[R-meta] Difference between univariate and multivariate parameterization

Fri, Aug 20, 2021 6:18 AM

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 at 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 at 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 at 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 at 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 at 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 at 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

[R-meta] Difference between univariate and multivariate parameterization

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