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

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 at r-project.org] On
Behalf Of Dr. Gerta R?cker
Sent: Tuesday, 08 March, 2022 20:30
To: Thiago Roza
Cc: r-sig-meta-analysis at 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 R?cker:

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