[R-meta] Calculating covariances in multivariate meta-analysis

[James Pustejovsky]

With the data structure you've described, is the full multi-variate
meta-analysis really worth the trouble? Just thinking out loud...what about
the following?

1. Pool correlations across studies only for pairs of measures that are
exactly aligned. Pool using FE meta-analysis of z-transformed correlations,
or pool the covariances directly (with weights Nj - 1).
2. Display this correlation matrix, with rows and columns sorted so that
measures of the same construct are close together.
3. If you want to further summarize this (synthetic) correlation matrix by
averaging entries together, use weighted averages (maybe of z-transformed
correlation?) with weights based on the total sample size going in to each
correlation.
4. To get CIs for any summaries from (3), just bootstrap the whole thing.

This is probably simplistic but I wonder whether it would be more
transparent than using the MV meta-analytic average.

James

On Wed, Jan 17, 2018 at 12:02 PM, Mark White <markhwhiteii at gmail.com> wrote:

> James,
>
> Thank you—I have looked at that citation (and similar ones mentioned in
> the documentation for metafor::rma.mv); yes, they do appear to be
> somewhat of a pain.
>
> The goal here is simple: All I want is the overall, meta-analytic
> correlation and confidence interval. That is, a multivariate estimate for y
> ~ 1 (i.e., metafor::rma.mv(yi, V)).
>
> Mark
>
> On Wed, Jan 17, 2018 at 11:30 AM, James Pustejovsky <jepusto at gmail.com>
> wrote:
>
>> Mark,
>>
>> The formulas needed to calculate the covariances are given in the
>> following reference:
>>
>> Olkin, I., & Finn, J. (1990). Testing correlated correlations.
>>> Psychological Bulletin, 108(2), 330–333.
>>
>>
>> Unfortunately they're a bit complicated, a pain in the rear to program,
>> and sometimes return non-positive definite covariance matrices that create
>> problems at the meta-analysis stage. If you've got the raw data, a cleaner
>> approach would be to use a basic bootstrap (i.e., re-sampling cases) for
>> the set of correlations you want to meta-analyze.
>>
>> But a larger question might be relevant here: what is the goal of
>> conducting a multi-variate meta-analysis on these correlations? Is it to
>> come up with a synthetic correlation matrix? To understand heterogeneity
>> across studies in the correlations? Depending on your answer--and given
>> that you have access to the raw data--other statistical approaches (other
>> than MV meta-analysis) might be equally or better suited for the problem.
>>
>> James
>>
>> On Wed, Jan 17, 2018 at 9:17 AM, Mark White <markhwhiteii at gmail.com>
>> wrote:
>>
>>> Hello all,
>>>
>>> I have 8 studies in my dissertation; I want to meta-analyze the
>>> correlation
>>> between focal variable X and outcome Y. Let variables for Study 1 be x1
>>> and
>>> y1, Study 2 be x2 and y2, etc. However, I also have *various
>>> measurements *of
>>> each construct in some studies. For example, in Study 1, I have the
>>> correlation between x1_1 and y1_1, as well as x1_2 and y1_2. And in Study
>>> 2, I have the correlation between x2_1 and y2_1 as well as x2_2 and y2_2.
>>> In Study 3, I have these all the way up to x3_10 and y3_10.
>>>
>>> I want to perform a multivariate meta-analysis, since I have all of the
>>> raw
>>> data. My question: How do I calculate the covariates between these
>>> correlations? I know I want to end up with a covariance matrix where the
>>> diagonal is the variance, off-diagonal the covariances (with all zeros
>>> where they are from different studies). In the analysis examples on the
>>> metafor website, these are already calculated for the user. How do I
>>> calculate these from my raw data?
>>>
>>> Thank you,
>>> Mark
>>>
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>>>
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>>
>>
>

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