[R-meta] Difference between univariate and multivariate parameterization

[Viechtbauer, Wolfgang (SP)]

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
>>
>>>-----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