Chris Guure writes:
I am trying to construct an informative prior by synthesizing or collecting some information from literature (meta-analysis) and then to apply that to a real data set (it is longitudinal data) for over 20 years follow-up.
In constructing the prior using the meta-analysis data, the issue of publication bias came up. I have tried looking to see if there is any literature on this but it seems almost all the articles on Bayesian meta-analysis do not actually account for this issue apart from one (Givens, Smith and Tweedie 1997).
My thinking was that I could assume a data augmentation approach by fitting a joint model with the assumption that the observed data are normally distributed and the unobserved studies probably exist but not included in my studies and can be thought of to be missing data (missing not at random or non-ignorable missingness). This way a Bernoulli distribution could be used to account for the missingness.
But according to Lesaffre and Lawson 2012, pp. 196; in hierarchical models, the data augmentation approach enters in a quite natural way via the latent (unobserved) random effects. This statement to me implies that my earlier idea may not be necessary and may even bias the posterior estimates.
My reply: You could certainly do this, build a model in which there are a bunch of latent unreported studies and then go from there. I don’t know how well this would work, though, for two reasons:
1. Estimating what’s missing based on the shape of the distribution—-that’s tough. Inferences will be so sensitive to all sorts of measurement and selection issues, and I’d be skeptical of whatever comes out.
2. You’re trying to adjust for unreported studies in a meta-analysis. But I’d be much more worried about choices in data processing and analysis in each of the studies you have. As I’ve written many times, I think the file-drawer problem is overrated and it’s nothing compared to the garden of forking paths.