# [R-meta] Meta-analysis of prevalence data: back-transformation and polytomous data

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Mon Feb 28 18:18:05 CET 2022

```Definitely would avoid the double arcsine transformation. It's just fraught with problems in this context. The regular arcsine transformation comes close anyway and doesn't have the same issues.

As for multinomial data:

You could go with a multilevel multinominal model. See this post on SO for some relevant packages: https://stackoverflow.com/questions/21082396/multinomial-logistic-multilevel-models-in-r

If the various events are not all rare, an alternative approach would be to use a 'normal' multivariate model with (logit-transformed) proportions corresponding to the various events. Say a study reports the number of cases for 3 different events. Let p1, p2, and p3 be the proportions for the three events and n the total number of cases. Under multinomial sampling, the (estimated) var-cov matrix of the proportions is then:

[p1*(1-p1)                    ]
1/n * [-p1*p2    p2*(1-p2)          ]
[-p1*p3    -p2*p3    p3*(1-p3)]

So this is the 3x3 part of the V matrix for this study. Actually, I would suggest to logit-transform the proportions, in which case the var-cov matrix is:

[1/(p1*(1-p1))                                      ]
1/n * [-1/((1-p1)*(1-p2)) 1/(p2*(1-p2))                   ]
[-1/((1-p1)*(1-p3)) -1/((1-p2)*(1-p3)) 1/(p3*(1-p3))]

Same for other studies. Let y be the vector with all the logit-transformed proportions and V the block-diagonal V matrix with all the var-cov matrices. Then we are back to rma.mv(y, V, ...), adding whatever moderators and random effects deemed necessary.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of CHAPPELL Francesca
>Sent: Thursday, 24 February, 2022 12:37
>To: Röver, Christian; jakub.ruszkowski using gumed.edu.pl; r-sig-meta-analysis using r-
>project.org
>Subject: Re: [R-meta] Meta-analysis of prevalence data: back-transformation and
>polytomous data
>
>This paper argues for modelling the within-study variance directly as binomial
>(https://pubmed.ncbi.nlm.nih.gov/18083461/)  and provides a SAS program to do so.
>I think R can do it too with glmer, though I don't have a handy program. But see
>https://journals.lww.com/epidem/Fulltext/2020/09000/Meta_analysis_of_Proportions_
>Using_Generalized.16.aspx , specifically the appendices for R code. I haven't
>come across a multinomial version that doesn't use WinBUGS.
>
>Francesca
>
>-----Original Message-----
>From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org> On Behalf
>Of Röver, Christian
>Sent: 24 February 2022 08:43
>To: jakub.ruszkowski using gumed.edu.pl; r-sig-meta-analysis using r-project.org
>Subject: Re: [R-meta] Meta-analysis of prevalence data: back-transformation and
>polytomous data
>
>This email was sent to you by someone outside the University.
>You should only click on links or attachments if you are certain that the email
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>
>Dear Jakub,
>
>I think Schwarzer et al. (2019; https://doi.org/10.1002/jrsm.1348) have a valid
>point, and that the double arcsine transform is not really suitable for meta-
>analysis purposes. The approach by Barendregt et al.
>(2013; https://doi.org/10.1136/jech-2013-203104) seems to me more like a kind of
>workaround, and I am not sure whether it will actually work generally, or would
>only "fix" the issue (or at least won't fail
>immediately) in some cases.
>
>I guess a quick and simple solution might be to go for the ("simple") arcsine
>transformation instead, or otherwise check out one of the more appropriate
>alternative approaches that were pointed out by Schwarzer et al. (2019).
>
>Cheers,
>
>Christian
>
>On Wed, 2022-02-23 at 12:06 +0100, Jakub Ruszkowski wrote:
>> Dear Community,
>>
>> I am trying to do a meta-analysis of prevalence according to the
>> recommendations arising from the current literature. I have two
>> problems that I cannot handle on my own.
>>
>> 1. I found that there are controversies about a back-transformation
>> method for the Freeman-Tukey double arcsine transformation (Schwarzer
>> et al., doi:
>> 10.1002/jrsm.1348). However, there is a probable resolution that
>> incorporates inverse variance instead of harmonic mean (Barendregt-Doi
>> implementation, clearly explained in Supplementary Materials in doi:
>> 10.1111/jebm.12445;
>> older version introducing it: 10.1136/jech-2013-203104).
>> Unfortunately, I am
>> not proficient in programming, so I am not sure how to implement this
>> solution on my own. Is there an R implementation of Barendregt-Doi
>> back-transformation available or is it possible to add this method to
>> the metafor?
>>
>> 2. Are there any available examples of R code to meta-analyze
>> ordinal/multinomial prevalence data (e.g., mild, moderate, severe
>> severity)?
>> I found one method implemented in MetaXL that used double arcsine
>> transformation (mentioned earlier doi: 10.1136/jech-2013-203104), and
>> one Bayesian method using the Dirichlet-multinomial model (doi:
>> 10.1080/03610918.2021.1887229). Unfortunately, the R code is not
>> supplemented with the latter article.
>>
>> Kind regards
```