[R-meta] Construct the covariance-matrices of different effect sizes
Hi Tzlil, Some responses to your follow-ups: (1) For the pearson correlations and ICCs, I think the data might need to include some sort of label that distinguishes unique measures and time-points. So for example, submax-T1, submax-T2, max-T1, max-T2. Also, if you could provide a reproducible example with data that someone could easily read in to R (using e.g., dput()), it will increase the probability that others will chime in with coding/debugging suggestions. There are some links to how to do this here: 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