Designs for the simultaneous inference of concentration–response curves

BACKGROUND: An important problem in toxicology in the context of gene expression data is the simultaneous inference of a large number of concentration–response relationships. The quality of the inference substantially depends on the choice of design of the experiments, in particular, on the set of different concentrations, at which observations are taken for the different genes under consideration. As this set has to be the same for all genes, the efficient planning of such experiments is very challenging. We address this problem by determining efficient designs for the simultaneous inference of a large number of concentration–response models. For that purpose, we both construct a D-optimality criterion for simultaneous inference and a K-means procedure which clusters the support points of the locally D-optimal designs of the individual models. RESULTS: We show that a planning of experiments that addresses the simultaneous inference of a large number of concentration–response relationships yields a substantially more accurate statistical analysis. In particular, we compare the performance of the constructed designs to the ones of other commonly used designs in terms of D-efficiencies and in terms of the quality of the resulting model fits using a real data example dealing with valproic acid. For the quality comparison we perform an extensive simulation study. CONCLUSIONS: The design maximizing the D-optimality criterion for simultaneous inference improves the inference of the different concentration–response relationships substantially. The design based on the K-means procedure also performs well, whereas a log-equidistant design, which was also included in the analysis, performs poorly in terms of the quality of the simultaneous inference. Based on our findings, the D-optimal design for simultaneous inference should be used for upcoming analyses dealing with high-dimensional gene expression data.