# [R-meta] Inner|outer model vs multiple random id terms in rma.mv

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Fri May 1 15:23:35 CEST 2020

```The relationship between the two formulations is as follows:

tau2 = sigma^2_inner + sigma^2_outer
rho =  sigma^2_inner / (sigma^2_inner + sigma^2_outer)

Since sigma^2_inner is constrained to be >= 0, this equivalence only holds when rho is estimated to be >= 0 under the multivariate parameterization. If rho is estimated to be negative, then this doesn't automatically imply that the multilevel parameterization is wrong. It could be that sigma^2_inner is in reality equal to 0 (or slightly positive), but since variance and correlation components are not precisely estimated when the number of studies (and estimates within studies) is small, the negative estimate of rho could also be an underestimate. You will find that

confint(res2, rho=1)

gives a very wide CI for rho, so also positive values of rho are quite compatible with these data in the second example (and in the first example, the CI essentially goes from -1 to 1).

Best,
Wolfgang

>-----Original Message-----
>From: Divya Ravichandar [mailto:divya using secondgenome.com]
>Sent: Wednesday, 29 April, 2020 21:09
>To: Viechtbauer, Wolfgang (SP)
>Cc: r-sig-meta-analysis using r-project.org
>Subject: Re: [R-meta] Inner|outer model vs multiple random id terms in
>rma.mv
>
>By more applicable, I was trying to get clarity around how to interpret the
>results of the two model when rho is < 0.
>
>The ~inner| outer in this example estimated a negative rho value and
>profiling indicated that this was indeed the correct estimate. In that case,
>does using `~1| inner , ~1|outer` formulation on this dataset
>invalid considering the assumption of rho>0 is not in line with the
>estimated value of rho < 0 by the inner|outer model
>
>On Wed, Apr 29, 2020 at 11:32 AM Viechtbauer, Wolfgang (SP)
><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>Yes, negative rho means negative correlation (rho is the correlation between
>random effects that have the same level of the 'outer' variable).
>
>I don't know what you mean with "more applicable". The '~1| inner ,
>~1|outer' formulation implicitly assumes that the correlation is >= 0. If
>this is what you would like to assume, then you can use this formulation.
>
>Best,
>Wolfgang
>
>>-----Original Message-----
>>From: Divya Ravichandar [mailto:divya using secondgenome.com]
>>Sent: Wednesday, 29 April, 2020 20:15
>>To: Viechtbauer, Wolfgang (SP)
>>Cc: r-sig-meta-analysis using r-project.org
>>Subject: Re: [R-meta] Inner|outer model vs multiple random id terms in
>>rma.mv
>>
>>Thank you Prof.Wolfgang. I was wondering how one would interpret negative
>>rho (does this imply there is negative correlation between the inner
>>levels?)
>>Also for a case where rho is negative is there a preference on whether the
>>`~inner| outer` or  `~1| inner , ~1|outer` is more applicable?
>>
>>On Wed, Apr 29, 2020 at 10:45 AM Viechtbauer, Wolfgang (SP)
>><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
>>Hi Divya,
>>
>>These two formulations will only yield the same results when rho is
>>estimated to be >= 0 (which is not the case in the second example).
>>
>>Best,
>>Wolfgang
>>
>>>-----Original Message-----
>>>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-
>>project.org]
>>>On Behalf Of Divya Ravichandar
>>>Sent: Wednesday, 29 April, 2020 19:00
>>>To: r-sig-meta-analysis using r-project.org
>>>Subject: Re: [R-meta] Inner|outer model vs multiple random id terms in
>>>rma.mv
>>>
>>>Hi all
>>>
>>>easier, I have reformatted the above example to avoid using an external
>csv
>>>file and such.
>>>
>>>Hi all,
>>>
>>>I am trying to understand why results from running a model of the form
>>>~lvl1|lv2 are not comparable to results of running ~1 | lvl1 ,~ 1 | lvl2
>>>
>>>In a simple example (case_simple in code below),results of the 2 models
>are
>>>comparable as expected.
>>>However, when running the 2 models on a more complex example
>(case_complex)
>>>markedly different results are obtained with ~ Dataset | Cohort estimating
>>>a pvalue of .02 and list(~ 1 | Dataset,~ 1 | Cohort) estimating a pvalue
>of
>>>.2
>>>
>>>Thank you
>>>
>>>*Reproducible example*
>>>library(metafor)
>>># example where results of the 2 models agree
>>>case_simple <- data.frame(Dataset=
>>>c("a","b","c","d"),Cohort=c("c1","c1","c2","c3"), Tech=
>>>c("a1","a2","a1","a1"),Effect_size=c(-1.5,-
>>>3,1.5,3),Standard_error=c(.2,.4,.2,.4))
>>>res1 = rma.mv(Effect_size, Standard_error^2, random = list(~ 1 | Dataset,~
>>>1 | Cohort), data=case_simple)
>>>res2=rma.mv(Effect_size, Standard_error^2, random = ~ Dataset | Cohort,
>>>data=case_simple)
>>>
>>># example where results of the 2 models DONT agree
>>>case_complex <-
>>>data.frame(Dataset=c("Dt1","Dt2","Dt3","Dt4","Dt5","Dt5","Dt6","Dt7","Dt8"
>,
>>"
>>>Dt9"),Cohort=c("C1","C2",rep("C3",5),rep("C4",2),"C5"),
>>>
>>>Effect_size=c(-0.002024454,-0.003915314,-0.042282757,-1.43826175,-
>>>0.045423574,-0.17682309,-21.72691245,-2.559727204,-0.091972279,-
>>>0.763332081),
>>>
>>>Standard_error=c(0.15283972,0.117452325,0.262002289,0.555230971,0.70891791
>2
>>,
>>>0.682989908,2.704749864,1.40514335,0.735696048,0.713557015))
>>>res1 = rma.mv(Effect_size, Standard_error^2, random = list(~ 1 | Dataset,~
>>>1 | Cohort), data=case_complex)
>>>res2=rma.mv(Effect_size, Standard_error^2, random = ~ Dataset | Cohort,
>>>data=case_complex)
>>>
>>>On Wed, Apr 22, 2020 at 9:51 AM Divya Ravichandar <divya using secondgenome.com>
>>>wrote:
>>>
>>>> Hi all,
>>>>
>>>> I am trying to understand why results from running a model of the form
>>>> ~lvl1|lv2 are not comparable to results of running ~1 | lvl1 ,~ 1 | lvl2
>>>>
>>>> In a simple example case below,results of the 2 models are comparable as
>>>> expected.
>>>>
>>>> ```case <- data.frame(Dataset=
>>>> c("a","b","c","d"),Cohort=c("c1","c1","c2","c3"), Tech=
>>>> c("a1","a2","a1","a1"),Effect_size=c(-1.5,-
>>>3,1.5,3),Standard_error=c(.2,.4,.2,.4))
>>>> res1 = rma.mv(Effect_size, Standard_error^2, random = list(~ 1 |
>>>> Dataset,~ 1 | Cohort), data=case)
>>>> res2=rma.mv(Effect_size, Standard_error^2, random = ~ Dataset | Cohort,
>>>> data=case)
>>>> ```
>>>> However, when running the 2 model on a more complex example [attached]
>>>> markedly different results are obtained with ~ Dataset | Cohort
>>>> estimating a pvalue of .02 and list(~ 1 | Dataset,~ 1 | Cohort)
>>>> estimating a pvalue of .2
>>>> --
>>>> *Divya Ravichandar*
>>>> Scientist
>>>> Second Genome
```