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

In repeated measures settings, modeling the correlation pattern of the data can be immensely important for proper analyses. Accurate inference requires proper choice of the correlation model. Optimal efficiency of the estimation procedure demands a parsimonious parameterization of the correlation structure, with sufficient sensitivity to detect the range of correlation patterns that may occur. Many repeated measures settings have within-subject correlation decreasing exponentially in time or space. Among the variety of correlation patterns available for this context, the continuous-time first-order autoregressive correlation structure, denoted AR(1), sees the most utilization. Despite its wide use, the AR(1) structure often poorly gauges within-subject correlations that decay at a slower or faster rate than required by the AR(1) model. To address this deficiency we propose a two-parameter generalization of the continuous-time AR(1) model, termed the linear exponent autoregressive (LEAR) correlation structure, which accommodates much slower and much faster decay patterns. Special cases of the LEAR family include the AR(1), compound symmetry, and first-order moving average correlation structures. Excellent analytic, numerical, and statistical properties help make the LEAR structure a valuable addition to the suite of parsimonious correlation models for repeated measures data. Both medical imaging data concerning neonate neurological development and longitudinal data concerning diet and hypertension [DASH (Dietary Approaches to Stop Hypertension) study] exemplify the utility of the LEAR correlation structure.

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A linear exponent AR(1) family of correlation structures

On the Properties of U-Statistics when the Observations are not Independent: Part One Estimation of Non-Serial Parameters in Some Stationary Stochastic Process

A note on the joint distribution of correlated quadratic forms

Commonly used kernel density estimators may not provide admissible values of the density or its functionals at the boundaries for densities with restricted support. For smoothing the empirical distribution a generalization of the Hille's lemma, considered here, alleviates some of the problems of kernel density estimator near the boundaries. For nonnegative random variables which crop up in reliability and survival analysis, the proposed procedure is
thoroughly explored; its consistency and asymptotic distributional results are established under appropriate regularity assumptions. Methods of obtaining smoothing parameters
through cross-validation are given, and graphical illustrations of the estimator for continuous
(at zero) as well as discontinuous densities are provided.

A New Smooth Density Estimator for Non-Negative Random Variables

Time-sequential nonparametrics Miscellaneous time-sequential procedures Change-Point nonparametrics Recursive residual M-tests Nonparametrics of sequential estimation Sequential confidence regions Nonparametric case Nonparametric repeated significance tests Repeated multiple comparisons tests Nonparametrics of sequential tests Pitman efficiency: sequential case Nonparametric tests with Power One.

Pranab Kumar Sen

Papers

A linear exponent AR(1) family of correlation structures

On the Properties of U-Statistics when the Observations are not Independent: Part One Estimation of Non-Serial Parameters in Some Stationary Stochastic Process

A note on the joint distribution of correlated quadratic forms

A New Smooth Density Estimator for Non-Negative Random Variables

Theory and Applications of Sequential Nonparametrics