[R-meta] correct tau interpretation three-level meta-analysis

James Pustejovsky jepu@to @end|ng |rom gm@||@com
Tue Jul 6 23:16:12 CEST 2021

Hi Filippo,

Please keep the listserv cc'd. Responses below.


On Tue, Jul 6, 2021 at 3:57 PM Filippo Gambarota <
filippo.gambarota using gmail.com> wrote:

> thank you James, this is extremely clear. If I understand correctly, my
> model under some situations could be not appropriate for variance
> estimation.
> Do you think that a bayesian model could partially mitigate the variance
> estimation? especially for the within-cluster variance?

A Bayesian model would still require making some assumption about the
correlation between effect size estimates, so it doesn't necessarily solve
the problem (although, it could be a good approach for all the usual
reasons that Bayesian inference is useful).

> In general yes, I have let's say 10 studies where the same sample is
> tested with several response variables. The three-level seemed the most
> appropriate to me (especially compared to the multivariate approach).

To get improved estimates of variance components, I think the best thing to
do would be to try and collect information about the correlations between
outcomes and use that to specify a variance-covariance matrix for effect
size estimates, as has been discussed in several previous exchanges on the
listserv. Short of that, you could specify an approximate
variance-covariance matrix, making some simplified assumption about the
unknown correlations, and then do sensitivity analysis across a range of

> Do you suggest the robust variance estimation approach?

RVE can be helpful for improving the coverage properties of confidence
intervals and the calibration of hypothesis tests *for average effect
sizes*, especially when there is concern about potential model
mis-specification. Fernandez-Castilla and colleagues (citation below) also
found that it can be helpful to use RVE in combination with 3LMA for this
purpose. But RVE does not help with estimation of variance components.


> On Tue, Jul 6, 2021, 11:11 PM James Pustejovsky <jepusto using gmail.com> wrote:
>> Hi Filippo,
>> To add to Wolfgang's response, one note of caution regarding
>> interpreting the variance components in the three-level meta-analysis
>> (3LMA) model is that the variance component estimates are somewhat
>> sensitive to assumptions. If your data structure involves multiple,
>> correlated effect size estimates (i.e., estimates based on the same sample
>> of participants, so that the sampling errors of the estimates are
>> correlated), then the 3LMA model involves some degree of model
>> mis-specification. Currently available evidence suggests that the 3LMA may
>> be fairly robust with respect to inferences about *average effect
>> sizes*---that is, even though the model is mis-specified, hypothesis tests
>> and confidence intervals based on the model still have calibration rates
>> that are close to correct.
>> This robustness property does NOT extend to estimation of variance
>> components. If the model is mis-specified, then there will generally be
>> some degree of systematic bias in the variance component estimates. For
>> instance, say that the true correlation between effect size estimates from
>> the same sample is around r = 0.6. Using the 3LMA is equivalent to assuming
>> r = 0.0. As far as I understand, this will lead to estimates of
>> within-study heterogeneity that are systematically *too small* and
>> estimates of between-study heterogeneity that are systematically *too
>> large*. How strong the biases are depends on the structure of your data, so
>> it's hard to say much further here.
>> To your other question:
>>> Do we interpret it as an average variability within each cluster among
>>> clusters? Or we are assuming that each cluster has the same
>>> within-cluster
>>> variability?
>> I would say that the answer is "both." As formulated, the 3LMA model does
>> make the assumption that each cluster has the same within-cluster variance
>> component (i.e., homogeneity of variance within clusters). But even if this
>> assumption is incorrect, the estimated within-cluster variance will be some
>> sort of weighted average of the within-cluster variances, at least at an
>> approximate level. In principle, you could estimate cluster-specific
>> variances using the following (assuming that every value of outcome is
>> unique across studies):
>> ```
>> rma.mv(yi, vi, random = list (~1|study, ~ study | outcome, struct =
>> "DIAG")
>> ```
>> But this probably isn't a good idea unless you have a lot of estimates
>> from every cluster. And the comments above regarding model
>> mis-specification apply here as well.
>> Kind Regards,
>> James

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