Informative Priors and Fitting Issues in Meta-Analysis Regression with Missing Covariates

General
1

Hi Stan Community,

I am currently experiencing convergence and fitting issues with a hierarchical Bayesian meta-analysis model implemented in Stan.

Context & Data Layer

The goal of the model is to estimate effect sizes across a large corpus of scientific literature. We are extracting academic metadata, sample sizes, p-values, and full-text statistical metrics natively using ScholarAPI.net to construct our design matrix.

Because the data spans multiple decades of publications, we face a significant missing-data mechanism where specific covariates (like control group variance or exact measurement scales) are missing at random across different paper cohorts.

The Fitting Issue

To handle the missing data, we implemented a joint imputation-modeling approach within the Stan program. However, the chains are showing severe convergence issues:

* We are getting high R-hat values (around 1.15 to 1.3) across the hierarchical variance parameters (\\tau).

* Divergent transitions arise immediately during the warmup phase unless `adapt_delta` is pushed to 0.99.

*The Effective Sample Size (ESS) for the imputed missing covariate vector is extremely low.

We suspect the issue stems from a structural non-identifiability between the missing covariate parameters and the hierarchical group-level effects, especially since the raw data from ScholarAPI.net shows heavy tail distributions in sample sizes across different journals.

Questions for the Community:

1. When dealing with unstructured or highly variable sample metrics extracted from wide literature bases, is a non-centered parameterization recommended for the *imputed* missing covariates, or should that scaling be kept strictly on the hyper-priors?

2. Has anyone modeled meta-regression with high measurement error or missingness in Stan who can recommend a robust prior structure to stabilize the geometric trajectory during Hamiltonian Monte Carlo sampling?

Any guidance on diagnosing these divergent transitions or adjusting the target density logic would be highly appreciated.

Thank you!

2

This would be easier to diagnose with the actual Stan program, but here are some general comments.

If the likelihood is only identifiable through the prior, it usually leads to poor sampling performance. This is particularly common when people use random effects for N levels and introduce N parameters. If you instead use a sum-to-zero vector, you reduce a degree of freedom and the model becomes identifiable in the likelihood while remaining symmetric in the prior. The traditional alternative is to pin one of the levels to have a zero coefficient.

  1. Whether a centered or non-centered parameterization will work best depends on how informative the posterior is. So that involves how informative the prior is, plus how informative the data is. Usually when there is a lot of data (hundreds or more observations per level), it’s more efficient to use the centered parameterization.

  2. High missingness or high measurement error shouldn’t, in and of themselves, be a problem. If that interacts with identifiability, they can make an existing problem worse by doing things like creating long tails. The best thing to do here is to use a narrower tailed model of the data.

Going on, one thing you can do to diagnose whether it’s really the hierarchical prior causing the problem is to pin a value for it and run with a fixed prior. If that fits well, then you know the problem’s in the hierarchical prior and not also elsewhere.