[R-meta] Difference between univariate and multivariate parameterization

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 at gmail.com]
Sent: Friday, 20 August, 2021 16:06
To: Viechtbauer, Wolfgang (SP)
Cc: Farzad Keyhan; R meta
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 at 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 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