## The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems (1996)

Venue: | Journal of the American Statistical Association |

Citations: | 16 - 3 self |

### BibTeX

@ARTICLE{Clyde96theequivalence,

author = {Merlise Clyde and Kathryn Chaloner},

title = {The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems},

journal = {Journal of the American Statistical Association},

year = {1996},

volume = {91},

pages = {1236--1244}

}

### OpenURL

### Abstract

Several competing objectives may be relevant in the design of an experiment. The competing objectives may not be easy to characterize in a single optimality criterion. One approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria. An alternative approach has been to optimize one criterion subject to constraints on the other criteria. An equivalence theorem is presented for the Bayesian constrained design problem. Equivalence theorems are essential in verifying optimality of proposed designs, especially when, as in most nonlinear design problems, numerical optimization is required. This theorem is used to show that the results of Cook and Wong on the equivalence of the weighted and constrained problems also apply much more generally. The results are applied to Bayesian nonlinear design problems with several objectives. KEY WORDS: Bayesian design, regression, nonlinear design 1. INTRODUCTION An experimen...

### Citations

3278 | Convex analysis - Rockafellar - 1970 |

1263 | Decisions with Multiple Objectives: Preferences and Value Tradeoffs - Keeney, Raiffa - 1988 |

1245 |
Statistical decision theory and Bayesian analysis
- Berger
- 1985
(Show Context)
Citation Context ...approximation to using a negative weighted quadratic loss function as a utility function. The quadratic loss function may also be a useful approximation to the true loss function in other situations (=-=Berger, 1985 page 61-=-). Note that if the prior distribution is point mass on one value of `, this special case corresponds to local optimality. The criteria OE D (��) and OE A (��) are typically concave over the s... |

175 | Bayesian experimental design: A review - Chaloner, Verdinelli - 1995 |

167 |
Optimal Design of Experiments
- Pukelsheim
- 2006
(Show Context)
Citation Context ...roblem. In many experimental design problems, equivalence theorems are essential for verifying optimality of a proposed design (see for example Silvey, 1980, Chaloner, 1984, Chaloner and Larntz 1989, =-=Pukelsheim, 1993-=-). This is particularly important in many Bayesian and nonlinear design problems where numerical methods must be used to find optimal designs. For recent examples of Bayesian and nonlinear design prob... |

82 |
User’s guide for NPSOL (version 4.0): A Fortran package for nonlinear programming
- Gill, Murray, et al.
- 1986
(Show Context)
Citation Context ...find an optimal design. There are several approaches that can be used for finding constrained optimal designs. The programs in this paper for numerical optimization use the NPSOL FORTRAN subroutines (=-=Gill et al., 1986-=-) for optimization of a nonlinear function subject to nonlinear constraints. This approach can also be used when the different criteria are not concave, although we no longer are assured that the solu... |

82 | LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics - Tierney - 1990 |

71 | Expected information as expected utility - Bernardo - 1979 |

64 |
Optimal Design
- Silvey
- 1980
(Show Context)
Citation Context ...ry constrained problem there is an equivalent weighted problem. In many experimental design problems, equivalence theorems are essential for verifying optimality of a proposed design (see for example =-=Silvey, 1980-=-, Chaloner, 1984, Chaloner and Larntz 1989, Pukelsheim, 1993). This is particularly important in many Bayesian and nonlinear design problems where numerical methods must be used to find optimal design... |

45 |
Optimal Bayesian Design Applied to Logistic Regression Experiments
- Chaloner, Larntz
- 1989
(Show Context)
Citation Context ...s an equivalent weighted problem. In many experimental design problems, equivalence theorems are essential for verifying optimality of a proposed design (see for example Silvey, 1980, Chaloner, 1984, =-=Chaloner and Larntz 1989-=-, Pukelsheim, 1993). This is particularly important in many Bayesian and nonlinear design problems where numerical methods must be used to find optimal designs. For recent examples of Bayesian and non... |

37 |
Locally optimal designs for estimating parameters
- Chernoff
- 1953
(Show Context)
Citation Context ..., optimizing over the set of information matrices can be applied. This method has been called "local optimality" as the designs should be reasonably efficient for values of ` close to the be=-=st guess (Chernoff, 1953-=-; White, 1973). If a prior distribution is put on ` and if a criterion is used which is the expectation, over the prior distribution, of a local optimality criterion, then methods for linear models ca... |

34 |
Bayesian Experimental Design
- Chaloner, Verdinelli
- 1995
(Show Context)
Citation Context ...o in efficient estimates of the parameters for a given model. Other factors such as minimizing costs or maximizing the output may also be important in selecting a design (Verdinelli and Kadane, 1992; =-=Verdinelli, 1992-=-). One approach to the design problem is to create a new objective function based on a weighted average of several design criteria and has been termed either a compound or a weighted design problem (L... |

19 | Sequential Experiments - DeGroot, Information - 1962 |

18 | Optimal design via curve fitting of Monte carlo experiment
- Müeller, Parmigiani
- 1995
(Show Context)
Citation Context ...constrained or weighted optimal designs (Clyde, 1993) and to plot directional derivatives. Under the weighted design formulation, other optimization methods for finding optimal designs (Haines, 1987; =-=Muller and Parmigiani, 1995-=-) can also be used directly. 3. EXAMPLES 3.1 Bayes A-optimality Bayesian c--optimality, as in (2), is justified as approximate expected square error loss for the estimation of some function g(`). If s... |

17 |
Bayesian Estimation and Experimental Design in Linear Regression Models
- Pilz
- 1991
(Show Context)
Citation Context ...nal derivatives and so equivalence theorems can be derived to verify optimality of a design (Whittle, 1973; Chaloner and Larntz, 1989). An exact design can be obtained by "rounding" a design=-= measure (Pilz, 1989-=- page 183; Pukelsheim and Reider, 1992). Of course, if the optimal design measure corresponds to an exact design it is optimal for the exact optimal design problem. In linear design problems the equiv... |

15 |
The application of the annealing algorithm to construction of exact optimal designs for linear regression models, Technometrics 29
- Haines
- 1987
(Show Context)
Citation Context ...PSTAT to find constrained or weighted optimal designs (Clyde, 1993) and to plot directional derivatives. Under the weighted design formulation, other optimization methods for finding optimal designs (=-=Haines, 1987-=-; Muller and Parmigiani, 1995) can also be used directly. 3. EXAMPLES 3.1 Bayes A-optimality Bayesian c--optimality, as in (2), is justified as approximate expected square error loss for the estimatio... |

14 |
Optimal Bayesian Experimental Design for Linear Models
- Chaloner
- 1984
(Show Context)
Citation Context ... problem there is an equivalent weighted problem. In many experimental design problems, equivalence theorems are essential for verifying optimality of a proposed design (see for example Silvey, 1980, =-=Chaloner, 1984-=-, Chaloner and Larntz 1989, Pukelsheim, 1993). This is particularly important in many Bayesian and nonlinear design problems where numerical methods must be used to find optimal designs. For recent ex... |

12 | Optimal experimental designs for properties of a compartmental model - Atkinson, Chaloner, et al. - 1993 |

12 |
Bayesian Designs for Maximizing Information and Outcome
- Verdinelli, Kadane
- 1992
(Show Context)
Citation Context ...nonlinear design 1. INTRODUCTION An experiment may often have several, competing objectives which cannot be easily characterized by only one of the standard optimality criteria (Box and Draper, 1975; =-=Verdinelli and Kadane, 1992-=-). This is particularly true in nonlinear models, where one may be interested in both discriminating between two or more competing models, and also in efficient estimates of the parameters for a given... |

10 | On the equivalence of constrained and compound optimal designs - Cook, Wong - 1994 |

9 |
Robust Designs
- Box, Draper
- 1975
(Show Context)
Citation Context ...n design, regression, nonlinear design 1. INTRODUCTION An experiment may often have several, competing objectives which cannot be easily characterized by only one of the standard optimality criteria (=-=Box and Draper, 1975-=-; Verdinelli and Kadane, 1992). This is particularly true in nonlinear models, where one may be interested in both discriminating between two or more competing models, and also in efficient estimates ... |

9 |
An object-oriented system for Bayesian nonlinear design using xlisp-stat
- CLYDE
- 1993
(Show Context)
Citation Context ...(or the 's) from the output can be used to plot the directional derivatives to verify the optimality of a design. Functions are available in XLISPSTAT to find constrained or weighted optimal designs (=-=Clyde, 1993-=-) and to plot directional derivatives. Under the weighted design formulation, other optimization methods for finding optimal designs (Haines, 1987; Muller and Parmigiani, 1995) can also be used direct... |

9 | On optimal screening ages - PARMIGIANI - 1993 |

9 |
Optimal experimental design for polynomial regression
- Stigler
- 1971
(Show Context)
Citation Context ...90). An alternative approach is to optimize one primary design criterion, subject to constraints for a specified minimum efficiency of other criteria. This is known as the constrained design problem (=-=Stigler, 1971-=-). These two approaches have been considered separately, as two different approaches, as ways of dealing with multiple objectives in the optimal design problem. Pukelsheim and Rosenberger (1993), for ... |

7 | Experimental designs for model discrimination - Pukelsheim, Rosenberger - 1993 |

7 |
An extension of the general equivalence theorem to nonlinear models
- WHITE
- 1973
(Show Context)
Citation Context ...r the set of information matrices can be applied. This method has been called "local optimality" as the designs should be reasonably efficient for values of ` close to the best guess (Cherno=-=ff, 1953; White, 1973-=-). If a prior distribution is put on ` and if a criterion is used which is the expectation, over the prior distribution, of a local optimality criterion, then methods for linear models can no longer b... |

6 |
Bayesian design for estimating the turning point of a quadratic regression
- Chaloner
- 1989
(Show Context)
Citation Context ...be �� = [\Gamma1; 1]. Standard methods show the optimal design is to take a proportion (7\Gamma2 p 7)=6 of the observations at both +1 and \Gamma1 and (2 p 7 \Gamma 4)=3 of the observations at 0 (=-=see Chaloner, 1989-=-, for example). The D--optimal design is to take 1/3 of the observations at-1, 0 and +1. As it may be of interest to both estimate g and also the parameter vector ` T = (` 0 ; ` 1 ; ` 2 ), some combin... |

6 | Cost-Utility Analysis of Alternative Strategies in Screening for Breast Cancer - Parmigiani, Kamlet - 1992 |

6 |
Efficient rounding of approximate designs
- Pukelsheim, Rieder
- 1992
(Show Context)
Citation Context ...o equivalence theorems can be derived to verify optimality of a design (Whittle, 1973; Chaloner and Larntz, 1989). An exact design can be obtained by "rounding" a design measure (Pilz, 1989 =-=page 183; Pukelsheim and Reider, 1992-=-). Of course, if the optimal design measure corresponds to an exact design it is optimal for the exact optimal design problem. In linear design problems the equivalence theorems are usually stated in ... |

5 |
Model-robust, linear-optimal designs
- Cook, Nachtsheim
- 1982
(Show Context)
Citation Context ... the design problem is to create a new objective function based on a weighted average of several design criteria and has been termed either a compound or a weighted design problem (Lauter 1974, 1976; =-=Cook and Nachtsheim 1982-=-; Dette 1990). An alternative approach is to optimize one primary design criterion, subject to constraints for a specified minimum efficiency of other criteria. This is known as the constrained design... |

5 | A geometric approach to optimal design for one-parameter non-linear models - HAINES - 1995 |

4 |
A generalization of D- and D 1 -optimal designs in polynomial regression
- Dette
- 1990
(Show Context)
Citation Context ...create a new objective function based on a weighted average of several design criteria and has been termed either a compound or a weighted design problem (Lauter 1974, 1976; Cook and Nachtsheim 1982; =-=Dette 1990-=-). An alternative approach is to optimize one primary design criterion, subject to constraints for a specified minimum efficiency of other criteria. This is known as the constrained design problem (St... |

4 | Constrained optimal designs - Lee - 1988 |

3 | Optimal multipurpose designs for regression models - Lauter - 1976 |

3 | Constrained Optimal Designs for Regression Models - Lee - 1987 |

2 | Comparison of Experiments and Information Measures - Goel, DeGroot - 1979 |

2 |
Experimental Planning in a Class of Models," Mathematische Operationsforschung und Statistik
- Lauter
- 1974
(Show Context)
Citation Context ...2). One approach to the design problem is to create a new objective function based on a weighted average of several design criteria and has been termed either a compound or a weighted design problem (=-=Lauter 1974-=-, 1976; Cook and Nachtsheim 1982; Dette 1990). An alternative approach is to optimize one primary design criterion, subject to constraints for a specified minimum efficiency of other criteria. This is... |

2 | Towards Models of Screening for Early Detection of Disease - Shahani - 1977 |

1 | D-optimum designs for heteroskedastic linear models - Atkinson, Cook - 1995 |

1 | Changes in utility as information," Bayesian Decision Analysis - DeGroot - 1984 |

1 |
Necessary Conditions for an Extremum. Translated by K
- Pshenichnyi
- 1971
(Show Context)
Citation Context ...re continuous in x. Under these assumptions \Xi is compact in the weak convergence topology. In the following theorems, these conditions ensure the existence of a solution to the constrained problem (=-=Pshenichnyi, 1971-=-) and a straightforward method of checking the optimality of designs by plotting directional derivatives. Many optimality criteria that are used satisfy these conditions, such as the Bayesian design c... |