The investigator identifies optimal and efficient designs for non-linear models. The focus is on (1) generalized linear models (GLMs) for binary data or count data; and (2) non-linear models for Event Related functional Magnetic Resonance Imaging (ER-fMRI) experiments. For the first of these, recent results are mostly restricted to models with a single covariate. The investigator studies common GLMs, such as logistic, probit and loglinear models, with multiple covariates and higher order terms. He develops novel theory and computational tools for identifying locally optimal designs under various optimality criteria as well as for identifying robust designs. For the second problem, the investigator identifies optimal and efficient designs under more realistic non-linear models for the combined objectives of estimation of the hemodynamic response function (HRF) and detection of brain activity. Traditionally, two separate linear models have been used for these disparate objectives. The use of a single non-linear model for modeling the hemodynamic response facilitates the simultaneous pursuit of both objectives. This approach provides not only a more natural formulation of design optimality criteria, but also results in better designs for ER-fMRI experiments.
Binary data and count data are very common in many scientific fields, such as drug discovery, clinical trials, social sciences, marketing, etc. While models and methods of analysis for such data are well established, the study of optimal design for the efficient use of available resources lags considerably. For example, when planning a dose-response study, it is important to know which dose levels of a drug should be used in the study, and how many subjects should be assigned to these levels in order to get the most information for questions that are of scientific interest. Recent advances and new tools developed by the investigator and his collaborators make it possible to derive optimal designs for a variety of commonly used models. For a second part of the project, the investigator finds efficient designs for ER-fMRI experiments. These experiments are part of a cutting edge approach for studying brain activity caused by certain simple tasks. A subject in an MRI scanner is presented with a series of tasks, each of them repeated multiple times, and the hemodynamic response is measured. The investigator identifies optimal and efficient orders for presenting the tasks to a subject in order to gain as much information as possible for the scientific goals of the experiment.
Experiments are an integral part of the discovery process in scientific studies. It is how we begin to understand relationships of variables of interest, including causal relationships. As an example, in a pharmaceutical study, an experiment may help to understand how increasing the dose of an active ingredient in a drug affects the body, such as in terms of concentration of the drug in the bloodstream after a certain amount of time, or in terms of blood pressure, or in terms of another variable that is of interest in the study. Such studies are also called dose-response studies. Designed experiments also help in the manufacturing process to make products of higher quality, such as better cars, better refrigerators, and so on. When we conduct an experiment, we would like to get the maximum amount of information, at a given cost, about questions of interest for the study (for example related to relationships between variables). Using a dose-response study as an example, in an experiment choices will have to be made for the number of subjects to include in the study and the doses that will be used for these subjects. Typically, the more subjects that are included, the more information that one can obtain. Cost considerations may however limit the number of subjects that can be used. If, for simplicity, 10 subjects can be used, we will not learn much about the relationship between the dose and a response variable of interest if we give each subject in the study the same dose. Part of such an experiment consists of making choices for the different doses that are used. These need not be 10 different doses, nor need they be equally spaced over a region of interest. Which doses to use depends on the parametric model that is used to describe the relationship between the dose and response variable. For linear models, this is a well studied problem; it is not nearly as well understood for generalized linear and nonlinear models. In this project, the PI developed tools to identify the best possible designs, according to an appropriate measure of information, for a wide variety of different generalized linear and nonlinear models. The tools consist of a combination of theoretical and computational results. Not only do they enable experimenters to use the best possible designs under these models and measures of information, but they also facilitate an evaluation of other designs, which an experimenter may prefer for practical reasons or because he or she is unsure about the correct dose-response model. Generalized linear models and nonlinear models are used frequently, and not just in the dose-response context, which is merely used as an example here. Knowledge about finding the best possible design in terms of maximizing information obtained from the experiment is therefore critically important for a wide variety of applications. One of the key results in this project is that it is shown that, again with reference to the dose-response example, for most of the reasonable measures of information, one does not have to consider designs with an arbitrary number of doses, but one can limit the choice often to 3 or 4 doses, depending on the model that is used to describe the dose-response relationship. Precise answers for the maximum number of doses that need to be considered for optimal designs are provided in the research articles for many different models. Since this maximum number is always small, it then becomes much easier to find the best design since it is only necessary to search among all designs with no more than this maximum number of doses. Surprisingly, the PI and his collaborators were able to establish that results apply equally for so called fixed effects models as for mixed effects models. All of this should help greatly in assessing the quality of designs used for experiments, resulting in the end in experiments that provide more information at a fixed cost.
