Annals of Applied Statistics

Bayesian aggregation of average data: An application in drug development

Sebastian Weber, Andrew Gelman, Daniel Lee, Michael Betancourt, Aki Vehtari, and Amy Racine-Poon

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Throughout the different phases of a drug development program, randomized trials are used to establish the tolerability, safety and efficacy of a candidate drug. At each stage one aims to optimize the design of future studies by extrapolation from the available evidence at the time. This includes collected trial data and relevant external data. However, relevant external data are typically available as averages only, for example, from trials on alternative treatments reported in the literature. Here we report on such an example from a drug development for wet age-related macular degeneration. This disease is the leading cause of severe vision loss in the elderly. While current treatment options are efficacious, they are also a substantial burden for the patient. Hence, new treatments are under development which need to be compared against existing treatments.

The general statistical problem this leads to is meta-analysis, which addresses the question of how we can combine data sets collected under different conditions. Bayesian methods have long been used to achieve partial pooling. Here we consider the challenge when the model of interest is complex (hierarchical and nonlinear) and one data set is given as raw data while the second data set is given as averages only. In such a situation, common meta-analytic methods can only be applied when the model is sufficiently simple for analytic approaches. When the model is too complex, for example, nonlinear, an analytic approach is not possible. We provide a Bayesian solution by using simulation to approximately reconstruct the likelihood of the external summary and allowing the parameters in the model to vary under the different conditions. We first evaluate our approach using fake data simulations and then report results for the drug development program that motivated this research.

Article information

Ann. Appl. Stat., Volume 12, Number 3 (2018), 1583-1604.

Received: July 2017
Revised: October 2017
First available in Project Euclid: 11 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Meta-analysis hierarchical modeling Bayesian computation pharmacometrics Stan


Weber, Sebastian; Gelman, Andrew; Lee, Daniel; Betancourt, Michael; Vehtari, Aki; Racine-Poon, Amy. Bayesian aggregation of average data: An application in drug development. Ann. Appl. Stat. 12 (2018), no. 3, 1583--1604. doi:10.1214/17-AOAS1122.

Export citation


  • Ambati, J. and Fowler, B. J. (2012). Mechanisms of age-related macular degeneration. Neuron 75 26–39.
  • Augood, C. A., Vingerling, J. R., de Jong, P. T., Chakravarthy, U., Seland, J., Soubrane, G., Tomazzoli, L., Topouzis, F., Bentham, G., Rahu, M., Vioque, J., Young, I. S. and Fletcher, A. E. (2006). Prevalence of age-related maculopathy in older Europeans. Arch. Ophthalmol. 124 529–535.
  • Betancourt, M. (2016). Diagnosing suboptimal cotangent disintegrations in Hamiltonian Monte Carlo. Preprint. Available at arXiv:1604.00695 [stat].
  • Brown, D. M., Kaiser, P. K., Michels, M., Soubrane, G., Heier, J. S., Kim, R. Y., Sy, J. P. and Schneider, S. (2006). Ranibizumab versus Verteporfin for Neovascular age-related macular degeneration. N. Engl. J. Med. 355 1432–1444.
  • Buschini, E., Piras, A., Nuzzi, R. and Vercelli, A. (2011). Age related macular degeneration and drusen: Neuroinflammation in the retina. Prog. Neurobiol. 95 14–25.
  • Caro, J. J. and Ishak, K. J. (2010). No head-to-head trial? Simulate the missing arms. PharmacoEcon. 28 957–967.
  • Dominici, F., Parmigiani, G., Wolpert, R. L. and Hasselblad, V. (1999). Meta-analysis of migraine headache teatments: Combining information from heterogeneous designs. J. Amer. Statist. Assoc. 94 16–28.
  • Gelman, A. (2004). Parameterization and Bayesian modeling. J. Amer. Statist. Assoc. 99 537–545.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press, Boca Raton, FL.
  • HARRIER. Efficacy and Safety of RTH258 Versus Aflibercept - Study 2 - Available at
  • Hart, W. M., ed. (1992). Adler’s Physiology of the Eye: Clinical Application, 9th ed. Mosby, St. Louis.
  • HAWK. Efficacy and Safety of RTH258 Versus Aflibercept - Available at
  • Heier, J. S., Brown, D. M., Chong, V., Korobelnik, J.-F., Kaiser, P. K., Nguyen, Q. D., Kirchhof, B., Ho, A., Ogura, Y., Yancopoulos, G. D., Stahl, N., Vitti, R., Berliner, A. J., Soo, Y., Anderesi, M., Groetzbach, G., Sommerauer, B., Sandbrink, R., Simader, C. and Schmidt-Erfurth, U. (2012). Intravitreal Aflibercept (VEGF trap-eye) in wet age-related macular degeneration. Ophthalmology 119 2537–2548.
  • Higgins, J. P. T. and Green, S. (2011). Cochrane Handbook for Systematic Reviews of Interventions, Version 5.1.0 ed. The Cochrane Collaboration.
  • Higgins, J. P. T. and Whitehead, A. (1996). Borrowing strength from external trials in a meta-analysis. Stat. Med. 15 2733–2749.
  • Hoffman, M. D. and Gelman, A. (2014). The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res. 15 1593–1623.
  • Ishak, K. J., Proskorovsky, I. and Benedict, A. (2015). Simulation and matching-based approaches for indirect comparison of treatments. PharmacoEcon. 33 537–549.
  • Jusko, W. J. and Ko, H. C. (1994). Physiologic indirect response models characterize diverse types of pharmacodynamic effects. Clin. Pharmacol. Ther. 56 406–419.
  • Khandhadia, S., Cipriani, V., Yates, J. R. W. and Lotery, A. J. (2012). Age-related macular degeneration and the complement system. Immunobiology 217 127–146.
  • Kinnunen, K., Petrovski, G., Moe, M. C., Berta, A. and Kaarniranta, K. (2012). Molecular mechanisms of retinal pigment epithelium damage and development of age-related macular degeneration. Acta Ophthalmol. 90 299–309.
  • Pocock, S. J. (1976). The combination of randomized and historical controls in clinical trials. J. Chronic Dis. 29 175–188.
  • Rosenfeld, P. J., Brown, D. M., Heier, J. S., Boyer, D. S., Kaiser, P. K., Chung, C. Y. and Kim, R. Y. (2006). Ranibizumab for neovascular age-related macular degeneration. N. Engl. J. Med. 355 1419–1431.
  • Schmidt-Erfurth, U., Eldem, B., Guymer, R., Korobelnik, J.-F., Schlingemann, R. O., Axer-Siegel, R., Wiedemann, P., Simader, C., Gekkieva, M. and Weichselberger, A. (2011). Efficacy and safety of monthly versus quarterly Ranibizumab treatment in neovascular age-related macular degeneration: The EXCITE study. Ophthalmology 118 831–839.
  • Sheiner, L. B. (1997). Learning versus confirming in clinical drug development. Clin. Pharmacol. Ther. 61 275–291.
  • Signorovitch, J. E., Wu, E. Q., Yu, A. P., Gerrits, C. M., Kantor, E., Bao, Y., Gupta, S. R. and Mulani, P. M. (2010). Comparative effectiveness without head-to-head trials. PharmacoEcon. 28 935–945.
  • Stan Development Team (2017). Stan: A C$++$ library for probability and sampling.
  • Weber, S., Carpenter, B., Lee, D., Bois, F. Y., Gelman, A. and Racine, A. (2014). Bayesian drug disease model with Stan: Using published longitudinal data summaries in population models, Population Approach Group Europe Meeting 2014, Alicante, Spain. Available at
  • Weber, S., Gelman, A., Lee, D., Betancourt, M., Vehtari, A. and Racine-Poon, A. (2018). Supplement to “Bayesian aggregation of average data: An application in drug development.” DOI:10.1214/17-AOAS1122SUPP.
  • Xu, L., Lu, T., Tuomi, L., Jumbe, N., Lu, J., Eppler, S., Kuebler, P., Damico-Beyer, L. A. and Joshi, A. (2013). Pharmacokinetics of Ranibizumab in patients with neovascular age-related macular degeneration: A population approach. Investig. Ophthalmol. Vis. Sci. 54 1616–1624.

Supplemental materials

  • Supplement: Program sources. Source code of R and Stan programs of simulation studies and drug-disease model.