Seminar on Nonlinear Waves
Brown University Center for Statistical Sciences Seminar
Abstract: Competing risk problems arise when the failure of an individual may be one of several distinct types or causes. The Cox model can be utilized to estimate the cause-specific hazard function. However the Cox model may not fit the data well. Although the Kaplan-Meier curve is frequently used in competing risk settings, it is often unsuitable since the cumulative cause-specific hazard has no direct clinical interpretation. An alternative estimator is the cumulative incidence function. We developed an inference procedure for quantifying the difference between two cumulative incidence functions using a class of semi-parametric linear transformation models under which an unknown transformation of the survival time is linearly related to the covariates with completely specified error distributions. Pointwise and simultaneous confidence intervals for cumulative incidence functions are constructed. A graphical method for choosing an appropriate model from this class is provided. All procedures are illustrated with data from a cancer clinical trial.
Stochastic Systems Seminar
Joint Applied Mathematics and Electrical Engineering Seminar (Pattern Theory/LEMS)
Abstract: Objects are everywhere - natural and man-made. With advances in technology, images in 2-D and 3-D provide easily accessible information on objects, especially their shapes. The field of shape analysis gives methods for the study of the shape of the objects where location, rotation and scale information can be removed. Assuming that a shape can be described by its landmarks, there have been significant statistical advances in this decade. It is in contrast with the historical work started in early 1900 by Karl Pearson where the measurements were mostly distances, measured by using callipers.
Some statistical aspects of the field have been summarized in the recent book on this topic: Dryden and Mardia (1998) Wiley. We will describe the lastest advances in statistical methodology to measure, describe and compare the shape of objects. To make this material generally accessible, we start from the analysis of triangles using Bookstein coordinates and then proceed to describe Kendall's coordinates Procrustes methods, tangent approximations, symmetry in shapes, growth data, image warping, averaging and object recognition.
Practical examples will be given from various fields including medical imaging, face analysis and biology. Open problems in the field will be also highlighted.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: Bounded-error scheme are schemes in which the error norm is bounded by a function of the time t, the mesh size h and the exact solution to the differential problem u ( typically a Sobolev norm of u ), i.e. the error norm is bounded by F(u,h,t), F is bounded for-all finite t, and F --> 0 as h --> 0. In practice we use error-boundness in a stricter sense. We require that the L_2 norm of the error, be bounded by a ``constant'' proportional to h^m (m being the spatial order of accuracy) for all finite t, or at most grow linearly in time, the time coefficient being proportional to h^m.
A methodology for constructing bounded error finite-difference semi-discrete schemes, for initial boundary value problems (IBVP), on complex, multi-dimensional shapes is presented. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite (N.D.) and bounded away from 0 by a constant independent of the size of the matrix, or is at least non-positive definite (N.P.D.). These properties, N.D. or N.P.D., enable us to prove that the scheme is error-bounded by a ``constant'', or error-bounded by linear growth in time, respectively. The differentiation matrix accounts for the boundary condition by imposing penalty terms.
This methodology was used to develop second and fourth order accurate approximations for second special derivative and a second order accurate approximation for first special derivative. Using these approximations error-bounded schemes were constructed for the one and multi-dimensional diffusion and linear advection-diffusion equations. Numerical examples show that the method is effective even where standard schemes, stable by traditional definitions, fail.
The methodology was adopted to construct error-bounded schemes for parabolic equations and systems containing mix-derivatives and for the wave equation Utt = Laplacian( U ).
Brown University Center for Statistical Sciences Seminar
Sponsored by: The Bruce M. Bigelow Class of 1955 Lecture Series, The Charles K. Colver Lectureships and Publication Fund and the Department of Diagnostic Imaging at Brown University and Rhode Island Hospital/Lifespan
Abstract: Mammographic screening is a radiological procedure for the detection of breast cancer in asymptomatic patients. Its purpose is to treat breast cancer at an early stage of development, in the hope of a more effective treatment. While mammographic screening is recommended in many countries, the modalities of screening are controversial. Issues include whether screening is beneficial for women under 50 years of age; the appropriate frequency of screening examinations; whether women who are at increased risk of breast cancer would benefit from more frequent screening; whether it is cost/effective to provide insurance coverage for screening mammograms; and what would be the impact of improved mammography technologies.
In this talk I will discuss an approach to addressing these questions based on comprehensive decision modeling. The goals of comprehensive decision models are to provide guidance to patients, physicians and policy makers on a range of different decision problems related to screening; bring to bear evidence from several sources, including the epidemiology and genetics of risk factors, relevant clinical trials of secondary prevention and treatment, and knowledge about tumor growth rates; give a realistic assessment of uncertainty about the relative merits of alternative choices; and provide the basis for a cost-utility analysis of specific interventions or the maximization of specific utility functions. The presentation will include the outline of a decision model of breast cancer screening, a comparison of alternative screening strategies under different scenarios of test sensitivity, and the optimization of an arbitrary age-dependent screening frequency. There will be digressions into specific statistical problems involved in developing components of the model and a final discussion of the strengths and limitations of trial-based and model-based approaches.
Department of Mathematics Colloquium
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