Showing papers in "Statistical Science in 2004"
TL;DR: The reasoning behind permutation methods for exact inference is discussed and situations when they are exact and distribution-free are described.
Abstract: The use of permutation methods for exact inference dates back to Fisher in 1935. Since then, the practicality of such methods has increased steadily with computing power. They can now easily be employed in many situations without concern for computing difficulties. We discuss the reasoning behind these methods and describe situations when they are exact and distribution-free. We illustrate their use in several examples.
640 citations
TL;DR: This article introduces network tomography, a new field which it is believed will benefit greatly from the wealth of statistical methods and algorithms including the application of pseudo-likelihood methods and tree estimation formulations.
Abstract: Today's Internet is a massive, distributed network which contin- ues to explode in size as e-commerce and related activities grow. The hetero- geneous and largely unregulated structure of the Internet renders tasks such as dynamic routing, optimized service provision, service level verification and detection of anomalous/malicious behavior extremely challenging. The problem is compounded by the fact that one cannot rely on the cooperation of individual servers and routers to aid in the collection of network traffic measurements vital for these tasks. In many ways, network monitoring and inference problems bear a strong resemblance to other "inverse problems" in which key aspects of a system are not directly observable. Familiar sig- nal processing or statistical problems such as tomographic image reconstruc- tion and phylogenetic tree identification have interesting connections to those arising in networking. This article introduces network tomography, a new field which we believe will benefit greatly from the wealth of statistical the- ory and algorithms. It focuses especially on recent developments in the field including the application of pseudo-likelihood methods and tree estimation formulations.
483 citations
TL;DR: In this article, the authors embark upon a rather idiosyncratic walk through some of the fundamental philosophical and pedagogical issues at stake in the Bayesian or frequentist paradigm. But they also recognize that each approach has a great deal to contribute to statistical practice and each is essential for full development of the other approach.
Abstract: Statistics has struggled for nearly a century over the issue of whether the Bayesian or frequentist paradigm is superior. This debate is far from over and, indeed, should continue, since there are fundamental philosophical and pedagogical issues at stake. At the methodological level, however, the debate has become considerably muted, with the recognition that each approach has a great deal to contribute to statistical practice and each is actually essential for full development of the other approach. In this article, we embark upon a rather idiosyncratic walk through some of these issues.
472 citations
TL;DR: For each inference problem, relevant nonparametric Bayesian models and approaches including Dirichlet process models and variations, Polya trees, wavelet based models, neural network models, spline regression, CART, dependent DP models and model validation with DP and Polya tree extensions of parametric models are reviewed.
Abstract: We review the current state of nonparametric Bayesian inference. The discussion follows a list of important statistical inference problems, including density estimation, regression, survival analysis, hierarchical models and model validation. For each inference problem we review relevant nonparametric Bayesian models and approaches including Dirichlet process (DP) models and variations, Polya trees, wavelet based models, neural network models, spline regression, CART, dependent DP models and model validation with DP and Polya tree extensions of parametric models.
416 citations
TL;DR: It is shown that alleged differences in the behavior of parameters in so-called marginal and conditional models are based on a failure to compare like with like, and these seemingly apparent differences are meaningless because they are mainly caused by preimposed unidentifiable constraints on the random effects in models.
Abstract: There has existed controversy about the use of marginal and conditional models, particularly in the analysis of data from longitudinal studies. We show that alleged differences in the behavior of parameters in so-called marginal and conditional models are based on a failure to compare like with like. In particular, these seemingly apparent differences are meaningless because they are mainly caused by preimposed unidentifiable constraints on the random effects in models. We discuss the advantages of conditional models over marginal models. We regard the conditional model as fundamental, from which marginal predictions can be made.
215 citations
TL;DR: An important aspect of the approach I advocate is modeling the relationship between a trial's primary endpoint and early indications of patient performance-auxiliary endpoints.
Abstract: The Bayesian approach is being used increasingly in medical research. The flexibility of the Bayesian approach allows for building designs of clinical trials that have good properties of any desired sort. Examples include maximizing effective treatment of patients in the trial, maximizing information about the slope of a dose–response curve, minimizing costs, minimizing the number of patients treated, minimizing the length of the trial and combinations of these desiderata. They also include standard frequentist operating characteristics when these are important considerations. Posterior probabilities are updated via Bayes’ theorem on the basis of accumulating data. These are used to effect modifications of the trial’s course, including stopping accrual, extending accrual beyond that originally planned, dropping treatment arms, adding arms, etc. An important aspect of the approach I advocate is modeling the relationship between a trial’s primary endpoint and early indications of patient performance—auxiliary endpoints. This has several highly desirable consequences. One is that it improves the efficiency of adaptive trials because information is available sooner than otherwise.
173 citations
AT&T1
TL;DR: The aims of this review are to explain the classical sampling methodology in the context of the Internet to readers who are not necessarily acquainted with either, to give an account of newer applications and sampling methods for passive measurement and to identify emerging areas that are ripe for the application of statistical expertise.
Abstract: Sampling has become an integral part of passive network measurement. This role is driven by the need to control the consumption of resources in the measurement infrastructure under increasing traffic rates and the demand for detailed measurements from applications and service providers. Classical sampling methods play an important role in the current practice of Internet measurement. The aims of this review are (i) to explain the classical sampling methodology in the context of the Internet to readers who are not necessarily acquainted with either, (ii) to give an account of newer applications and sampling methods for passive measurement and (iii) to identify emerging areas that are ripe for the application of statistical expertise.
170 citations
TL;DR: The paper reviews the various approaches taken to overcome the difficulty of closed-form Bayesian analysis in feed-forward neural networks, involving the use of Gaussian approximations, Markov chain Monte Carlo simulation routines and a class of non-Gaussian but "deterministic" approximation called variational approxIMations.
Abstract: Models such as feed-forward neural networks and certain other structures investigated in the computer science literature are not amenable to closed-form Bayesian analysis. The paper reviews the various approaches taken to overcome this difficulty, involving the use of Gaussian approximations, Markov chain Monte Carlo simulation routines and a class of non-Gaussian but “deterministic” approximations called variational approximations.
144 citations
TL;DR: In this paper, affine-invariant spatial sign and spatial rank vectors are used for multivariate nonparametric statistical tests of hypotheses for the one-sample location problem, the several sample location problem and the problem of testing independence between pairs of vectors.
Abstract: Multivariate nonparametric statistical tests of hypotheses are described for the one-sample location problem, the several-sample location problem and the problem of testing independence between pairs of vectors. These methods are based on affine-invariant spatial sign and spatial rank vectors. They provide affine-invariant multivariate generalizations of the univariate sign test, signed-rank test, Wilcoxon rank sum test, Kruskal–Wallis test, and the Kendall and Spearman correlation tests. While the emphasis is on tests of hypotheses, certain references to associated affine-equivariant estimators are included. Pitman asymptotic efficiencies demonstrate the excellent performance of these methods, particularly in heavy-tailed population settings. Moreover, these methods are easy to compute for data in common dimensions.
141 citations
TL;DR: Quantile and conditional quantile statistical thinking, as I have innovated it in my research since 1976, is outlined in this comprehensive survey and introductory course in quantile data analysis.
Abstract: Quantile and conditional quantile statistical thinking, as I have innovated it in my research since 1976, is outlined in this comprehensive survey and introductory course in quantile data analysis. We propose that a unification of the theory and practice of statistical methods of data modeling may be possible by a quantile perspective. Our broad range of topics of univariate and bivariate probability and statistics are best summarized by the key words. Two fascinating practical examples are given that involve positive mean and negative median investment returns, and the relationship between radon concentration and cancer.
132 citations
TL;DR: In this article, the authors argue that the Bayesian approach is best seen as providing additional tools for those carrying out health-care evaluations, rather than replacing their traditional methods, and make a distinction between those features that arise from the basic Bayesian philosophy and those that come from the modern ability to make inferences using very complex models.
Abstract: We argue that the Bayesian approach is best seen as providing additional tools for those carrying out health-care evaluations, rather than replacing their traditional methods. A distinction is made between those features that arise from the basic Bayesian philosophy and those that come from the modern ability to make inferences using very complex models. Selected examples of the former include explicit recognition of the wide cast of stakeholders in any evaluation, simple use of Bayes theorem and use of a community of prior distributions. In the context of complex models, we selectively focus on the possible role of simple Monte Carlo methods, alternative structural models for incorporating historical data and making inferences on complex functions of indirectly estimated parameters. These selected issues are illustrated by two worked examples presented in a standardized format. The emphasis throughout is on inference rather than decision-making.
TL;DR: In this paper, a generalized method of moments (GOM) is used to adjust the naive estimator to be consistent and asymptotically normal, and the objective function of this procedure is shown to be interpretable as an indirect likelihood.
Abstract: This article presents an exposition and synthesis of the theory and some applications of the so-called indirect method of inference. These ideas have been exploited in the field of econometrics, but less so in other fields such as biostatistics and epidemiology. In the indirect method, statistical inference is based on an intermediate statistic, which typically follows an asymptotic normal distribution, but is not necessarily a consistent estimator of the parameter of interest. This intermediate statistic can be a naive estimator based on a convenient but misspecified model, a sample moment or a solution to an estimating equation. We review a procedure of indirect inference based on the generalized method of moments, which involves adjusting the naive estimator to be consistent and asymptotically normal. The objective function of this procedure is shown to be interpretable as an “indirect likelihood” based on the intermediate statistic. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. This method is often more convenient computationally than maximum likelihood estimation when handling such model complexities as random effects and measurement error, for example, and it can also serve as a basis for robust inference and model selection, with less stringent assumptions on the data generating mechanism. Many familiar estimation techniques can be viewed as examples of this approach. We describe applications to measurement error, omitted covariates and recurrent events. A dataset concerning prevention of mammary tumors in rats is analyzed using a Poisson regression model with overdispersion. A second dataset from an epidemiological study is analyzed using a logistic regression model with mismeasured covariates. A third dataset of exam scores is used to illustrate robust covariance selection in graphical models.
TL;DR: It is observed that scoring functions resulting from proper posterior distributions, or approximations to such distributions, showed the best performance and can be used to improve upon existing motif-finding programs.
Abstract: The Bayesian approach together with Markov chain Monte Carlo techniques has provided an attractive solution to many important bioinformatics problems such as multiple sequence alignment, microarray analysis and the discovery of gene regulatory binding motifs. The employment of such methods and, more broadly, explicit statistical modeling, has revolutionized the field of computational biology. After reviewing several heuristics-based computational methods, this article presents a systematic account of Bayesian formulations and solutions to the motif discovery problem. Generalizations are made to further enhance the Bayesian approach. Motivated by the need of a speedy algorithm, we also provide a perspective of the problem from the viewpoint of optimizing a scoring function. We observe that scoring functions resulting from proper posterior distributions, or approximations to such distributions, showed the best performance and can be used to improve upon existing motif-finding programs. Simulation analyses and a real-data example are used to support our observation.
TL;DR: In contrast to tests for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality as discussed by the authors, which is why uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance.
Abstract: Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).
TL;DR: In this paper, the authors used the center-outward ranking induced by the notion of data depth to describe several nonparametric tests of location and scale differences for multivariate distributions.
Abstract: Multivariate statistics plays a role of ever increasing importance in the modern era of information technology. Using the center-outward ranking induced by the notion of data depth, we describe several nonparametric tests of location and scale differences for multivariate distributions. The tests for location differences are derived from graphs in the so-called DD plots (depth vs. depth plots) and are implemented through the idea of permutation tests. The proposed test statistics are scale-standardized measures for the location difference and they can be carried out without estimating the scale or variance of the underlying distributions. The test for scale differences introduced in Liu and Singh (2003) is a natural multivariate rank test derived from the center-outward depth ranking and it extends the Wilcoxon rank-sum test to the testing of multivariate scale. We discuss the properties of these tests, and provide simulation results as well as a comparison study under normality. Finally, we apply the tests to compare airlines’ performances in the context of aviation safety evaluations.
TL;DR: A falsification method for probabilistic or causal theories, based on “Borel criteria,” is described, and it is argued that this minimalist approach, free of any distracting metaphysical inputs, provides the essential support for the conduct and advance of Science.
Abstract: This article expounds a philosophical approach to Probability and Causality: a synthesis of the personalist Bayesian views of de Finetti and Popper’s falsificationist programme. A falsification method for probabilistic or causal theories, based on “Borel criteria,” is described. It is argued that this minimalist approach, free of any distracting metaphysical inputs, provides the essential support required for the conduct and advance of Science.
TL;DR: In this paper, the structural differences between ranked set samples and simple random samples are discussed, and properties of a ranked set sample analog of the Mann-Whitney-Wilcoxon statistic are presented.
Abstract: This paper is intended to provide the reader with an introduction to ranked set sampling, a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random samples. Methods for obtaining ranked set samples are described, and the structural differences between ranked set samples and simple random samples are discussed. Properties of the sample mean associated with a balanced ranked set sample are developed. A nonparametric ranked set sample estimator of the distribution function is discussed and properties of a ranked set sample analog of the Mann–Whitney–Wilcoxon statistic are presented.
TL;DR: In this article, the authors compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches, concluding that suffiency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction from the initial variable to a variable of the same dimension as the parameter.
Abstract: Sufficiency has long been regarded as the primary reduction pro- cedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, condi- tional procedures are almost as old and yet appear only occasionally in the central statistical literature. Recent likelihood theory examines the form of a general large sample statistical model and finds that certain natural condi- tional procedures provide, in wide generality, the definitive reduction from the initial variable to a variable of the same dimension as the parameter, a variable that can be viewed as directly measuring the parameter. We begin with a discussion of two intriguing examples from the literature that compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches. We then take two simple normal examples, with and with- out known scaling, and progressively replace the restrictive normal location assumption by more general distributional assumptions. We find that suffi- ciency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction for the analy- sis. We then examine the vector parameter case and find that the elimination of nuisance parameters requires a marginalization step, not the commonly profferred conditional calculation that is based on exponential model struc- ture. Some general conditioning and modelling criteria are then introduced. This is followed by a survey of common ancillary examples, which are then assessed for conformity to the criteria. In turn, this leads to a discussion of the place for the global repeated sampling principle in statistical inference. It is argued that the principle in conjunction with various optimality criteria has been a primary factor in the longstanding attachment to the sufficiency approach and in the related neglect of the conditioning procedures based di- rectly on available evidence.
TL;DR: This paper presented three lectures on a robust analysis of linear models, similar to the traditional least square-based analysis, which offers the user a unified methodology for inference procedures in general linear models.
Abstract: This paper presents three lectures on a robust analysis of linear models. One of the main goals of these lectures is to show that this analysis, similar to the traditional least squares-based analysis, offers the user a unified methodology for inference procedures in general linear models. This discussion is facilitated throughout by the simple geometry underlying the analysis. The traditional analysis is based on the least squares fit which minimizes the Euclidean norm, while the robust analysis is based on a fit which minimizes another norm. Several examples involving real data sets are used in the lectures to help motivate the discussion.
TL;DR: In this article, the authors introduced and illustrated methods for synthesizing evidence from case control and cohort studies, and controlled trials, accounting for differences among the studies in their design, length of follow-up and quality.
Abstract: Methods are introduced and illustrated for synthesising evidence from case control and cohort studies, and controlled trials, accounting for differences among the studies in their design, length of followup and quality. The methods, based on hierarchical but nonexchangeable Bayesian models, are illustrated in a synthesis of disparate information about the health effects of passive exposure to environmental tobacco smoke.
TL;DR: In this paper, statistical techniques for modeling data from cohort studies that examine long-term effects of air pollution on children's health by comparing data from multiple communities with a diverse pollution profile are discussed.
Abstract: In this article we discuss statistical techniques for modeling data from cohort studies that examine long-term effects of air pollution on children’s health by comparing data from multiple communities with a diverse pollution profile. Under a general multilevel modeling paradigm, we discuss models for different outcome types along with their connections to the generalized mixed effects models methodology. The model specifications include linear and flexible models for continuous lung function data, logistic and/or time-to-event models for symptoms data that account for misspecifications via hidden Markov models and Poisson models for school absence counts. The main aim of the modeling scheme is to be able to estimate effects at various levels (e.g., within subjects across time, within communities across subjects and between communities). We also discuss in detail various recurring issues such as ecologic bias, exposure measurement error, multicollinearity in multipollutant models, interrelationships between major endpoints and choice of appropriate exposure metrics. The key conceptual issues and recent methodologic advances are reviewed, with illustrative results from the Southern California Children’s Health Study, a 10-year study of the effects of air pollution on children’s respiratory health.
TL;DR: In this article, the authors show how the advances in Bayesian analysis and statistical computation are intermingled, and how they can be used together in the context of complex problems.
Abstract: The emergence in the past years of Bayesian analysis in many methodological and applied fields as the solution to the modeling of complex problems cannot be dissociated from major changes in its computational implementation. We show in this review how the advances in Bayesian analysis and statistical computation are intermingled.
TL;DR: In this article, the authors extend the notion of multivariate position to two-dimensional quantile-quantile plots, which can reveal outliers, differences in location and scale, and other differences between the distributions.
Abstract: Quantile–quantile (QQ) plots for comparing two distributions are constructed by matching like-positioned values (i.e., quantiles) in the two distributions. These plots can reveal outliers, differences in location and scale, and other differences between the distributions. A particularly useful application is comparing residuals from an estimated linear model to the normal. A robust estimate, such as the Sen–Theil estimate, of the regression line is important. Extensions to two-dimensional QQ plots are presented, relying on a particular notion of multivariate position.
TL;DR: This article is an abridgment of the full report of a workshop to discuss the future challenges and opportunities for the statistics community held at the National Science Foundation in May 2002.
Abstract: In May 2002 a workshop was held at the National Science Foundation to discuss the future challenges and opportunities for the statistics community. After the workshop the scientific committee produced an extensive report that described the general consensus of the community. This article is an abridgment of the full report.
TL;DR: In this paper, the authors present a biographical sketch of Thomas Bayes, from whom Bayes theorem takes its name, which includes his family background and education, as well as his scientific and theological work.
Abstract: Thomas Bayes, from whom Bayes theorem takes its name, was probably born in 1701, so the year 2001 marked the 300th anniversary of his birth. This biography was written to celebrate this anniversary. The current sketch of his life includes his family background and education, as well as his scientific and theological work. In contrast to some, but not all, biographies of Bayes, the current biography is an attempt to cover areas beyond Bayes’ scientific work. When commenting on the writing of scientific biography, Pearson [(1978). The History of Statistics in the 17th and 18th Centuries… . Charles Griffin and Company, London] stated, “it is impossible to understand a man’s work unless you understand something of his character and unless you understand something of his environment. And his environment means the state of affairs social and political of his own age.” The intention here is to follow this general approach to biography. There is very little primary source material on Bayes and his work. For example, only three of his letters and a notebook containing some sketches of his own work, almost all unpublished, as well as notes on the work of others are known to have survived. Neither the letters nor the notebook is dated, and only one of the letters can be dated accurately from internal evidence. This biography contains new information about Bayes. In particular, among the papers of the 2nd Earl Stanhope, letters and papers of Bayes have been uncovered that previously were not known to exist. The letters indirectly confirm the centrality of Stanhope in Bayes’ election to the Royal Society. They also provide evidence that Bayes was part of a network of mathematicians initially centered on Stanhope. In addition, the letters shed light on Bayes’ work in infinite series.
TL;DR: Nonparametric regression as mentioned in this paper is a popular area of research in statistical visualization and simulation, and a survey of the literature can be found in the introduction of this paper. But it is difficult for many students to have an initial intuition about what the techniques are and why they work.
Abstract: An introduction to nonparametric regression is accomplished with selected real data sets, statistical graphics and simulations from known functions. It is pedagogically effective for many to have some initial intuition about what the techniques are and why they work. Visual displays of small examples along with the plots of several types of smoothers are a good beginning. Some students benefit from a brief historical development of the topic, provided that they are familiar with other methodology, such as linear regression. Ultimately, one must engage the formulas for some of the linear curve estimators. These mathematical expressions for local smoothers are more easily understood after the student has seen a graph and a description of what the procedure is actually doing. In this article there are several such figures. These are mostly scatterplots of a single response against one predictor. Kernel smoothers have series expansions for bias and variance. The leading terms of those expansions yield approximate expressions for asymptotic mean squared error. In turn these provide one criterion for selection of the bandwidth. This choice of a smoothing parameter is done a rich variety of ways in practice. The final sections cover alternative approaches and extensions. The survey is supplemented with citations to some excellent books and articles. These provide the student with an entry into the literature, which is rapidly developing in traditional print media as well as on line.
TL;DR: In this article, a nonparametric confidence set for the unknown CMB spectrum was constructed, based on minimal assumptions, which closely matches the model-based estimates used by cosmologists, but can make a wide range of additional inferences.
Abstract: The cosmic microwave background (CMB), which permeates the entire Universe, is the radiation left over from just 380,000 years after the Big Bang. On very large scales, the CMB radiation field is smooth and isotropic, but the existence of structure in the Universe—stars, galaxies, clusters of galaxies, …—suggests that the field should fluctuate on smaller scales. Recent observations, from the Cosmic Microwave Background Explorer to the Wilkinson Microwave Anisotropy Probe, have strikingly confirmed this prediction. CMB fluctuations provide clues to the Universe’s structure and composition shortly after the Big Bang that are critical for testing cosmological models. For example, CMB data can be used to determine what portion of the Universe is composed of ordinary matter versus the mysterious dark matter and dark energy. To this end, cosmologists usually summarize the fluctuations by the power spectrum, which gives the variance as a function of angular frequency. The spectrum’s shape, and in particular the location and height of its peaks, relates directly to the parameters in the cosmological models. Thus, a critical statistical question is how accurately can these peaks be estimated. We use recently developed techniques to construct a nonparametric confidence set for the unknown CMB spectrum. Our estimated spectrum, based on minimal assumptions, closely matches the model-based estimates used by cosmologists, but we can make a wide range of additional inferences. We apply these techniques to test various models and to extract confidence intervals on cosmological parameters of interest. Our analysis shows that, even without parametric assumptions, the first peak is resolved accurately with current data but that the second and third peaks are not.
TL;DR: In this article, the authors provide a brief survey of recent developments in this area and present a procedure to test the assumption that the generating random field is Gaussian, which is of fundamental importance both to validate statistical inference procedures and to discriminate between competing scenarios for the Big Bang dynamics.
Abstract: Cosmic microwave background (CMB) radiation can be viewed as a snapshot of the Universe 13 billion years ago, when it had 0.002% of its current age. A flood of data on CMB is becoming available thanks to satellite and balloon-borne missions, and a number of statistical issues have been raised consequently. A very relevant issue is the characterization of the statistical distribution of CMB and, in particular, procedures to test the assumption that the generating random field is Gaussian. Gaussianity tests are of fundamental importance both to validate statistical inference procedures and to discriminate between competing scenarios for the Big Bang dynamics. Several procedures have been proposed in the cosmological literature. This article is an attempt to provide a brief survey of developments in this area.
TL;DR: A survey of modern Bayesian asymptotics is given in this article, where specific attention is paid to the Hellinger consistency of posterior distributions and the study of Bayes factors.
Abstract: A survey of modern Bayesian asymptotics is given. Specific attention is paid to the Hellinger consistency of posterior distributions and the asymptotic study of Bayes factors.
TL;DR: In this article, rank-based analyses of crossover studies are presented. But they do not consider the issue of statistical power in choosing an appropriate test when more than one nonparametric approach is available.
Abstract: We illustrate nonparametric, and particularly rank-based analyses of crossover studies, designs in which each subject receives more than one treatment over time. Principles involved in using the Wilcoxon rank sum test in the simple two-period, two-treatment crossover are described through theory and example. We then extend the ideas to two-treatment designs with more than two periods and to three-treatment, three-period designs. When more than one nonparametric approach is available, we consider the issue of statistical power in choosing an appropriate test.