Optimal Designs for Comparing Regression Curves: Dependence Within and Between Groups

We consider the problem of designing experiments for the comparison of two regression curves describing the relation between a predictor and a response in two groups, where the data between and within the group may be dependent. In order to derive efficient designs we use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in a corresponding continuous model. It is demonstrated that in general simultaneous estimation using the data from both groups yields more precise results than estimation of the parameters separately in the two groups. Using the BLUE from simultaneous estimation, we then construct an efficient linear estimator for finite sample size by minimizing the mean squared error between the optimal solution in the continuous model and its discrete approximation with respect to the weights (of the linear estimator). Finally, the optimal design points are determined by minimizing the maximal width of a simultaneous confidence band for the difference of the two regression functions. The advantages of the new approach are illustrated by means of a simulation study, where it is shown that the use of the optimal designs yields substantially narrower confidence bands than the application of uniform designs.

Being Uncertain in Chromatographic Calibration—Some Unobvious Details in Experimental Design

This is an introductory tutorial and review about the uncertainty problem in chromatographic calibration. It emphasizes some unobvious, but important details influencing errors in the calibration curve estimation, uncertainty in prediction, as well as the connections and dependences between them, all from various perspectives of uncertainty measurement. Nonuniform D-optimal designs coming from Fedorov theorem are computed and presented. As an example, all possible designs of 24 calibration samples (3–8, 4–6, 6–4, 8–3 and 12–2, both uniform and D-optimal) are compared in context of many optimality criteria. It can be concluded that there are only two independent (orthogonal, but slightly complex) trends in optimality of these designs. The conclusions are important, as the uniform designs with many concentrations are not the best choices, contrary to some intuitive perception. Nonuniform designs are visibly better alternative in most calibration cases.

The Construction of Effective Multi-Dimensional Computer Designs of Experiments Based on a Quasi-Random Additive Recursive Rd-sequence

Uniform multi-dimensional designs of experiments for effective research in computer modelling are highly demanded. The combinations of several one-dimensional quasi-random sequences with a uniform distribution are used to create designs with high homogeneity, but their optimal choice is a separate problem, the solution of which is not trivial. It is believed that now the best results are achieved using Sobol’s LPτ-sequences, but this is not observed in all cases of their combinations. The authors proposed the creation of effective uniform designs with guaranteed acceptably low discrepancy using recursive Rd-sequences and not requiring additional research to find successful combinations of vectors set distributed in a single hypercube. The authors performed a comparative analysis of both approaches using indicators of centred and wrap-around discrepancies, graphical visualization based on Voronoi diagrams. The conclusion was drawn on the practical use of the proposed approach in cases where the requirements for the designs allowed restricting to its not ideal but close to it variant with low discrepancy, which was obtained automatically without additional research.