[R-meta] Difference between univariate and multivariate parameterization

[Farzad Keyhan]

Dear Luke,

In the multivariate specification (model 2), tau^2 = sigma^2_between  +
sigma^2_within. You can confirm that by your two models' output as well.
Also, because rho = sigma^2_between / (sigma^2_between  +  sigma^2_within),
then, the off-diagonal elements of the matrix can be shown to be rho*tau^2
which again is equivalent to sigma^2_between in model 1's matrix.

Note that sampling errors in a two-estimate study could be different hence
appropriate subscripts will be needed to distinguish between them.

Finally, note that even a study with a single effect size estimate gets the
sigma^2_within, either directly (model 1) or indirectly (model 2) which
would mean that, that one-estimate study **could** have had more estimates
but it just so happens that it doesn't as a result of some form of
multi-stage sampling; first studies, and then effect sizes from within
those studies.

I actually raised this last point a while back on the list (
https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-July/002994.html)
as I found this framework a potentially unrealistic but in the end, it's
the best approach we have.

Cheers,
Fred



On Wed, Aug 18, 2021 at 1:30 PM Luke Martinez <martinezlukerm using gmail.com>
wrote:

> Dear Colleagues,
>
> Imagine I have two models.
>
> Model 1:
>
> random = ~1 | study / row_id
>
> Model 2:
>
> random = ~ row_id | study,  struct = "CS"
>
> I understand that the diagonal elements of the variance-covariance matrix
> of a study with two effect size estimates for each model will be:
>
> Model 1:
>
> VAR(y_ij) = sigma^2_between  +  sigma^2_within + e_ij
>
> Model 2:
>
> VAR(y_ij) = tau^2 + e_ij
>
> Question: In model 2's variance-covariance matrix, what fills the role of
> sigma^2_within (within-study heterogeneity) that exists in model 1's
> matrix?
>
> Thank you very much for your assistance,
> Luke Martinez
>
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>
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