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The question is in the title.

Meta-analysis with binomial distribution

I'm trying to do a meta-analysis on occurence of an event across different studies. For each study $i$ ($i=1,\dots,N$), I have the number of participants $n_i$ and the number of events $k_i$. I don't have individual data. The number of event thus follows a binomial distribution: $k_i\sim Bin(\theta_i,n_i)$, where I assume a random effect: $\theta_i\sim N(\theta,\tau)$.

Regression meta-analysis

Additionally, I'm doing a regression on $\theta$ using the variable $x_i$: $$\theta_i\sim N(\alpha+\beta x_i,\tau).$$

Multi-level regression meta-analysis

Some studies are split into groups. As they have the same baseline characteristics, I thought of using a multi-level model (first formulation): $$\theta_{i,j}\sim N(\alpha+\beta x_{i,j},\tau+\delta_j).$$ We could alternatively write it like this (although it is a bit different, second formulation): $$\theta_{i}\sim N(\alpha+\beta x_i,\tau).$$ $$\theta_{i,j}\sim N(\theta_i+\gamma_i x_{ij},\delta_j)$$

Which function to use in R

I thought of doing it with the rma.mv() function from the metareg package, but the issue is that you cannot specify the distribution of the data. As far as I understood, it assumes normally distributed data. I thought of applying a logit transformation to the probability of success $p_i:=k_i/n_i$, but it's not possible to the some zero counts ($y_i=0$ for some studies i). Any other function that could solve that? Moreover, I'm not sure if such models follow my first or second formulation.

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closed as off-topic by Nick Cox, user158565, Michael Chernick, mkt, jpmuc Jul 17 at 7:36

This question appears to be off-topic. The users who voted to close gave this specific reason:

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If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ To be honest, I don't understand why it is off topic. I modified the question now, could anyone tell me if it fits better to CrossValidated or if not, where I should post it? $\endgroup$ – Anthony Hauser Jul 25 at 12:03
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It's not clear to me why general purpose GLMM (e.g. lme4 R package) or Bayesian models (e.g. brms, rstanarm, or rstan R packages) would not work. Surely, you can specify the model of interest as a GLMM with a binomial outcome and a trial main effect with whatever random effects you require?

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