[R-meta] Difference between univariate and multivariate parameterization

[Norman DAURELLE]

Dear all, 

I write this e-mail to state that " To help people reading this (n = 0;-)" is not a true statement. 

I like reading the e-mails I get from this mailing list when I manage to take the time to do so, and I especially like reading them when I feel like I grasp what is being written. 

So : thank you for the exchange, I'm glad I am reading this thread. 

Best wishes, 
Norman 


De: "Luke Martinez" <martinezlukerm using gmail.com> 
�: "Wolfgang Viechtbauer, SP" <wolfgang.viechtbauer using maastrichtuniversity.nl> 
Cc: "r-sig-meta-analysis" <r-sig-meta-analysis using r-project.org> 
Envoy�: Vendredi 20 Ao�t 2021 16:06:08 
Objet: 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 
> >> 
> >>>-----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 
> 

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