Modeling Heterogeneity in Bayesian Meta-Analysis

Meta-analysis has been frequently used to combine findings from independent studies in many areas. Bayesian methods are an important set of tools for performing meta-analyses. They avoid some potentially unrealistic assumptions that are required by conventional frequentist methods. More importantly, meta-analysts can incorporate prior information from many sources, including experts' opinions and prior meta-analyses. Nevertheless, Bayesian methods are used less frequently than conventional frequentist methods, primarily because of the need for nontrivial statistical coding, while frequentist approaches can be implemented via many user-friendly software packages. This thesis divide into three parts, the first part aims at providing a practical review of implementations for Bayesian meta-analyses with various prior distributions. We present Bayesian methods for meta-analyses with the focus on a univariate meta-analysis of odds ratio for binary outcomes. We summarize various commonly used prior distribution choices for the between-studies heterogeneity variance, a critical parameter in meta-analyses. They include the inverse-gamma, uniform, and half-normal distributions, as well as evidence-based informative log-normal priors. Five real-world examples are presented to illustrate their performance. Under certain circumstances, Bayesian methods can produce markedly different results from those by frequentist methods, including a change in decision on statistical significance. When data information is limited, the choice of priors may have a large impact on meta-analytic results, in which case sensitivity analyses are recommended. Moreover, the algorithm for implementing Bayesian analyses may not converge for extremely sparse data; caution is needed in interpreting respective results. As such, convergence should be routinely examined. When select statistical assumptions made by conventional frequentist methods are violated, Bayesian methods provide a reliable alternative to perform a meta-analysis. A critical issue of meta-analysis is that the combined studies could be heterogeneous because of differences in study populations, design conducts, etc. Thus, heterogeneity is an issue that can affect the validity of meta-analytic results. In the existence of heterogeneity, a random-effects meta-analysis is considered an appropriate model to estimate a summary effect and a between-study variance. For several studies, conventional methods could be used to estimate the random-effects meta-analysis model, but the estimate of between-study heterogeneity could be imprecise in the case of few studies, and this imprecision is not acknowledged. In this situation, a Bayesian random-effects meta-analysis is advantageous in accounting for all sources of uncertainty. The second part of this thesis uses Bayesian methods to examine the impact of meta-analysis characteristics on between-study heterogeneity by analyzing data from a large collection of meta-analyses with binary outcomes. We classify and examine all included meta-analyses by outcome types and intervention comparison types. Then, based on these outcome and intervention classifications, we derive a set of predictive distributions for the extent of between-study variance expected in a future meta-analysis in different effect measures. These findings may help researchers to use them as informative prior distributions in new meta-analyses of binary outcomes. The last part of this thesis quantifies the between-study heterogeneity by using prediction intervals. By reporting the prediction interval, we show how often there is a contradiction in the conclusion compared to the conclusion based on the confidence interval. We also show the benefits of reporting the prediction interval in a meta-analysis, which has a straightforward clinical meaning that may guide researchers to expect the true effects in future settings. In addition, we use a large collection of meta-analyses from the Cochrane Database of Systematic Reviews to evaluate the real-world performance of the prediction interval. We assess whether the previous meta-analysis successfully predicts a future study or a study not included in the meta-analysis based on publication years. This assessment is illustrated using a case study. In the existence of heterogeneity, a random-effects meta-analysis is considered an appropriate model to estimate a summary effect and a between-study variance. For many studies, conventional methods could be used to estimate the random-effects meta-analysis model, but the estimate of between-study heterogeneity could be imprecise in the case of few studies as well as this imprecision is not acknowledged. In this situation, a Bayesian random-effects meta-analysis is advantageous in accounting for all sources of uncertainty. The second part of this thesis uses Bayesian methods to examine the impact of meta-analysis characteristics on between-study heterogeneity by analyzing data from a large collection of meta-analyses with binary outcomes. We classify and examine all included meta-analyses by outcome types and intervention comparison types. We derive a set of predictive distributions for the extent of between-study variance expected in a future meta-analysis in various effect measures. These findings may help researchers to use them as informative prior distributions in new meta-analyses of binary outcomes. The last part of this thesis quantifies the between-study heterogeneity by using prediction intervals. By reporting the prediction interval, we show how often there is a contradiction in the conclusion compared to the conclusion based on the confidence interval. We also show the benefits of reporting the prediction interval in a meta-analysis, which has a straightforward clinical meaning that may guide researchers to expect the true effects in future settings. In addition, we assess whether the previous meta-analysis successfully predicts a future study or a study not included in the meta-analysis based on publication years. This assessment is illustrated using a case study.