# [R-meta] Difference between univariate and multivariate parameterization

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Fri Aug 20 16:14:37 CEST 2021

```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)
>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)
>>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?
>>
>>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)
>>>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
```