# [R-meta] Questions about the use of metaprop for the pooling of proportions

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Tue Mar 8 22:58:42 CET 2022

```Hi Gerta,

Under homogeneity, we have X_i ~ Binomial(n_i, pi), in which case sum(X_i) ~ Binomial(sum(n_i), pi) and hence

sum(out1)/sum(n)
plogis(coef(glm(out1/n ~ 1, weights = n, family = binomial)))

or using metaprop() / rma.glmm()

plogis(metaprop(out1, n)\$TE.fixed)
plogis(coef(rma.glmm(measure="PLO", xi=out1, ni=n, method="EE")))

are all identical. It goes to show how the logistic regression approach gives an 'exact' model, based on the exact distributional properties of binomial counts.

As for Thiago's data: I think this is fine. But essentially he has multinomial data. I recently described in a post how such data could be addressed if one would want to analyze them all simultaneously:

https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-February/003878.html

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of Dr. Gerta Rücker
>Sent: Tuesday, 08 March, 2022 20:30
>To: Thiago Roza
>Cc: r-sig-meta-analysis using r-project.org
>Subject: Re: [R-meta] Questions about the use of metaprop for the pooling of
>proportions
>
>Dear Thiago,
>
>I found that, apparently, the result presented by the common effect
>model (=fixed effect model) is simply the sum of all entries/events over
>all studies, divided by the total sample size (summed up over all
>studies). You see this by typing the following after the code in my last
>e-mail:
>
>all.equal(sum(out1)/sum(n), plogis(m1\$TE.fixed))
>all.equal(sum(out2)/sum(n), plogis(m2\$TE.fixed))
>all.equal(sum(out3)/sum(n), plogis(m3\$TE.fixed))
>
>This means that the method is equivalent to considering the data as a
>contingency table where the rows correspond to the studies and the
>columns to the outcomes. The meta-analytic result corresponds to the
>percentages in the column sums, and of course these add to 100%. In fact
>this is the easiest way to deal with this kind of data.
>
>@Guido, @Wolfgang: I couldn't find thisinformation on the metaprop or
>the rma.glmm help pages. Do you see any problem with interpreting
>Thiago's data as a contingency table? I think that, by contrast to
>pairwise comparison data, confounding/ecological bias is not an issue here.
>
>Best,
>
>Gerta
>
>Am 08.03.2022 um 19:30 schrieb Dr. Gerta Rücker:
>> Dear Thiago,
>>
>> So you have proportions of several mutually exclusive outcomes. Of
>> course, these are dependent because the sum is always the total
>> numbers of cases in the study (corresponding to 100% in that study).
>> Nevertheless, I don't see any reason why not pooling each outcome
>> separately using metaprop(). In fact, depending on the transformation,
>> the resulting average proportion will not generally sum up to 100%,
>> particularly not when using no transformation at all. This raises the
>> question which transformation to choose. The default in metaprop() is
>> random intercept logistic regression model with transformation logit.
>>
>> I made an observation that I have to think about, and you may try
>> this. If I use the default, the sum of the pooled percentages over all
>> outcomes is indeed always 1 for the fixed effect estimate. I used code
>> like this (here for 3 outcomes):
>>
>> #### Random data ####
>> out1 <- rbinom(10,100,0.1)
>> out2 <- rbinom(10,100,0.5)
>> out3 <- rbinom(10,100,0.9)
>> n <- out1 + out2 + out3
>> m1 <- metaprop(out1, n)
>> m2 <- metaprop(out2, n)
>> m3 <- metaprop(out3, n)
>> plogis(m1\$TE.fixed) + plogis(m2\$TE.fixed) + plogis(m3\$TE.fixed)
>>
>> (plogis is the inverse of the logit transformation, often called
>> "expit": plogis(x) = exp(x)/(1 + exp(x).) These seem to sum up to 1
>> for the fixed effect estimates, but not in general for the random
>> effects estimates, only in case of small heterogeneity (which is
>> rarely the case with proportions).
>>
>> I am interested to hear whether this works with your data. (And I have
>> to prove that this holds in general ...)
>>
>> Best,
>>
>> Gerta
```