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

Publisher Summary This chapter reviews some recent results in finite population asymptotics. In finite population sampling (FPS), the theory of objective or probabilistic sampling plays a fundamental role in statistical inference. In survey sampling, the sampling frame defines the units and the size of the survey population from which the sample is taken unambiguously. It also reconstructs a population having an uncountable (or infinite) number of natural units in terms of a finite population by redefining suitable sampling units. Resampling methods, including the jackknife and the bootstrap, can provide standard error estimates and nonparametric confidence intervals for the parameters of interest or approximate sampling distributions of statistics. The estimation of the total size of a population including, in particular, mobile populations such as the number of fish in a lake is of great importance in a variety of biological environmental and ecological investigations.

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12 Asymptotics in finite population sampling

A method is given for the comparison of two linear forms of a common random vector under the criterion of Pitman's measure of closeness. Assuming multivariate normality of the random vector, one can determine exact expressions for the closeness probabilities. The applicability of the theory is illustrated on the comparison of ridge regression estimators.

Comparison of Linear Estimators Using Pitman's Measure of Closeness

Asymptotically Efficient Tests by the Method of N Rankings

For a sequence of random variables forming an $m$-dependent stochastic process (not necessarily stationary), asymptotic distribution and other convergence properties of the extremum of certain functions of the empirical distribution are studied. In this context, it is shown that the asymptotic probability of the classical Kolmogorov-Smirnov statistic exceeding any positive real number provides an upper bound for the corresponding probability when the underlying random variables are not necessarily identically distributed. The theory is specifically applied to the study of the limiting distribution, strong convergence and convergence of the first moment of the strength of a bundle of parallel filaments (which is shown to be the extremum of a function of the empirical distribution).

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Limiting Behavior of the Extremum of Certain Sample Functions

Two results of Billingsley [Convergence of Probability Measures (1968), Wiley, New York] on weak convergence of empirical distribution functions for sequences of $\phi$-mixing random variables are shown to hold under weaker conditions.

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Pranab Kumar Sen

Papers

12 Asymptotics in finite population sampling

Comparison of Linear Estimators Using Pitman's Measure of Closeness

Asymptotically Efficient Tests by the Method of N Rankings

Limiting Behavior of the Extremum of Certain Sample Functions

A Note on Weak Convergence of Empirical Processes for Sequences of $\phi$- Mixing Random Variables