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

For non-identically distributed random variables, some inequalities on the median and the tails of the distribution of a sample order statistic are derived, and a simple condition for the existence of its moments is studied.

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A Note on Order Statistics for Heterogeneous Distributions

From Finite Sample to Asymptotic Methods in Statistics: Stochastic Processes: An Overview

For a continuous and diagonally symmetric multivariate distribution, incorporating the idea of preliminary test estimators, a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and $U$-statistics. In an asymptotic setup, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.

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On Some Shrinkage Estimators of Multivariate Location

An appraisal of some aspects of statistical inference under inequality constraints

Introduction In this paper the conditions for the existence of the moments of the sample quantile as well as the asymptotic convergence of these moments to· those of the asymptotic distribution of the sample quantile have been studied. It is well known that the distribution of the quantile tends to the normal law, under certain conditions, as the sample size increases. That result, ·however, does not imply the convergence of the moments to those of the normal law under the same conditions. There arises the problem of specifying the further conditions sufficient for this convergence of the moments. Chu and Hotelling (1955) investigated the case of the sample median and have shown that the k-th central moment of the sample m-edian converges to its asymptotic value provided that the parent population admits of the existence of the k-th order moment and certain other conditions pertaining to the density function are true. We will show here that the existence of the moments of the sample quantiles follows under much less restrictive assumptionson the parent density function and the existence of the 6-th order moment for some o>O is sufficient, at least, for the asymptotic convergence of the moments. Further, we will extend the theorem of Chu and Hotelling to the class of almost all quantiles and arrive at parallel results under certain relaxations of the conditions imposed by them.

Pranab Kumar Sen

Papers

A Note on Order Statistics for Heterogeneous Distributions

From Finite Sample to Asymptotic Methods in Statistics: Stochastic Processes: An Overview

On Some Shrinkage Estimators of Multivariate Location

An appraisal of some aspects of statistical inference under inequality constraints

On the Moments of the Sample Quantiles