[R] metafor package: effect sizes are not fully independent

Viechtbauer Wolfgang (STAT) Wolfgang.Viechtbauer at STAT.unimaas.nl
Mon Feb 8 10:22:37 CET 2010

Hello Gang,

Mike has already given you some excellent advice and references. I just want to add some information about the metafor package.

To be precise, the meta-analytic mixed-effects model is given by:

y_i = X_i * beta_i + u_i + e_i

where X_i and beta_i have the usual interpretation, u_i is a random effect (for "residual heterogeneity"), and e_i is sampling error. Var(e_i) = v_i is assumed to be known for each study. Moreover, assume Cov(e_i, e_i') = 0, Cov(u_i, u_i') = 0, and Cov(u_i, e_i) = 0. This is the model underlying the rma() function.

Correlation can arise between the e_i's (i.e., Cov(e_i, e_i') != 0), between the u_i's (i.e., Cov(u_i, u_i') != 0), or both. For example, if you have multiple estimates from the same study calculated from the same sample of subjects, then the e_i's for that study will certainly be correlated and it usually makes sense then to assume that the corresponding u_i's are correlated as well. For the correlation between the e_i's, there are often equations that you can use to calculate those (or to be precise: to estimate those correlations). So, instead of just assuming that v_i is known, we then have an entire variance-covariance matrix of the e_i's that is assumed to be known. The tricky thing then is to figure out what structure to use for the variance-covariance matrix of correlated u_i's.

Currently, the metafor package cannot handle situations where some of the effect size estimates are correlated. I am currently working on functions that will allow for correlated estimates, but it will take some time before those will become part of the package.

If it makes sense to do so in your case, you could consider aggregating dependent estimates into a single estimate and then using that for the meta-analysis.


Wolfgang Viechtbauer                        http://www.wvbauer.com/
Department of Methodology and Statistics    Tel: +31 (43) 388-2277
School for Public Health and Primary Care   Office Location:
Maastricht University, P.O. Box 616         Room B2.01 (second floor)
6200 MD Maastricht, The Netherlands         Debyeplein 1 (Randwyck)

----Original Message----
From: r-help-bounces at r-project.org
[mailto:r-help-bounces at r-project.org] On Behalf Of Gang Chen Sent:
Friday, February 05, 2010 23:08 To: r-help at r-project.org
Subject: [R] metafor package: effect sizes are not fully independent

> In a classical meta analysis model y_i = X_i * beta_i + e_i, data
> {y_i} are assumed to be independent effect sizes. However, I'm
> encountering the following two scenarios:
> (1) Each source has multiple effect sizes, thus {y_i} are not fully
> independent with each other. (2) Each source has multiple effect
> sizes, each of the effect size from a source can be categorized as
> one of a factor levels (e.g., happy, sad, and neutral). Maybe better
> denote the data as y_ij, effect size at the j-th level from the i-th
> source. I can code the levels with dummy variables into the X_i
> matrix, but apparently the data from the same source are correlated
> with each other. In this case, I would like to run a few tests one of
> which is, for example, whether there is any difference across all the
> levels of the factor.
> Can metafor handle these two cases?
> Thanks,
> Gang
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