Bayes Optimal Designs for Two- and Three-Level Factorial Experiments

Abstract Two- and three-level factorial designs are often used for exploratory experiments with many independent variables. The purpose of such experiments is to estimate the effects of all active independent variables and their interactions. For reasons of economy or other constraints, only a fraction of the full number of experimental settings of the independent variables may be run, precluding the separate data-based estimation of all the effects. In this situation of relatively sparse data and many unknown parameters, the benefits of reliable prior information, used in the estimation and experimental design, can be especially significant. In this article, exact Bayes-optimal designs of two- and three-level fractional factorial experiments and of blocked two- and three-level factorial experiments are derived. The optimality criteria used in this process are versions of the Bayes ψ criterion introduced by Duncan and DeGroot. It is shown that the classical fractional factorial designs for the 2k experime...

[1]  W. David Kelton,et al.  Statistical design and analysis , 1986, WSC '86.

[2]  Alessandra Giovagnoli,et al.  Bayes D-optimal and E-optimal block designs , 1983 .

[3]  William G. Cochran,et al.  Experimental Designs, 2nd Edition , 1950 .

[4]  W. J. Studden,et al.  Robust Bayesian Experimental Designs in Normal Linear Models , 1991 .

[5]  Peter W. M. John.  Statistical Methods in Engineering and Quality Assurance , 1990 .

[6]  Jürgen Pilz,et al.  Bayesian estimation and experimental design in linear regression models , 1992 .

[7]  P. John.  Statistical Design and Analysis of Experiments , 1971 .

[8]  S. R. Searle.  Linear Models , 1971 .

[9]  W. J. Studden,et al.  Bayesian Optimal Designs for Linear Regression Models , 1991 .

[10]  George E. P. Box,et al.  Empirical Model‐Building and Response Surfaces , 1988 .

[11]  K. Chaloner.  Optimal Bayesian Experimental Design for Linear Models , 1984 .

[12]  James R. Thompson.  Empirical Model Building , 1989 .

[13]  Paul H. Garthwaite,et al.  Quantifying Expert Opinion in Linear Regression Problems , 1988 .

[14]  I. Verdinelli,et al.  A note on Bayes designs for inference using a hierarchical linear model , 1980 .

[15]  F. Yates.  Design and Analysis of Factorial Experiments , 1958 .

[16]  F. Kolyva-Machera.  D-optimality in 3k-designs for N ≡ 1 mod 9 observations , 1989 .

[17]  H. Wynn,et al.  G-majorization with applications to matrix orderings , 1985 .

[18]  R. J. Owen,et al.  The Optimum Design of a Two-Factor Experiment Using Prior Information , 1970 .

[19]  Wayne S. Smith,et al.  Interactive Elicitation of Opinion for a Normal Linear Model , 1980 .

[20]  Blaza Toman,et al.  Bayesian optimal experimental design for treatment—control comparisons in the presence of two-way heterogeneity , 1991 .

[21]  Y. L. Tong.  The multivariate normal distribution , 1989 .