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

We derive sample size formulas for the many-one test of Steel (1959) when the all-pairs power is preassigned. In this large sample approach we replace, similar to Noether (1987), the unknown variances and also the unknown correlation coefficients in the power expressions by their known values under the null hypotheses. We then obtain least favorable configurations for one-and two-sided comparisons. The reliability of our formulas is examined in computer simulations for different alternatives with various distributions.

Sample size determination of steel's nonparametric many-one test

On inadmissibility of Hotelling T 2 -tests for restricted alternatives

: Asympototic normality of a class of one sample Chernoff-Savage type of rank order statistics has been established by Govindarajulu ((1960), Central limit theorems and asymptotic efficiency for one sample nonparametric procedures). His proof appears to be quite lengthy and cumbersome. The object of this note is to provide a greatly shortened and simplified proof of the same theorem. (Author)

On the Asymptotic Normality of One Sample Chernoff-Savage Test Statistics

The object of the present investigation is to generalize the results of Greenberg (1966) to a wider class of robust estimators which includes her estimator as a special case. As in Greenberg (1966), we consider an incomplete block design $D$ consisting of $J$ blocks of (constant) size $b$ to which $c(> b)$ treatments are applied, there being $n_j$ replications of the $j$th block, for $j = 1, \cdots, J$. Let $n = \sum^J_{j=1} n_j$ and let $S_j$ consist of the numbers of the $b$ treatments applied in the $j$th block, for $j = 1, \cdots , J$. The observable random variables are then \begin{align}\tag{1.1}X_{ij\alpha} = v + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha}, \\ \alpha = 1, \cdots, n_j; \quad i \varepsilon S_j; \quad j = 1, \cdots, J,\end{align} where $\xi_i$ is the $i$th treatment-effect, $\mu_j$ the $j$th replication effect, $\beta_{j\alpha}$ the effect of the $\alpha$th block in the $j$th replication set and $U_{ij\alpha}$'s are independent and identically distributed residual error components with a common distribution $F(u)$. We may set (without any loss of generality) that \begin{equation*}\tag{1.2}\sum^c_{i=1} \xi_i = 0, \quad \sum^J_{j=1} \mu_j = 0; \quad \sum^{n_j}_{\alpha=1} \beta_{j\alpha} = 0 \quad\text{for all} j = 1, \cdots, J.\end{equation*} Our intention is to provide some robust estimators of contrasts among $\xi_i$'s and to study their various properties.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177698715

On Robust Estimation in Incomplete Block Designs

Publisher Summary This chapter highlights the asymptotic representations and interrelations of robust estimators and their applications. Various types of robust estimators of location, regression, and scale parameters have been proposed and studied. These include the L-estimators based on linear functions of order statistics, maximum likelihood type (or M-estimators), R-estimators based on appropriate rank statistics, statistical functionals including Hoeffding's U-statistics and von Mises' differentiable statistical functions, and others. Regression quantiles and regression rank scores estimators are the hybrid of robust estimators, such as minimum-distance estimators, Pitman-type (or P-) estimators, and Bayes-type (or B-) estimators. Typically, such robust estimators are nonlinear; they are often defined implicitly in terms of some estimating functions, whose solution may require an iterative procedure. Moreover such classes of robust estimators are not necessarily disjoint, though they have been mostly studied by using apparently different analysis tools. To appreciate fully the robustness picture, the general asymptotic properties and interrelations of various robust estimators in a systematic and unified manner should be looked into.

Pranab Kumar Sen

Papers

Sample size determination of steel's nonparametric many-one test

On inadmissibility of Hotelling T 2 -tests for restricted alternatives

On the Asymptotic Normality of One Sample Chernoff-Savage Test Statistics

On Robust Estimation in Incomplete Block Designs

17 Asymptotic representations and interrelations of robust estimators and their applications