[R-meta] Construct the covariance-matrices of different effect sizes

https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis

(2) Yes, impute_covariance_matrix() works just fine with raw mean
differences. The correlation you input into the function corresponds
exactly to the correlation between outcomes.

(3) Thanks for linking to the earlier post. The formula Wolfgang gave is
probably derived by the delta method.

Just to illustrate how using these formulas would work, if assume a
correlation between outcomes of r = 0.7, then that means that the
correlation between raw mean differences estimates should also be 0.7, but
that the correlation between SDLN estimates should be r^2 = 0.49. And the
correlation between ICCs would be something else yet again, according to
the formulas implemented in rcalc(). One implication of this is that if you
want to conduct sensitivity analyses from r = 0.0 to r = 0.9, then the
sensitivity analyses for the raw mean difference estimates should be over
the range r = 0.0 to r = 0.9, but the sensitivity for SDLN values should
run r = 0.0, r = 0.01, r = 0.04, r = 0.09,..., r = 0.81.

James



On Tue, Jan 19, 2021 at 10:02 PM Tzlil Shushan <tzlil21092 at gmail.com>
wrote:

Hi James,

Thank you so much for the response and further explanations around the
merits associated with V-matrices in correlated datasets. I think that both
you and Wolfgang have already provided a thorough insight on this in
previous threads in the mailing list and papers as this recent one
https://osf.io/x8yre/
However, I struggle to be confident with the formulas I use for each
effect index, and since my math background isn't wide, I found it hard to
extract them from papers and I'll be super grateful to get your help here :)

Since I don't have enough information on the correlation between
outcomes, I'm using 'guesstimate' values and back it up with sensitivity
analysis (at first glance, with my current formulas for V-matrices, values
range from 0 to 0.9 correlation did not yield changes in the results).

(1) I might start with my Pearson correlation and ICC, because they are
probably the tricky ones, and up to now I don't really have something to
support on besides impute_covariance_matrix formula. For both datasets I
used r to Fisher's z transformation to approximate normal distributed
datasets.

ICC is simply the ICC between test-retest (submaximal heart rate values).
ICCs from the same sample can be derived from the number of test-retests
(i.e. more than one paired tests), different intensities, different test
duration etc., which I investigate  separately as meta-regression. However,
I'm first interested in the average effect.

Pearson correlations are derived from the association between heart rate
at submaximal intensity(ies) and criterion measure of aerobic fitness
(let's say max test). In theory, we expect an inverse relationship?lower
heart rate at submaximal intensity associated with higher result in max
test. Similarly to ICC, these effects are obtained from different
(submaximal) intensities, time-points across the season etc. I want to
extract the average effect, and then continue to moderator analyses.

## these are datasets examples

es.id study.id  n  icc     yi     vi
1     1         25 0.90 1.4722 0.0455
2     1         25 0.84 1.2212 0.0455
3     2         16 0.72 0.9076 0.0769
4     2         16 0.85 1.2562 0.0769
5     2         16 0.83 1.1881 0.0769
6     3         38 0.92 1.5890 0.0286

es.id study.id  n     r      yi           vi
1           1       25 -0.61 -0.7089 0.0455
2           1       25 -0.58 -0.6625 0.0455
3           2       16 -0.58 -0.6625 0.0769
4           2       16 -0.60 -0.6931 0.0769
5           2       16 -0.56 -0.6328 0.0769
6           3       38 -0.55 -0.6184 0.0286

I tried to implement rcalc() with my dataset, but the thing is that I
have two variables that are constant (submax and submax tests for ICC and
submax and max tests for *r*) that were examined repeatedly within
samples, while the formula requires different paired variables
within sample/studies as the example here
https://wviechtb.github.io/metafor/reference/rcalc.html Is it right? I'm
quite lost here!

(2) As you mentioned, I construct the correlation between raw mean
difference using impute_covariance_matrix(). However, as I understand, it
is more useful to use with standardised mean difference effects? The same
is true for raw mean difference?

(3) Yes, my effect size index for SDs is the the log transformed SD (plus
bias correction for sample size) using the method proposed by Nakagawa
et.al 2015

https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.12309 ("SDLN"
in escal function) whereby the sampling variance is constructed solely
based on sample size 1/2(n-1). To construct the V-matrix I used Wolfgang
response in the mailing list regarding a similar question (see below)
 https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2019-July/001630.html
Is this the right (or a fair) way to approach this? Does the Delta method
work better here? If so, can you refer me to something that I can work with.

## this is a small example of my dataset

es.id study.id  n   SD     yi     vi
1         1         25 1.45 0.3924 0.0208
2         1         25 1.41 0.3644 0.0208
3         2         16 1.05 0.0821 0.0333
4         2         16 1.20 0.2157 0.0333
5         2         16 1.27 0.2724 0.0333
6         3          38 1.24 0.2286 0.0135

I'll be grateful to get any insight from you and others on the mailing
list..

Thanks and Kind regards,

Tzlil Shushan | Sport Scientist, Physical Preparation Coach

BEd Physical Education and Exercise Science
MSc Exercise Science - High Performance Sports: Strength &
Conditioning, CSCS
PhD Candidate Human Performance Science & Sports Analytics



??????? ??? ??, 20 ????? 2021 ?-1:53 ??? ?James Pustejovsky?? <?
jepusto at gmail.com??>:?

Hi Tzlil,

Apologies for the long delay in responding to your query. You've raised
some excellent questions about rather subtle issues. To your question (1),
I would say that it is NOT compulsory to use a sampling variance-covariance
matrix (the "V-matrix") for the effect sizes of each type. Omitting the
V-matrix amounts to assuming a correlation of zero. If your goal is
primarily to understand average effect sizes (of each type), then using
robust standard errors/hypothesis tests/confidence intervals will work even
if you have a mis-specified assumption about the correlation among effect
sizes.

That said, there are at least two potential benefits to using a V-matrix
based on more plausible assumptions about the correlation between effect
sizes. First, using a working model that is closer to the true dependence
structure will yield a more precise estimate of the average effect. If
you're just estimating an average effect of each type, the gain in
precision will probably be pretty small. If you're estimating a
meta-regression with a more complex set of predictors, the gains can be
more substantial.

The second potential benefit is that using a plausible V-matrix will
give you better, more defensible estimates of the variance components
(between-study variance and within-study variance). Whether based on REML
or some other estimation method, the variance component estimates are NOT
robust to mistaken assumptions about the sampling correlation structure.
They'll be biased unless you have the sampling correlation structure
approximately correct. So to the extent that understanding heterogeneity is
important, I think it's worth working on building in a V-matrix.

To your question (2), I like the approach you've outlined, where you use
different V-matrices for each of the effect indices you're looking at. I
think ideally, you would start by making a single assumption about the
degree of correlation between the *outcomes*, and then using that to
derive the appropriate degree of correlation between each of the indices:
* For raw mean differences, the correlation between outcomes will
translate directly into the correlation between mean differences.
* For SDs, I'm not sure exactly what your ES index is. Is it the log
ratio of SDs? How did you arrive at the formula for the correlation between
effect sizes? I don't know of a source for this, but it could be derived
via the delta method.
* For ICCs and pearson correlations, using Wolfgang's function would be
the way to go. Perhaps if you can provide a small example of your data and
syntax that you've attempted, folks on the list can provide guidance about
applying the function.

Kind Regards,
James

On Thu, Jan 7, 2021 at 5:16 PM Tzlil Shushan <tzlil21092 at gmail.com>
wrote:

Dear Wolfgang and James,

Apologise for the long assay in advance..

In my meta-analysis I obtained different effect sizes coming
from test-retest and correlational designs. Accordingly, I performed 4
different meta-analyses for each effect size:

Raw mean difference of test-retest
Standard deviation (using Nakagawa et al. 2015 approach) of test-retest
Intraclass correlation (transformed to z fisher values) of test-retest
Pearson correlation coefficient (transformed to z fisher
values) derived from the same test against criterion measure.

Because many studies meeting inclusion criteria provided more than one
effect size through various ways of repeated measures (for example,
multiple intensities of the test, repeated measures across the year), which
all based on a common sample of participants, I treated each unique sample
as an independent study (NOTE: this approach serves our purposes on the
best way and adding further level will results in low number of
clusters?which I don't want, given the use of RVE).

Thanks to the great discussions in this group, we've done the following:
(1) used rma.mv() to estimate the overall average estimate and the
variance in hierarchical working model. The same for meta-regressions we
performed.

(2) Compute robust variance estimates with robust() and coef_test()
functions, clustering at the level of studies (the same is true for both
overall models and meta-regression).

However, after reading some threads in the groups in the last weeks

https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-January/002565.html

https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-February/000647.html
and more...I think that one step further is to
provide variance-covariance matrices for each meta-analysis before step 1
and 2 noted above.

In this regard I have some other questions:

(1) Is it compulsory to create (an estimate) variance-covariance given
the structure of my dataset?

(2) IF YES, I'm not sure if I can use the same covariance formulas for
all effect sizes. For example, impute_covariance_matrix() from clubSandwich
can work fine with all effect sizes (mean diff, SD, icc etc.)? or should I
estimate the covariance-matrix with a unique function for each effect size?

Based on reading and suggestions:
? I used impute_covariance_matrix() for mean difference.

? For standard deviation I constructed the formula below:
calc.v <- function(x) {
v <- matrix(r^2/(2*x$ni[1]-1), nrow=nrow(x), ncol=nrow(x))
diag(v) <- x$vi
v
}
V <- bldiag(lapply(split(dat, dat$study), calc.v))
http://www.metafor-project.org/doku.php/analyses:gleser2009

? for icc and pearson correlation I've looked at this
https://wviechtb.github.io/metafor/reference/rcalc.html but I couldn't
create something which is appropriate to my dataset (I don't really know
how to specify var1 and var2).

With this regard, I created a sensitivity analysis (with 0.3, 0.5, 0.7
and 0.9) which revealed similar overall estimates (also similar to the
working models without covariance-matrix), albeit, changed a bit the
magnitude of sigma2.1 and sigma2.2

I'll be thankful to get any thoughts..

Kind regards and thanks in advance!

Tzlil

Tzlil Shushan | Sport Scientist, Physical Preparation Coach

BEd Physical Education and Exercise Science
MSc Exercise Science - High Performance Sports: Strength &
Conditioning, CSCS
PhD Candidate Human Performance Science & Sports Analytics