[R-meta] Dealing with effect size dependance with a small number of studies

To construct an approximate 'V' matrix, something like this should work:

impute_covariance_matrix(MA_dat_raw$SV, cluster=paste0(MA_dat_raw$IDstudy, ".", MA_dat_raw$IDsubsample), r = 0.6)

I paste together the study and sample ID variables, but this is only necessary if IDsubsample is coded in such a way that the same value of IDsubsample might be used for two different studies. For example, if:

IDstudy IDsubsample IDOutcome IDeffect
1       1           outcomeA  1
1       1           outcomeA  2
1       1           outcomeB  1
1       1           outcomeB  2
1       2           outcomeA  1
1       2           outcomeA  2
1       2           outcomeB  1
1       2           outcomeB  2
2       1           outcomeA  1
2       1           outcomeB  1

then setting cluster = IDsubsample would not make sense, since it would allow for correlation across studies. On the other hand, if the last two lines were:

2       3           outcomeA  1
2       3           outcomeB  1

then using IDsubsample as the cluster variable would be fine.

Of course, the r = 0.6 is something that you need to think about.

Also, this assumes a single correlation for pairs of estimates, regardless of whether the two estimates are for the same outcome or for different outcomes. Usually, I would expect a stronger correlation for two estimates of the same outcome. Constructing a V matrix that reflects this is more tricky.

However, in the end, what you are doing here is just trying to make the 'working model' somewhat more realistic (by not assuming 0 for the correlation, which is in essence what you do when you use V=SV). The cluster-robust inference approach then takes this working model as input and computes the standard errors of the fixed effects in such a way that even if the model is misspecified, the estimated standard errors are (asymptotically) correct.

Best,
Wolfgang

-----Original Message-----
From: Danka Puric [mailto:djaguard at gmail.com]
Sent: Tuesday, 05 January, 2021 13:28
To: Viechtbauer, Wolfgang (SP)
Cc: James Pustejovsky; R meta
Subject: Re: [R-meta] Dealing with effect size dependance with a small number of
studies

Dear Wolfgang,

message received (both times) :)

I'll respond inline as well.

On Tue, Jan 5, 2021 at 12:16 PM Viechtbauer, Wolfgang (SP)
<wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:

Ok, I think I am replying to the right post now ...

Responses again below.

Best,
Wolfgang

-----Original Message-----
From: Danka Puric [mailto:djaguard at gmail.com]
Sent: Tuesday, 05 January, 2021 10:36
To: Viechtbauer, Wolfgang (SP)
Cc: James Pustejovsky; R meta
Subject: Re: [R-meta] Dealing with effect size dependance with a small
number of studies

Dear James, Wolfgang,

Thanks a lot for the quick and informative responses!

1. I guess I made things unnecessarily complicated :) The thing is, I know
that all these models are essentially the same:

notationa <- rma.mv(ES_corrected, SV, random = ~ factor(IDeffect) | IDstudy,
data=MA_dat_raw)
notationb <- rma.mv(ES_corrected, SV, random = ~ 1 | IDstudy/IDeffect,
data=MA_dat_raw)
notationc <- rma.mv(ES_corrected, SV, random = ~ IDeffect | IDstudy,
data=MA_dat_raw)

but I read in the Konstantopoulos (2011) example that they only deal with
the dependence arising from effect sizes coming from the same studies, but
NOT with dependence arising from multiple ES coming from the same group of
participants. I then erroneously concluded that in order to deal with this
type of dependence I would need to use struct = "UN", but I understand now
that's not the case.

Yes, the 'struct' part is a different issue.

If you directly want to account for dependency due to multiple effect size

estimates coming from the same group of subjects, you would need to calculate
the covariance between the sampling errors and include this in the 'V' matrix
(the second argument in rma.mv(), to which you are passing 'SV'). In addition,
we then also want to use a model like the one above to account for possible
dependency in the underlying true effects. That is in fact how things are done
in the Berkey example (and since 'outcome' is meaningful there, we can use
struct="UN" to have an estimate of tau^2 for the two different outcomes).

Thanks for the additional clarification! We will try to create "V"
using the impute_covariance_matrix function James suggested and then
use it instead of SV in our syntax:
model <- rma.mv(ES_corrected, V, random =  ~ 1 | IDstudy / IDsubsample
/ IDeffect, data=MA_dat_raw)

Also, indeed, IDeffect does not refer to the type of outcome in a study.
Actually, we do have an outcome variable DV which could be used instead of
IDeffect, but sometimes it has the same value for several ESs in the same
group of participants, so it didn't seem appropriate to use it in this case.
I did realize the model with IDeffect was not structured like Berkey at al.
but thought it would be a better option, as IDeffect variables have unique
values across IDstudy.

As for sigma1.1 and sigma2.1. it's quite possible I just got something mixed
up when I compared different notations (I may have plotted the wrong model),
but anyway, this model is definitely wrong for the data, so I'll just leave
it at that.

Agreed.

2. So, just to be sure I got this right, the following model

model <- rma.mv(ES_corrected, SV, random =  ~ 1 | IDstudy / IDsubsample/
IDeffect, data=MA_dat_raw)

in combination with clubSandwich robust estimates will yield adequate effect
size estimates for the situation where the same group of participants
provided more than one ES? That's actually the model I fit first, but then
thought wasn't appropriate after all.

Yes, this looks like a sensible approach. In principle, since you mentioned it

above, you could even consider:

random =  ~ 1 | IDstudy / IDsubsample / IDOutcome / IDeffect

where IDOutcome is the id variable for the the different outcomes. This model

is in principle possible, since you mentioned that sometimes, within a
particular subsample, there are multiple effects for the same outcome (if this
were not the case, then IDOutcome and IDeffect would not be uniquely
identifiable). However, this may be pushing things a bit with k=69 estimates. In
essence, this is a five-level model, so two levels more than the 'three-level
model' described by Konstantopoulos (2011):

http://www.metafor-project.org/doku.php/analyses:konstantopoulos2011

(in multilevel model parlance, the standard random-effects model would be

considered a two-level model, so the number of levels is 1 + the number of
hierarchical levels you are adding via 'random').

Yes, adding an IDoutcome variable would actually be a great way to
approach the issue of same/different outcomes coming from the same
subsample. But I also agree that such a model might be too complex for
the data and I would expect both IDoutcome and IDeffect to have
variances very close to zero. Still, we can give it a try and see if
it makes sense.

I will also now look into inputting the covariance matrices and see if it's
possible to implement with the data we have. Thanks for suggesting this,
James.

In this case, covariances in the sampling errors will occur only within

subsamples. So, the 'V' matrix will be block-diagonal with blocks corresponding
to the subsamples. For example, suppose we have

IDstudy IDsubsample IDOutcome IDeffect
1       1           outcomeA  1
1       1           outcomeA  2
1       1           outcomeB  1
1       1           outcomeB  2
1       2           outcomeA  1
1       2           outcomeA  2
1       2           outcomeB  1
1       2           outcomeB  2

so a study with two groups and in each group both outcomes (A and B) were

measured in two different ways (e.g., using two different scales), leading to
two different effects. Then the V matrix for this study would be an 8x8 matrix
that is composed of two 4x4 blocks.

Again, thank you for clarifying this. I assumed this was how it
worked, I'm just still not sure how to create V from the data we have,
but I'll get onto that straight away!

Thanks so much!
Danka