Inference for the Optimum Using Linear Regression Models with Discrete Inputs

Abstract

We present a multiple-comparison-with-the-best procedure to provide inference for the optimum from regression models with discrete inputs. Two applications are given to illustrate the methodology: two-level factorial designs to identify the best drug combination and order-of-addition experiments where the primary objective is to identify the sequence with the largest mean response. The methods easily accommodate restrictions limiting the inference set of conditions. We use simulation to determine the critical values. While the methods apply to any linear regression model, we identify cases that require just a single critical value, and we also show where approximations and upper bounds mitigate the need for intensive computation. We tabulate the required critical values for a variety of common applications: the main-effect model and two-factor interaction model estimated by certain two-level factorial designs, and the pairwise order model and several component-position models for estimation based on optimal order-of-addition designs. Our work greatly simplifies the problem of rigorous inference for the optimum from regression models with discrete inputs.

Keywords:

• Component-position model
• Experimental design
• Factorial design
• Multiple comparison with the best
• Order-of-addition
• Pairwise order model

Supplementary Materials

In the online supplementary materials of this article, we provide the three nonequivalent covariance matrices for Example 1, the MCB inference for Example 1 with alternative models, MCB for Example 2 with the candidate set limited to combinations with only 3 or 4 drugs, proofs of Theorems 2 and 3 plus additional theorems and corollaries for component-position models, the critical values corresponding to Figure 1, and MCB inference for further order-of-addition examples. Additionally, the zip file contains R code for performing MCB inference, as well as an R Markdown file for replicating the results in Tables 2–4.

Acknowledgments

The insightful comments of the editor, associate editor and a referee helped us improve the clarity of this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors are especially grateful for the China Scholarship Council’s funding that provided for them to work together during Dr. Li’s 8-month visit at the University of Tennessee. The authors acknowledge National Natural Science Foundation of China (grant nos. 12131001, 12226343, 12022115, 11971465) and the National Ten Thousand Talents Program.