Methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed. When a test is based on an observed variable that lies on a continuous or graded scale, an assessment of the overall value of the test can be made through the use of a receiver operating characteristic (ROC) curve. The curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates. When two or more empirical curves are constructed based on tests performed on the same individuals, statistical analysis on differences between curves must take into account the correlated nature of the data. This paper presents a nonparametric approach to the analysis of areas under correlated ROC curves, by using the theory on generalized U-statistics to generate an estimated covariance matrix.

Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach.

THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

An essential textbook for any student or researcher in biology needing to design experiments, sample programs or analyse the resulting data The text begins with a revision of estimation and hypothesis testing methods, covering both classical and Bayesian philosophies, before advancing to the analysis of linear and generalized linear models Topics covered include linear and logistic regression, simple and complex ANOVA models (for factorial, nested, block, split-plot and repeated measures and covariance designs), and log-linear models Multivariate techniques, including classification and ordination, are then introduced Special emphasis is placed on checking assumptions, exploratory data analysis and presentation of results The main analyses are illustrated with many examples from published papers and there is an extensive reference list to both the statistical and biological literature The book is supported by a website that provides all data sets, questions for each chapter and links to software

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Experimental Design and Data Analysis for Biologists

Benjamini and Hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems. A simple procedure was given there as an FDR controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate. We prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses. This condition for positive dependency is general enough to cover many problems of practical interest, including the comparisons of many treatments with a single control, multivariate normal test statistics with positive correlation matrix and multivariate $t$. Furthermore, the test statistics may be discrete, and the tested hypotheses composite without posing special difficulties. For all other forms of dependency, a simple conservative modification of the procedure controls the false discovery rate. Thus the range of problems for which a procedure with proven FDR control can be offered is greatly increased.

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The control of the false discovery rate in multiple testing under dependency

The least squares estimator of a regression coefficient β is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of β based on Kendall's [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Yj - Yi )/(tj-ti ) joining pairs of points with ti ≠ ti , and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators.

Estimates of the Regression Coefficient Based on Kendall's Tau

Wassily Hoeffding earned his Ph.D. degree in mathematics from the Berlin University in 1940 for a dissertation in correlation theory which dealt with some aspects of bivariate probability distributions that are invariant under monotone transformations of the marginals. This dissertation was primarily devoted to some (descriptive) studies of certain measures of rank correlations. With the impending Second World War, for Wassily, living in Berlin in the early forties was not that comfortable. Nevertheless, he managed to advance his basic research on nonparametric correlation theory. It was only after his eventual migration to the United States (in the Fall of 1946) that he started to appreciate the full depth of probability theory and statistics (during his sojourn at the Columbia University, New York), and most of his pioneering work emerged during his longtime residence at Chapel Hill (1947–1991). He felt that “…probability and statistics were very poorly represented in Berlin at that time (1936–1945) …”. Notwithstanding this, his early work on correlation theory was not just a landmark in nonparametrics; it also endowed him with a career-long zeal and affection for the pursuit of the most fundamental research in mathematical statistics, probability theory, numerical analysis and a variety of other related areas. In this respect, nonparametrics was indisputedly the “jewel in the crown” of Wassily’s creativity and ingenuity in research. Wassily Hoeffding indeed played a seminal role in stimulating basic research in a broad domain of mathematical statistics and probability theory, and his “collected work” in this volume reflects the genuine depth and immense breadth of his research contributions. In this article, I shall mainly confine myself to describing the profound impact of his work in the general area of nonparametrics, with occasional detours to some other related areas.

The Impact of Wassily Hoeffding’s Research on Nonparametrics

For a multiple regression model, bearing the plausibility of a subset of the regression parameters being close to a pivot, for the complementary subset, based on the usual James-Stein rule, a general formulation of shrinkage R-estimation is considered. In the light of asymptotic distributional risks of estimators, performance characteristics ( under local alternatives) of the classical R-est-imator and its preliminary test and shrinkage versions (all based on the common score function ) are studied. These shed light on the relative dominance picture in a meaningful asymptotic setup.

On shrinkage r-estimation in a multiple regression model

Subhypotheses testing against restricted alternatives for the Cox regression model

Combination of statistical tests for multivariate hypotheses against restricted alternatives

Background and Purpose— Leukocyte count is an independent predictor of stroke. We investigated the association between leukocyte count and progression of aortic atheroma over 12 months in stroke/transient ischemic attack (TIA) patients. Methods— Consecutive ischemic stroke and transient ischemic attack patients underwent 12-month sequential transesophageal echocardiography and were assessed for total and differential leukocyte counts on admission. Paired aortic plaque images were assessed for several parameters, including changes in grade, intimal-medial thickness (IMT), and cross-sectional area. Multivariate linear and logistic regressions were used to calculate the effect of leukocyte count on the change in aortic atheromas over 12 months. Results— Of the 115 participants (mean±SD age, 64.6±11.9 years; 53.1% men; 73.4% white, 24.2% black, and 2.3% Asian), 45 (35%) showed clinically significant progression of aortic atheromas (maximal change in IMT >0.70 mm over 12 months). The mean admission leukocyte c...

Pranab Kumar Sen

Papers

The Impact of Wassily Hoeffding’s Research on Nonparametrics

On shrinkage r-estimation in a multiple regression model

Subhypotheses testing against restricted alternatives for the Cox regression model

Combination of statistical tests for multivariate hypotheses against restricted alternatives

Association of Leukocyte Count With Progression of Aortic Atheroma in Stroke/Transient Ischemic Attack Patients