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


  • Dungang Liu
  • Regina Y. Liu
  • Minge Xie


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 nonnegligible loss of information. This article introduces a meta-analysis 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 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 a unifying approach for combining summary statistics, subsuming many of the existing meta-analysis methods as special cases.

Suggested Citation

  • Dungang Liu & Regina Y. Liu & Minge Xie, 2015. "Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 326-340, March.
  • Handle: RePEc:taf:jnlasa:v:110:y:2015:i:509:p:326-340
    DOI: 10.1080/01621459.2014.899235

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    References listed on IDEAS

    1. Ryan Martin & Chuanhai Liu, 2013. "Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 301-313, March.
    2. D. Y. Lin & D. Zeng, 2010. "On the relative efficiency of using summary statistics versus individual-level data in meta-analysis," Biometrika, Biometrika Trust, vol. 97(2), pages 321-332.
    3. Min-ge Xie & Kesar Singh, 2013. "Confidence Distribution, the Frequentist Distribution Estimator of a Parameter: A Review," International Statistical Review, International Statistical Institute, vol. 81(1), pages 3-39, April.
    4. Thomas Mathew & Kenneth Nordstrom, 1999. "On the Equivalence of Meta-Analysis Using Literature and Using Individual Patient Data," Biometrics, The International Biometric Society, vol. 55(4), pages 1221-1223, December.
    5. Xie, Minge & Singh, Kesar & Strawderman, William E., 2011. "Confidence Distributions and a Unifying Framework for Meta-Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 320-333.
    6. A. E. Ades & A. J. Sutton, 2006. "Multiparameter evidence synthesis in epidemiology and medical decision-making: current approaches," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 169(1), pages 5-35.
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    1. repec:bla:biomet:v:72:y:2016:i:4:p:1378-1386 is not listed on IDEAS
    2. repec:eee:stapro:v:134:y:2018:i:c:p:98-105 is not listed on IDEAS

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