Inference for the Optimum Using Linear Regression Models with Discrete Inputs

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.