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

In the broad range WHO interpretation of Quality of Life, albeit made from an individualistic perspective, there are numerous qualitative factors along with some relatively more quantitative ones which are useful in the context of health related quality of life assessment problems. Though item analysis is commonly used in practice for (quantitative) risk assessment, for drawing valid conclusions, statistical reasoning is essential. Quality of life, survival time and quality adjusted life are important (population-based) measures that need to be appraised in light of statistical and health-related undercurrents. This paper addresses some basic statistical issues prevailing in this context.

Measures of quality adjusted life and quality of life deficiency: statistical perspectives

In this paper we consider a class of rank order tests for the identity of two multiple regression surfaces 

$$X_i^{\left( j \right)} = \beta _0^{\left( j \right)} + \sum\limits_{k = 1}^p {\beta _k^{\left( j \right)} C_i^{\left( k \right)} + Z_i^{\left( j \right)} ,{\text{ }}j = 1,2,....}$$

(1)

(0.1) Here Xi= (Xi(1),Xi(2), i=1,..., N are the observable random variables, cii(1),..., ci(p), i=1, ...,N are the vectors of known constants, Β's are the regression parameters, and Zi=(Z)i(1), Zzi(2), i=1, ..., N are independent and identically distributed random variables. It is assumed that (Zi(1), Zi(2)) are either interchangeable random variables or their joint distribution is diagonally symmetric about (0, 0). We wish to test the hypothesis H0: Βk(1)=Βk(2), k=0, 1,...,p, p≧1 (0.2) against the alternative that at least one of the p+1 equalities above is not true. If we make the transformation Xi=Xi(1)-Xi(2)Zi=Zi(1)-Zi(2), Βk=Βk(1)-Βk(2), i=1, ...,N, k=0,1, ...,p then the above problem reduces to that of testing H′0: Βk=0, k=0,1,...,p (0.3) against the alternative that Βk0 for at least one k. A class of permutationally distribution free rank order tests is proposed for this problem. Using the methods of Hajek (1962), based on the concept of contiguity of probability distributions, the asymptotic properties of the proposed tests are studied. These results are used to derive general formulas for the asymptotic relative efficiencies of these tests with respect to one another and to the least squares procedure.

On a class of rank order tests for the identity of two multiple regression surfaces

The classical jackknifing based on a (without replacement) resampling scheme with deletion of one observation at a time serves the dual purpose of bias reduction and variance estimation. Delete-k jackknifing, for some k > 1, is a variant of this scheme. For not so smooth functionals, such delete-k jackknifing schemes may provide a consistent variance estimator while the classical jackknifing may fail to do so. On the other hand, in delete-k jackknifing, there may be a far greater computational complication. For these reasons, it is quite natural to inquire whether for a smooth statistical functionals, delete-k jackknifing is that desirable or not? In the light of second order asymptotic distributional representations, it is shown that for second order (compact) differentiable statistical functionals, for any k, small compared to n (the sample size), delete-k jackknifing behaves very similar to the classical one. Thus, for smooth functionals there may not be any incentive for delete-k jackknifing.

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Whither delete-k jackknifing for smooth statistical functionals?

For experiments involvingm factors (A
 1,…,A
 m), each at 2 levels (1, 2), and replicated inn(≧2) blocks, a class of nonparametric procedures for estimating and testing the various main effects and interactions are considered. The procedures are based on a simple alignment process and involve the use of some well known rank statistics. Their performance characteristics are compared with those of the standard (normal-theory) parametric procedures. Extensions to confounded or partially confounded designs are also considered.

Pranab Kumar Sen

Papers

Measures of quality adjusted life and quality of life deficiency: statistical perspectives

On a class of rank order tests for the identity of two multiple regression surfaces

Whither delete-k jackknifing for smooth statistical functionals?

Nonparametric inference in n replicated 2 m factorial expriments

Chapter 4 – Selected rank tests