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

[James Pustejovsky]

Hi Danka,

Responses inline below.

Kind Regards,
James


On Mon, Jan 4, 2021 at 5:41 AM Danka Puric <djaguard using 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).
> 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).
> 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)

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.


> 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.

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