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

e • Nonparametric (as well as jackknifed) estimators are developed for the 'availability' of an equipment supported by a single spare and a repair facility, where down-time occurs whenever no spare/repaired unit is available at the point of failure of an operating unit. Asymptotic properties of the natural and the jack-knifed estimators of availability are studied (without assuming the stochastic independence of life and repair times). Fixed sample size as well as sequential procedures are considered , and progressively censored schemes are also introduced in this context.

Nonparametric estimators of availability under provisions of spare and repair

Some nonparametric generalizations of Tukey’s [9]T-method of multiple comparisons are considered for randomized blocks and the allied efficiency results are studied. For this, the distribution theory of aligned rank order statistics developed in [6], [7] is extended for multiple comparisons along the lines of [5] which deals with one-way layouts.

On nonparametric T-method of multiple comparisons for randomized blocks

Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such information might involve direct knowledge of the actual form of the true model or might be represented by a set of differential equations that the model must satisfy. We develop M-procedures for estimating parameters and testing hypotheses of interest about these parameters in nonlinear regression models for repeated measurement data. Under regularity conditions, the asymptotic properties of the M-procedures are presented, including the uniform linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed with iterative procedures, similar to Newton–Raphson and Fisher's scoring methods. The methodology is illustrated by using a multivariate logistic regression model with real data, along with a simulation study.

A Robust Procedure in Nonlinear Models for Repeated Measurements

The present paper deals with the asymptotic theory of sequential confidence intervals of prescribed width $2d (d > 0)$ and prescribed coverage probability $1 - \alpha (0 < \alpha < 1)$ for the (unknown, per unit) strength of bundle of parallel filaments. In this context, certain useful convergence results on the empirical distribution and on the bundle strength of filaments are established and incorporated in the proofs of the main theorems. The results are the sequential counterparts of some fixed sample size results derived in a concurrent paper of Sen, Bhattacharyya and Suh [9].

https://projecteuclid.org/download/pdf_1/euclid.aos/1176342418

On fixed size confidence bands for the bundle strength of filaments

A challenging and crucial issue in clinical studies in cancer involving gene microarray experiments is the discovery, among a large number of genes, of a relatively small panel of genes whose elements are associated with a relevant clinical outcome variable such as time-to-death or time-to-recurrence of disease. A semiparametric approach, using dependence functions known as copulas, is considered to quantify and estimate the pairwise association between the outcome and each gene expression. These time-to-event type endpoints are typically subject to censoring as not all events are realized at the time of the analysis. Furthermore, given that the total number of genes is typically large, it is imperative to control a relevant error rate in any gene discovery procedure. The proposed method addresses the two aforementioned issues by direct incorporation of the censoring mechanism and by appropriate statistical adjustment for multiplicity. The performance of the proposed method is studied through simulation and illustrated with an application using a case study in lung cancer.

Pranab Kumar Sen

Papers

Nonparametric estimators of availability under provisions of spare and repair

On nonparametric T-method of multiple comparisons for randomized blocks

A Robust Procedure in Nonlinear Models for Repeated Measurements

On fixed size confidence bands for the bundle strength of filaments

A copula approach for detecting prognostic genes associated with survival outcome in microarray studies.