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

The Spearman footrule and a Markov chain property

For certain basic problems in sequential analysis,it is shown that a piecewise sequential methodology resting on replicated piecewise stopping numbers provides a natural estimate of the variance of the overall stopping variable without compromising much on the performancecharacteristics of the estimation rules.

Replicated piecewise stopping numbers and sequential analysis

Publisher Summary This chapter focuses on nonparametric simultaneous inference for some multivariate analysis of variance (MANOVA) models and presents simultaneous inference procedures based on general rank order statistics and derived estimators. One of the basic reasons for advocating the use of rank statistics and estimates is their good robustness against outliers and gross-errors. For nonparametric procedures considered in the chapter, any specific form of the underlying distributions (only continuity or sometimes symmetry suffices) is not required, and hence they enjoy a broader scope of applicability. They are robust. But, to meet this scope in full, the exact percentile points of the null distributions of various nonparametric statistics on which simultaneous procedures rest should be computed. For general rank statistics, these distributions can be enumerated by reference to suitable permutational invariance structures. The distributions of various nonparametric statistics are usually dominated by their asymptotic forms so that the use of asymptotic percentile points results in conservatism and does not affect the validity of these procedures. On the other hand, it leads to some loss in efficiency (or power); this loss is usually very small when sample sizes are large.

22 Nonparametric simultaneous inference for some MANOVA models

The theory of locally most powerful rank tests and the union intersection principle are incorporated in the formulation of some distribution-free rank tests for ordered alternatives in some simple linear models (including the classical one-way layout as a special case). Some approximations for the null-hypothesis distributions of the test statistics ( for finite sample sizes) are considered, and the theory is supplemented by some numerical and simulation results.

Union-intersection rank tests for ordered alternatives in some simple linear models

In the context of time-sequential studies, progressively censored tests for a simple regression model based on weighted empirical distributions are considered for ungrouped as well as grouped data situations. Early decision rules based on such tests are formulated. The asymptotic theory of the proposed tests rests on a construction of suitable empirical processes and their convergence (in distribution) to appropriate Gaussian functions. Critical values of the proposed test statistics are obtained by simulation, For a hypothetical example (of practical interest), a comparative study is made for the empirical powers and stopping times for some rival tests.

Pranab Kumar Sen

Papers

The Spearman footrule and a Markov chain property

Replicated piecewise stopping numbers and sequential analysis

22 Nonparametric simultaneous inference for some MANOVA models

Union-intersection rank tests for ordered alternatives in some simple linear models

Progressively censored tests for clinical experiments and life testing problems based on weighted empirical distributions