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

[Dr. Gerta Rücker]

Hi Wolfgang,

Thank you! Indeed I just saw that the ML estimate under the binomial 
model and the assumption of homogeneity gives (sum r_i)/(sum n_i). In 
fact this seems equivalent to logistic regression. Probably it works 
also under the multinomial model, I didn't write this down. I admit that 
I never had thought about this :(

Best,

Gerta

Am 08.03.2022 um 22:58 schrieb Viechtbauer, Wolfgang (SP):
> 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

-- 

Dr. rer. nat. Gerta Rücker, Dipl.-Math.

Guest Scientist
Institute of Medical Biometry and Statistics,
Faculty of Medicine and Medical Center - University of Freiburg

Zinkmattenstr. 6a, D-79108 Freiburg, Germany

Mail:     ruecker using imbi.uni-freiburg.de
Homepage: https://www.uniklinik-freiburg.de/imbi-en/employees.html?imbiuser=ruecker