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When fitting a multiple meta-regression in metafor, the output shows multiple coefficients, for each predictor.

To make an example, similar to what is done here: https://www.metafor-project.org/doku.php/tips:meta_regression_with_log_rr

dat <- escalc(measure="OR", ai=cpos, bi=cneg, ci=tpos, di=tneg, data=dat.bcg)
res <- rma(yi, vi, mods = ~ alloc+ablat, data=dat)

where ablat is a continuous moderator, and alloc a categorical moderator.

The output shows:

Mixed-Effects Model (k = 13; tau^2 estimator: REML)

tau^2 (estimated amount of residual heterogeneity):     0.1402 (SE = 0.1109)
tau (square root of estimated tau^2 value):             0.3744
I^2 (residual heterogeneity / unaccounted variability): 68.79%
H^2 (unaccounted variability / sampling variability):   3.20
R^2 (amount of heterogeneity accounted for):            58.50%

Test for Residual Heterogeneity:
QE(df = 9) = 24.3281, p-val = 0.0038

Test of Moderators (coefficients 2:4):
QM(df = 3) = 12.4943, p-val = 0.0059

Model Results:

                 estimate      se     zval    pval    ci.lb   ci.ub 
intrcpt           -0.2609  0.4017  -0.6495  0.5160  -1.0483  0.5265     
allocrandom        0.1801  0.3482   0.5170  0.6051  -0.5025  0.8626     
allocsystematic   -0.1461  0.3766  -0.3881  0.6979  -0.8842  0.5919     
ablat              0.0294  0.0091   3.2222  0.0013   0.0115  0.0473  ** 

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Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Now, we know that exponentiation of the coefficients for allocrandom and allocsystemic will gave us the "ratio of Odds Ratio", compared to the "alternate" location; similar, we know that the exponentiation of the coefficient for ablat will gave us the average odds ratio for one-unit increase in absolute latitude (Let's skip for a moment the proble of the exponentiation as a non-linear transformation, for simplicity).

Now, the question is related to the formal interpretation of these coefficients in a multiple meta-regression model like the one reported above.

It is right to say that, according to this multiple meta-regression model, the change in average odds ratio for one-unit increase in absolute latitude is 1.03 (which is equal to the exponentiation of the coefficient 0.0294), independently from the type of allocation? Or do I miss something?

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On the log scale, the model essentially implies that there are three parallel lines for the relationship between ablat and the log odds ratios, that is, the lines all have the same slope (i.e., 0.0294), but different intercepts, depending on the alloc level. People will use different terminology to describe this in words. Some might say that 0.0294 is the slope "controlling for alloc". Some might say that it is the slope when "holding alloc constant". I guess one could also say it is the slope "independently of alloc". So, in that sense, the answer to your question is yes, you can phrase the findings as you did.

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  • $\begingroup$ Wolfgang, thank you! I guess this also applies whether in a model I have two continuous moderators (the coefficient of one is the slope of the line when I "hold constant" the other continuous moderator). Besides - I do want to explore more the topic of multiple meta-regression. Do you have any reference paper to which you may want to point me, which provide guidance for building, interpretation and reporting of multiple meta-regressions? Thanks in advance! $\endgroup$ Mar 15, 2021 at 8:11
  • $\begingroup$ @pankevedmo have you perhaps seen the metafor home page? There is much material there metafor-project.org/doku.php $\endgroup$
    – mdewey
    Mar 15, 2021 at 14:18
  • $\begingroup$ @mdewey Yeah for sure, and I think it is very useful. Just to know if there is any additional reference paper/book particularly focused on multiple meta-regression. $\endgroup$ Mar 15, 2021 at 15:46

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