Primera Cohorte

Corollary 2.3.7 For a two point design d(T1, t₁; T₂, t₂), with xed positive values for T1, T2 and ti (i = 1, 2), if the value of tj (j = 1, 2) j 6= i is constrained to be within [tmin, tmax], then the value of tj = t_opt that maximises the determinant of the information matrix M is

t_opt =







tmin if tTj < tmin

t_max if tTj > t_max.

Proof

WLOG let j = 1 and i = 2. The proof follows from the proof of Lemma 2.3.4, as A = ctθ1exp(−lθ₂/T ). Thus if tT1 > t_max then ^{∂|M |}_∂t1 > 0 and tmax is the optimal choice. If tT1 < t_min then ^{∂|M |}_∂t1 < 0 and tmin is the optimal choice.

Bibliography

Asprey, S. P. and Macchietto, S. (2002) Designing robust optimal dynamic experiments. Journal of Process Control, 12, 545556.

Atkinson, A. C. (2003) Horwitz's rule, transforming both sides and the design of experiments for mechanistic models. Journal of the Royal Statistical Society.

Series C (Applied Statistics), 52, 261278.

 (2004) Some Bayesian optimum designs for response transformation in nonlinear models with nonconstant variance. MODA 7 - Advances in Model-Oriented Design and Analysis, Heidelberg: Physica-Verlag, 1322.

 (2005a) Eciencies for optimum designs when transforming the response in nonlinear models with nonconstant variance. Metrika, 62, 127138.

 (2005b) Robust optimum designs for transformation of the responses in a multivariate chemical kinetic model. Technometrics, 47, 478487.

 (2008a) DT -optimum designs for model discrimination and parameter estimation.

Journal of Statistical Planning and Inference, 138, 5664.

 (2008b) Examples of the use of an equivalence theorem in constructing optimum experimental designs for random-eects nonlinear regression models. Journal of Statistical Planning and Inference, 138, 25952606.

Atkinson, A. C. and Bogacka, B. (1997) Compound D- and DS-optimum designs for determining the order of a chemical reaction. Technometrics, 39, 347356.

 (2002) Compound and other optimum designs for systems of non-linear dierential equations arising in chemical kinetics. Chemometrics and Intelligent Laboratory Systems, 61, 1733.

Atkinson, A. C., Chaloner, K., Herzberg, A. M. and Juritz, J. (1993) Optimum experimental designs for properties of a compartmental model. Biometrics, 49, 325337.

Atkinson, A. C., Donev, A. N. and Tobias, R. D. S. (2007) Optimum Experimental Designs, with SAS. Oxford University Press.

Atkinson, A. C. and Fedorov, V. V. (1975) The design of experiments for discriminating between two rival models. Biometrika, 62, 5770.

Barz, T., Arellano-Garcia, H. and Wozny, G. (2010) Handling uncertainty in model-based optimal experimental design. Industrial and Engineering Chemistry Research, 49, 57025713.

Bates, D. and Watts, D. (1988) Nonlinear Regression Analysis and its Applications.

John Wiley & Sons.

Bauer, I., Bock, H., Körkel, H. and Schlöder, J. (2000) Numerical methods for optimum experimental design in DAE systems. Journal of Computational and Applied Mathematics, 120, 125.

Bogacka, B., Patan, M., Johnson, P. J., Youdim, K. and Atkinson, A. C. (2011) Optimum design of experiments for enzyme inhibition kinetic models. Journal of Biopharmaceutical Statistics, 21, 555572.

Bogacka, B. and Wright, F. J. (2005) Non-linear design problem in a chemical kinetic model with non-constant error variance. Journal of Statistical Planning and Inference, 128, 633648.

Braess, D. and Dette, H. (2007) On the number of support points of maximin and Bayesian optimal designs. Annals of Statistics, 35, 772792.

Buzzi-Ferraris, G., Forzatti, P., Emig, G. and Hofmann, H. (1984) Sequential experimental design procedure for model discrimination in the case of multiple responses. Chemical Engineering Science, 39, 8185.

Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995) A limited memory algorithm for bound constrained optimization. SIAM Journal of Scientic Computing, 16, 11901208.

Cassity, C. (1965) Abscissas, coecients, and error term for the generalized gauss-laguerre quadrature formula using the zero ordinate. Mathematics of Computation, 19, 287296.

Chaloner, K. and Verdinelli, I. (1995) Bayesian experimental design: a review.

Statistical Science, 10, 273304.

Cheng, C. (1995) Optimal regression designs under random block-eects models.

Statistica Sinica, 5, 485497.

Chu, Y. and Hahn, J. (2008) Integrating parameter selection with experimental design under uncertainty for nonlinear dynamic systems. AIChE, 54, 23102320.

Dette, H. and Biedermann, S. (2003) Robust and ecient designs for the Michaelis-Menten model. Journal of the American Statistical Association, 98, 679686.

Dette, H., Bretz, F., Pepelyshev, A. and Pinheiro, J. (2008) Optimal designs for dose-nding studies. Journal of the American Statistical Association, 103, 1225

1237.

Dette, H., Melas, V. B., Pepelyshev, A. and Strigul, N. (2005) Robust and ecient design of experiments for the Monod model. Journal of Theoretical Biology, 234, 537550.

Dette, H., Pepelyshev, A. and Holland-Letz, T. (2010) Optimal designs for random eect models with correlated errors with applications in population pharmacokinetics. The Annals of Applied Statistics, 4, 14301450.

Dette, H., Pepelyshev, A., Melas, V. B. and Strigul, N. (2003) Ecient design of experiments in the Monod model. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, 725742.

Dragalin, V., Fedorov, V. V. and Wu, Y. (2008) Adaptive designs for selecting drug combinations based on ecacy-toxicity response. Journal of Statistical Planning and Inference, 138, 352373.

Duul, S. B., Mentré, F. and Aarons, L. (2001) Optimal design of a population pharmacodynamic experiment for ivabradine. Pharmaceutical Research, 18, 83

89.

Elfving, G. (1952) Optimum allocation in linear regression theory. Annals of Mathematical Statistics, 23, 255262.

Fang, K. T., Li, R. and Sudjianto, A. (2006) Design and Modelling for Computer Experiments. Chapman and Hall.

Fedorov, V. V. and Leonov, S. (2007) Population pharmacokinetic measures, their estimation and selection of sampling times. Journal of Biopharmaceutical Statistics, 17, 919941.

Firth, D. and Hinde, J. P. (1997) On Bayesian D-optimum design criteria and the equivalence theorem in non- linear models. Journal of the Royal Statistical Society.

Series B (Methodological), 59, 793797.

Franceschini, G. and Macchietto, S. (2008a) Model-based design of experiments for parameter precision: State of the art. Chemical Engineering Science, 63, 4846

4872.

 (2008b) Novel anticorrelation criteria for model-based experiment design: theory and formulations. AIChE, 54, 10101024.

Gagnon, R. and Leonov, S. (2005) Optimal population designs for PK models with serial sampling. Journal of Biopharmaceutical statistics, 15, 143163.

Geudj, J., Bazzoli, C., Neumann, A. U. and Mentré, F. (2011) Design evaluation and optimization for models of hepatitis C viral dynamics. Statistics in Medicine, 30, 10451056.

Goos, P. and Vandebroek, M. (2004) Outperforming completely randomized designs.

Journal of Quality Technology, 36, 1226.

Gotwalt, C. M., Jones, B. A. and Steinberg, D. M. (2009) Fast computation of designs robust to parameter uncertainty for nonlinear settings. Technometrics, 51, 8895.

Gregson, J. (2009) Pseudo-Bayesian Design of Experiments. Master's thesis, School of Mathematics, University of Southampton.

Gueorguieva, I., Aarons, L., Ogungbenro, K., Joga, K. M., Rodgers, T. and Rowland, M. (2006) Optimal design for multivariate response pharmacokinetic models. Journal of Pharmacokinetics and Pharmacodynamics, 33, 97124.

Hamada, M., Martz, H. F., Reese, C. S. and Wilson, A. G. (2001) Finding near-optimal Bayesian experimental designs via genetic algorithms. The American Statistician, 55, 175181.

Hughes-Oliver (1998) Optimal designs for nonlinear models with correlated errors.

IMS Lecture Notes- Monograph Series, 34, 163174.

Ju, H. L. and Lucas, J. M. (2002) L^k factorial experiments with hard-to-change and easy-to-change factors. Journal of Quality Technology, 34, 411421.

Kiefer, J. and Wolfowitz, J. (1960) The equivalence of two extremum problems.

Canadian Journal of Mathematics, 12, 363366.

Laidler, K. J. (1984) The development of the Arrhenius equation. Journal of Chemical Education, 61, 494.

Lischer, P. (1999) Good statistical practice in analytical chemistry. In Probability Theory and Mathematical Statistics, 112. Dordrecht: VSP.

López-Fidalgo, J., Tommasi, C. and Trandar, P. C. (2008) Optimal designs for discriminating between some extensions of the Michaelis-Menten model. Journal of Statistical Planning and Inference, 138, 37973804.

Mannaswamy, A., Munson-McGee, S. H. and Anersen, P. K. (2010) D-optimal designs for the cross viscosity model applied to guar gum mixtures. Journal of Food Engineering, 97, 403409.

Matthews, J. N. S. and Allcock, G. C. (2004) Optimal designs for Michaelis-Menten kinetic studies. Statistics in Medicine, 23, 477491.

Matthews, J. N. S. and James, P. W. (2005) Restricted optimal design in the measurement of cerebral blood ow using the Kety-Schmidt technique. In Applied Optimal Designs, chap. 8. John Wiley & Sons.

McGree, J. M., Eccleston, J. A. and Duull, S. B. (2008) Compound optimal design criteria for nonlinear models. Journal of Biopharmaceutical Statistics, 18, 646

661.

McKay, M. D., Conover, W. and Beckman, R. (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21, 239240.

Mentré, F., Mallet, A. and Baccar, D. (1997) Optimal design in random-eects regression models. Biometrika, 84, 429442.

Monahan, J. and Genz, A. (1997) Spherical-radial integration rules for Bayesian computation. Journal of the American Statistical Association, 92, 664674.

Ogungbenro, K. and Aarons, L. (2008) Optimisation of sampling windows design for population pharmacokinetic experiments. Journal of Pharmacokinetic Phar-macodynamics, 35, 465482.

Owen, M., Luscombe, C., Lai, L., Godbert, S. Crookes, D. and Emiabata-Smith, D.

(2001) Eciency by design: Optimisation in process research. Organic Process Research & Development, 5, 308323.

Patan, M. and Bogacka, B. (2007) Optimum experimental designs for dynamic systems in the presence of correlated errors. Computational Statistics and Data Analysis, 51, 65445661.

Pinheiro, J. and Bates, D. (2004) Mixed Eects Models in S and S-Plus. Springer.

Press, W., Teuklosky, S., Vetterling, W. and Flannery, B. (1992) Numerical Recipes in C (2nd edition). New York: Cambridge University Press.

Pronzato, L. (2010) Penalised optimal designs for dose nding. Journal of Statistical Planning and Inference, 140, 283296.

Pukelsheim, F. and Rieder, S. (1992) Ecient rounding of approximate designs.

Biometrika, 79, 763770.

Retout, S., Comets, E., Bazzoli, C. and Mentré, F. (2009) Design optimization in nonlinear mixed eects models using cost functions: application to a joint model of iniximab and methotrexate pharmacokinetics. Communications in Statistics-Theory and Methods, 38, 33513368.

Retout, S., Comets, E., Samson, A. and Mentré, F. (2007) Design in nonlinear mixed eects models: Optimization using the Fedorov-Wynn algorithm and power of the Wald test for binary covariates. Statistics in Medicine, 26, 51625179.

Retout, S. and Mentré, F. (2003) Further developments of the Fisher information matrix in nonlinear mixed eects models with evaluation in population pharma-cokinetics. Journal of Biopharmaceutical Statistics, 13, 209227.

Rodriguez-Aragón, L. J. and López-Fidalgo, J. (2005) Optimal designs for the Arrhenius equation. Chemometrics and Intelligent Laboratory Systems, 77, 131

138.

Rodríguez-Díaz, J. M. and Santos-Martín, M. T. (2009) Study of the best designs for modications of the Arrhenius equation. Chemometrics and Intelligent Laboratory Systems, 95, 199208.