[R-meta] multi-level meta-analysis with dependent effect sizes - Gleser and Olkin formulae?

Andrew Guerin Andrew@Guerin @ending from newc@@tle@@c@uk
Fri Sep 14 17:10:28 CEST 2018

Hi James,

Thanks for your help so far. I think I have got the hang of building a vcov matrix based on the code from the worked example here: http://www.metafor-project.org/doku.php/analyses:gleser2009

Just a couple of questions remain:
- as in the example above I have used the Gleser and Olkin formula 19.19 (p365 in G&O 2009) to calculate the covariance: 1/nc + d1*d2/2ntotal. However, in the book this is given as the formula for standardised mean differences, whereas I am using a simple raw mean difference. Does the same formula apply? The 1994 edition of the G&O chapter makes it sound like this version of the formula is specifically for situations where the standardised mean difference, d, is calculated using the pooled SD. G&O (1994) give a slightly different formula for situations where the pooled sd is not used (ie. the effect size is Glass' D) : cov = (1+ 1/2d1*d2)/nc. This leaves me with the impression that neither of these formulae apply to raw mean differences. 

- For a follow-up part of my analysis, I plan to look at whether treatments affect sample variability - the data structure will be identical but the effect size will be lnVR (variability ratio) - this is available in the metafor escalc() function. Is anyone aware of an appropriate formula for calculating the covariance for this type of data? G&O 2009 do give some formulae for ratios, but these seem to be specific to cases where the original data are proportions. 

Best Regards


From: James Pustejovsky <jepusto using gmail.com>
Sent: 11 September 2018 19:47
To: Andrew Guerin
Cc: r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] multi-level meta-analysis with dependent effect sizes


Your description of the data structure is very clear. (Bravo for that! Many meta-analyses that I have reviewed---even quite recent ones---don't do a great job of describing the hierarchical/multivariate structure of the data, even though it's quite important.)  Based on your responses, I think the best thing to do would be to use the Gleser & Olkin approach to try and approximate the covariances among effect size estimates that share a common control condition. The reason I think it's best is that you should have  all the information you need in order to calculate the covariances (when the effect sizes are differences in means, all that is needed are means, SDs, and Ns per cell), and so there's not a strong rationale for using other approximations. For example, using  impute_covariance_matrix() with r = 0.7 is likely to be a bit too high. Consider an example where two treatment cells A and B, and the control cell C all have equal sample sizes n, and the SDs are also all equal to S. Then 

Var(A - C) =  S^2 * 2 / n

Var(B - C) = S^2 * 2 / n
Cov(A - C, B - C) = S^2 / n
And so Cor(A - C, B - C) = 0.5, rather than 0.7. 

A further reason for going this route is that using good estimates of sampling covariances is the only way I know of to get defensible, interpretable estimates of the random effects variance components of the model. If you use something along the lines  of your model D,

nmtD <- rma.mv(yi=md, V=vcvndat,  mods = ~ factor(sampletype) + factor(A) ... + factor(H), random = ~1|sid/contrast, data=ndat, method="REML") 

(but with vcvndat based on Gleser & Olkin), then you would be able to interpret the random effects vairances on sid and contrast. You noted that one of the variance components (not sure which one is sigma2.1 versus sigma2.2) does seem to be sensitive to  the assumed r, so Gleser & Olkin would be a way to pin that down. It would also let you explore other potential specifications of the random effects structure, such as allowing variance components to differ across levels of sampletype (or other moderators).

All that said, computing the full Gleser & Olkin vcov matrix is a bit inconvenient because you have to work with block diagonal matrices. If you send a smallish example dataset, I can try to provide some example code for how to do these calculations.


On Tue, Sep 11, 2018 at 10:54 AM Andrew Guerin <Andrew.Guerin using newcastle.ac.uk> wrote:

Hi James,

Thanks for the rapid response. 

The study is looking at how a preparation technique (acid treatment) affects particular measurements (stable isotope signatures) in sample materials. Experiments compare the isotope signatures of replicates with and  without acid treatment - so it is the differences, rather than the absolute values, which are of interest (so of your two suggestions, it would have to be option 1).

The hierarchical dependence arises as you have guessed. All the included studies are explicit tests of acid treatment. While some studies look at the effects on just one type of sample material (say,  crab muscle or fish scales)  others might test a selection of materials from various sources. So there can be multiple values of 'sampletype' within a study. The other moderators (A-H) concern potentially influential differences  in the experimental methods used (eg. type of acid, method of application, how samples were treated before acid exposure, etc). 'Sampletype' is the main moderator of interest - the idea is to be able to say whether or not acid treatment is desirable/necessary  for a particular type of material (an overall estimate of effect size for all types of sample combined is not very interesting). The other moderators are less fundamentally interesting, they are there mainly to account for differences in the methods used in  the studies included in the analysis, though of course it is interesting to be able to report whether they make any difference. 

In terms of variation in sampletype and the other moderators, you could put the studies into 4 groups:

1) Single sample type, fixed protocol (all moderators A-H constant; 10 studies)
2) Single sample type, multiple protocols (generally one or two moderators will vary, others will remain constant; 2 studies)
3) Multiple sample types, fixed protocol (the most common - 25 studies)
4) Multiple sample types, multiple protocols (such studies are a lot of work, so they are generally limited to 4 or fewer sample types, and a small number of variations in one or two moderators;  7 studies)

So there is a bit of both between-study and within-study variation in the moderators. For the two most common types of study (1 and 3) since there is only one protocol there will be multiple independent  treatment/control pairs, so no 'multiple treatments' type dependence (though still potential hierarchical dependence). For type 2 there will be one group of treatments per study, but all compared to a single control. Studies in group 4 will presumably exhibit  both types of dependence.



From: James Pustejovsky <jepusto using gmail.com>
Sent: 11 September 2018 15:23
To: Andrew Guerin
Cc: r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] multi-level meta-analysis with dependent effect sizes


In your study, how does the hierarchical dependence arise? Is it that some studies report results from multiple (presumably independent) samples? 

If so, then I would argue that the best thing to do might be one of the following:

1. Go full Gleser-Olkin. If multiple-treatments dependence is the *only* type of sampling dependence in your data, then it would be better just to work through the calculation of the covariances. This would let you then treat the sampling variance-covariance  matrix as known, and you could avoid the need for cluster-robust standard errors all together (unless there are other types of mis-specification to worry about, in which case clustered SEs could still be useful). 

2. Since you have a common outcome scale, you could also consider using the mean outcome as the ES. If different treatment groups within a study are independent (i.e., a between-subjects design), then modeling the means directly would let you avoid dealing  with the Gleser-Olkin business. 

Generally, which of these (or the other approaches you described) is most defensible will depend on the goals of your analysis and, in particular, whether the moderators of interest involve between-study variation, within-study variation, or both. For  moderators that vary between-study, I have found (based on personal experience) that the approach to modeling/handling dependence tends not to matter much. But for moderators where there is within-study variation, it is more salient. 

Does sampletype vary within study? And what about the other factors?


On Tue, Sep 11, 2018 at 8:30 AM Andrew Guerin <Andrew.Guerin using newcastle.ac.uk> wrote:
  I am new to meta-analysis, and have benefited greatly from the excellent documentation and resources for meta-analysis in R, especially the websites for metafor and clubSandwich. I am planning some extensive future work using meta-analysis, so thought it would  be useful to get some experience of the whole process by starting with something relatively simple (haha) and trying to work it up for a publication. I would like some advice on whether I am following the correct approach.

I selected a topic with which I am quite familiar and which has a relatively limited literature: I did some thorough literature searching and came up with ~70 relevant studies, which I will use for a few models looking at different aspects (I'll focus on one  here). Once I had coded all the data it became clear that the analysis was not going to be as simple as I had hoped, since the data have two kinds of dependence that should be accounted for:

1. Hierarchical dependence - a lot of the studies contribute multiple effect sizes.

2. Many of the effect sizes share controls - different treatments are compared against a common 'untreated' control - Gleser and Olkin's (2009) 'multiple treatments' dependence.

For this analysis, there are a total of 302 effect sizes, grouped into 244 'contrast' groups which share a control, from 43 studies. The 244 contrast groups contain 1-8 effect sizes.

There are various moderators. The most important is 'sampletype' - expected to be significant, as it is expected that treatment will have different effects on different sample types. The other factors all relate to variations in the experimental treatments  used, to keep things general I'll just call them "A" to "H".

ndat = data frame

md = effect size measure (raw mean difference, since all data are on a common scale, with a limited range)

mdv = sampling variation for mean difference, obtained using escalc().

sid = study id (a simple numerical identifier)

contrast = identifier showing which effect size measures / treatments share a control (eg. study one contributes 5 effect sizes, with values of 'contrast' 1A, 1B, 1B, 1C, 1D, indicating that of the 5 effect sizes in study 1, 2 share a control).

As I see it I have the following options for how to proceed:

A) conduct a completely 'cluster naive' mixed effects meta-analysis

nmtA <- rma(md, mdv,  mods = ~ factor(sampletype) + factor(A) ... + factor(H), data=ndat, method="REML")

anova(nmtA, btt=20:21) ## example hypothesis test for moderator 'D'

B) ignore the hierarchical dependence (ie. assume that any samples with different values of 'contrast' within the same study are independent), but account for the correlated errors within clusters / contrast groups by RVE in robumeta, using clubSandwich for  the hypothesis tests.

nmtB <- robu(md ~ factor(sampletype) + factor(A) ... + factor(H), data=ndat, var.eff.size=mdv, rho = 0.8, modelweights="CORR", studynum=contrast)

Wald_test(nmtB, constraints = 20:21, vcov="CR2") ## example hypothesis test for moderator 'D'

C) still ignoring the hierarchical dependence, but now explicitly specifying the random effects in  rma.mv, after imputing a variance-covariance matrix. Robust parameter estimates and hypothesis test are then carried out using clubSandwich.

vcvndat <- impute_covariance_matrix(vi = ndat$mdv, cluster=ndat$contrast, r=0.7)

nmtC <-  rma.mv(yi=md, V=vcvndat,  mods = ~ factor(sampletype) + factor(A) ... + factor(H), random = ~1|contrast, data=ndat, method="REML")

nmtC_robust <- coef_test(nmtC, vcov="CR2")

Wald_test(nmtC, constraints = 20:21, vcov="CR2") ## example hypothesis test for moderator 'D'

D) attempting to account for the hierarchical and multiple-treatments dependence via a multi-level model, setting random effect structure as random = ~ 1| sid/contrast, and then using clubSandwich for RVE and hypothesis tests.

vcvndat <- impute_covariance_matrix(vi = ndat$mdv, cluster=ndat$contrast, r=0.7)

nmtD <-  rma.mv(yi=md, V=vcvndat,  mods = ~ factor(sampletype) + factor(A) ... + factor(H), random = ~1|sid/contrast, data=ndat, method="REML")

nmtD_robust <- coef_test(nmtD, vcov="CR2", cluster=ndat$contrast)

Wald_test(nmtD, constraints = 20:21, vcov="CR2", cluster=ndat$contrast) ## example hypothesis test for moderator 'D'

My main question really is which (if any) of these approaches is the best way forward? For this particular dataset, the outcomes (at least in terms of which moderators are significant/non-significant, and most parameter estimates) are pretty similar for all  the options (including some other options that I tried that I have not mentioned here). All find 'sampletype' highly significant, and find similar sets of the other moderators to be non-significant (except for the naive analysis, A which finds more moderators  to be significant). B, C, and D are all very similar. Although D is the only one that explicitly accounts for the hierarchical dependence, does the fact that the outcomes from B and C are substantially similar mean that this could be safely overlooked for  this analysis?

I have not yet attempted to build my own vcov matrix using the Gleser and Olkin (2009) formula. Is this worth doing instead of using impute_covariance_matrix? I picked 0.7 as a plausible value for r in impute_convariance_matrix(), but I have raw data from one  study that might allow me to derive a better estimate. I did investigate the effect of varying r from 0-1. It does not seem to have much impact on parameter estimates, the significance/non-significance of moderators, or the estimate of sigma^2.2 in the multi-level  model. The intercept moves about a little, but the main effect is that when r is less than about 0.5, sigma^2.1 is basically 0. Is this indicating a problem?

Finally I was wondering whether it is wise to account for multiple comparisons when evaluating the hypothesis tests. Since I have several factors, I repeat Wald_test() several times. Models nmtB, nmtC, and nmtD all have moderators  (aside from 'sampletype')  that are significant at p < 0.05, but with p values only just under 0.05. Adjustment for multiple comparisons (eg. using p.adjust(method="fdr")) shifts the outcomes of all these tests to p > 0.05.

I would be grateful for any advice on the above.

Many thanks,


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