Meta-analysis (MA) combines multiple studies to estimate a quantity of interest. Some existing MA models have shortcomings in the form of 1) inappropriate inference targets, 2) strong assumptions about how studies are sampled, and 3) prior distributions for variance parameters with inadequate shrinkage.

In Chapter 2 we build a three-random effect (3RE) Bayesian random effects MA model for observational contingency table data as an extension of the standard two-random effect (2RE) model. We add a random effect for the log-odds of having a risk factor with random effects for the log-odds of an event and for the log-odds ratio of the event for those with or without the risk factor. The 3RE model allows for calculation of more statistics than the 2RE model, and we define a novel estimand for statistics calculated from contingency tables -- the expected value of a statistic for a new study given the hyperparameters. The new estimand shows less bias and higher 95% credible interval coverage as compared with a naive plug-in estimator. We apply the model to a dataset of studies on patients presenting to the emergency department with syncope. In Chapter 3 we propose a new approach to combining multiple selection models for publication bias using Bayesian stacking of posterior distributions. We demonstrate the effectiveness of stacking selection models through simulations and real datasets that exhibit symptoms of publication bias.

Chapter 4 proposes a new class of prior distributions for the covariance matrix of random effects in MA. The new priors allow random effects variances to be shrunk towards zero and for shrinkage of correlations between random effects. We show through both synthetic and real data examples that the new prior distributions lead to less diffuse posterior distributions and shorter 95% credible intervals in a 3RE MA model for observational data and an arm-based network meta-analysis (AB-NMA) model for randomized controlled trials.