The investigators study the statistical theory underlying control of error rates for adaptive analysis of sparse factorial designs. Considered are orthogonal and nonorthogonal saturated designs, nearly saturated designs, and supersaturated designs. Methods under consideration rely upon the presence of an adequate but unknown level of effect sparsity, for lack of an adequate variance estimator independent of the effect estimates, and adapt to the level of effect sparsity suggested by the data. Much of the difficulty and intrigue lies in the apparently circular nature of the required arguments, since relatively large effect estimates are in some sense adaptively set aside for variance estimation, then the resulting variance estimate is used to make inferences about the effects. Lack of an adequate independent variance estimator gives rise to a variety of problems that are important and technically challenging. The investigators bring to bear on these problems a geometric perspective and the rigor and perspective of the multiple comparisons field, seeking methods of inference that control error rate strongly, while using the data adaptively and efficiently.
The investigators study the probabilistic foundations of data analysis for sophisticated experiments fundamental to the development and production of high-quality, low-cost products. Critical characteristics often depend in unknown ways on an unknown subset of a larger collection of variables or factors. The resulting necessity to study many factors simultaneously in small, economical experiments--essentially to learn a lot from but a little data--presents many statistical challenges. While great progress has been made in the development of designs or plans for conducting such experiments, fundamental problems concerning the corresponding data analysis remain. The investigators study the theoretical foundations for the analysis of data from such experiments, and consequently propose innovative methods of data analysis, justifying them mathematically. This entails seeking solutions to a variety of related problems in probability pertinent to the analysis of data from such factorial experiments. Results have direct applications in engineering product and process design and in statistical process control and improvement, enhancing efforts to achieve competitive economic advantage.
