[R-meta] Difference between univariate and multivariate parameterization

[Luke Martinez]

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
> >
> >>-----Original Message-----
> >>From: R-sig-meta-analysis [mailto:
> r-sig-meta-analysis-bounces using r-project.org] On
> >>Behalf Of Farzad Keyhan
> >>Sent: Wednesday, 18 August, 2021 21:32
> >>To: Luke Martinez
> >>Cc: R meta
> >>Subject: Re: [R-meta] Difference between univariate and multivariate
> >>parameterization
> >>
> >>Dear Luke,
> >>
> >>In the multivariate specification (model 2), tau^2 = sigma^2_between  +
> >>sigma^2_within. You can confirm that by your two models' output as well.
> >>Also, because rho = sigma^2_between / (sigma^2_between  +
> sigma^2_within),
> >>then, the off-diagonal elements of the matrix can be shown to be
> rho*tau^2
> >>which again is equivalent to sigma^2_between in model 1's matrix.
> >>
> >>Note that sampling errors in a two-estimate study could be different
> hence
> >>appropriate subscripts will be needed to distinguish between them.
> >>
> >>Finally, note that even a study with a single effect size estimate gets
> the
> >>sigma^2_within, either directly (model 1) or indirectly (model 2) which
> >>would mean that, that one-estimate study **could** have had more
> estimates
> >>but it just so happens that it doesn't as a result of some form of
> >>multi-stage sampling; first studies, and then effect sizes from within
> >>those studies.
> >>
> >>I actually raised this last point a while back on the list (
> >>https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-July/002994.html
> )
> >>as I found this framework a potentially unrealistic but in the end, it's
> >>the best approach we have.
> >>
> >>Cheers,
> >>Fred
> >>
> >>On Wed, Aug 18, 2021 at 1:30 PM Luke Martinez <martinezlukerm using gmail.com>
> >>wrote:
> >>
> >>> Dear Colleagues,
> >>>
> >>> Imagine I have two models.
> >>>
> >>> Model 1:
> >>>
> >>> random = ~1 | study / row_id
> >>>
> >>> Model 2:
> >>>
> >>> random = ~ row_id | study,  struct = "CS"
> >>>
> >>> I understand that the diagonal elements of the variance-covariance
> matrix
> >>> of a study with two effect size estimates for each model will be:
> >>>
> >>> Model 1:
> >>>
> >>> VAR(y_ij) = sigma^2_between  +  sigma^2_within + e_ij
> >>>
> >>> Model 2:
> >>>
> >>> VAR(y_ij) = tau^2 + e_ij
> >>>
> >>> Question: In model 2's variance-covariance matrix, what fills the role
> of
> >>> sigma^2_within (within-study heterogeneity) that exists in model 1's
> >>> matrix?
> >>>
> >>> Thank you very much for your assistance,
> >>> Luke Martinez
>

	[[alternative HTML version deleted]]