[R-meta] Difference between univariate and multivariate parameterization

[Viechtbauer, Wolfgang (SP)]

Please see below.

Best,
Wolfgang

>-----Original Message-----
>From: Luke Martinez [mailto:martinezlukerm using gmail.com]
>Sent: Friday, 20 August, 2021 16:41
>To: Viechtbauer, Wolfgang (SP)
>Cc: Farzad Keyhan; R meta
>Subject: Re: [R-meta] Difference between univariate and multivariate
>parameterization
>
>Sure. So, for other data structures (below) where paste0(dat$study, ".",
>dat$sample) alone can't produce unique rows (perhaps due to another nested
>grouping variable like "grp"), the "within study heterogeneity" is given by
>paste0(dat$study, ".", dat$sample)  and "within sample heterogeneity" is given by
>paste0(dat$study, ".", dat$sample, dat$grp), correct?

Correct, that is,

rma.mv(yi, vi, random = ~ 1 | study/sample/grp)

would be identical to

dat$studysample    <- paste0(dat$study, dat$sample)
dat$studysamplegrp <- paste0(dat$study, dat$sample, dat$grp)

rma.mv(yi, vi, random = list(~ 1 | study, ~ 1 | studysample, ~ 1 | studysamplegrp))

>Which then would mean that Fred's comment about studies with a single estimate
>getting the same sigma^2_within that they shouldn't becomes studies with a single
>sample and a single group getting two sigma^2_withins (one for study, one for
>sample) that they shouldn't, correct?

I am not able to fully parse your question. But yes, in the models above, the diagonal of the marginal (model-implied) var-cov matrix is

sigma^2_study + sigma^2_studysample + sigma^2_studysamplegrp + v_ijk

(three subscripts on v for studies, samples, and groups), so irrespective of the type of study (whether it has one or multiple samples) and irrespective of the type of sample (whether it contains one or multiple groups), all three variance-components (plus the sampling variance) are being added. If one deems this not to be sensible, one could extent the idea I described previously to constrain the studysample variance component to 0 for single-sample studies and the studysamplegrp variance component to 0 for single-group studies.

>Thank you, Luke
>
>study sample grp
>1     1      1
>1     1      2
>1     2      1
>1     2      2
>2     1      1
>2     1      2
>
>On Fri, Aug 20, 2021 at 9:14 AM Viechtbauer, Wolfgang (SP)
><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>Note that:
>
>dat$sampleinstudy <- paste0(dat$studyid, ".", dat$sampleid)
>
>was used to create this variable (and indeed, it takes on a unique value per row).
>
>This is in fact in essence what happens when you fit the model:
>
>rma.mv(yi, vi, random = ~ 1 | studyid/sampleid, data=dat)
>
>and hence this is the same model:
>
>rma.mv(yi, vi, random = list(~ 1 | studyid, ~ 1 | sampleinstudy), data=dat)
>
>And the latter can then be extended with:
>
>rma.mv(yi, vi, random = list(~ 1 | studyid, ~ multsample | sampleinstudy),
>struct="DIAG", data=dat)
>
>as discussed.
>
>Best,
>Wolfgang
>
>>-----Original Message-----
>>From: Luke Martinez [mailto:martinezlukerm using gmail.com]
>>Sent: Friday, 20 August, 2021 16:06
>>To: Viechtbauer, Wolfgang (SP)
>>Cc: Farzad Keyhan; R meta
>>Subject: Re: [R-meta] Difference between univariate and multivariate
>>parameterization
>>
>>Ah!! "sampleinstudy" just so happens to be equivalent to a "row_id" **in this
>>particular dataset**. And in this particular case, the second random term (~
>>multsample | sampleinstudy) in reality is modeling the row_id (within-study
>>heterogeneity).
>>
>>To help people reading this (n = 0;-), when you say, "just like in the standard
>>multilevel structure", you mean a standard 3-level model of the form
>>(~1|study_id/row_id).
>>
>>I guess the one other time that this confusion happened to me was when I was
>>looking at "dat.konstantopoulos2011", demonstrating that the data structure
>should
>>always take precedence over the syntax!
>>
>>Super clear and helpful as always,
>>Luke
>>
>>On Fri, Aug 20, 2021 at 8:18 AM Viechtbauer, Wolfgang (SP)
>><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>>For reference, we are discussing this:
>>
>>list(~ 1 | studyid, ~ multsample | sampleinstudy), struct="DIAG"
>>
>>where the data structure is like this:
>>
>>studyid  sampleinstudy  multsample
>>1        1              1
>>1        2              1
>>2        3              0
>>3        4              1
>>3        5              1
>>3        6              1
>>4        7              0
>>5        8              1
>>5        9              1
>>
>>~ 1 | studyid adds a random effect corresponding to the study level. This is to
>>account for 'between-study heterogeneity'.
>>
>>~ multsample | sampleinstudy adds a random effect to the sampleinstudy level. For
>>rows where sampleinstudy is the same, rows where multsample = 0 versus 1 would
>get
>>different but correlated random effects. However, since there is just one row per
>>sampleinstudy, this never happens. So, each row is gettings its own random effect
>>(just like in the standard multilevel structure). With struct="DIAG", we allow
>for
>>a different tau^2 for multsample = 0 versus 1. So this models 'within-study
>>heterogeneity' and allows this variance component to differ for single versus
>>multisample studies (and one can then constrain the former to 0 if one likes).
>>
>>Best,
>>Wolfgang
>>
>>>-----Original Message-----
>>>From: Luke Martinez [mailto:martinezlukerm using gmail.com]
>>>Sent: Friday, 20 August, 2021 14:37
>>>To: Viechtbauer, Wolfgang (SP)
>>>Cc: Farzad Keyhan; R meta
>>>Subject: Re: [R-meta] Difference between univariate and multivariate
>>>parameterization
>>>
>>>Dear Wolfgang,
>>>
>>>Many thanks.
>>>
>>>>>>> "In res5, the two tau^2 values can be thought of as sigma^2_within for
>>single
>>>vs multi sample studies."
>>>
>>>I believe my question was why/how in res5 (and res4) models, tau^2 values
>>>represent only sigma^2_within?
>>>
>>>Is it because we have eliminated the off-diagonal elements (by struct="DIAG") in
>>>"~ multsample | sampleinstudy" or because we have previously defined the
>>>sigma^2_between studies using "~ 1 | studyid" and thus tau^2 values in "~
>>>multsample | sampleinstudy" can't represent anything other than sigma^2_within
>>>samples nested in studies?
>>>
>>>I appreciate your clarification,
>>>Luke
>>>
>>>PS. On the other hand, my understanding is that "sigma^2_between" and
>>>"sigma^2_within" are unique to each grouping variable so we can have
>>>"sigma^2_between_studies" and "sigma^2_between_study_sample_combinations" and
>the
>>>same is true for "sigma^2_withins".
>>>
>>>On Fri, Aug 20, 2021 at 6:31 AM Viechtbauer, Wolfgang (SP)
>>><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>>>Dear Luke,
>>>
>>>tau^2 doesn't mean the same thing across different models. In res5, the two
>tau^2
>>>values can be thought of as sigma^2_within for single vs multi sample studies.
>>>Whether we call something tau^2, sigma^2, or chicken^2 doesn't carry any
>inherent
>>>meaning.
>>>
>>>For example:
>>>
>>>dat <- dat.crede2010
>>>dat <- escalc(measure="ZCOR", ri=ri, ni=ni, data=dat, subset=criterion=="grade")
>>>
>>>dat$studyid.copy <- dat$studyid
>>>dat$sampleid.copy <- paste0(dat$studyid, ".", dat$sampleid)
>>>rma.mv(yi, vi, random = ~ 1 | studyid/sampleid, data=dat)
>>>rma.mv(yi, vi, random = list(~ studyid | studyid.copy, ~ sampleid |
>>>sampleid.copy), struct=c("ID","ID"), data=dat)
>>>
>>>are identical models, but in the first we have two sigma^2 values and in the
>>other
>>>we have tau^2 and gamma^2 (a bit of a silly example, but just to illustrate the
>>>point).
>>>
>>>Best,
>>>Wolfgang
>>>
>>>>-----Original Message-----
>>>>From: Luke Martinez [mailto:martinezlukerm using gmail.com]
>>>>Sent: Thursday, 19 August, 2021 5:05
>>>>To: Viechtbauer, Wolfgang (SP)
>>>>Cc: Farzad Keyhan; R meta
>>>>Subject: Re: [R-meta] Difference between univariate and multivariate
>>>>parameterization
>>>>
>>>>Dear Wolfgang,
>>>>
>>>>Thanks for your reply. But, if in the multivariate specification: tau^2 =
>>>>sigma^2_between  +  sigma^2_within, then in your suggested "res5" model where
>>you
>>>>fixed tau2 = 0 for single sample studies, you have killed both sigma^2_between
>+
>>>>sigma^2_within, and not just sigma^2_within?
>>>>
>>>>Am I missing something?
>>>>
>>>>Thank you very much,
>>>>Luke
>>>>
>>>>On Wed, Aug 18, 2021 at 3:01 PM Viechtbauer, Wolfgang (SP)
>>>><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>>>>It is also possible to formulate a model where sigma^2_within is *not* added
>for
>>>>'single sample/estimate studies'. Let's consider this example:
>>>>
>>>>library(metafor)
>>>>
>>>>dat <- dat.crede2010
>>>>dat <- escalc(measure="ZCOR", ri=ri, ni=ni, data=dat,
>subset=criterion=="grade")
>>>>
>>>>table(dat$studyid) # most studies are single sample studies
>>>>
>>>># multilevel model
>>>>res1 <- rma.mv(yi, vi, random = ~ 1 | studyid/sampleid, data=dat)
>>>>res1
>>>>
>>>># multivariate parameterization
>>>>res2 <- rma.mv(yi, vi, random = ~ factor(sampleid) | studyid, data=dat)
>>>>res2
>>>>
>>>># as a reminder, the multilevel model is identical to this formulation
>>>>dat$sampleinstudy <- paste0(dat$studyid, ".", dat$sampleid)
>>>>res3 <- rma.mv(yi, vi, random = list(~ 1 | studyid, ~ 1 | sampleinstudy),
>>>>data=dat)
>>>>res3
>>>>
>>>># logical to indicate for each study whether it is a multi sample study
>>>>dat$multsample <- ave(dat$studyid, dat$studyid, FUN=length) > 1
>>>>
>>>># fit model that allows for a different sigma^2_within for single vs multi
>>sample
>>>>studies
>>>>res4 <- rma.mv(yi, vi, random = list(~ 1 | studyid, ~ multsample |
>>>sampleinstudy),
>>>>struct="DIAG", data=dat)
>>>>res4
>>>>
>>>># fit model that forces sigma^2_within = 0 for single sample studies
>>>>res5 <- rma.mv(yi, vi, random = list(~ 1 | studyid, ~ multsample |
>>>sampleinstudy),
>>>>struct="DIAG", tau2=c(0,NA), data=dat)
>>>>res5
>>>>
>>>>So this is all possible if you like.
>>>>
>>>>Best,
>>>>Wolfgang