Statistical Science

Design Issues for Generalized Linear Models: A Review

André I. Khuri, Bhramar Mukherjee, Bikas K. Sinha, and Malay Ghosh

Full-text: Open access

Enhanced PDF (376 KB)

Abstract

Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and building of an adequate model. However, one major problem that handicaps the construction of a GLM design is its dependence on the unknown parameters of the fitted model. Several approaches have been proposed in the past 25 years to solve this problem. These approaches, however, have provided only partial solutions that apply in only some special cases, and the problem, in general, remains largely unresolved. The purpose of this article is to focus attention on the aforementioned dependence problem. We provide a survey of various existing techniques dealing with the dependence problem. This survey includes discussions concerning locally optimal designs, sequential designs, Bayesian designs and the quantile dispersion graph approach for comparing designs for GLMs.

Article information

Source
Statist. Sci., Volume 21, Number 3 (2006), 376-399.

Dates
First available in Project Euclid: 20 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.ss/1166642442

Digital Object Identifier
doi:10.1214/088342306000000105

Mathematical Reviews number (MathSciNet)
MR2339137

Zentralblatt MATH identifier
1246.62168

Keywords
Bayesian design dependence on unknown parameters locally optimal design logistic regression response surface methodology quantal dispersion graphs sequential design

Citation

Khuri, André I.; Mukherjee, Bhramar; Sinha, Bikas K.; Ghosh, Malay. Design Issues for Generalized Linear Models: A Review. Statist. Sci. 21 (2006), no. 3, 376--399. doi:10.1214/088342306000000105. https://projecteuclid.org/euclid.ss/1166642442


Export citation

References

  • Abdelbasit, K. M. and Plackett, R. L. (1983). Experimental design for binary data. J. Amer. Statist. Assoc. 78 90--98.
    Mathematical Reviews (MathSciNet): MR0696852
    Digital Object Identifier: doi:10.2307/2287114
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0501.62071
  • Abdelhamid, S. N. (1973). Transformation of observations in stochastic approximation. Ann. Statist. 1 1158--1174.
    Mathematical Reviews (MathSciNet): MR0348944
    Digital Object Identifier: doi:10.1214/aos/1176342564
    Project Euclid: euclid.aos/1176342564
    Zentralblatt MATH: 0303.62065
  • Anbar, D. (1973). On optimal estimation methods using stochastic approximation procedures. Ann. Statist. 1 1175--1184.
    Mathematical Reviews (MathSciNet): MR0351001
    Digital Object Identifier: doi:10.1214/aos/1176342565
    Project Euclid: euclid.aos/1176342565
    Zentralblatt MATH: 0277.62064
  • Ashton, W. D. (1972). The Logit Transformation with Special Reference to Its Uses in Bioassay. Hafner, New York.
    Mathematical Reviews (MathSciNet): MR0381220
    Zentralblatt MATH: 0244.62003
  • Atkinson, A. C., Chaloner, K., Herzberg, A. M. and Juritz, J. (1993). Optimum experimental designs for properties of a compartmental model. Biometrics 49 325--337.
  • Atkinson, A. C., Demetrio, C. G. B. and Zocchi, S. S. (1995). Optimum dose levels when males and females differ in response. Appl. Statist. 44 213--226.
  • Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs. Clarendon, Oxford.
  • Atkinson, A. C. and Haines, L. M. (1996). Designs for nonlinear and generalized linear models. In Design and Analysis of Experiments (S. Ghosh and C. R. Rao, eds.) 437--475. North-Holland, Amsterdam.
    Mathematical Reviews (MathSciNet): MR1492576
    Zentralblatt MATH: 0910.62070
  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR0804611
    Zentralblatt MATH: 0572.62008
  • Bernardo, J.-M. (1979). Expected information as expected utility. Ann. Statist. 7 686--690.
    Mathematical Reviews (MathSciNet): MR0527503
    Digital Object Identifier: doi:10.1214/aos/1176344689
    Project Euclid: euclid.aos/1176344689
    Zentralblatt MATH: 0407.62002
  • Berry, D. A. and Fristedt, B. (1985). Bandit Problems: Sequential Allocation of Experiments. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR0813698
    Zentralblatt MATH: 0659.62086
  • Berry, D. A. and Pearson, L. M. (1985). Optimal designs for clinical trials with dichotomous responses. Statistics in Medicine 4 497--508.
    Zentralblatt MATH: 0972.62107
  • Box, G. E. P. and Draper, N. R. (1987). Empirical Model-Building and Response Surfaces. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR0861118
    Zentralblatt MATH: 0614.62104
  • Box, G. E. P. and Lucas, H. L. (1959). Design of experiments in non-linear situations. Biometrika 46 77--90.
    Mathematical Reviews (MathSciNet): MR0102155
    Zentralblatt MATH: 0086.34803
    JSTOR: links.jstor.org
  • Brooks, R. J. (1972). A decision theory approach to optimal regression designs. Biometrika 59 563--571.
    Mathematical Reviews (MathSciNet): MR0334426
    Zentralblatt MATH: 0246.62008
    Digital Object Identifier: doi:10.1093/biomet/59.3.563
    JSTOR: links.jstor.org
  • Brooks, R. J. (1974). On the choice of an experiment for prediction in linear regression. Biometrika 61 303--311.
    Mathematical Reviews (MathSciNet): MR0368333
    Zentralblatt MATH: 0282.62066
    Digital Object Identifier: doi:10.1093/biomet/61.2.303
    JSTOR: links.jstor.org
  • Brooks, R. J. (1976). Optimal regression designs for prediction when prior knowledge is available. Metrika 23 221--230.
    Mathematical Reviews (MathSciNet): MR0440810
    Digital Object Identifier: doi:10.1007/BF01902868
    Zentralblatt MATH: 0346.62054
  • Brooks, R. J. (1977). Optimal regression design for control in linear regression. Biometrika 64 319--325.
    Mathematical Reviews (MathSciNet): MR0488525
    Zentralblatt MATH: 0362.62070
    Digital Object Identifier: doi:10.1093/biomet/64.2.319
    JSTOR: links.jstor.org
  • Burridge, J. and Sebastiani, P. (1992). Optimal designs for generalised linear models. J. Italian Statist. Soc. 1 183--202.
  • Burridge, J. and Sebastiani, P. (1994). $D$-optimal designs for generalised linear models with variance proportional to the square of the mean. Biometrika 81 295--304.
    Mathematical Reviews (MathSciNet): MR1294893
    Zentralblatt MATH: 0807.62055
    JSTOR: links.jstor.org
  • Chaloner, K. (1984). Optimal Bayesian experimental designs for linear models. Ann. Statist. 12 283--300.
    Mathematical Reviews (MathSciNet): MR0733514
    Digital Object Identifier: doi:10.1214/aos/1176346407
    Project Euclid: euclid.aos/1176346407
    Zentralblatt MATH: 0554.62063
  • Chaloner, K. (1987). An approach to design for generalized linear models. In Model-Oriented Data Analysis (V. V. Fedorov and H. Läuter, eds.) 3--12. Springer, Berlin.
  • Chaloner, K. (1993). A note on optimal Bayesian design for nonlinear problems. J. Statist. Plann. Inference 37 229--235.
    Mathematical Reviews (MathSciNet): MR1243800
    Digital Object Identifier: doi:10.1016/0378-3758(93)90091-J
    Zentralblatt MATH: 0786.62073
  • Chaloner, K. and Larntz, K. (1988). Software for logistic regression experiment design. In Optimal Design and Analysis of Experiments (Y. Dodge, V. V. Fedorov and H. P. Wynn, eds.) 207--211. North-Holland, Amsterdam.
  • Chaloner, K. and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference 21 191--208.
    Mathematical Reviews (MathSciNet): MR0985457
    Digital Object Identifier: doi:10.1016/0378-3758(89)90004-9
    Zentralblatt MATH: 0666.62073
  • Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273--304.
    Mathematical Reviews (MathSciNet): MR1390519
    Digital Object Identifier: doi:10.1214/ss/1177009939
    Project Euclid: euclid.ss/1177009939
    Zentralblatt MATH: 0955.62617
  • Chen, K. (2000). Optimal sequential designs of case-control studies. Ann. Statist. 28 1452--1471.
    Mathematical Reviews (MathSciNet): MR1805792
    Digital Object Identifier: doi:10.1214/aos/1015957402
    Project Euclid: euclid.aos/1015957402
    Zentralblatt MATH: 1105.62364
  • Chernoff, H. (1953). Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 586--602.
    Mathematical Reviews (MathSciNet): MR0058932
    Digital Object Identifier: doi:10.1214/aoms/1177728915
    Project Euclid: euclid.aoms/1177728915
  • Chernoff, H. (1972). Sequential Analysis and Optimal Design. SIAM, Philadelphia.
    Mathematical Reviews (MathSciNet): MR0309249
    Zentralblatt MATH: 0265.62024
  • Clyde, M. A. (1993a). Bayesian optimal designs for approximate normality. Ph.D. dissertation, Univ. Minnesota.
  • Clyde, M. A. (1993b). An object-oriented system for Bayesian nonlinear design using xlisp-stat. Technical Report 587, School of Statistics, Univ. Minnesota.
  • Clyde, M. A. and Chaloner, K. (1996). The equivalence of constrained and weighted designs in multiple objective design problems. J. Amer. Statist. Assoc. 91 1236--1244.
    Mathematical Reviews (MathSciNet): MR1424621
    Digital Object Identifier: doi:10.2307/2291742
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0883.62079
  • Clyde, M. A. and Chaloner, K. (2002). Constrained design strategies for improving normal approximations in nonlinear regression problems. J. Statist. Plann. Inference 104 175--196.
    Mathematical Reviews (MathSciNet): MR1900525
    Digital Object Identifier: doi:10.1016/S0378-3758(01)00239-7
    Zentralblatt MATH: 0988.62043
  • Clyde, M. A., Müller, P. and Parmigiani, G. (1995). Optimal designs for heart defibrillators. Case Studies in Bayesian Statistics II. Lecture Notes in Statist. 105 278--292. Springer, New York.
  • Das, P., Mandal, N. K. and Sinha, B. K. (2003). de la Garza phenomenon in linear regression with heteroscedastic errors. Technical report, Dept. Statistics, Calcutta Univ.
  • DasGupta, A. (1996). Review of optimal Bayes designs. In Design and Analysis of Experiments (S. Ghosh and C. R. Rao, eds.) 1099--1147. North-Holland, Amsterdam.
    Mathematical Reviews (MathSciNet): MR1492591
    Zentralblatt MATH: 0911.62066
  • DasGupta, A., Mukhopadhyay, S. and Studden, W. J. (1992). Compromise designs in heteroscedastic linear models. J. Statist. Plann. Inference 32 363--384.
    Mathematical Reviews (MathSciNet): MR1190204
    Digital Object Identifier: doi:10.1016/0378-3758(92)90017-M
    Zentralblatt MATH: 0766.62043
  • DasGupta, A. and Studden, W. J. (1991). Robust Bayesian experimental designs in normal linear models. Ann. Statist. 19 1244--1256.
    Mathematical Reviews (MathSciNet): MR1126323
    Digital Object Identifier: doi:10.1214/aos/1176348247
    Project Euclid: euclid.aos/1176348247
    Zentralblatt MATH: 0744.62099
  • Dawid, A. and Sebastiani, P. (1999). Coherent dispersion criteria for optimal experimental design. Ann. Statist. 27 65--81.
    Mathematical Reviews (MathSciNet): MR1701101
    Digital Object Identifier: doi:10.1214/aos/1018031101
    Project Euclid: euclid.aos/1018031101
    Zentralblatt MATH: 0948.62057
  • DeGroot, M. H. (1986). Concepts of information based on utility. In Recent Developments in the Foundations of Utility and Risk Theory (L. Daboni, A. Montesano and M. Lines, eds.) 265--275. Reidel, Dordrecht.
    Zentralblatt MATH: 0663.90001
  • de la Garza, A. (1954). Spacing of information in polynomial regression. Ann. Math. Statist. 25 123--130.
    Mathematical Reviews (MathSciNet): MR0060777
    Digital Object Identifier: doi:10.1214/aoms/1177728851
    Project Euclid: euclid.aoms/1177728851
  • Dette, H. and Neugebauer, H.-M. (1996). Bayesian optimal one-point designs for one-parameter nonlinear models. J. Statist. Plann. Inference 52 17--30.
    Mathematical Reviews (MathSciNet): MR1391681
    Digital Object Identifier: doi:10.1016/0378-3758(95)00104-2
    Zentralblatt MATH: 0960.62529
  • Dette, H. and Sperlich, S. (1994). A note on Bayesian $D$-optimal designs for a generalization of the exponential growth model. South African Statist. J. 28 103--117.
    Mathematical Reviews (MathSciNet): MR1333863
  • Diaz, M. P., Barchuk, A. H., Luque, S. and Oviedo, C. (2002). Generalized linear models to study spatial distribution of tree species in Argentinean arid Chaco. J. Appl. Statist. 29 685--694.
    Mathematical Reviews (MathSciNet): MR1888680
    Digital Object Identifier: doi:10.1080/02664760120098748
    Zentralblatt MATH: 05198724
  • Dixon, W. J. and Mood, A. M. (1948). A method for obtaining and analyzing sensitivity data. J. Amer. Statist. Assoc. 43 109--126.
  • Dobson, A. J. (2002). An Introduction to Generalized Linear Models, 2nd ed. Chapman and Hall/CRC, Boca Raton, FL.
    Mathematical Reviews (MathSciNet): MR2068205
    Zentralblatt MATH: 1008.62067
  • Draper, N. R. and Hunter, W. G. (1967). The use of prior distributions in the design of experiments for parameter estimation in non-linear situations: Multiresponse case. Biometrika 54 662--665.
    Mathematical Reviews (MathSciNet): MR0223028
    JSTOR: links.jstor.org
  • Dubov, E. L. (1977). $D$-optimal designs for nonlinear models under the Bayesian approach. In Regression Experiments 103--111. Moscow Univ. Press. (In Russian.)
  • DuMouchel, W. and Jones, B. (1994). A simple Bayesian modification of $D$-optimal designs to reduce dependence on an assumed model. Technometrics 36 37--47.
  • Etzioni, R. and Kadane, J. B. (1993). Optimal experimental design for another's analysis. J. Amer. Statist. Assoc. 88 1404--1411.
    Mathematical Reviews (MathSciNet): MR1245377
    Digital Object Identifier: doi:10.2307/2291284
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0792.62062
  • Fabian, V. (1968). On asymptotic normality in stochastic approximation. Ann. Math. Statist. 39 1327--1332.
    Mathematical Reviews (MathSciNet): MR0231429
    Digital Object Identifier: doi:10.1214/aoms/1177698258
    Project Euclid: euclid.aoms/1177698258
  • Fabian, V. (1983). A local asymptotic minimax optimality of an adaptive Robbins--Monro stochastic approximation procedure. Mathematical Learning Models---Theory and Algorithms. Lecture Notes in Statist. 20 43--49. Springer, New York.
    Mathematical Reviews (MathSciNet): MR0731719
    Zentralblatt MATH: 0533.62075
  • Fedorov, V. V. (1972). Theory of Optimal Experiments. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR0403103
  • Flournoy, N. (1993). A clinical experiment in bone marrow transplantation: Estimating a percentage point of a quantal response curve. Case Studies in Bayesian Statistics. Lecture Notes in Statist. 83 324--336. Springer, New York.
  • Ford, I., Torsney, B. and Wu, C. F. J. (1992). The use of a cannonical form in the construction of locally optimal designs for nonlinear problems. J. Roy. Statist. Soc. Ser. B 54 569--583.
    Mathematical Reviews (MathSciNet): MR1160483
    JSTOR: links.jstor.org
  • Freeman, P. R. (1970). Optimal Bayesian sequential estimation of the median effective dose. Biometrika 57 79--89.
    Mathematical Reviews (MathSciNet): MR0293780
    Digital Object Identifier: doi:10.1093/biomet/57.1.79
    JSTOR: links.jstor.org
  • Frees, E. W. and Ruppert, D. (1990). Estimation following a sequentially designed experiment. J. Amer. Statist. Assoc. 85 1123--1129.
    Mathematical Reviews (MathSciNet): MR1134509
    Digital Object Identifier: doi:10.2307/2289610
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0741.62075
  • Gill, P. E., Murray, W., Saunders, M. A. and Wright, M. H. (1986). User's guide for NPSOL (version 4.0): A Fortran package for nonlinear programming. Technical Report SOL 86-2, Dept. Operations Research, Stanford Univ.
  • Haines, L. (1995). A geometric approach to optimal design for one-parameter non-linear models. J. Roy. Statist. Soc. Ser. B 57 575--598.
    Mathematical Reviews (MathSciNet): MR1341325
    JSTOR: links.jstor.org
  • Hatzis, C. and Larntz, K. (1992). Optimal design in nonlinear multiresponse estimation: Poisson model for filter feeding. Biometrics 48 1235--1248.
  • Heise, M. A. and Myers, R. H. (1996). Optimal designs for bivariate logistic regression. Biometrics 52 613--624.
  • Hern, A. and Dorn, S. (2001). Statistical modelling of insect behavioral responses in relation to the chemical composition of test extracts. Physiological Entomology 26 381--390.
  • Jewell, N. P. and Shiboski, S. (1990). Statistical analysis of HIV infectivity based on partner studies. Biometrics 46 1133--1150.
  • Khan, M. K. and Yazdi, A. A. (1988). On $D$-optimal designs for binary data. J. Statist. Plann. Inference 18 83--91.
    Mathematical Reviews (MathSciNet): MR0926416
    Digital Object Identifier: doi:10.1016/0378-3758(88)90018-3
    Zentralblatt MATH: 0629.62071
  • Khuri, A. I. (1993). Response surface methodology within the framework of GLM. J. Combin. Inform. System Sci. 18 193--202.
    Mathematical Reviews (MathSciNet): MR1317235
  • Khuri, A. I. and Cornell, J. A. (1996). Response Surfaces, 2nd ed. Dekker, New York.
    Mathematical Reviews (MathSciNet): MR1447628
    Zentralblatt MATH: 0953.62073
  • Kiefer, J. and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statist. 23 462--466.
    Mathematical Reviews (MathSciNet): MR0050243
    Digital Object Identifier: doi:10.1214/aoms/1177729392
    Project Euclid: euclid.aoms/1177729392
  • Kuo, L. (1983). Bayesian bioassay design. Ann. Statist. 11 886--895.
    Mathematical Reviews (MathSciNet): MR0707938
    Digital Object Identifier: doi:10.1214/aos/1176346254
    Project Euclid: euclid.aos/1176346254
    Zentralblatt MATH: 0521.62086
  • Kuo, L., Soyer, R. and Wang, F. (1999). Optimal design for quantal bioassay via Monte Carlo methods. In Bayesian Statistics VI (J.-M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 795--802. Oxford Univ. Press, New York.
  • Lai, T. L. and Robbins, H. (1979). Adaptive design and stochastic approximation. Ann. Statist. 7 1196--1221.
    Mathematical Reviews (MathSciNet): MR0550144
    Digital Object Identifier: doi:10.1214/aos/1176344840
    Project Euclid: euclid.aos/1176344840
    Zentralblatt MATH: 0426.62059
  • Läuter, E. (1974). Experimental design in a class of models. Math. Oper. Statist. 5 379--396.
    Mathematical Reviews (MathSciNet): MR0440812
  • Läuter, E. (1976). Optimal multipurpose designs for regression models. Math. Oper. Statist. 7 51--68.
    Mathematical Reviews (MathSciNet): MR0413385
  • Lee, Y. J. and Nelder, J. A. (2002). Analysis of ulcer data using hierarchical generalized linear models. Statistics in Medicine 21 191--202.
  • Lindley, D. V. (1956). On a measure of the information provided by an experiment. Ann. Math. Statist. 27 986--1005.
    Mathematical Reviews (MathSciNet): MR0083936
    Digital Object Identifier: doi:10.1214/aoms/1177728069
    Project Euclid: euclid.aoms/1177728069
  • Lindley, D. V. and Singpurwalla, N. D. (1991). On the evidence needed to reach agreed action between adversaries, with application to acceptance sampling. J. Amer. Statist. Assoc. 86 933--937.
    Mathematical Reviews (MathSciNet): MR1146341
    Digital Object Identifier: doi:10.2307/2290509
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0850.62129
  • Lindsey, J. K. (1997). Applying Generalized Linear Models. Springer, New York.
    Zentralblatt MATH: 0883.62074
  • Liski, E. P., Mandal, N. K., Shah, K. R. and Sinha, B. K. (2002). Topics in Optimal Design. Lecture Notes in Statist. 163. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1933941
    Zentralblatt MATH: 0985.62057
  • Marks, B. L. (1962). Some optimal sequential schemes for estimating the mean of a cumulative normal quantal response curve. J. Roy. Statist. Soc. Ser. B 24 393--400.
  • Markus, R. A., Frank, J., Groshen, S. and Azen, S. P. (1995). An alternative approach to the optimal design of an LD50 bioassay. Statistics in Medicine 14 841--852.
  • Mathew, T. and Sinha, B. K. (2001). Optimal designs for binary data under logistic regression. J. Statist. Plann. Inference 93 295--307.
    Mathematical Reviews (MathSciNet): MR1822403
    Digital Object Identifier: doi:10.1016/S0378-3758(00)00173-7
    Zentralblatt MATH: 0965.62061
  • Matthews, J. N. S. (1999). Effect of prior specification on Bayesian design for two-sample comparison of a binary outcome. Amer. Statist. 53 254--256.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR727836
    Zentralblatt MATH: 0588.62104
  • McCulloch, C. E. and Searle, S. R. (2001). Generalized, Linear and Mixed Models. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1884506
    Zentralblatt MATH: 0964.62061
  • Mehrabi, Y. and Matthews, J. N. S. (1998). Implementable Bayesian designs for limiting dilution assays. Biometrics 54 1398--1406.
  • Minkin, S. (1987). Optimal designs for binary data. J. Amer. Statist. Assoc. 82 1098--1103.
    Mathematical Reviews (MathSciNet): MR0922174
    Digital Object Identifier: doi:10.2307/2289386
    JSTOR: links.jstor.org
  • Minkin, S. (1993). Experimental design for clonogenic assays in chemotherapy. J. Amer. Statist. Assoc. 88 410--420.
    Mathematical Reviews (MathSciNet): MR1224368
    Digital Object Identifier: doi:10.2307/2290319
    JSTOR: links.jstor.org
  • Mukhopadhyay, S. and Haines, L. (1995). Bayesian $D$-optimal designs for the exponential growth model. J. Statist. Plann. Inference 44 385--397.
    Mathematical Reviews (MathSciNet): MR1332682
    Digital Object Identifier: doi:10.1016/0378-3758(94)00056-2
    Zentralblatt MATH: 0811.62073
  • Müller, P. (1999). Simulation-based optimal design. In Bayesian Statistics VI (J.-M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 459--474. Oxford Univ. Press, New York.
    Mathematical Reviews (MathSciNet): MR1723509
  • Müller, P. and Parmigiani, G. (1995). Optimal design via curve fitting of Monte Carlo experiments. J. Amer. Statist. Assoc. 90 1322--1330.
    Mathematical Reviews (MathSciNet): MR1379474
    Digital Object Identifier: doi:10.2307/2291522
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0867.62065
  • Myers, R. H. (1999). Response surface methodology---current status and future directions. J. Quality Technology 31 30--44.
  • Myers, R. H., Khuri, A. I. and Carter, W. H. (1989). Response surface methodology: 1966--1988. Technometrics 31 137--157.
    Mathematical Reviews (MathSciNet): MR1007291
    Digital Object Identifier: doi:10.2307/1268813
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0693.62062
  • Myers, R. H. and Montgomery, D. C. (1995). Response Surface Methodology. Wiley, New York.
  • Myers, R. H., Montgomery, D. C. and Vining, G. G. (2002). Generalized Linear Models. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1863403
    Zentralblatt MATH: 0996.62065
  • Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. Computer J. 7 308--313.
  • Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models. J. Roy. Statist. Soc. Ser. A 135 370--384.
  • Owen, R. J. (1970). The optimum design of a two-factor experiment using prior information. Ann. Math. Statist. 41 1917--1934.
    Mathematical Reviews (MathSciNet): MR0279943
    Digital Object Identifier: doi:10.1214/aoms/1177696693
    Project Euclid: euclid.aoms/1177696693
  • Owen, R. J. (1975). A Bayesian sequential procedure for quantal response in the context of adaptive mental testing. J. Amer. Statist. Assoc. 70 351--356.
    Mathematical Reviews (MathSciNet): MR0381185
    Digital Object Identifier: doi:10.2307/2285821
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0324.62061
  • Palmer, J. L. and Müller, P. (1998). Bayesian optimal design in population models of hematologic data. Statistics in Medicine 17 1613--1622.
  • Parmigiani, G. (1993). On optimal screening ages. J. Amer. Statist. Assoc. 88 622--628.
    Mathematical Reviews (MathSciNet): MR1224388
    Digital Object Identifier: doi:10.2307/2290344
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0776.62085
  • Parmigiani, G. and Berry, D. A. (1994). Applications of Lindley information measure to the design of clinical experiments. In Aspects of Uncertainty (P. R. Freeman and A. F. M. Smith, eds.) 329--348. Wiley, Chichester.
    Mathematical Reviews (MathSciNet): MR1309700
    Zentralblatt MATH: 0843.62103
  • Parmigiani, G. and Müller, P. (1995). Simulation approach to one-stage and sequential optimal design problems. In MODA-$4$---Advances in Model Oriented Data Analysis (C. P. Kitsos and W. G. Müller, eds.) 37--47. Physica, Heidelberg.
    Mathematical Reviews (MathSciNet): MR1366881
    Zentralblatt MATH: 0850.62629
  • Pilz, J. (1991). Bayesian Estimation and Experimental Design in Linear Regression Models, 2nd ed. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1118380
    Zentralblatt MATH: 0745.62068
  • Pronzato, L. and Walter, E. (1987). Robust experiment designs for nonlinear regression models. In Model-Oriented Data Analysis (V. V. Fedorov and H. Laüter, eds.) 77--86. Springer, Berlin.
    Zentralblatt MATH: 0907.00035
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1211416
    Zentralblatt MATH: 0834.62068
  • Ridout, M. S. (1995). Three-stage designs for seed testing experiments. Appl. Statist. 44 153--162.
  • Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400--407.
    Mathematical Reviews (MathSciNet): MR0042668
    Digital Object Identifier: doi:10.1214/aoms/1177729586
    Project Euclid: euclid.aoms/1177729586
  • Robbins, H. and Siegmund, D. (1971). A convergence theorem for non-negative almost positive supermartingales and some applications. In Optimizing Methods in Statistics (J. S. Rustagi, ed.) 233--257. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR343355
    Zentralblatt MATH: 0286.60025
  • Robinson, K. S. and Khuri, A. I. (2003). Quantile dispersion graphs for evaluating and comparing designs for logistic regression models. Comput. Statist. Data Anal. 43 47--62.
    Mathematical Reviews (MathSciNet): MR2005386
  • Rosenberger, W. F. and Grill, S. E. (1997). A sequential design for psychophysical experiments: An application to estimating timing of sensory events. Statistics in Medicine 16 2245--2260.
  • Ruppert, D., Reish, R. L., Deriso, R. B. and Carroll, R. J. (1984). Optimization using stochastic approximation and Monte Carlo simulation (with application to harvesting of Atlantic menhaden). Biometrics 40 535--545.
  • Sebastiani, P. and Settimi, R. (1997). A note on $D$-optimal designs for a logistic regression model. J. Statist. Plann. Inference 59 359--368.
    Mathematical Reviews (MathSciNet): MR1450507
    Digital Object Identifier: doi:10.1016/S0378-3758(96)00111-5
    Zentralblatt MATH: 0900.62410
  • Sebastiani, P. and Settimi, R. (1998). First-order optimal designs for non-linear models. J. Statist. Plann. Inference 74 177--192.
    Mathematical Reviews (MathSciNet): MR1665126
    Digital Object Identifier: doi:10.1016/S0378-3758(98)00078-0
    Zentralblatt MATH: 0924.62081
  • Sielken, R. L. (1973). Stopping times for stochastic approximation procedures. Z. Wahrsch. Verw. Gebiete 26 67--75.
    Mathematical Reviews (MathSciNet): MR0341781
    Digital Object Identifier: doi:10.1007/BF00533961
    Zentralblatt MATH: 0244.62062
  • Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR0606742
    Zentralblatt MATH: 0468.62070
  • Sitter, R. R. (1992). Robust designs for binary data. Biometrics 48 1145--1155.
    Mathematical Reviews (MathSciNet): MR1212858
    Digital Object Identifier: doi:10.2307/2532705
    JSTOR: links.jstor.org
  • Sitter, R. R. and Forbes, B. E. (1997). Optimal two-stage designs for binary response experiments. Statist. Sinica 7 941--955.
    Mathematical Reviews (MathSciNet): MR1488652
    Zentralblatt MATH: 1067.62562
  • Sitter, R. R. and Torsney, B. (1995). Optimal designs for binary response experiments with two design variables. Statist. Sinica 5 405--419.
    Mathematical Reviews (MathSciNet): MR1347597
  • Sitter, R. R. and Wu, C.-F. J. (1993). Optimal designs for binary response experiments: Fieller, $D$, and $A$ criteria. Scand. J. Statist. 20 329--341.
    Mathematical Reviews (MathSciNet): MR1276697
  • Sitter, R. R. and Wu, C.-F. J. (1999). Two-stage design of quantal response studies. Biometrics 55 396--402.
  • Smith, D. M. and Ridout, M. S. (1998). Locally and Bayesian optimal designs for binary dose-response models with various link functions. In COMPSTAT 98 (R. Payne and P. Green, eds.) 455--460. Physica, Heidelberg.
  • Smith, D. M. and Ridout, M. S. (2003). Optimal designs for criteria involving log(potency) in comparative binary bioassays. J. Statist. Plann. Inference 113 617--632.
    Mathematical Reviews (MathSciNet): MR1965132
    Digital Object Identifier: doi:10.1016/S0378-3758(02)00112-X
    Zentralblatt MATH: 1014.62088
  • Smith, D. M. and Ridout, M. S. (2005). Algorithms for finding locally and Bayesian optimal designs for binary dose-response models with control mortality. J. Statist. Plann. Inference 133 463--478.
    Mathematical Reviews (MathSciNet): MR2205343
    Digital Object Identifier: doi:10.1016/j.jspi.2004.01.017
    Zentralblatt MATH: 1065.62136
  • Spears, F. M., Brown, B. W. and Atkinson, E. N. (1997). The effect of incomplete knowledge of parameter values on single- and multiple-stage designs for logistic regression. Biometrics 53 1--10.
  • Steinberg, D. M. (1985). Model robust response surface designs: Scaling two-level factorials. Biometrika 72 513--526.
    Mathematical Reviews (MathSciNet): MR0817565
    Zentralblatt MATH: 0605.62086
    Digital Object Identifier: doi:10.1093/biomet/72.3.513
    JSTOR: links.jstor.org
  • Storer, B. E. (1989). Design and analysis of phase I clinical trials. Biometrics 45 925--937.
    Mathematical Reviews (MathSciNet): MR1029610
    Digital Object Identifier: doi:10.2307/2531693
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0715.62241
  • Storer, B. E. (1990). A sequential phase II/II trial for binary outcomes. Statistics in Medicine 9 229--235.
  • Stroup, D. F. and Braun, H. I. (1982). On a new stopping rule for stochastic approximation. Z. Wahrsch. Verw. Gebiete 60 535--554.
    Mathematical Reviews (MathSciNet): MR0665744
    Digital Object Identifier: doi:10.1007/BF00535715
  • Sun, D., Tsutakawa, R. K. and Lu, W.-S. (1996). Bayesian design of experiment for quantal responses: What's promised versus what's delivered. J. Statist. Plann. Inference 52 289--306.
    Mathematical Reviews (MathSciNet): MR1401025
    Digital Object Identifier: doi:10.1016/0378-3758(95)00120-4
    Zentralblatt MATH: 0877.62073
  • Tierney, L. (1990). LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics. Wiley, New York.
  • Toman, B. and Gastwirth, J. L. (1993). Robust Bayesian experimental design and estimation for analysis of variance models using a class of normal mixtures. J. Statist. Plann. Inference 35 383--398.
    Mathematical Reviews (MathSciNet): MR1229251
    Digital Object Identifier: doi:10.1016/0378-3758(93)90024-Z
    Zentralblatt MATH: 0772.62043
  • Toman, B. and Gastwirth, J. L. (1994). Efficiency robust experimental design and estimation using a data-based prior. Statist. Sinica 4 603--615.
    Mathematical Reviews (MathSciNet): MR1309431
  • Tsutakawa, R. K. (1972). Design of experiment for bioassay. J. Amer. Statist. Assoc. 67 584--590.
  • Tsutakawa, R. K. (1980). Selection of dose levels for estimating a percentage point of a logistic quantal response curve. Appl. Statist. 29 25--33.
  • Venter, J. H. (1967). An extension of the Robbins--Monro procedure. Ann. Math. Statist. 38 181--190.
    Mathematical Reviews (MathSciNet): MR0205396
    Digital Object Identifier: doi:10.1214/aoms/1177699069
    Project Euclid: euclid.aoms/1177699069
  • Wei, C. Z. (1985). Asymptotic properties of least-squares estimates in stochastic regression models. Ann. Statist. 13 1498--1508.
    Mathematical Reviews (MathSciNet): MR0811506
    Digital Object Identifier: doi:10.1214/aos/1176349751
    Project Euclid: euclid.aos/1176349751
    Zentralblatt MATH: 0582.62062
  • Wetherill, G. B. (1963). Sequential estimation of quantal response curves (with discussion). J. Roy. Statist. Soc. Ser. B 25 1--48.
    Mathematical Reviews (MathSciNet): MR56895
    JSTOR: links.jstor.org
  • White, L. V. (1973). An extension of the general equivalence theorem to nonlinear models. Biometrika 60 345--348.
    Mathematical Reviews (MathSciNet): MR0326948
    Zentralblatt MATH: 0262.62037
    Digital Object Identifier: doi:10.1093/biomet/60.2.345
    JSTOR: links.jstor.org
  • White, L. V. (1975). The optimal design of experiments for estimation in nonlinear models. Ph.D. dissertation, Univ. London.
  • Whittle, P. (1973). Some general points in the theory of optimal experimental design. J. Roy. Statist. Soc. Ser. B 35 123--130.
    Mathematical Reviews (MathSciNet): MR0356388
    JSTOR: links.jstor.org
  • Wijesinha, M. C. and Khuri, A. I. (1987a). The sequential generation of multiresponse $D$-optimal designs when the variance--covariance matrix is not known. Comm. Statist. Simulation Comput. 16 239--259.
    Mathematical Reviews (MathSciNet): MR0883130
    Digital Object Identifier: doi:10.1080/03610918708812589
    Zentralblatt MATH: 0616.62104
  • Wijesinha, M. C. and Khuri, A. I. (1987b). Construction of optimal designs to increase the power of the multiresponse lack of fit test. J. Statist. Plann. Inference 16 179--192.
    Mathematical Reviews (MathSciNet): MR0895758
    Digital Object Identifier: doi:10.1016/0378-3758(87)90067-X
    Zentralblatt MATH: 0629.62073
  • Wu, C.-F. J. (1985). Efficient sequential designs with binary data. J. Amer. Statist. Assoc. 80 974--984.
    Mathematical Reviews (MathSciNet): MR0819603
    Digital Object Identifier: doi:10.2307/2288563
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0588.62133
  • Wu, C.-F. J. (1988). Optimal design for percentile estimation of a quantal response curve. In Optimal Design and Analysis of Experiments (Y. Dodge, V. V. Federov and H. P. Wynn, eds.) 213--223. North-Holland, Amsterdam.
  • Yan, Z. W., Bate, S., Chandler, R. E., Isham, V. and Wheater, H. (2002). An analysis of daily maximum wind speed in northwestern Europe using generalized linear models. J. Climate 15 2073--2088.
  • Zacks, S. (1977). Problems and approaches in design of experiments for estimation and testing in non-linear models. In Multivariate Analysis IV (P. R. Krishnaiah, ed.) 209--223. North-Holland, Amsterdam.
    Mathematical Reviews (MathSciNet): MR0455232
    Zentralblatt MATH: 0368.62052
  • Zhu, W., Ahn, H. and Wong, W. K. (1998). Multiple-objective optimal designs for the logit model. Comm. Statist. A---Theory Methods 27 1581--1592.
    Mathematical Reviews (MathSciNet): MR1835128
    Digital Object Identifier: doi:10.1214/lnms/1215456187
  • Zhu, W. and Wong, W. K. (2001). Bayesian optimal designs for estimating a set of symmetric quantiles. Statistics in Medicine 20 123--137.
  • Zocchi, S. S. and Atkinson, A. C. (1999). Optimum experimental designs for multinomial logistic models. Biometrics 55 437--444.
    Mathematical Reviews (MathSciNet): MR1705110
    Digital Object Identifier: doi:10.1111/j.0006-341X.1999.00437.x
    JSTOR: links.jstor.org
    Zentralblatt MATH: 1059.62580

The Institute of Mathematical Statistics

Statistical Science

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more