[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). 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.