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

https://projecteuclid.org/download/pdf_1/euclid.aop/1176996943

An Almost Sure Invariance Principle for Mutivariate Kolmogorov-Smirnov Statistics

Based on an al.ost sure Wiener process approxi.ation for linear functions of order statistics along with the strong consistency of their variance esti.at.s, a class of sequential tests is considered. These test ter.inate with probability'. Asyaptotic OC and ASN functions are also studied. AHS 1970 Classification No. 62 L 10, 62 G 10.

Asymptotic theory of sequential tests based on linear functions of order statistics

Fingerprint analysis comparing the polychlorinated dibenzo-p-dioxin and dibenzofuran (PCDD/F) congener profile patterns of collected samples with those of potential dioxin emission source(s) is an important tool for identifying environmental dioxin pollution. The constraint that the proportions of the 17 PCDD/F congeners comprising a fingerprint sum up to one motivates a multivariate gamma distribution, which leads to a Dirichlet distribution. Because of the complexity in restricted likelihood ratio tests and typical sample size limitations resulting from laboratory analysis costs, permutation test procedures are employed for hypothesis testing of the homogeneity of congener profiles. Pearson-type chi-squared tests based on the Dirichlet distribution (DM) assumption, the generalized form of DM using the arithmetic mean and geometric mean of the proportions, and the robust aligned rank test, are proposed and compared through simulations. PCDD/F samples collected from the stack of a local municipal solid waste incinerator and from ambient air near the municipal solid waste incinerator in Taiwan were illustrated as an example. The simulation results showed that the aligned rank test, followed by the DM-based test, was generally robust to distributional assumptions and had high statistical power. The arithmetic-mean-based and geometric-mean-based tests outperformed one another in different conditions, dependent on the underlying distribution. Copyright © 2012 John Wiley & Sons, Ltd.

Robust permutation tests for homogeneity of fingerprint patterns of dioxin congener profiles

For a sequence of stochastic vectors forming either a forward or a reverse martingale, a Hajek-Renyi type inequality is derived, and its applications in some problems of simultaneous confidence regions are stressed.

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A Hajek-Renyi Type Inequality for Stochastic Vectors with Applications to Simultaneous Confidence Regions

We discuss the estimation of the expected value of the quality-adjusted survival, based on multistate models We generalize an earlier work, considering the sojourn times in health states are not identically distributed, for a given vector of covariates Approaches based on semiparametric and parametric (exponential and Weibull distributions) methodologies are considered A simulation study is conducted to evaluate the performance of the proposed estimator and the jackknife resampling method is used to estimate the variance of such estimator An application to a real data set is also included

Pranab Kumar Sen

Papers

An Almost Sure Invariance Principle for Mutivariate Kolmogorov-Smirnov Statistics

Asymptotic theory of sequential tests based on linear functions of order statistics

Robust permutation tests for homogeneity of fingerprint patterns of dioxin congener profiles

A Hajek-Renyi Type Inequality for Stochastic Vectors with Applications to Simultaneous Confidence Regions

A semi-Markov multistate model for estimation of the mean quality-adjusted survival for non-progressive processes