Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness.

Meta-analysis has been widely used to synthesize evidence from multiple studies for common hypotheses or parameters of interest. However, it has not yet been fully developed for incorporating heterogeneous studies, which arise often in applications due to different study designs, populations or outcomes. For heterogeneous studies, the parameter of interest may not be estimable for certain studies, and in such a case, these studies are typically excluded from conventional meta-analysis. The exclusion of part of the studies can lead to a non-negligible loss of information. This paper introduces a metaanalysis for heterogeneous studies by combining the confidence density functions derived from the summary statistics of individual studies, hence referred to as the CD approach. It includes all the studies in the analysis and makes use of all information, direct as well as indirect. Under a general likelihood inference framework, this new approach is shown to have several desirable properties, including: i) it is asymptotically as efficient as the maximum likelihood approach using individual participant data (IPD) from all studies; ii) unlike the IPD analysis, it suffices to use summary statistics to carry out the CD approach. Individual-level data are not required; and iii) it is robust against misspecification of the working covariance structure of the parameter estimates. Besides its own theoretical significance, the last property also substantially broadens the applicability of the CD approach. All the properties of the CD approach are further confirmed by data simulated from a randomized clinical trials setting as well as by real data on aircraft landing performance. Overall, one obtains an unifying approach for combining summary statistics, subsuming many of the existing meta-analysis methods as special cases.