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Optimal design to discriminate between rival copula models for a bivariate binary response

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Abstract

We consider a bivariate logistic model for a binary response, and we assume that two rival dependence structures are possible. Copula functions are very useful tools to model different kinds of dependence with arbitrary marginal distributions. We consider Clayton and Gumbel copulae as competing association models. The focus is on applications in testing a new drug looking at both efficacy and toxicity outcomes. In this context, one of the main goals is to find the dose which maximizes the probability of efficacy without toxicity, herein called P-optimal dose. If the P-optimal dose changes under the two rival copulae, then it is relevant to identify the proper association model. To this aim, we propose a criterion (called PKL) which enables us to find the optimal doses to discriminate between the rival copulae, subject to a constraint that protects patients against dangerous doses. Furthermore, by applying the likelihood ratio test for non-nested models, via a simulation study we confirm that the PKL-optimal design is really able to discriminate between the rival copulae.

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Notes

  1. In non-nested hypotheses, neither model can be obtained from the other by imposing a parametric restriction.

  2. Observe that \({\widehat{\theta }}_{Cl}\) and \(\widehat{\theta }_{G}\) are referred to as QML (quasi maximum likelihood) estimators when they are obtained under the not true hypotheses \(H_{CL}\) and \(H_G\).

References

  • Atkinson AC, Fedorov VV (1975a) The design of experiments for discriminating between two rival models. Biometrika 62(1):57–70

    Article  MathSciNet  MATH  Google Scholar 

  • Atkinson AC, Fedorov VV (1975b) Optimal design: experiments for discriminating between several models. Biometrika 62(2):289–303

    MathSciNet  MATH  Google Scholar 

  • Cook R, Wong W (1994) On the equivalence of constrained and compound optimal designs. J Am Stat Assoc 89(426):687–692

    Article  MathSciNet  MATH  Google Scholar 

  • Cox D (1961) Tests of separate families of hypotheses. In: Proceedings of the Fourth Berkeley symposium on mathematical statistic and probability. University of California Press, Berkeley

  • Cox D (1962) Further results on tests of separate families of hypotheses. J R Stat Soc B 24:406–424

    MathSciNet  MATH  Google Scholar 

  • Deldossi L, Osmetti SA, Tommasi C (2016) PKL-Optimality Criterion in Copula Models for efficacy-toxicity response. In: Kunert J, Muller CH, Atkinson AC (eds) mODa 11—dvances in model-oriented design and analysis: proceedings of the 11th international workshop in model-oriented design and analysis. Springer, Heidelberg

  • Denman N, McGree J, Eccleston J, Duffull S (2011) Design of experiments for bivariate binary responses modelled by Copula functions. Comput Stat Data Anal 55(4):1509–1520

    Article  MathSciNet  MATH  Google Scholar 

  • Dette H, Titoff S (2009) Optimal discrimination designs. Ann Stat 37(4):2056–2082

    Article  MathSciNet  MATH  Google Scholar 

  • Dragalin V, Fedorov V (2006) Adaptive designs for dose-finding based on efficacy-toxicity response. J Stat Plan Inference 136(6):1800–1823

    Article  MathSciNet  MATH  Google Scholar 

  • Dragalin V, Fedorov V, Wu Y (2008) A two-stage design for dose-finding that accounts for both efficacy and safety. Stat Med 27(25):5156–5176

    Article  MathSciNet  Google Scholar 

  • Drovandi CC, McGree JM, Pettitt AN (2014) A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design. J Comput Gr Stat 23(1):3–24

    Article  MathSciNet  Google Scholar 

  • Fedorov V, Hackl P (1997) Model-oriented design of experiments. Springer, New York

    Book  MATH  Google Scholar 

  • Fedorov V, Leonov S (2014) Optimal design for nonlinear response models. Chapman and Hall, Boca Raton

    MATH  Google Scholar 

  • Gao L, Rosenberger W (2013) Adaptive Bayesian design with penalty based on toxicity-efficacy response. In: Ucínski D, Atkinson AC, Patan M (eds) mODa 10—advances in model-oriented design and analysis. Springer, Heidelberg

    Google Scholar 

  • Kim S, Flournoy N (2015) Optimal experimental design for systems with bivariate failures under a bivariate Weibull function. J Roy Stat Soc: Ser C (Appl Stat) 64(3):413–432

    Article  MathSciNet  Google Scholar 

  • Ponce de Leon AC, Atkinson AC (1991) Optimum experimental design for discriminating between two rival models in the presence of prior information. Biometrika 78(3):601–608

    Article  MathSciNet  MATH  Google Scholar 

  • López-Fidalgo J, Tommasi C, Trandafir P (2007) An optimal experimental design criterion for discriminating between non-normal models. J R Stat Soc: Ser B (Stat Methodol) 69(2):231–242

    Article  MathSciNet  MATH  Google Scholar 

  • López-Fidalgo J, Tommasi C, Trandafir PC (2008) Optimal designs for discriminating between some extensions of the Michaelis–Menten model. J Stat Plan Inference 138(12):3797–3804

    Article  MathSciNet  MATH  Google Scholar 

  • McGree J, Eccleston J (2008) Probability-based optimal design. Aust N Z J Stat 50(1):13–28

    Article  MathSciNet  Google Scholar 

  • Monfardini C (2003) An illustration of Cox’s non-nested testing procedure for logit and probit models. Comput Stat Data Anal 42(3):425–444

    Article  MathSciNet  MATH  Google Scholar 

  • Nelsen R (2006) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  • Perrone E, Müller W (2016) Optimal designs for copula models. Statistics 50(4):917–929

    Article  MathSciNet  MATH  Google Scholar 

  • Perrone E, Rappold A, Müller WG (2017) Ds-optimality in copula models. Stat Methods Appl 26(3):403–418

    Article  MathSciNet  MATH  Google Scholar 

  • Pesaran H, Weeks M (2001) Non-nested hypothesis testing: an overview. In: Baltagi BH (ed) A companion to theoretical econometrics. Blackwell Publishing Ltd, Malden

    Google Scholar 

  • Pukelsheim F (2006) Optimal design of experiments. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Tao Y, Liu J, Li Z, Lin J, Lu T, Yan F (2013) Dose-finding based on bivariate efficacy-toxicity outcome using archimedean copula. PLoS ONE 8(11):1–6

    Google Scholar 

  • Thall P (2012) Bayesian adaptive dose-finding based on efficacy and toxicity. J Stat Res 46:187–202

    MathSciNet  Google Scholar 

  • Thall P, Cook J (2004) Dose-finding based on efficacy-toxicity trade-offs. Biometrics 60(3):684–693

    Article  MathSciNet  MATH  Google Scholar 

  • Tommasi C (2007) Optimal designs for discriminating among several non-normal models. In: López-Fidalgo J, Rodrguez-Daz JM, Torsney B (eds) mODa 8—advances in model-oriented design and analysis: Proceedings of the 8th international workshop in model-oriented design and analysis. Springer, Heidelberg

  • Tommasi C (2009) Optimal designs for both model discrimination and parameter estimation. J Stat Plan Inference 139(12):4123–4132

    Article  MathSciNet  MATH  Google Scholar 

  • Uciński D, Bogacka B (2005) T-optimum designs for discrimination between two multiresponse dynamic models. J R Stat Soc Ser B (Stat Methodol) 67(1):3–18

    Article  MathSciNet  MATH  Google Scholar 

  • Verbeek M (2008) A Guide to modern econometrics. Wiley, Chichester

    MATH  Google Scholar 

  • Yuan Y, Guosheng Y (2009) Bayesian dose finding by jointly modelling toxicity and efficacy as time-to-event outcomes. J Roy Stat Soc: Ser C (Appl Stat) 58(5):719–736

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Dipartimento di Scienze Statistiche, Università Cattolica del Sacro Cuore, Largo Gemelli, 1, 20123, Milan, Italy

    Laura Deldossi & Silvia Angela Osmetti

  2. Dipartimento di Economia, Management e Metodi Quantitativi, Università degli Studi di Milano, via Conservatorio, 7, 20122, Milan, Italy

    Chiara Tommasi

Authors
  1. Laura Deldossi
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  2. Silvia Angela Osmetti
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  3. Chiara Tommasi
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Correspondence to Laura Deldossi.

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Deldossi, L., Osmetti, S.A. & Tommasi, C. Optimal design to discriminate between rival copula models for a bivariate binary response. TEST 28, 147–165 (2019). https://doi.org/10.1007/s11749-018-0595-1

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