[R-meta] Different estimates for multilevel meta-analysis when using rma.mv with mods or subset

Hi Janis,

There are potentially two things going on here. First, when estimating separate models within each subgroup, you allow the heterogeneity (variance components) to differ by subgroup. This in turn leads to a different weighting of the individual effect size estimates in the calculation of the overall average effect sizes than the weighting used in the moderator analysis. That difference in weighting might be enough to account for the swings in the average ES for each category. To determine which model is more appropriate, you could use fit statistics or a likelihood ratio test. To get such information, you'll need to find a way to express the subgroup analyses in terms of a single model. You can do this as follows:

1. Make a factor variable from targetcommitment so that you don't have to repeatedly calculate it:
cc_d$tc_fac <- factor(cc_d$targetcommitment)

2. If you created the V matrix using metafor::vcalc(), set the subgroup argument to the moderator variable:
Vsub <- vcalc(..., subgroup = cc_d$tc_fac, ...)

3. Fit a model using this new V matrix, specifying a random effects structure that allows independent effects in each subgroup:
rma.mv(yi, V = Vsub,
              mods = ~ tc_fac - 1,
              random = list(~ tc_fac | id, ~ tc_fac | id:esid), struct = c("ID","ID"),
              data = cc_d, method = "REML")
Note that I've omitted the middle level of random effects because there doesn't seem to be any variance there after accounting for the id and esid levels. The results of (3) should be identical (or nearly so) to the results from the subgroup analysis. But now they're embedded within one model, so you can get fit statistics or conduct a likelihood ratio test to compare this model against the model that assumes homogeneous variance components across subgroups.

The second thing that might be going on is that your data seems to include a few publications that have both levels of the targetcommitment variable. Based on the reported counts from your output, it looks like there might be four publications (reporting a total of nine studies) that have both levels. Because of this structure, the first model you use for moderator analysis will calculated average effect size estimates for each level of targetcommitment based in part on the effect size estimates *for the other category* from the four publications / 9 studies that include both levels. This has the effect of moving the averages towards each other---that is -.175 and .241 get pulled inward to  -.155 and .188.

If you're particularly interested in figuring out the difference between levels of targetcommitment, the question is then whether this "pulling in" is a reasonable thing to do. You could consider this on a conceptual level: is it reasonable to put special emphasis on the results from these four publications in estimating the difference between levels of targetcommitment? If there's not a clear conceptual rationale, then you might consider fitting a slightly different model, which includes the study-mean variable and the study-mean-centered variable as separate terms. For targetcommitment, this would mean calculating the average of the dummy variable (targetcommitment == 2) for each study (this gives you the first predictor, call it tc_study) and then subtracting the average from the original dummy variable (this gives you the second predictor, call it tc_within). The model should then use the moderators:
~ 1 + tc_study + tc_within
Tanner-Smith and Tipton (2014; https://doi.org/10.1002/jrsm.1091) recommend this as a generic strategy for meta-regression of dependent effect sizes.

James


On Wed, Feb 8, 2023 at 2:40 PM Janis Zickfeld via R-sig-meta-analysis <r-sig-meta-analysis at r-project.org> wrote:

Hi all,

my question has already been raised before (e.g.,
https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-April/000774.html),
but the answers did not help me understanding the main problem or
solve differences in the output, so that is why I'm posting it again.

We are conducting a meta-analysis and have effect sizes ("esid")
nested within studies or samples ("study") nested within publications
("id"). There are 344 effect sizes in total across 110 publications
and 160 study - publication combinations. There are 48.75% of studies
including more than one effect size and of these 26.82% are dependent
effects because they either use the same sample or the same control
treatment used to calculate the effect. Therefore, we have constructed
an approximate variance-covariance matrix to account for this. Some of
the moderators only use a subset of this data, as they have missing
values (as in the example below).

My main problem is now that I get different estimates when running the
rma.mv model with factorial moderators using 'mods' or when subsetting
them via the 'subset' command. I have seen this discussed here
(http://www.metafor-project.org/doku.php/tips:comp_two_independent_estimates)
and it seems that the main difference is that 'mods' uses the same
residual heterogeneity, whereas 'subset' allows for different levels
of tau^2. However, this example does not discuss a multilevel
meta-analysis.

When running 'mods' using the following syntax:

rma.mv(yi, V, random = ~ 1 | id/study/esid, data = cc_d, method =
"REML", mods = ~factor(target_commitment) - 1)

I get:

Multivariate Meta-Analysis Model (k = 302; method: REML)

Variance Components:

            estim    sqrt  nlvls  fixed        factor
sigma^2.1  0.0346  0.1861     96     no            id
sigma^2.2  0.0000  0.0000    138     no      id/study
sigma^2.3  0.0356  0.1887    302     no  id/study/esid

Test for Residual Heterogeneity:
QE(df = 300) = 2213.7041, p-val < .0001

Test of Moderators (coefficients 1:2):
QM(df = 2) = 54.1962, p-val < .0001

Model Results:

                                             estimate         se
zval        pval        ci.lb    ci.ub
factor(target_commitment)1   -0.1551  0.0315  -4.9258  <.0001  -0.2169
 -0.0934  ***
factor(target_commitment)5    0.1878  0.0420   4.4770  <.0001   0.1056
  0.2701  ***


However, running:

rma.mv(yi, V, random = ~ 1 | id/study/ID2, data = cc_d, method =
"REML", subset=target_commitment==1)

I obtain:

Multivariate Meta-Analysis Model (k = 215; method: REML)

Variance Components:

                   estim    sqrt        nlvls  fixed        factor
sigma^2.1  0.0401  0.2002     67     no            id
sigma^2.2  0.0000  0.0000     93     no      id/study
sigma^2.3  0.0459  0.2142    215     no  id/study/esid

Test for Heterogeneity:
Q(df = 214) = 1876.2588, p-val < .0001

Model Results:

                                estimate      se        tval
df      pval       ci.lb    ci.ub
targetcommitment1 -0.1747  0.0352  -4.9607  214  <.0001  -0.2441  -0.1053  ***


Multivariate Meta-Analysis Model (k = 87; method: REML)

Variance Components:

            estim    sqrt  nlvls  fixed        factor
sigma^2.1  0.0294  0.1713     33     no            id
sigma^2.2  0.0000  0.0000     54     no      id/study
sigma^2.3  0.0116  0.1077     87     no  id/study/esid

Test for Heterogeneity:
Q(df = 86) = 302.2812, p-val < .0001

Model Results:

                                 estimate      se    tval  df    pval
 ci.lb   ci.ub
  targetcommitment5 0.2407  0.0397  6.0606  86  <.0001  0.1618  0.3197  ***


So there is a difference between -.155 and -.175, as well as .188 and
.241. These are not the biggest differences, but I have other
moderators for which the differences are stronger.

I tried to follow the suggestions of one of the previous questions
(https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-April/000774.html),
but this didn't reduce the difference for me.

My main question would be which of the two approaches would be the
preferable one (and if there is any empirical basis for this)? Is it
more a choice of preference here or is one approach superior in the
present case?

I could also post a link to the data if that would be helpful for
reproducing the findings.

I apologize for double posting this issue and hope that maybe someone
can clarify which option I should choose.

Best wishes,
Janis