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

Although measures of diversity and inequality have been extensively proposed in socio-economic and health perspectives, violation of a subtle monotonicity criterion (under stochastic ordering) diminishes their rationality and utility in the context of poverty and other ecomomic indexes. A modification is proposed here to develop certain stochastic ordering monotonicity preserving diversity measures that overcome this drawback. Such measures are shown to be invariant under increasing transformation, and thereby appropriate for (partially) ordered categorical response data models.AMS{2000) Subject Classification: 62H17, 62-07.

http://csa.sagepub.com/content/50/3-4/137.full.pdf

On Stochastic Ordering and a General Class of Poverty Indexes

Various methods have been proposed to estimate intra-cluster correlation coefficients (ICCs) for correlated binary data, and many are very sensitive to the type of design and underlying distributional assumptions. We proposed a new method to estimate ICC and its 95% confidence intervals based on resampling principles and U-statistics, where we resampled with replacement pairs of individuals from within and between clusters. We concluded from our simulation study that the resampling-based estimates approximate the population ICC more precisely than the analysis of variance and method of moments techniques for different event rates, varying number of clusters, and cluster sizes.

Resampling method to estimate intra-cluster correlation for clustered binary data

Asymptotic normality of the extremum of certain sample functions

Professor Calyampudi Radhakrishna Rao has emerged as one of the most outstanding mathematical statisticians of our time, and his fundamental research contributions covering a wider spectrum of mathematical statistics, design of experiment, combinatorial mathematics, multivariate analysis, information theory, sample surveys, biometry, econometrics, and a variety of other fields, span over a period of little more than 50 years. While C.R. Rao, currently in his mid-seventies, has been embarking on new research projects, the classical work he accomplished in the 1940s is gaining new momentum through vigorous extensions, fruitful applications in diverse fields, and novel interpretations. In Breakthroughs in Statistics, Volume 1, Dr. P.K. Pathak has written an Introduction to another outstanding (1945) article of C.R. Rao, relating to the classical Cramer—Rao Inequality, wherein he has also made detailed comments on Rao’s significant contributions in related fields. In view of this, we will avoid the duplications, and mainly confine ourselves to the scientific aspects of the current article and its profound impact on up-to-date developments in statistical inference and its applications.

Introduction to Rao (1948) Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation

: By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(>1)-dimensional stochastic vectors.

Pranab Kumar Sen

Papers

On Stochastic Ordering and a General Class of Poverty Indexes

Resampling method to estimate intra-cluster correlation for clustered binary data

Asymptotic normality of the extremum of certain sample functions

Introduction to Rao (1948) Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation

On Weak Convergence of Empirical Processes for Random Number of Independent Stochastic Vectors