Richard F. Gunst’s research while affiliated with Southern Methodist University and other places

Various statistical models have been proposed to analyze fMRI data. The usual goal is to make inferences about the effects that are related to an external stimulus. The primary focus of this paper is on those statisti-cal methods that enable one to detect 'significantly activated' regions of the brain due to event-related stimuli. Most of these methods share a common property, requiring estimation of the hemodynamic response function (HRF) as part of the deterministic component of the statistical model. We propose and investigate a new approach that does not require HRF fits to detect 'activated' voxels. We argue that the method not only avoids fitting a specific HRF, but still takes into account that the unknown response is delayed and smeared in time. This method also adapts to differential re-sponses of the BOLD response across different brain regions and experimen-tal sessions. The maximum cross-correlation between the kernel-smoothed stimulus sequence and shifted (lagged) values of the observed response is the proposed test statistic. Using our recommended approach we show through realistic simulations and with real data that we obtain better sensitivity than simple correlation methods using default values of SPM2. The simulation experiment incor-porates different HRFs empirically determined from real data. The noise models are also different AR(3) fits and fractional Gaussians estimated from real data. We conclude that our proposed method is more powerful than simple correlation procedures, because of its robustness to variation in the HRF.

Paul D. Minton graduated with Bachelor of Science and Master of Science degrees in Mathematics from Southern Methodist University (SMU). Following the completion of his doctoral degree in Statistics from the University of North Carolina (Chapel Hill) in 1951, he returned to SMU along with a vision of a unique Statistics Department within a university that had as its primary mission a broad-based liberal arts undergraduate education. Minton's vision focused on his belief that statisticians were collaborators and that there was no limit on opportunities for collaboration in Statistics, regardless of the nature and mission of an academic institution. The key to achieving this vision was the collaborative merging of interests and opportunities within the university and throughout the local region. © 2013 Springer Science+Business Media New York. All rights reserved.

To properly formulate functional magnetic resonance imaging (fMRI) experiments with complex mental activity, it is advantageous to permit great flexibility in the statistical components of the design of these studies. The length of an experiment, the placement of various stimuli and the modeling approach used all affect the ability to detect mental activity. Major advances in understanding the implications of various designs of fMRI experiments have taken place over the last decade. Nevertheless, new and increasingly difficult issues relating to the modeling of hemodynamic responses and the detection of activated brain regions continue to arise because of the increasing complexity of the experiments. In this article, the D-optimality criterion is used in conjunction with a genetic algorithm to create probability-based design generators for the selection of designs in event-related fMRI experiments where the hemodynamic response function is modeled with a function that is nonlinear in the parameters. The designs produced by these generators are shown to perform well compared with locally D-optimal designs and provide insight into optimal design characteristics that investigators can utilize in the selection of interstimulus intervals. Designs with these characteristics are shown to be applicable to fMRI studies involving one or two stimulus types. The designs are also shown to be robust with respect to misspecification of an AR(1) error autocorrelation and compare favorably with a maximin procedure. Copyright © 2012 John Wiley & Sons, Ltd.

Several authors have proposed nonparametric semivariogram estimators. Shapiro and Botha (1991) did so by application of Bochner’s theorem and Cherry et al. (1996) further investigated this technique where it performed favorably against parametric estimators even when data were generated under the parametric model. While the former makes allowances for a prescribed nugget and the latter outlines a possible approach, neither of these demonstrate nugget estimation in practice, which is essential to spatial modeling and proper statistical inference. We propose a modified form of this method, which admits practical nugget estimation and broadens the basis. This is achieved by a simple change to the basis and an appropriate restriction of the node space as dictated by the first root of the Bessel function of the first kind of order ν. The efficacy of this new unsupervised semiparametric method is demonstrated via application and simulation, where it is shown to be comparable with correctly specified parametric models while outperforming misspecified ones. We conclude with remarks about selecting the appropriate basis and node space definition.

Collinearities can occur among the error-free (true) predictors in measurement-error models just as they occur among predictors in traditional regression models. The coefficient estimators for measurement-error models suffer the same ill effects of collinearities as do least squares estimators. For linear measurement-error models, collinearities are a property of the secondorder moment matrix of the unobservable error-free predictors. For nonlinear models, they are a property of the second-order moment matrix of derivatives of the nonlinear function. In this article, collinearities among the predictors in measurement-error models are defined, and diagnostics for their detection are discussed. The collinearity diagnostics introduced are similar to those used in traditional regression models, but they are applied to estimated second-order moment matrices of the error-free predictors rather than to the second-order moment matrix of the observed predictors. Examples are discussed, one of which demonstrates the masking of collinearities that can result from measurement errors.

Preface. Introduction. Building Empirical Models. Two--Level Factorial Designs. Two--Level Fractional Factorial Designs. Process Improvement with Steepest Ascent. The Analysis of Second--Order Response Surfaces. Experimental Designs for Fitting Response Surfaces--I. Experimental Designs for Fitting Response Surfaces--II. Advanced Response Surface Topics I. Advanced Response Surface Topics II. Robust Parameter Design and Process Robustness Studies. Experiments with Mixtures. Other Mixture Design and Analysis Techniques. Continuous Process Improvement with Evolutionary Operation. Appendix 1: Variable Selection and Model--Building in Regression. Appendix 2: Multicollinearity and Biased Estimation in Regression. Appendix 3: Robust Regression. Appendix 4: Some Mathematical Insights into Ridge Analysis. Appendix 5: Moment Matrix of a Rotatable Design. Appendix 6: Rotatability of a Second--Order Equiradial Design. Appendix 7: Relationship Between D--Optimality and the Volume of a Joint Confidence Ellipsoid on s. Appendix 8: Relationship Between Maximum Prediction Variance in a Region and the Number of Parameters. Appendix 9: The Development of Equation (8.21). Appendix 10: Determination of Data Augmentation Result (Choice of xr+1 for the Sequential Development of a D--Optimal Design). Index.