An integer linear programing approach to find trend-robust run orders of experimental designs

Abstract

When a multifactor experiment is carried out over a period of time, the responses may depend on a time trend. Unless the tests of the experiment are conducted in proper order, the time trend has a negative impact on the precision of the estimates of the main effects, the interaction effects, and the quadratic effects. A proper run order, called a trend-robust run order, minimizes the confounding between the effects’ contrast vectors and the time trend’s linear, quadratic, and cubic components. Finding a trend-robust run order is essentially a permutation problem. We develop a multistage approach based on integer programing to find a trend-robust run order for any given design. The multistage nature of our algorithm allows us to prioritize the trend robustness of the main-effect estimates. In the literature, most of the methods used are tailored to specific designs and are not applicable to an arbitrary design. Additionally, little attention has been paid to trend-robust run orders of response surface designs, such as central composite designs, Box–Behnken designs, and definitive screening designs. Our algorithm succeeds in identifying trend-robust run orders for arbitrary factorial designs and response surface designs with two up to six factors.

Keywords:

• factorial designs
• mixed integer program
• multistage approach
• orthogonality
• response surface designs
• trend robustness

Additional information

Notes on contributors

José Núñez Ares

José Núñez Ares completed his doctoral studies at the Faculty of BioScience Engineering of the University of Leuven (KU Leuven) in 2018 with a PhD thesis titled “Trend-robust and minimally aliased response surface designs by means of integer linear programming” under the supervision of Prof. Peter Goos. He then moved to the University of Wisconsin-Madison to pursue a PostDoc with Prof. Jeff Linderoth. There he worked on several topics in theory and applications of integer programming optimization. Since October 2018 he is working as a Postdoctoral researcher in design of experiments and optimization at the Faculty of BioScience Engineering of the University of Leuven (KU Leuven), under the guide of Prof. Peter Goos and Dr. Bart de Ketelaere.

Peter Goos

Peter Goos is a full professor at the Faculty of Bio-Science Engineering of the University of Leuven and at the Faculty of Applied Economics of the University of Antwerp, where he teaches various introductory and advanced courses on statistics and probability. His main research area is the statistical design and analysis of experiments. Besides numerous influential articles in various kinds of scientific journals, he published the books The Optimal Design of Blocked and Split-Plot Experiments, Optimal Experimental Design: A Case-Study Approach, Statistics with JMP: Graphs, Descriptive Statistics and Probability and Statistics with JMP: Hypothesis Tests, ANOVA and Regression. For his work, Peter Goos has received two Shewell Awards and two Lloyd S. Nelson Awards of the American Society for Quality, the Ziegel Award and the Statistics in Chemistry Award from the American Statistical Association, and the Young Statistician Award of the European Network for Business and Industrial Statistics (ENBIS).