I don't think a GEE is sensible. It doesn't cope that well with varying cluster sizes as far as I recall. The single linear regression is a terrible idea in my view as you are just inflating effective sample size and the large clusters will dominate the analysis.

I think this is where a multilevel model is least likely to be problematic as the estimates are partially pooled. So the estimates are shrunk towards the grand mean without being completely dominated by the large clusters. I don't see any indication in your original post that the estimates are unstable - so I'm not sure why you think this. If you are worried about that maybe estimate it as a Bayesian multilevel model with weakly informative priors to check.

Thank you so much for your answer. When I fit the LMM, a few variables that were significant in the simple linear regression became non-significant. But if I use GEE, with an independent correlation structure, the significance stays. Since the main focus is more on the fixed effects, would it be reasonable to use GEE (instead of LMM) to adjust only for clustering by country? As GEE provides robust standard errors, which help account for within-cluster correlation.

Thank you!

I'm not an expert on GEE but I've read a few sources where it suggests they are not great when there is this kind of imbalance. For instance I did a quick search and found this blog post:

https://wildlifesnpits.wordpress.com/2014/10/24/dealing-with-ugly-data-generalized-estimating-equations-gee/

which says "These models perform poorly when there are many observations from a handful of subjects."

Personally I'd use the linear mixed model as its also the more conservative approach - if you pick models based on which gives you more significant results then that's a poor strategy (it inflates Type I error and some would regard it as p-hacking).

There is also the question of what is estimated. GEE is a population average model that estimates that quantity. The LMM is trying to model the variation across the individual countries rather than treating it as a nuisance variable. So I'd probably prefer that for your case as it would be potentially interesting in most cases given that countries are not homogenous. I'd worry that the covariance structure isn't correctly estimated with this kind of inequality in cluster size.

Thank you so much for your answer. I think the statement above refers to the following: If you have only a few clusters but lots of data per cluster, GEE perform poorly.

Since I have 50 clusters, I will go by GEE with exchangeable correlation.

Yes, in LMM, there is concern about the covariance structure; it may not be correctly estimated due to the inequality in cluster sizes. To address this, we can use empirical (robust) standard errors to be safer.

Alternatively, GEE inherently provides robust standard errors.

or even can do a sensitivity analysis to include both ways.

I think the issue with GEEs seems to be imbalance in the cluster sizes not just few clusters.

Yes, that’s correct, when there are substantial differences in cluster sizes, GEE might give more weight to larger clusters, potentially affecting the accuracy of estimates. However, since I have a good number of clusters, 50 clusters, GEE should still be robust.

if I had only a few clusters, and one was massive (unequal sample size), that large cluster could dominate the results and make the estimates less stable.

The reason I picked GEE is that it can address small individual cluster sizes. (in my case, half of the clusters have a few respondents). This is a strength of GEE, especially when compared to LMM, that might require larger numbers within each cluster.

I was thinking the advantages of GEE ; its ability to handle those small clusters(22 clusters with only one respondent) and provide robust standard errors, outweigh its limitations (unequal sample size).