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The rescaled Pólya urn: local reinforcement and chi-squared goodness-of-fit test. ADV APPL PROBAB 2022. [DOI: 10.1017/apr.2021.56] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
AbstractMotivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.
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Aletti G, Crimaldi I, Ghiglietti A. Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means. BERNOULLI 2020. [DOI: 10.3150/19-bej1143] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Aletti G, Crimaldi I, Ghiglietti A. Networks of reinforced stochastic processes: Asymptotics for the empirical means. BERNOULLI 2019. [DOI: 10.3150/18-bej1092] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Aletti G, Ghiglietti A, Vidyashankar AN. Dynamics of an adaptive randomly reinforced urn. BERNOULLI 2018. [DOI: 10.3150/17-bej926] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ghiglietti A, Scarale MG, Miceli R, Ieva F, Mariani L, Gavazzi C, Paganoni AM, Edefonti V. Urn models for response-adaptive randomized designs: a simulation study based on a non-adaptive randomized trial. J Biopharm Stat 2018; 28:1203-1215. [PMID: 29565749 DOI: 10.1080/10543406.2018.1452024] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
Abstract
Recently, response-adaptive designs have been proposed in randomized clinical trials to achieve ethical and/or cost advantages by using sequential accrual information collected during the trial to dynamically update the probabilities of treatment assignments. In this context, urn models-where the probability to assign patients to treatments is interpreted as the proportion of balls of different colors available in a virtual urn-have been used as response-adaptive randomization rules. We propose the use of Randomly Reinforced Urn (RRU) models in a simulation study based on a published randomized clinical trial on the efficacy of home enteral nutrition in cancer patients after major gastrointestinal surgery. We compare results with the RRU design with those previously published with the non-adaptive approach. We also provide a code written with the R software to implement the RRU design in practice. In detail, we simulate 10,000 trials based on the RRU model in three set-ups of different total sample sizes. We report information on the number of patients allocated to the inferior treatment and on the empirical power of the t-test for the treatment coefficient in the ANOVA model. We carry out a sensitivity analysis to assess the effect of different urn compositions. For each sample size, in approximately 75% of the simulation runs, the number of patients allocated to the inferior treatment by the RRU design is lower, as compared to the non-adaptive design. The empirical power of the t-test for the treatment effect is similar in the two designs.
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Affiliation(s)
- Andrea Ghiglietti
- a Dipartimento di Matematica "F. Enriques" , Università degli Studi di Milano , Milano , Italy
| | - Maria Giovanna Scarale
- b Laboratorio di Statistica Medica, Biometria, ed Epidemiologia "G. A. Maccacaro", Dipartimento di Scienze Cliniche e di Comunità , Università degli Studi di Milano , Milano , Italy.,c Unit of Biostatistics, Poliambulatorio "Giovanni Paolo II" , IRCCS Casa Sollievo della Sofferenza , San Giovanni Rotondo , Italy
| | - Rosalba Miceli
- d Struttura Semplice di Epidemiologia Clinica e Organizzazione Trials , Fondazione IRCCS Istituto Nazionale Tumori , Milano , Italy
| | - Francesca Ieva
- e MOX - Modellistica e Calcolo Scientifico, Dipartimento di Matematica , Politecnico di Milano , Milano , Italy
| | - Luigi Mariani
- d Struttura Semplice di Epidemiologia Clinica e Organizzazione Trials , Fondazione IRCCS Istituto Nazionale Tumori , Milano , Italy
| | - Cecilia Gavazzi
- f Struttura Semplice Dipartimentale di Terapia Nutrizionale , Fondazione IRCCS Istituto Nazionale dei Tumori , Milano , Italy
| | - Anna Maria Paganoni
- e MOX - Modellistica e Calcolo Scientifico, Dipartimento di Matematica , Politecnico di Milano , Milano , Italy
| | - Valeria Edefonti
- b Laboratorio di Statistica Medica, Biometria, ed Epidemiologia "G. A. Maccacaro", Dipartimento di Scienze Cliniche e di Comunità , Università degli Studi di Milano , Milano , Italy
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Aletti G, Crimaldi I, Ghiglietti A. Synchronization of reinforced stochastic processes with a network-based interaction. ANN APPL PROBAB 2017. [DOI: 10.1214/17-aap1296] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ghiglietti A, Vidyashankar AN, Rosenberger WF. Central limit theorem for an adaptive randomly reinforced urn model. ANN APPL PROBAB 2017. [DOI: 10.1214/16-aap1274] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ghiglietti A, Paganoni AM. An urn model to construct an efficient test procedure for response adaptive designs. STAT METHOD APPL-GER 2015. [DOI: 10.1007/s10260-015-0314-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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