Optimal designs for nonlinear regression models with respect to non-informative priors

In nonlinear regression models the Fisher information depends on the parameters of the model. Consequently, optimal designs maximizing some functional of the information matrix cannot be implemented directly but require some preliminary knowledge about the unknown parameters. Bayesian optimality criteria provide an attractive solution to this problem. These criteria depend sensitively on a reasonable specication of a prior distribution for the model parameters which might not be available in all applications. In this paper we investigate Bayesian optimality criteria with non-informative prior distributions. In particular, we study the Jereys and the Berger-Bernardo prior for which the corresponding optimality criteria are not necessarily concave. Several examples are investigated where optimal designs with respect to the new criteria are calculated and compared to Bayesian optimal designs based on a uniform and a functional uniform prior.

[1]  Holger Dette.  Discrimination designs for polynomial regression on compact intervals , 1994 .

[2]  W. J. Studden,et al.  Optimal Designs for Trigonometric and Polynomial Regression Using Canonical Moments , 1985 .

[3]  Mei-Mei Zen,et al.  Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models , 2004 .

[4]  John Hinde,et al.  On Bayesian D‐optimum Design Criteria and the Equivalence Theorem in Non‐linear Models , 1997 .

[5]  A. S. Hedayat,et al.  LOCALLY D-OPTIMAL DESIGNS BASED ON A CLASS OF COMPOSED MODELS RESULTED FROM BLENDING EMAX AND ONE-COMPARTMENT MODELS ⁄ , 2008, 0803.2108.

[6]  W. J. Studden,et al.  OPTIMAL DESIGNS FOR WEIGHTED POLYNOMIAL REGRESSION USING CANONICAL MOMENTS , 1982 .

[7]  Byron Jones,et al.  Design of cross-over trials for pharmacokinetic studies , 1999 .

[8]  Purushottam W. Laud,et al.  On Bayesian Analysis of Generalized Linear Models Using Jeffreys's Prior , 1991 .

[9]  Zhide Fang,et al.  D-optimal designs for weighted polynomial regression , 2003 .

[10]  P. Sprent,et al.  Nonlinear Regression Modeling-A Unified Practical Approach. , 1985 .

[11]  Holger Dette,et al.  Optimal Designs for Dose-Finding Studies , 2008 .

[12]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[13]  M. Ghosh,et al.  Design Issues for Generalized Linear Models: A Review , 2006, math/0701088.

[14]  W. J. Studden,et al.  On an extremal problem of Feje´r , 1988 .

[15]  W. J. Studden.  Some Robust-Type D-Optimal Designs in Polynomial Regression , 1982 .

[16]  Holger Dette,et al.  The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis , 1997 .

[17]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[18]  W. J. Studden.  $D_s$-Optimal Designs for Polynomial Regression Using Continued Fractions , 1980 .

[19]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[20]  Holger Dette,et al.  Optimal designs for the emax, log-linear and exponential models , 2010 .

[21]  John Stufken,et al.  SUPPORT POINTS OF LOCALLY OPTIMAL DESIGNS FOR NONLINEAR MODELS WITH TWO PARAMETERS , 2009, 0903.0728.

[22]  Holger Dette,et al.  Bayesian D-optimal designs on a fixed number of design points for heteroscedastic polynomial models , 1998 .

[23]  Nicholas G. Polson.  On the Expected Amount of Information from a Non‐Linear Model , 1992 .

[24]  K. Chaloner,et al.  Bayesian Experimental Design: A Review , 1995 .

[25]  Larry Wasserman,et al.  Noninformative priors and nuisance parameters , 1993 .

[26]  John Hinde,et al.  Parameter Neutral Optimum Design for Non‐linear Models , 1997 .

[27]  Fu-Chuen Chang.  D-OPTIMAL DESIGNS FOR WEIGHTED POLYNOMIAL REGRESSION A FUNCTIONAL-ALGEBRAIC APPROACH , 2005 .

[28]  Fu-Chuen Chang,et al.  D-optimal designs for polynomial regression with exponential weight function , 2009 .

[29]  Holger Dette,et al.  Bayesian D-optimal designs for exponential regression models , 1997 .

[30]  Holger Dette,et al.  Optimal Designs for Identifying the Degree of a Polynomial Regression , 1995 .

[31]  Holger Dette,et al.  On the number of support points of maximin and Bayesian D-optimal designs in nonlinear regression models , 2004 .

[32]  K. Chaloner,et al.  Optimal Bayesian design applied to logistic regression experiments , 1989 .

[33]  K. Chaloner,et al.  Optimum experimental designs for properties of a compartmental model. , 1993, Biometrics.

[34]  J. Kiefer.  General Equivalence Theory for Optimum Designs (Approximate Theory) , 1974 .

[35]  H. Chernoff.  Locally Optimal Designs for Estimating Parameters , 1953 .

[36]  Holger Dette,et al.  On the number of support points of maximin and Bayesian optimal designs , 2007, 0708.1901.

[37]  V. Melas,et al.  Robust T-optimal discriminating designs , 2013, 1309.4652.

[38]  Holger Dette,et al.  A note on the de la Garza phenomenon for locally optimal designs , 2011, 1105.3575.

[39]  Lorens A. Imhof,et al.  E-optimal designs for rational models , 1996 .

[40]  Min Yang,et al.  On the de la Garza phenomenon , 2009, 0912.3861.

[41]  Blaza Toman,et al.  Bayesian Experimental Design , 2006 .

[42]  Kathryn Chaloner,et al.  D- and c-optimal designs for exponential regression models used in viral dynamics and other applications , 2003 .

[43]  E. Walter,et al.  Robust experiment design via stochastic approximation , 1985 .

[44]  Russell C. Eberhart,et al.  A new optimizer using particle swarm theory , 1995, MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science.

[45]  Linda M. Haines,et al.  A Geometric Approach to Optimal Design for One‐Parameter Non‐Linear Models , 1995 .

[46]  B. Bornkamp.  Functional Uniform Priors for Nonlinear Modeling , 2011, Biometrics.

[47]  Drew Seils,et al.  Optimal design , 2007 .