Showing papers in "Statistical Science in 2005"

Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling

Abstract: In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. While MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps underappreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to so-called label switching in the MCMC output. This means that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints, relabelling algorithms and label invariant loss functions. We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.

Experiments in Stochastic Computation for High-Dimensional Graphical Models

Abstract: We discuss the implementation, development and performance of methods of stochastic computation in Gaussian graphical models. We view these methods from the perspective of high-dimensional model search, with a particular interest in the scalability with dimension of Markov chain Monte Carlo (MCMC) and other stochastic search methods. After reviewing the structure and context of undirected Gaussian graphical models and model uncertainty (covariance selection), we discuss prior specifications, including new priors over models, and then explore a number of examples using various methods of stochastic computation. Traditional MCMC methods are the point of departure for this experimentation; we then develop alternative stochastic search ideas and contrast this new approach with MCMC. Our examples range from low (12–20) to moderate (150) dimension, and combine simple synthetic examples with data analysis from gene expression studies. We conclude with comments about the need and potential for new computational methods in far higher dimensions, including constructive approaches to Gaussian graphical modeling and computation.

On Model Expansion, Model Contraction, Identifiability and Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables

Abstract: When a candidate model for data is nonidentifiable, conventional wisdom dictates that the model must be simplified somehow so as to gain identifiability. We explore two scenarios involving mismeasured variables where, in fact, model expansion, as opposed to model contraction, might be used to obtain identifiability. We compare the merits of model contraction and model expansion. We also investigate whether it is necessarily a good idea to alter the model for the sake of identifiability. In particular, estimators obtained from identifiable models are compared to those obtained from nonidentifiable models in tandem with crude prior distributions. Both asymptotic theory and simulations with Markov chain Monte Carlo-based estimators are used to draw comparisons. A technical point which arises is that the asymptotic behavior of a posterior mean from a nonidentifiable model can be investigated using standard asymptotic theory, once the posterior mean is described in terms of the identifiable part of the model only.

Strong, Weak and False Inverse Power Laws

Abstract: Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated data sets, and if one looks more carefully in the lower range of values that was originally excluded, the power law behavior usually breaks down at some point. This breakdown seems to fall into two broad cases, called here (1) weak and (2) false inverse power laws (WIPL and FIPL, resp.). Case 1 refers to the situation where the sample data fit a distribution that has an approximate inverse power form only in some upper range of values. Case 2 refers to the situation where a highly truncated sample from certain exponential-type (and in particular, “lognormal-like”) distributions can convincingly mimic a power law. The main objectives of this paper are (a) to show how the discovery of Pareto–Zipf-type laws is closely associated with truncated data sets; (b) to elaborate on the categories of strong, weak and false inverse power laws; and (c) to analyze FIPLs in some detail. I conclude that many, but not all, Pareto–Zipf examples are likely to be FIPL finite mixture distributions and that there are few genuine instances of SIPLs.

A Selective Overview of Nonparametric Methods in Financial Econometrics

Abstract: This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.

Semiparametric Estimation of Treatment Effect in a Pretest-Posttest Study with Missing Data

Abstract: The pretest–posttest study is commonplace in numerous applications. Typically, subjects are randomized to two treatments, and response is measured at baseline, prior to intervention with the randomized treatment (pretest), and at prespecified follow-up time (posttest). Interest focuses on the effect of treatments on the change between mean baseline and follow-up response. Missing posttest response for some subjects is routine, and disregarding missing cases can lead to invalid inference. Despite the popularity of this design, a consensus on an appropriate analysis when no data are missing, let alone for taking into account missing follow-up, does not exist. Under a semiparametric perspective on the pretest–posttest model, in which limited distributional assumptions on pretest or posttest response are made, we show how the theory of Robins, Rotnitzky and Zhao may be used to characterize a class of consistent treatment effect estimators and to identify the efficient estimator in the class. We then describe how the theoretical results translate into practice. The development not only shows how a unified framework for inference in this setting emerges from the Robins, Rotnitzky and Zhao theory, but also provides a review and demonstration of the key aspects of this theory in a familiar context. The results are also relevant to the problem of comparing two treatment means with adjustment for baseline covariates.

Data Dissemination and Disclosure Limitation in a World Without Microdata: A Risk-Utility Framework for Remote Access Analysis Servers

Abstract: Given the public’s ever-increasing concerns about data confidentiality, in the near future statistical agencies may be unable or unwilling, or even may not be legally allowed, to release any genuine microdata—data on individual units, such as individuals or establishments. In such a world, an alternative dissemination strategy is remote access analysis servers, to which users submit requests for output from statistical models fit using the data, but are not allowed access to the data themselves. Analysis servers, however, are not free from the risk of disclosure, especially in the face of multiple, interacting queries. We describe these risks and propose quantifiable measures of risk and data utility that can be used to specify which queries can be answered, and with what output. The risk-utility framework is illustrated for regression models.

Control variates for quasi-Monte Carlo

Abstract: Quasi-Monte Carlo (QMC) methods have begun to displace ordinary Monte Carlo (MC) methods in many practical problems. It is natural and obvious to combine QMC methods with traditional variance reduction techniques used in MC sampling, such as control variates. There can, however, be some surprises. The optimal control variate coecient for QMC methods is not in general the same as for MC. Using the MC formula for the control variate coecient can worsen the performance of QMC methods. A good control variate in QMC is not necessarily one that correlates with the target integrand. Instead, certain high frequency parts or derivatives of the control variate should correlate with the corresponding quantities of the target. We present strategies for applying control variate coecients with QMC, and illustrate the method on a 16 dimensional integral from computational nance. We also include a survey of QMC aimed at a statistical readership.

Fisher and Regression

Abstract: In 1922 R. A. Fisher introduced the modern regression model, synthesizing the regression theory of Pearson and Yule and the least squares theory of Gauss. The innovation was based on Fisher’s realization that the distribution associated with the regression coefficient was unaffected by the distribution of X. Subsequently Fisher interpreted the fixed X assumption in terms of his notion of ancillarity. This paper considers these developments against the background of the development of statistical theory in the early twentieth century.

Fuzzy and Randomized Confidence Intervals and P-Values

Abstract: The optimal hypothesis tests for the binomial distribution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two-tailed randomized tests. Conventional confidence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of UMP and UMPU tests. We also introduce a new P-value notion, called fuzzy P-values or abstract randomized P-values, that also inherits the same exactness and optimality.

How to Lie with Bad Data

Abstract: As Huff’s landmark book made clear, lying with statistics can be accomplished in many ways. Distorting graphics, manipulating data or using biased samples are just a few of the tried and true methods. Failing to use the correct statistical procedure or failing to check the conditions for when the selected method is appropriate can distort results as well, whether the motives of the analyst are honorable or not. Even when the statistical procedure and motives are correct, bad data can produce results that have no validity at all. This article provides some examples of how bad data can arise, what kinds of bad data exist, how to detect and measure bad data, and how to improve the quality of data that have already been collected.

Lying with maps

Abstract: Darrell Huff’s How to Lie with Statistics was the inspiration for How to Lie with Maps, in which the author showed that geometric distortion and graphic generalization of data are unavoidable elements of cartographic representation. New examples of how ill-conceived or deliberately contrived statistical maps can greatly distort geographic reality demonstrate that lying with maps is a special case of lying with statistics. Issues addressed include the effects of map scale on geometry and feature selection, the importance of using a symbolization metaphor appropriate to the data and the power of data classification to either reveal meaningful spatial trends or promote misleading interpretations.

Fisher in 1921

Abstract: Ronald A. Fisher’s 1921 article on mathematical statistics (submitted and read in 1921; published in 1922) was arguably the most influential article on that subject in the twentieth century, yet up to that time Fisher was primarily occupied with other pursuits. A number of previously published documents are examined in a new light to argue that the origin of that work owes a considerable (and unacknowledged) debt to a challenge issued in 1916 by Karl Pearson.

From Unit Root to Stein’s Estimator to Fisher’s k Statistics: If You Have a Moment, I Can Tell You More

Abstract: Any general textbook that discusses moment generating functions (MGFs) shows how to obtain a moment of positive-integer order via differentiation, although usually the presented examples are only illustrative, because the corresponding moments can be calculated in more direct ways. It is thus somewhat unfortunate that very few textbooks discuss the use of MGFs when it becomes the simplest, and sometimes the only, approach for analytic calculation and manipulation of moments. Such situations arise when we need to evaluate the moments of ratios and logarithms, two of the most common transformations in statistics. Such moments can be obtained by differentiating and integrating a joint MGF of the underlying untransformed random variables in appropriate ways. These techniques are examples of multivariate Laplace transform methods and can also be derived from the fact that moments of negative orders can be obtained by integrating an MGF. This article reviews, extends and corrects various results scattered in the literature on this joint-MGF approach, and provides four applications of independent interest to demonstrate its power and beauty. The first application, which motivated this article, is for the exact calculation of the moments of a well-known limiting distribution under the unit-root AR(1) model. The second, which builds on Stigler’s Galtonian perspective, reveals a straightforward, non-Bayesian constructive derivation of the Stein estimator, as well as convenient expressions for studying its risk and bias. The third finds an exceedingly simple bound for the bias of a sample correlation from a bivariate normal population, namely the magnitude of the relative bias is not just of order n−1, but actually is bounded above by n−1 for all sample sizes n≥2. The fourth tackles the otherwise intractable problem of studying the finite-sample optimal bridge in the context of bridge sampling for computing normalizing constants. A by-product of the joint-MGF approach is that positive-order fractional moments can be easily obtained from an MGF without invoking the concept of fractional differentiation, a method used by R. A. Fisher in his study of k statistics 45 years before it reappeared in the probability literature.

How to Confuse with Statistics or: The Use and Misuse of Conditional Probabilities

Abstract: This article shows by various examples how consumers of statistical information may be confused when this information is presented in terms of conditional probabilities. It also shows how this confusion helps others to lie with statistics, and it suggests both confusion and lies can be exposed by using alternative modes of conveying statistical information. Contains a discussion of traffic safety, violence, and intimate partner abuse

Lies, Calculations and Constructions: Beyond How to Lie with Statistics

Abstract: Darrell Huff’s How to Lie with Statistics remains the best-known, nontechnical call for critical thinking about statistics. However, drawing a distinction between statistics and lying ignores the process by which statistics are socially constructed. For instance, bad statistics often are disseminated by sincere, albeit innumerate advocates (e.g., inflated estimates for the number of anorexia deaths) or through research findings selectively highlighted to attract media coverage (e.g., a recent study on the extent of bullying). Further, the spread of computers has made the production and dissemination of dubious statistics easier. While critics may agree on the desirability of increasing statistical literacy, it is unclear who might accept this responsibility.

Comment: Randomized Confidence Intervals and the Mid-P Approach

Abstract: We enjoyed reading the interesting, thought-provok ing article by Geyer and Meeden. In our comments we will try to place their work in perspective rela tive to the original proposals for exact and random ized confidence intervals for the binomial parameter. We propose a fuzzy version of the original binomial randomized confidence interval, due to Stevens (1950). Our approach motivates an existing nonrandomized confidence interval based on inverting a test using the mid-P value. The mid-P confidence interval provides a sensible compromise that mitigates the effects of con servatism of exact methods, yet provides results that are more easily understandable to the scientist.

Darrell Huff and Fifty Years of How to Lie with Statistics

Abstract: Over the last fifty years, How to Lie with Statistics has sold more copies than any other statistical text. This note explores the factors that contributed to its success and provides biographical sketches of its creators: author Darrell Huff and illustrator Irving Geis.

Comment: A Selective Overview of Nonparametric Methods in Financial Econometrics

Abstract: These comments concentrate on two issues arising from Fan’s overview. The first concerns the importance of finite sample estimation bias relative to the specification and discretization biases that are emphasized in Fan’s discussion. Past research and simulations given here both reveal that finite sample effects can be more important than the other two effects when judged from either statistical or economic viewpoints. Second, we draw attention to a very different nonparametric technique that is based on computing an empirical version of the quadratic variation process. This technique is not mentioned by Fan but has many advantages and has accordingly attracted much recent attention in financial econometrics and empirical applications.

Rejoinder: Fuzzy and Randomized Confidence Intervals and P-Values

Abstract: Agresti and Gottard propose an equal-tailed fuzzy interval they attribute to Stevens (1950), although, of course, the notion of a fuzzy confidence interval was not exactly what Stevens proposed. This is the fuzzy confidence interval with membership function given by (1.1b) of our article, where is the critical function of the equal-tailed randomized test. Brown, Cai and DasGupta propose a fuzzy interval they attribute to Pratt (1961), although, of course, the notion of a fuzzy confidence interval was not exactly what Pratt proposed. This is the fuzzy confidence inter val with membership function given by (1.1b), where (/)(-, a, 9) is the critical function of the most pow erful randomized simple-versus-simple test with null hypothesis that the data are Binomial(n, 9) and alterna tive hypothesis that the data have the discrete uniform distribution on {0,... ,n}. Figure 1 herein shows these two new fuzzy intervals along with the UMPU fuzzy intervals we proposed. Clearer and larger figures for more values of x are given on the web (www. s tat. umn. edu/geyer / fuzz). From the figure it can be seen that the Pratt (Brown-Cai-DasGupta) intervals are not unimodal, a point noted by Pratt (1961) and by Brown, Cai, and DasGupta in their comments. These fuzzy intervals arise from an optimality argument we think shows a fundamental misunderstanding of fuzzy confidence in tervals (which, of course, we cannot anachronistically blame Pratt for). From our point of view, what they actually do is optimally test against an alternative (dis crete uniform) that we cannot imagine will ever be of interest in applications. Nevertheless, we say the more the merrier. If one likes these fuzzy intervals, then use them. The equal-tailed (Agresti-Gottard) tests are more reasonable. There is little practical difference between their proposal and ours. As they say, their intervals look more reasonable for x in the middle of the range and ours look more reasonable elsewhere, but good fre quentists cannot think this way (however natural it may be), since any frequentist property depends on averag ing overalls.

Comment: Fuzzy and Bayesian p-Values and u-Values

Abstract: It is a pleasure to discuss this fascinating paper, which presents a new way to express uncertainty in p-values for composite null hypotheses. I particularly like the graphical display above equation (1.2). I have some brief comments about the confidence interval for binomial proportions and then some more to say about the relationship to Bayesian p-values. Geyer and Meeden’s fuzzy p-values are related to Bayesian p-values in that they recognize the uncertainty inherent in testing a composite null hypothesis. Fuzzy and Bayesian p-values also both allow for test statistics to depend on missing data, latent data and parameters, as well as on observed data. There is the potential to generalize these ideas to graphical test statistics and connect to exploratory data analysis.

Rejoinder: A selective overview of nonparametric methods in financial econometrics

Abstract: I am fully aware that financial econometrics has grown into a vast discipline itself and that it is impossible for me to provide an overview within a reasonable length. Therefore, I greatly appreciate what all discussants have done to expand the scope of discussion and provide additional references. They have also posed open statistical problems for handling nonstationary and/or non-Markovian data with or without market noise. In addition, statistical issues on various versions of capital asset pricing models and their related stochastic discount models [15, 19], the efficient market hypothesis [44] and risk management [17, 45] have barely been discussed. These reflect the vibrant intersection of the interfaces between statistics and finance. I will make some further efforts in outlining econometric problems where statistics plays an important role after brief response to the issues raised by the discussants.

In Search of the Magic Lasso: The Truth About the Polygraph

Abstract: In the wake of controversy over allegations of espionage by Wen Ho Lee, a nuclear scientist at the Department of Energy’s Los Alamos National Laboratory, the department ordered that polygraph tests be given to scientists working in similar positions. Soon thereafter, at the request of Congress, the department asked the National Research Council (NRC) to conduct a thorough study of polygraph testing’s ability to distinguish accurately between lying and truth-telling across a variety of settings and examinees, even in the face of countermeasures that may be employed to defeat the test. This paper tells some of the story of the work of the Committee to Review the Scientific Evidence on the Polygraph, its report and the reception of that report by the U.S. government and Congress.

Comment: A Selective Overview of Nonparametric Methods in Financial Econometrics

Abstract: Professor Fan should be congratulated for his review that convincingly demonstrates the usefulness of non parametric techniques to financial econometric prob lems. He is mainly concerned with financial models given by stochastic differential equations, that is, dif fusion processes. I will therefore complement his se lective review by discussing some important problems and useful methods for diffusion models that he has

How to Accuse the Other Guy of Lying with Statistics

Abstract: We’ve known how to lie with statistics for 50 years now. What we really need are theory and praxis for accusing someone else of lying with statistics. The author’s experience with the response to The Bell Curve has led him to suspect that such a formulation already exists, probably imparted during a secret initiation for professors in the social sciences. This article represents his best attempt to reconstruct what must be in it.

A Conversation with John Hartigan

Abstract: John Anthony Hartigan was born on July 2, 1937 in Sydney, Australia. He attended the University of Sydney, earning a B.Sc. degree in mathematics in 1959 and an M.Sc. degree in mathematics the following year. In 1960 John moved to Princeton where he studied for his Ph.D. in statistics under the guidance of John Tukey and Frank Anscombe. He completed his Ph.D. in 1962, and worked as an Instructor at Princeton in 1962–1963, and as a visiting lecturer at the Cambridge Statistical Laboratory in 1963–1964. In 1964 he joined the faculty at Princeton. He moved to Yale as Associate Professor with tenure in 1969, became a Professor in 1972 and, in 1983, became Eugene Higgins Professor of Statistics at Yale—a position previously held by Jimmie Savage. He served as Chairman of the Statistics Department at Yale from 1973 to 1975 and again from 1988 to 1994. John was instrumental in the establishment of the Social Sciences Statistical Laboratory at Yale and served as its Director from 1985 to 1989 and again from 1991 to 1993. He served as Chairman of the National Research Council Committee on the General Aptitude Test Battery from 1987 to 1990. John’s research interests cover the foundations of probability and statistics, classification, clustering, Bayes methods and statistical computing. He has published over 80 journal papers and two books: Clustering Algorithms in 1975 and Bayes Theory in 1983. He is a Fellow of the American Statistical Association and of the Institute of Mathematical Statistics. He served as President of the Classification Society from 1978 to 1980 and as Editor of Statistical Science from 1984 to 1987. John married Pamela Harvey in 1960. They have three daughters and three grandchildren.

A Public Health Controversy in 19th Century Canada

Abstract: From 1858 to 1860, the English naturalist and social activist Philip P. Carpenter toured North America. In April of 1859 he visited Montreal, Canada. Shocked by the sanitary conditions of the city, he wrote a paper that used statistical arguments to call for health reforms. Six years later he settled in Montreal and quickly became an active promoter of this cause. He began accumulating additional numerical evidence in support of his views. In the aftermath of a cholera scare in 1866, Carpenter became the driving force behind the creation of the Montreal Sanitary Association. That same year he published a second, more detailed article that took advantage of the 1861 census data to analyze mortality rates in Montreal. He made further statistical investigations in 1869. Unfortunately, Carpenter did not understand some of the subtleties as- sociated with the analysis of vital statistics. An obscure bookkeeper, Andrew A. Watt, made a scathing public attack on both Carpenter's data and his interpretation thereof. In a series of newspaper articles, Watt scrutinized systematically all of Carpenter's writings, showing his faults and correcting them wherever he could. Although Watt's arguments were correct, the public was slow to under- stand them. The controversy continued through 1870. When the nature of Watt's criticisms finally became better understood and Carpenter persisted with statistical arguments, the latter lost credibility and was abandoned by his own association.

Comment: Fuzzy and Randomized Confidence Intervals and P-Values

Abstract: Geyer and Meeden are to be congratulated for a ma jor idea on how to express uncertainty in classical fre quentist statistics. Their development of the trinity of fuzzy test functions, confidence intervals and P -values brings a new coherence to the relationship among these statistical entities when the test is randomized. In Geyer and Meeden, the uncertainty or randomization is due to the discreteness of data random variables, but another form of uncertainty or randomness arises in la tent variable problems and may be treated in an analo gous fashion. In this discussion, I will focus on fuzzy P -values in that context. Following the notation of Geyer and Meeden, let X denote the observed data random variables, but sup pose there are latent variables W of scientific interest, so that the ideal test statistic is t(X, W), a function of both X and W. Many examples of this situation arise in the analysis of genetic data, where W may be un observable DNA types or paths of descent of DNA to the observed individuals of a pedigree or population. Indeed, in this case the hypothesis of interest often concerns the probability distribution of W and hence the ideal test statistic is a function of W alone. The data random variable X is only of interest for the in formation it provides about W through some proba bility model Pr(X|W). For convenience, we consider here the case where the latent test statistic t(W) is a function only of W and assume the random variable W to be continuous. The discrete case is considered by Thompson and Geyer (2005). In the area of statistical genetic methodology, a stan dard procedure has been to average over the latent variable uncertainty and form test statistics E(t (W)|X) (Whittemore and Halpern, 1994; Kruglyak, Daly, Reeve-Daly and Lander, 1996). However, there appear to be neither theoretical justification nor optimality properties for such a proceeding. Moreover, the dis tribution of such a test statistic is not only hard or im possible to obtain, but depends on the distribution of X given W, which may itself be subject to consider able uncertainty. The second key point made by Geyer and Meeden is that the distribution function of the ab

Comment: Fuzzy and Randomized Confidence Intervals and P-Values

Abstract: We thank Professor Geyer and Professor Meeden for their thought-provoking article. We hope to also be thought-provoking in response, for we pretty much dis agree with their position. The fuzzy procedures proposed by the authors result from examining the test function, (?)(x,a,9) in three different ways, as a function of each of the three vari ables. This is an interesting exercise, which has not been done before in this way, and the authors are to be commended for their innovation. However, we think the resulting procedures will be of limited practical in terest. The authors start with the belief that discontinuous