Optimal Data Augmentation Strategies for Additive Models

Consider an experiment where the factors are measured on a continuous scale, and suppose that the experimenter is permitted to augment the existing observations with one or more new data points. Bondar's universal optimality criterion (U optimality) suggests that the problem is best studied in the eigenvector coordinate system. We proceed by showing how to construct new points that first equate and then jointly increase the smaller eigenvalues of the crossproduct of the independent variables. We discuss the limitations of design augmentation strategies based solely on the crossproduct matrix. Our goals are to equate and minimize the variances of the estimated regression coefficients, keep the new points constrained in a prespecified experimental region, use as much of the information in the original points as possible, and keep the number of required new points as small as possible. We offer several data augmentation strategies to meet these requirements. As U optimality subsumes each of the D, A, E, and (M, S) optimality criteria, our strategies guarantee the U, D, A, E, and (M, S) optimality of the set of new points. The recommended number of additional points depends on how nearly optimal is the layout based on the existing observations and on the tightness of the regional constraints. We illustrate our strategies with a well-known experimental data set. For an additive model, our U-optimal solutions to the data augmentation problem are superior to other solutions available in the literature.