# Re: Random Effect Model in Binomial Proportions Meta-analysis

I have used a random effects model for the meta-analysis of binomial proportions and have encountered several queries.

In one meta-analysis, I have four studies. Sample size of 98, 58, 107, 62

Weight: 26.1%, 23.6%, 26.4%, 23.9%

I understand that a random effect model doesn't take into consideration the sample size, and therefore despite the widely-varying sample size, the weight may not reflect this. However, from my (limited) understanding, random effects model for binomial proportions, the within-study variance is considered. And according to Ma et al., the paper states that " The within-study variance σ2i is considered known and usually estimated from a normal distribution. Specifically, for the outcome of proportion, the effect size is taken as the logitYi=log(Pi1−Pi) where Pi represents the proportion of event within the ith study. The within-study variance of Yi is then estimated by

σˆ2i=1xi+1ni−xi,

where xi and ni denote the number of events and sample size in the ith study, i = 1,…,K.

Although my forte is not in statistics or maths, it seems to be that the sample size is indeed relevant, and therefore the weights should not be so similar to each other despite having different sample sizes?

I would be grateful for any advice on this matter.

P.S. The link for the paper that I referenced: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5010877/

• Pay attention to formula (6), (9) and (13) in your linked paper. Could you get the idea what distributions that $p_i$ follow? – user158565 Jun 11 at 22:16

When you have estimated the within study variance ($$v_i$$) using whatever method is appropriate then in a fixed effects analysis the weights are just its reciprocal $$w_i = \frac{1}{v_i}$$
However in random effects you also estimate the between study variance $$\tau^2$$ and use as weights $$w_i = \frac{1}{v_i + \tau^2}$$. As $$\tau^2$$ becomes large relative to the individual $$v_i$$ it will make them more equal until in extreme cases they are effectively equal however variable the individual study variabilities were. This means that small studies may end up with as much weight as larger ones.