# [R-meta] Multivariate meta-analysis with metafor: Should I adjust sample sizes/variances for multiple groups ('double counting') when combined with multiple endpoints?

James Pustejovsky jepusto at gmail.com
Fri Jun 16 16:03:57 CEST 2017

```Emily,

I would offer a couple of suggestions for different ways to approach this.
I think the main question is whether, for the studies with multiple
intervention groups, do you really care (scientifically, with respect to
conditions? If not---if they're really just a nuisance that you need to
find a way to smooth over---then two simple approaches to handling them
might be attractive:

1. Pick the single condition that best represents the treatment construct
of interest.
2. Average the treatment conditions together, and then take the difference
between the averaged treatment condition and the single control condition.
Say that you have treatment conditions q, r, s, with sample means yq, yr,
ys, sample standard deviations sq sr, ss, and sample sizes nq, nr, ns.
Calculate the average sample mean y_avg = (nq * yq + nr * yr + ns * ys) /
(nq + nr + ns). Say the control condition has sample mean, sd, and size
given by yc, sc, and nc. You can then calculate a d statistic as

d = (y_avg - yc) / sp,

where sp^2 = ((nq - 1) * sq^2 + (nr - 1) * sr^2 + (ns - 1) * ss^2 + (nc -
1) * sc^2) / (nq + nr + ns + nc - 4)). The variance of d is (approximately)
Vd = 1 / nq + 1 / nr + 1 / ns + 1 / nc + d^2 / (nq + nr + ns + nc - 4).
You can also use a Hedges-g correction with J(nq + nr + ns + nc - 4), where
J(x) = 1 - 3 / (4 x - 1).

Option (2) will give more precise treatment effects (because of increased
sample size), but might muddy the water (or be harder to explain in a
paper) if the treatment conditions are really distinct. But if the
meta-regression model that you want to estimate does not make any
distinction between the treatment conditions, then option (2) is actually
very close or even identical to the more complex option described below.

On the other hand, if you really care about the distinctions between
treatment conditions, as you would if the covariates you are examining have
variation within a given study depending on which treatment condition
you're looking at, then you would probably want to

3. Calculate the full sampling variance-covariance matrix of all
combinations of effects and feed this into metafor as part of the V matrix.

Here's a blog post with the relevant formulas: http://jepusto.github.io/
Correlations-between-SMDs

Cheers,
James

On Fri, Jun 16, 2017 at 7:46 AM, Emily Finne <emily.finne at uni-bielefeld.de>
wrote:

> Dear all,
>
> as I am seemingly the first to post a question on this list, I hope my
> question is not a silly one.
>
> First of all I'd like to thank Wolfgang Viechtbauer for all the
> to conduct different kinds of meta-analyses with metafor.
> I've already learned a lot so far!
> All these bits of code are really helpful and appreciated, since I am
> relatively new to working with R (and in doing meta-analysis).
>
> There is, however, one point I am still confused about. I try to explain
> my analysis first and then the question:
>
> I have 30 RCTs matching our inclusion criteria and I use Hedges g as
> effect size. The aim is to analyze different intervention techniques
> (coded as present or absent) as potential moderators of effect sizes.
> All studies included a self-report measure of the outcome, some
> additionally reported results for an objective measure of the same
> outcome. I would like to include both outcomes in a multivariate model.
> There are also a few studies with multiple treatment groups all compared
> to the same control condition. Since the groups differ in the techniques
> they used and are therefore of interest, information from all
> intervention groups should be included.
>
> Initially I wanted to compute two separate univariate models for the two
> outcome measures (subjective and objective), and because of the shared
> control groups within some trials I split the sample size of the
> controls (with two interventions compared to the same group of, say 40
> people, I included two comparions with n=20 each) to avoid double
> counting (that's what the Cochrane Handbook recommends in this case).
>
> But after starting to work through the different options, I came to the
> conclusion that the multivariate model would be more appropriate for
> this analysis.
> So, the model I want to fit looks like this:
>
> library(metafor)
>
> MA1 <- rma.mv(yi=Hedgesg, V,  random = ~ Outcome | trial, struct="UN",
> data=datMA, test="t", mods=~Outcome)
>
> or for one overall effect size  (because both outcomes did not differ
> significantly):
>
> MA2 <- rma.mv(yi=Hedgesg, V,  random = ~ Outcome | trial, struct="UN",
> data=datMA, test="t")
>
> for the overall effect and then for the meta-regression model:
>
> MA3 <- rma.mv(yi=Hedgesg, V,  random = ~ Outcome | trial, struct="UN",
> data=datMA, test="t", mods=~ technique1)
>
> My model is most similar to the example given here:
> http://www.metafor-project.org/doku.php/analyses:berkey1998
>
> V is the variance-covariance matrix based on the variances and estimated
> covariances between the effects of both outcome measures within a study
> (as explained in the linked example above).
>
> Trial is the study ID.
>
> BUT besides these 2 outcomes I have these studies with multiple
> intervention groups. There is one trial with even 6 effect sizes (2
> outcomes * 3 interventions).
>
> I wonder, what to do with the splitting up of control groups now. For
> the two outcomes measured within the same persons, I am quite sure that
> I don't have to adjust any sample sizes (i.e., variances), because the
> model 'knows' that these outcomes both are from the same persons .
> But what about the multiple groups? They are of course also nested
> within trials, but I didn't estimate a covariance between these effect
> multilevel variant - or did I? (My idea is to additionally use the
> robust estimation (with cluster = trial)).
>
> Is it right then to use the original sample size/ variance from the
> control groups although some were used in multiple comparisons? Or
> should the affected CGs be splitted up within this model as in the
> univariate model? Will  metafor account for the nesting of different
> interventions within a trial when computing an overall pooled effect
> size with the specified multivariate model?
> Which variant would yield the correct pooled effect size, whithout
> 'double counting'?
>
> I think his is mainly a question on how the metafor 'rma.mv' weighs the
> effect sizes to arrive at the pooled effect when using the random = ~
> inner | outer factor argument.
>
>
> I tried to find out by looking at the results of both variants but I
> couldn't suss it out...
>
>
> Any help would be appreciated. Many thanks!
>
> Best,
> Emily
>
>
>
> --
> Dr. Emily Finne, Dipl.-Psych.
>
> Universität Bielefeld
> Fakultät für Gesundheitswissenschaften
> AG 4: Prävention und Gesundheitsförderung
> Postfach 10 01 31
>
> D-33501 Bielefeld
>
>
> Mail:emily.finne at uni-bielefeld.de
> http://www.uni-bielefeld.de/gesundhw/ag4
>
>
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>
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