Abstract
In the design of experiments, urn models have been widely used as randomization devices to allocate subjects to treatments and incorporate ethical constraints. We propose a new adaptive randomly reinforced urn design, in a clinical trial context. The design consists of a randomly reinforced urn wherein a sequential allocation of patients to treatments is performed and the associated responses are collected. The model is based on two stochastic sequences representing random and time-dependent thresholds for the urn proportion process. These thresholds are defined as functions of the estimators of unknown parameters modeling the response distributions, so that they change any time a new response is collected. First and second-order asymptotic results under different conditions have been investigated. Specifically, we present the limit, the rate of convergence and the asymptotic distribution of the proportion of subjects assigned to the treatments.
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References
Aletti, G., May, C., Secchi, P.: A central limit theorem, and related results, for two-color randomly reinforced urn. Ann. Appl. Probab. 41, 829–844 (2009)
Aletti, G., May, C., Secchi, P.: A functional equation whose unknown is P([0; 1]) valued. J. Theor. Probab. 25, 1207–1232 (2012)
Aletti, G., Ghiglietti, A., Paganoni, A.M.: A modified randomly reinforced urn design. J. Appl. Probab. 50, 486–498 (2013)
Aletti, G., Ghiglietti, A., Vidyashankar, A.N.: Dynamics of an Adaptive Randomly Reinforced Urn. Technical report (2015). arXiv:1508.02629
Atkinson, A.C., Biswas, A.: Randomised Response-Adaptive Designs in Clinical Trials. Chapman and Hall/CRC, Boca Raton (2013)
Baldi Antognini, A., Giannerini, S.: Generalized Polya urn designs with null balance. J. Appl. Probab. 44, 661–669 (2007)
Durham, S.D., Flournoy, N., Li, W.: A sequential design for maximazing the probability of a favourable response. Can. J. Stat. 26 (3), 479–495 (1998)
Ghiglietti, A., Paganoni, A.M.: Statistical properties of two-color randomly reinforced urn design targeting fixed allocations. Electron. J. Stat. 8, 708–737 (2014)
Ghiglietti, A., Paganoni, A.M.: An urn model to construct an efficient test procedure for response adaptive designs. Stat. Methods Appl. (2014). doi:10.1007/s10260-015-0314-y
Ghiglietti, A., Vidyashankar, A.N., Rosenberger, W.F.: Central Limit Theorem for an Adaptive Ranldomly Reinforced Urn Model. Technical report (2015). arXiv:1502.06130
Hall, P., Heyde, C.C.: Martingale Limit Theory and its Application. Academic Press, New York (1980)
Hu, F., Rosenberger, W.F.: The Theory of Response-Adaptive Randomization in Clinical Trials. Wiley, New York (2006)
Lachin, J.M., Rosenberger, W.F.: Randomization in Clinical Trials: Theory and Practice. Wiley, New York (2002)
May, C., Flournoy, N.: Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn. Ann. Stat. 37 (2), 1058–1078 (2009)
Melfi, F., Page, C.: Estimation after adaptive allocation. J. Plan. Stat. Inference 87, 353–363 (2000)
Muliere, P., Paganoni, A.M., Secchi, P.: A randomly reinforced urn. J. Stat. Plan. Inference 136, 1853–1874 (2006)
Zhang, L.-X., Hu, F., Cheung, S.H., Chan, W.S.: Immigrated urn models: theoretical properties and applications. Ann. Stat. 39, 643–671 (2011)
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Ghiglietti, A. (2016). Asymptotic Properties of an Adaptive Randomly Reinforced Urn Model. In: Kunert, J., Müller, C., Atkinson, A. (eds) mODa 11 - Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-31266-8_14
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DOI: https://doi.org/10.1007/978-3-319-31266-8_14
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