[R-meta] Var-cov structure in multilevel/multivariate meta-analysis

[Fabian Schellhaas]

I actually have one more question related to this: If I want to compute
leave-one-out statistics for outlier and influential case diagnostics
(e.g., deleted residuals, Cook's distance) at cluster level, would it make
more sense to do so at the level at which the variance-covariance was
imputed (here: samples) or at the level at which cluster-robust standard
errors were estimated (here: papers)? Or both?

Thanks!
Fabian


On Sat, Mar 23, 2019 at 2:10 PM James Pustejovsky <jepusto using gmail.com> wrote:

> Yes exactly!
>
> Sent from my iPhone
>
> On Mar 23, 2019, at 12:42 PM, Fabian Schellhaas <
> fabian.schellhaas using yale.edu> wrote:
>
> Hi James,
>
> Awesome, that makes so much more sense now! So the crux in our initial
> approach was simply in how we specified clustering in
> `clubSandwich::coef_test`.
>
> As a general rule of thumb one would thus go through the following steps:
> 1. Impute the variance-covariance matrix at the highest level of
> clustering at which residuals are non-independent.
> 2. Fit the model, adding random effects as necessary to account for
> hierarchical dependence.
> 3. Compute the cluster-robust errors at the outermost level of clustering
> from step 1 or 2, whichever is higher.
>
> Thanks so much for your help -- really appreciated.
> Fabian
>
> ---
> Fabian M. H. Schellhaas | Ph.D. Candidate | Department of Psychology |
> Yale University
>
>
> On Sat, Mar 23, 2019 at 1:30 PM James Pustejovsky <jepusto using gmail.com>
> wrote:
>
>> Aha I see. It’s not necessary to impute the correlation matrix at the
>> same level as you cluster the standard errors. I would suggest imputing the
>> correlation matrix at the sample level, because it is only at this level
>> that there is dependence in the sampling errors (conditional on true
>> effects for the sample). At higher levels, the sampling errors will be
>> independent but there may be dependence between true effects (hierarchical
>> dependence) due to use of common procedures, interventions, or whatnot.
>>
>> But then for purposes of clustering, we need to “catch” all forms of
>> dependence (both sampling dependence and hierarchical dependence) and so
>> would cluster at the highest level.
>>
>> James
>>
>> On Mar 23, 2019, at 11:39 AM, Fabian Schellhaas <
>> fabian.schellhaas using yale.edu> wrote:
>>
>> Hi James,
>>
>> I suppose the variance components change because we're now imputing the
>> variance-covariance matrix at a higher level of clustering. My
>> understanding has been that the vcov imputation needs to be done at the
>> same level of clustering used for the estimation of cluster-robust standard
>> errors later, but I'm keen to hear what your take is. I included more
>> details below.
>>
>> Initially, the model was (incorrectly) fitted as follows:
>> vcv <- clubSandwich::impute_covariance_matrix(vi = data$vi, cluster =
>> data$sample_id, r = 0.7)
>> m <- metafor::rma.mv(yi, V = vcv, random = ~ 1 | paper_id/comp_id/es_id)
>>
>> Variance components: sigma^2.1 = 0.0068, sigma^2.2 = 0.0087, sigma^2.3 =
>> 0.0137
>> Comparing this model to one model with random effects only for
>> comparisons and estimates: LRT = 15.11, p < .001
>>
>> Now, fitting the model with the vcov matrix imputed at the level of
>> papers:
>> vcv <- clubSandwich::impute_covariance_matrix(vi = data$vi, cluster =
>> data$paper_id, r = 0.7)
>> m <- metafor::rma.mv(yi, V = vcv, random = ~ 1 | paper_id/comp_id/es_id)
>>
>> Variance components: sigma^2.1 = 0.0012, sigma^2.2 = 0.0131, sigma^2.3 =
>> 0.0142
>> Comparing this model to one model with random effects only for
>> comparisons and estimates: LRT = 0.59, p = .443
>>
>> Many thanks,
>> Fabian
>>
>> ---
>> Fabian M. H. Schellhaas | Ph.D. Candidate | Department of Psychology |
>> Yale University
>>
>>
>> On Sat, Mar 23, 2019 at 12:18 PM James Pustejovsky <jepusto using gmail.com>
>> wrote:
>>
>>> Fabian,
>>>
>>> I don’t follow the new results you’ve just described. I meant clustering
>>> as in the clustering variable specified when computing robust standard
>>> errors. Switching this to a higher level should not affect your variance
>>> component estimates at all—only the SEs of the fixed effects will change.
>>> So I’m not sure what you’ve modified that leads to changes in the variance
>>> component estimates.
>>>
>>> James
>>>
>>> On Mar 23, 2019, at 11:10 AM, Fabian Schellhaas <
>>> fabian.schellhaas using yale.edu> wrote:
>>>
>>> Hi James,
>>>
>>> Thanks a lot for the quick reply -- this makes sense. I just checked and
>>> clustering at a higher level instead (e.g., study) actually reduces the
>>> variance component for papers so much that retaining it does not improve
>>> model fit, so I suppose that random effect would no longer be needed? More
>>> generally, is it a correct rule of thumb to specify clustering at the
>>> highest/outermost level, and then model hierarchical dependence at lower
>>> levels by adding random effects as needed?
>>>
>>> Thanks!
>>> Fabian
>>>
>>>
>>> On Sat, Mar 23, 2019 at 11:41 AM James Pustejovsky <jepusto using gmail.com>
>>> wrote:
>>>
>>>> Fabian,
>>>>
>>>> That will not work. Cluster-robust standard errors require that the
>>>> clusters are independent, and so clustering at the level of the sample will
>>>> fail to capture dependence at higher levels (studies and papers). I think
>>>> the appropriate approach here is to cluster at the highest level (papers).
>>>>
>>>> James
>>>>
>>>>
>>>> > On Mar 22, 2019, at 5:41 PM, Fabian Schellhaas <
>>>> fabian.schellhaas using yale.edu> wrote:
>>>> >
>>>> > Hi Wolfgang and mailing list,
>>>> >
>>>> > I would like to follow up on point (3) from this old thread. Quick
>>>> > refresher of the data structure: We have (approximately) 650 effects
>>>> from
>>>> > 400 treatment-control comparisons, which come from 325 independent
>>>> samples,
>>>> > nested in 275 studies from 200 papers. Many of the samples include
>>>> more
>>>> > than one treatment-control comparison and evaluate their effect on
>>>> more
>>>> > than one outcome measure, resulting in correlated residuals clustered
>>>> at
>>>> > the level of independent samples.
>>>> >
>>>> > First, we model the hierarchical dependence of the true effects. LRTs
>>>> > indicate that model fit is improved significantly by adding random
>>>> effects
>>>> > for treatment-control comparisons (relative to a single-level model)
>>>> and
>>>> > further improved by adding random effects for papers (relative to the
>>>> > two-level model). Adding random effects for samples and studies did
>>>> not
>>>> > improve on the two-level model. So in short, we include random
>>>> effects for
>>>> > papers, treatment-control comparisons, and individual estimates,
>>>> skipping
>>>> > studies and samples. Second, to deal with the nonindependent
>>>> residuals, due
>>>> > to multiple comparisons and multiple outcome measures, we use
>>>> > cluster-robust variance estimation. In our data, residuals are
>>>> clustered at
>>>> > the level of independent samples.
>>>> >
>>>> > As such, we could fit a model as follows:
>>>> > vcv <- clubSandwich::impute_covariance_matrix(vi = data$vi, cluster =
>>>> > data$sample_id, r = 0.7)
>>>> > m <- metafor::rma.mv(yi, V = vcv, random = ~ 1 |
>>>> paper_id/comp_id/es_id)
>>>> > clubSandwich::coef_test(m, cluster = data$sample_id, vcov = "CR2")
>>>> >
>>>> > Are there any problems with computing robust standard errors at a
>>>> level of
>>>> > clustering (here: samples) that does not correspond to the levels at
>>>> which
>>>> > hierarchical dependence of the true effects are modeled (here: papers
>>>> and
>>>> > treatment-control comparisons)? If so, what would be a better
>>>> approach?
>>>> >
>>>> > Many thanks!
>>>> > Fabian
>>>> >
>>>> >
>>>> > On Fri, Oct 5, 2018 at 11:37 AM Viechtbauer, Wolfgang (SP) <
>>>> > wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>>>> >
>>>> >> Hi Fabian,
>>>> >>
>>>> >> (very clear description of the structure, thanks!)
>>>> >>
>>>> >> 1) Your approach sounds sensible.
>>>> >>
>>>> >> 2) If you are going to use cluster-robust inference methods in the
>>>> end
>>>> >> anyway, then getting the var-cov matrix of the sampling errors
>>>> 'exactly
>>>> >> right' is probably not crucial. It can be a huge pain constructing
>>>> the
>>>> >> var-cov matrix, especially when dealing with complex data structures
>>>> as you
>>>> >> describe. So, sticking to the "best guess" approach is probably
>>>> defensible.
>>>> >>
>>>> >> 3) It is difficult to give general advice, but it is certainly
>>>> possible to
>>>> >> add random effects for samples, studies, and papers (plus random
>>>> effects
>>>> >> for the individual estimates) here. One can probably skip a level if
>>>> the
>>>> >> number of units at a particular level is not much higher than the
>>>> number of
>>>> >> units at the next level (the two variance components are then hard to
>>>> >> distinguish). So, for example, 200 studies in 180 papers is quite
>>>> similar,
>>>> >> so one could probably leave out the studies level and only add random
>>>> >> effects for papers (plus for samples and the individual estimates).
>>>> You can
>>>> >> also run likelihood ratio tests to compare models to see if adding
>>>> random
>>>> >> effects at the studies level actually improves the model fit
>>>> significantly.
>>>> >>
>>>> >> Best,
>>>> >> Wolfgang
>>>> >>
>>>> >> -----Original Message-----
>>>> >> From: R-sig-meta-analysis [mailto:
>>>> >> r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Fabian
>>>> Schellhaas
>>>> >> Sent: Thursday, 27 September, 2018 23:55
>>>> >> To: r-sig-meta-analysis using r-project.org
>>>> >> Subject: [R-meta] Var-cov structure in multilevel/multivariate
>>>> >> meta-analysis
>>>> >>
>>>> >> Dear all,
>>>> >>
>>>> >> My meta-analytic database consists of 350+ effect size estimates,
>>>> drawn
>>>> >> from 240+ samples, which in turn were drawn from 200+ studies,
>>>> reported in
>>>> >> 180+ papers. Papers report results from 1-3 studies each, studies
>>>> report
>>>> >> results from 1-2 samples each, and samples contribute 1-6 effect
>>>> sizes
>>>> >> each. Multiple effects per sample are possible due to (a) multiple
>>>> >> comparisons, such that more than one treatment is compared to the
>>>> same
>>>> >> control group, (b) multiple outcomes, such that more than one
>>>> outcome is
>>>> >> measured within the same sample, or (c) both. We coded for a number
>>>> of
>>>> >> potential moderators, which vary between samples, within samples, or
>>>> both.
>>>> >> I included an example of the data below.
>>>> >>
>>>> >> There are two main sources of non-independence: First, there is
>>>> >> hierarchical dependence of the true effects, insofar as effects
>>>> nested in
>>>> >> the same sample (and possibly those nested in the same study and
>>>> paper) are
>>>> >> correlated. Second, there is dependence arising from correlated
>>>> sampling
>>>> >> errors when effect-size estimates are drawn from the same set of
>>>> >> respondents. This is the case whenever a sample contributes more
>>>> than one
>>>> >> effect, i.e. when there are multiple treatments and/or multiple
>>>> outcomes.
>>>> >>
>>>> >> To model these data, I start by constructing a “best guess” of the
>>>> var-cov
>>>> >> matrices following James Pustejovsky's approach (e.g.,
>>>> >>
>>>> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2017-August/000094.html
>>>> >> ),
>>>> >> treating samples in my database as independent clusters. Then, I use
>>>> these
>>>> >> var-cov matrices to construct the multilevel/multivariate
>>>> meta-analytic
>>>> >> model. To account for the misspecification of the var-cov structure,
>>>> I
>>>> >> perform all coefficient and moderator tests using cluster-robust
>>>> variance
>>>> >> estimation. This general approach has also been recommended on this
>>>> mailing
>>>> >> list and allows me (I think) to use all available data, test all my
>>>> >> moderators, and estimate all parameters with an acceptable degree of
>>>> >> precision.
>>>> >>
>>>> >> My questions:
>>>> >>
>>>> >> 1. Is this approach advisable, given the nature of my data? Any
>>>> problems I
>>>> >> missed?
>>>> >>
>>>> >> 2. Most manuscripts don’t report the correlations between multiple
>>>> >> outcomes, thus preventing the precise calculation of covariances for
>>>> this
>>>> >> type of dependent effect size. By contrast, it appears to be fairly
>>>> >> straightforward to calculate the covariances between
>>>> multiple-treatment
>>>> >> effects (i.e., those sharing a control group), as per Gleser and
>>>> Olkin
>>>> >> (2009). Given my data, is there a practical way to construct the
>>>> var-cov
>>>> >> matrices using a combination of “best guesses” (when correlations
>>>> cannot be
>>>> >> computed) and precise computations (when they can be computed via
>>>> Gleser
>>>> >> and Olkin)? I should note that I’d be happy to just stick with the
>>>> “best
>>>> >> guess” approach entirely, but as Wolfgang Viechtbauer pointed out (
>>>> >>
>>>> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2017-August/000131.html
>>>> >> ),
>>>> >> only a better approximation of the var-cov structure can improve
>>>> precision
>>>> >> of the fixed-effects estimates. That's why I'm exploring this option.
>>>> >>
>>>> >> 3. How would I best determine for which hierarchical levels to
>>>> specify
>>>> >> random effects? I certainly expect the true effects within the same
>>>> set of
>>>> >> respondents to be correlated, so would at least add a random effect
>>>> for
>>>> >> sample. Beyond that (i.e., study, paper, and so forth) I’m not so
>>>> sure.
>>>> >>
>>>> >> Cheers,
>>>> >>
>>>> >> Fabian
>>>> >>
>>>> >> ### Database example:
>>>> >>
>>>> >> Paper 1 contributes two studies - one containing just one sample,
>>>> the other
>>>> >> containing two samples – evaluating the effect of treatment vs.
>>>> control on
>>>> >> one outcome. Paper 2 contributes one study containing one sample,
>>>> >> evaluating the effect of two treatments (relative to the same
>>>> control) on
>>>> >> two separate outcomes each. The first moderator varies between
>>>> samples, the
>>>> >> second moderator varies both between and within samples.
>>>> >>
>>>> >> paper     study sample    comp es yi        vi mod1 mod2
>>>> >>
>>>> >> 1         1 1         1 1 0.x       0.x A A
>>>> >>
>>>> >> 1         2 2         2 2 0.x       0.x B B
>>>> >>
>>>> >> 1         2 3         3 3 0.x       0.x A B
>>>> >>
>>>> >> 2         3 4         4 4 0.x       0.x B A
>>>> >>
>>>> >> 2         3 4         4 5 0.x       0.x B C
>>>> >>
>>>> >> 2         3 4         5 6 0.x       0.x B A
>>>> >>
>>>> >> 2         3 4         5 7 0.x       0.x B C
>>>> >>
>>>> >> ---
>>>> >> Fabian Schellhaas | Ph.D. Candidate | Department of Psychology | Yale
>>>> >> University
>>>> >>
>>>> >
>>>> >    [[alternative HTML version deleted]]
>>>> >
>>>> > _______________________________________________
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>>>> > R-sig-meta-analysis using r-project.org
>>>> > https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis
>>>>
>>>

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