IDeffect | IDsubsample, but this
model did not even converge (I assume because the number of effect sizes
per subsample is even smaller than the number of ES per study).

This is probably again related to using struct="UN", which is (probably) not
appropriate here.

Are we correct in concluding that a multi-level model can not be properly
fit with the data that we have and an alternative approach (RVE or effect
size aggregation) is better suited to the data?

You have the wrong syntax here. If you want to specify a multi-level
meta-analysis model in which effecstares nested within studies, use the "/"
character to indicate nesting:
? nested_UN <-rma.mv(ES_corrected, SV, random = ~ IDstudy / IDeffect,
data=MA_dat_raw)
Or if you want to include sub-samples as an intermediate level:
? nested_UN <-rma.mv(ES_corrected, SV, random = ~ IDstudy / IDsubsample /
IDeffect, data=MA_dat_raw)

It should be: "random = ~ 1 | IDstudy / IDeffect" or "random = ~ 1 | IDstudy
/ IDsubsample / IDeffect".

Both of these will give estimates of average effect size and variance
component estimates. However, the corresponding standard errors of the
average effect sizes are based on the assumption that the entire model is
correctly specified. RVE relaxes that assumption. Thus, the decision to use
RVE or not should be based on a judgement about the plausibility of the
model's assumptions (rather than on whether you can get a model to
converge).

2. If we want to use RVE, would the following model which includes random
effects at all three levels (effect size, subsample, study) be

appropriate

in combination with clubSandwich package robust coefficient estimates?
model <-rma.mv(ES_corrected, SV, random =? ~ 1 | IDstudy / IDsubsample/
IDeffect, data=MA_dat_raw)
coef_test(model, vcov = "CR2")
Or should something else be done in order to adequately address the issue
of effect size dependence?

This seems fine. One step better would be to consider whether the effect
size estimates within a given sub-sample have correlated sampling errors.
This would be the case, for instance, if the effect sizes are for different
outcome measures (or measures of the same outcome at different points in
time), assessed on the same sub-sample of individual participants. Details
on how to do this can be found here:
https://www.jepusto.com/imputing-covariance-matrices-for-multi-variate-

meta-

analysis/

3. The variances for this model are:

Variance Components:
? ? ? ? ? ? ?estim? ? sqrt? nlvls? fixed? ? ? ? ? ? ? ? ? ? ? ? factor

sigma^2.1? 0.0589? 0.2427? ? ?20? ? ?no? ? ? ? ? ? ? ? ? ? ? ?IDstudy
sigma^2.2? 0.0250? 0.1583? ? ?53? ? ?no? ? ? ? ? ?IDstudy/IDsubsample
sigma^2.3? 0.0014? 0.0373? ? ?69? ? ?no? IDstudy/IDsubsample/IDeffect

In other words, there is very little variance at the level of IDeffect,
after Study and Subsample have been taken into account. The profile
likelihood plot for sigma^2.3 does, however, appear to peak at the
corresponding value when ?zoomed in? (with xlim=c(0,0.01)).
Should we consider this a satisfactory model, or is the variance at the
level of IDeffect too small to be meaningful? Presumably, this has to do
with the fact that the majority of subsamples (43 out of 53) only
contribute to the MA with one effect size, for 8 subsamples there are 2

ES

per subsample, and in two instances 5 ESs per subsample.

Would an acceptable alternative model be:
nested <- rma.mv(ES_corrected, SV, random = ~ 1 | IDstudy/IDeffect,
data=MA_dat_raw)
Here, we?ve excluded random effects at the subsample level, because it

made

more sense to include random effects at the level of individual effect
sizes and the two variables have a substantial overlap. The variances for
this model seem adequate (and their profile plots look fine, too).

Variance Components:
? ? ? ? ? ? ?estim? ? sqrt? nlvls? fixed? ? ? ? ? ? factor

sigma^2.1? 0.0678? 0.2604? ? ?20? ? ?no? ? ? ? ? ?IDstudy
sigma^2.2? 0.0150? 0.1223? ? ?69? ? ?no? IDstudy/IDeffect

The nice thing about RVE is that the standard errors for the average effect
are calculated in a way that does not require the correct specification of
the random effects structure. As a result, you should get very similar
standard errors regardless of whether you include random effects for all
three levels or whether you exclude a level. However, the variance
component estimates are still based on an assumption that the model is
correctly specified. I think it would therefore be preferable to use the
model that captures the theoretically relevant levels of variation, so in
this case, all three levels.

Agree. I would go with the IDstudy/IDsubsample/IDeffect model.

4. Finally, we are also interested in examining the effects of a

moderator

variable which defines different outcomes. So, in cases when one

subsample

produces more than one effect size ? sometimes these effect sizes belong

to

the same level of the moderator variable (same outcome under different
circumstances) and sometimes they belong to different levels of the
moderator (different outcomes). Theoretically, we would expect ?same-

level?

ESs to be more correlated than ?different-level? ones, but with the small
number of subsamples that report more than one ES this seems impossible

to

model. Does the use of clubSandwich robust coefficient already take care

of

this?

It depends on what you mean by "take care of" this issue. RVE does not
really solve the problem of how to model within- versus between-sample
variation in a predictor, but it does mean that you can be less worried
about getting the variance structure exactly correct. To address the issue
you raise, one thing you could do is include a version of the moderator
that is centered within each study, in addition to the study-level mean of
the moderator. This would let you parse out "same-level" versus
"different-level" variation in the moderator. However, with so few studies
that have more than one level of the moderator, the within-study version of
the predictor will have very little variation and so it will come with a
large standard error.