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 object of the present investigation is to extend the findings of Hajek [5J on asymptotically most powerful rank order tests (AMPROT) to grouped data where the underlying distributions are essentially continuous but I the observable random variables correspond to a finite or countable set of contiguous class intervals. In this context, the two sample problem for grouped data is considered and various efficiency results are also studied.

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Asymptotically most powerful rank order tests for grouped data

I General Distribution Theory and Inference.- 1 PIC: Power Divergence Information Criterion.- 1.1 Introduction.- 1.2 The Power Divergence Measures.- 1.3 Power-divergence Information Criterion (PIC).- 1.4 Count Data from the Framingham Study.- 1.5 Conclusions.- 1.6 References.- 2 Multivariate Student's t and Its Applications.- 2.1 Introduction.- 2.2 A Multivariate Student t Distribution.- 2.2.1 Definition and Probability Integral.- 2.2.2 Computing the Probability Integral for p > 2.- 2.2.3 An Approximation for the Unequal pij Case.- 2.3 Applications of Multivariate t.- 2.3.1 Two-stage Indifference Zone Selection.- 2.3.2 Subset Selection.- 2.3.3 Treatments Versus Control Multiple Comparisons.- 2.3.4 Testing Multiple Contrasts.- 2.3.5 Multiple Comparisons with the "Best".- 2.3.6 Additional Applications.- 2.4 References.- 3 Two Sets of Multivariate Bonferroni-type Inequalities.- 3.1 Introduction.- 3.2 The Results.- 3.3 Proofs.- 3.4 References.- 4 On the Proportion above Sample Mean for Symmetric Stable Laws.- 4.1 Introduction.- 4.2 The Result.- 4.3 A Partial Converse.- 4.4 Remarks and Extensions.- 4.5 References.- 5 The Relative Efficiency of Several Statistics Measuring Skewness.- 5.1 Introduction.- 5.2 Skewness Functional for X with Finite Support.- 5.3 Relative Efficiency of Skewness Estimators when X has Finite Support.- 5.4 Relative Efficiency of Skewness Estimators when X has Infinite Support.- 5.5 Summary.- 5.6 References.- 6 On a Class of Symmetric Nonnormal Distributions with a Kurtosis of Three.- 6.1 Introduction.- 6.2 Symmetric Mixtures with ?2 = 3.- 6.3 Examples.- 6.4 Limiting Distributions of the Extremes.- 6.5 Comments.- 6.6 References.- II Order Statistics - Distribution Theory.- 7 Moments of the Selection Differential from Exponential and Uniform Parents.- 7.1 Introduction.- 7.2 Moments of D from an Exponential Parent.- 7.3 Moments of D from a Uniform Parent.- 7.4 Asymptotes for the Moments of D.- 7.5 Convergence of the Moments: Exponential Parent.- 7.6 Convergence of the Moments: Uniform Parent.- 7.7 Tabular Analysis.- 7.7.1 Mean.- 7.7.2 Variance.- 7.7.3 Skewness.- 7.7.4 Kurtosis.- 7.8 Practical Implications.- 7.9 References.- 8 The Tallest Man in the World.- 8.1 Introduction.- 8.2 Some Useful Results on Branching Processes.- 8.3 The Tallest Man in the World.- 8.4 The Tallest Man in History.- 8.5 What Kind of Limit Laws can be Encountered?.- 8.6 References.- 9 Characterizing Distributions by Properties of Order Statistics - A Partial Review.- 9.1 Introduction.- 9.2 Independence of Linear Functions.- 9.3 Identical Distributions of Functions of Order Statistics.- 9.4 Moment Properties.- 9.5 Statistical Properties.- 9.6 Asymptotic Properties.- 9.7 References.- 10 Stochastic Ordering of the Number of Records.- 10.1 Introduction.- 10.2 The Results.- 10.2.1 Stochastic Ordering.- 10.2.2 Expectation of the Number of k-th Records.- 10.2.3 Integrated Likelihood Ratio Ordering.- 10.3 Proofs and an Example.- 10.4 References.- 11 Moments of Cauchy Order Statistics via Riemann Zeta Functions.- 11.1 Introduction.- 11.2 An Expression for the Mean.- 11.3 Expressions for Product Moments.- 11.4 References.- 12 Order Statistics of Bivariate Exponential Random Variables.- 12.1 Introduction.- 12.2 Freund, Marshall-Olkin, ?nd Raftery's BVE Distributions.- 12.3 Joint Distributions.- 12.4 Marginal Distributions.- 12.4.1 Properties of T1.- 12.4.2 Properties of T2.- 12.5 Copula Functions.- 12.6 References.- III Order Statistics in Inference and Applications.- 13 Maximum Likelihood Estimation of the Laplace Parameters Based on Type-II Censored Samples.- 13.1 Introduction.- 13.2 Maximum Likelihood Estimators.- 13.3 Efficiency Relative to BLUE's.- 13.4 References.- 14 The Impact of Order Statistics on Signal Processing.- 14.1 Introduction.- 14.2 Order Statistic Filters.- 14.2.1 Median and Rank-Order Filters.- 14.2.2 RO Filters.- 14.2.3 OS Filters.- 14.3 Generalizations.- 14.3.1 ? and Ll Filters.- 14.3.2 Permutation Filters.- 14.3.3 WMMR Filters.- 14.3.4 Stack Filters.- 14.3.5 Morphological Filters.- 14.4 Related Applications of Order Statistics.- 14.4.1 Edge Detection.- 14.4.2 Signal Enhancement and Restoration.- 14.5 Conclusions.- 14.6 References.- 15 A Nonlinear Ordered Rank Test to Detect Stochastic Ordering Between Two Distributions.- 15.1 Introduction.- 15.2 The Proposed Test.- 15.3 Simulation Study.- 15.4 Discussion.- 15.5 Asymptotic Properties.- 15.6 References.- 16 Estimation of Location and Scale Parameters of a Logistic Distribution Using a Ranked Set Sample.- 16.1 Introduction.- 16.2 Estimation of the Location Parameter.- 16.2.1 Best Linear Unbiased Estimator.- 16.2.2 Which Order Statistic?.- 16.3 Estimation of the Scale Parameter.- 16.4 Estimation of Quantiles.- 16.5 Proofs.- 16.6 References.- 17 Probability Models for an Employment Problem.- 17.1 Introduction.- 17.2 Definition of the Problem.- 17.2.1 Hiring Criteria.- 17.2.2 Notation.- 17.3 The Case of Couples and Single Applicants.- 17.3.1 Small Sample Results.- 17.3.2 Asymptotic Results.- 17.4 A Strategy for Couples.- 17.5 Conclusions.- 17.6 References.- IV Analysis of Variance and Experimental Design.- 18 On the Robustness of Bayes Estimators of the Variance Ratio in Balanced One-Way ANOVA Models with Covariates.- 18.1 Introduction.- 18.2 The Bayes Estimator and its Asymptotic Properties.- 18.3 Jackknifed Estimator of the Asymptotic Variance and Asymptotic Confidence Intervals.- 18.4 Simulation Results.- 18.5 References.- 19 Interchange Algorithms for Constructing Designs with Complex Blocking Structures.- 19.1 Historical Perspective.- 19.2 Optimality Criteria.- 19.3 Interchange Algorithms.- 19.4 Objective Functions.- 19.5 Discussion.- 19.6 References.- 20 Paired Comparisons for Multiple Characteristics: An ANOCOVA Approach.- 20.1 Introduction.- 20.2 ANOCOVAPC for Paired Characteristics.- 20.3 Probability Laws for Multiple Dichotomous Attributes.- 20.4 MANOCOVAPC Paired Comparisons Models and Analyses.- 20.5 Concluding Remarks.- 20.6 References.- V Biometry and Applications.- 21 On Assessing Multiple Equivalences with Reference to Bioequivalence.- 21.1 Introduction.- 21.2 Many-to-One Comparisons.- 21.3 Assessing Equivalence of k Formulations.- 21.4 Multiple Partial Equivalences.- 21.5 Example.- 21.6 References.- 22 Competing Risks.- 22.1 Introduction.- 22.2 Some Independent Competing Risk Results (After 1978).- 22.3 Non-identifiability Issues.- 22.4 Methods Assuming Informative Censoring (Dependent Competing Risks).- 22.5 Inference Using Estimable Quantities.- 22.6 Summary.- 22.7 References.- 23 Statistical Aspects of the Detection of Activation Effects in Cerebral Blood Flow and Metabolism by Positron Emission Tomography.- 23.1 Introduction: The Analysis of PET Data.- 23.2 The Activation Experiment and the Data.- 23.3 Activation Effects and a Simple Whole Brain Adjustment.- 23.4 Adjustment by the Analysis of Covariance.- 23.5 References.- 24 On Optimality of Q-Charts for Outliers in Statistical Process Control.- 24.1 Introduction.- 24.2 The Normal Mean Q-Chart.- 24.3 The Binomial Q-Chart.- 24.4 The Poisson Q-Chart.- 24.5 References.- VI Postscript.- 25 Herbert A. David.- 26 Conference Abstracts, Anecdotes, and Appreciation.

Statistical Theory and Applications: Papers in Honor of Herbert A. David

For a general (real) parameter, let Mnbe the M-estimator and Mn(1) be its one-step version (based on a suitable initial estimator Mn(0)). It is known that, under certain regularity conditions, n(Mn(1)-Mn)=Op(1). The asymptotic distribution of n(Mn(1)-Mn) is studied; it is typically non-normal and it reveals the role of the initial estimator Mn(0).

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

Estimation Following Sequential Tests

High-dimensional categorical data models, often with inadequately large sample sizes, crop up in many fields of application. The SARS epidemic, originating in southern China in 2002, had an identified single-stranded and positive-sense RNA virus with large genome size and moderate mutation rate. The present genomic study is used as a prime illustration for motivating appropriate statistical methodology for comprehending the genomic variation in such high-dimensional categorical data models. Because of underlying restraints, a pseudomarginal approach based on Hamming distance is considered in a constrained statistical inference setup. The union-intersection principle and jackknifing methods are incorporated in exploring appropriate statistical procedures.

Pranab Kumar Sen

Papers

Asymptotically most powerful rank order tests for grouped data

Statistical Theory and Applications: Papers in Honor of Herbert A. David

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

Estimation Following Sequential Tests

High-dimension, low-sample size perspectives in constrained statistical inference: The SARSCoV RNA genome in illustration