[R-meta] Handling dependencies among multiple independent and dependent variables

[Viechtbauer Wolfgang (SP)]

I have posted a function that allows for processing data like this. See here:

https://gist.github.com/wviechtb/700983ab0bde94bed7c645fce770f8e9

Take a look especially at the very last example (note: the bldiag() is in the metafor package, so load this first). The data here look like this:

   id     yi var1 var2
1   1  0.179  BMI  PUL
2   1  0.396  BMI  SBP
3   1  0.088  PUL  SBP
4   1  0.080  BMI  DPB
5   1 -0.042  DPB  PUL
6   1  0.719  DPB  SBP
7   2  0.179  BMI  PUL
8   2  0.396  BMI  SBP
9   2  0.088  PUL  SBP
10  2  0.080  BMI  DPB
11  2 -0.042  DPB  PUL
12  2  0.719  DPB  SBP

'yi' is the correlation between 'var1' and 'var2'. Then you can use:

sav <- rmat(yi ~ var1 + var2 | id, n=c(20,30), data=dat)
sav

(so with n, you denote the sample sizes of study 1 and study 2 and so on).

This will return the dataset plus the full var-cov matrix of the correlations (a 12x12 matrix here). You can then use rma.mv() for analyzing these data, setting argument 'V' equal to the var-cov matrix (sav$V).

So, the idea is this: Set up your dataset as above, with var1 and var2 indicating the types of variables. So, for the 4 cases you describe below, you would have:

id var1    var2    yi
1  orient1 prof1   .
2  orient1 prof1   .
2  orient1 prof2   .
2  prof1   prof2   .
3  orient1 prof1   .
3  orient2 prof1   .
3  orient1 orient2 .
4  orient1 prof1   .
4  orient1 prof2   .
4  orient2 prof1   .
4  orient2 prof2   .
4  prof1   prof2   .
4  orient1 orient2 .

Note: Even though you are only interested in cor(orient, prof), you need the prof-prof and orient-orient correlations, because those are needed to compute the covariances. Then use rmat() as above.

For the rma.mv() model, you actually do not want to distinguish between orient1 and orient2 and prof1 and prof2. So you want to expand your dataset so that it looks like this:

id var1    var2    yi  type
1  orient1 prof1   .   orient.prof
2  orient1 prof1   .   orient.prof
2  orient1 prof2   .   orient.prof
2  prof1   prof2   .   prof.prof
3  orient1 prof1   .   orient.prof
3  orient2 prof1   .   orient.prof
3  orient1 orient2 .   orient.orient
4  orient1 prof1   .   orient.prof
4  orient1 prof2   .   orient.prof
4  orient2 prof1   .   orient.prof
4  orient2 prof2   .   orient.prof
4  prof1   prof2   .   prof.prof
4  orient1 orient2 .   orient.orient

Then you can use:

rma.mv(yi, sav$V, mods = ~ type - 1, random = ~ type | id, struct="UN", data=dat)

This will give you the estimated average correlation for orient.prof, prof.prof, and orient.orient. You are interested in the first (although the other two types are also interesting).

By coding 'type' in this way, when a study provides multiple orient-prof correlations, they are in essence pooled together in the model fitting (while properly accounting for their covarances, since those are in sav$V).

One could also approach this from a 'MASEM' (meta-analytic structural equation modeling) framework. If you are interested in that approach, take a look at Mike Cheung's book "Meta-Analysis: A Structural Equation Modeling Approach" or Suszanne Jak's book "Meta-Analytic Structural Equation Modelling".

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-project.org] On Behalf Of Jens Schüler
Sent: Tuesday, 06 March, 2018 14:50
To: r-sig-meta-analysis at r-project.org
Subject: [R-meta] Handling dependencies among multiple independent and dependent variables

Hi all,

I am currently working on a meta-analysis on strategic orientations (IV) and firm performance (DV), using correlational data, and struggle with how I should best handle dependencies among effect sizes.

Short elaboration of the issue:
The strategic orientation I am interested in consists of three subdimensions and some studies report correlational data between the subdimensions and e.g. profitability, whereas other studies report only a correlation between the whole construct and profitability. Moreover, some studies use more than one profitability measure. I am interested in the overall aggregate effect between the "whole" strategic orientation and "whole" profitability.

This leaves me with the following cases:
1. Whole orientation and single profitability measure (trivial)
2. Whole orientation and multiple profitability measures
3. Multiple dimensions and single profitability measure
4. Multiple dimensions and multiple profitability measures 

Following the mailing list and the metaphor website, I could handle the 2nd case through nesting the multiple effects in studies and using the impute_covariance_matrix function of the clubSandwhich package. However, I am not exactly sure on how to appropriately handle case 3 and 4 and especially, how to combine 2 to 4 in order to run the analysis on the aggregate level (whole orientation - whole profitability).

Due to working with correlations, the respective studies report the correlations between the variables e.g. between the dimensions, between the profitability measures and the correlations between the dimensions and profitability measures.

Best
Jens