bluuebunny/crossref_metadata_2025 huggingface dataset:
C
hoi–
W
illiams Distribution
Authors: Castiglioni Paolo
Published: July 2005

The Choi–Williams distribution is a transform that represents the spectral content of nonstationary signal as a two‐dimensional time–frequency map. It largely avoids one of the main problems of the Wigner–Ville distribution: the presence of interference terms in regions where one would expect zero power values.
Discrimination and Clustering for Multivariate Time Series
Authors: Shumway Robert H.
Published: July 2005

This entry gives a summary of the time and frequency domain approaches to classification and clustering for biostatistical and medical data that have been collected as multivariate correlated time series.
Intervention Analysis in Time Series
Authors: Helfenstein Ulrich
Published: July 2005

When data are collected in the form of dependent time series data, important questions are concerned with “changes” in the series. Changes may be “man made” or they may arise “naturally”: How efficient was a preventive program to decrease the monthly number of accidents? How did the pattern of morbidity change after an environmental accident? Notifications of diseases or accidents and so on, are often collected in regular intervals. ARIMA models allow the stochastic dependence of consecutive data to be modeled flexibly. Intervention analysis is an extension of ARIMA modeling allowing the study of the structure of changes of ARIMA processes. Since the series may be nonstationary, large changes of the series could occur even when no intervention takes place. Intervention analysis may allow an investigator to distinguish between what can be expected due to nonstationarity alone and what cannot. Analogous questions of “change” may arise when studying time series data recorded in an individual patient. This article reviews basic concepts of intervention analysis. The methods are illustrated by medical applications.
Linear Mixed Effects Models for Longitudinal Data
Authors: Pinheiro José C.
Published: July 2005

Linear mixed‐effects (LME) models have become a popular tool for analyzing longitudinal data that arise in areas as diverse as clinical trials, epidemiology, agriculture, economics, and geophysics. The increasing popularity of these models is explained by the flexibility they offer in modeling the within‐subject correlation often present in longitudinal data, by the handling of both balanced and unbalanced data, and by the availability of reliable and efficient software for fitting them. This paper describes a general LME model for longitudinal Gaussian data, presenting the model definition and its underlying assumptions. Maximum likelihood and restricted maximum likelihood estimation methods are discussed, as well as asymptotic inference methods. Available software for fitting and analyzing LME models are briefly reviewed. The model and methods are illustrated through an example from a longitudinal growth study.
Lomb Periodogram
Authors: Castiglioni Paolo
Published: July 2005

The Lomb periodogram is a generalization of the periodogram for unequally spaced time series. Since traditional spectral estimators need even sampling rates, unevenly spaced series need to be interpolated and resampled before spectral analysis. By contrast, the Lomb method does not require interpolation and resampling.
Multiple Time Series
Authors: Davies Neville
Published: July 2005

In medical investigations, it is not uncommon to record observations on two or more time series. For example, blood pressure and weight may be regularly monitored on a sample of patients. The individual time series may be analyzed by methods appropriate for univariate series, but this article concentrates on methods of analysis that help in understanding the system that gives rise to the data.
Multivariate Methods for Binary Longitudinal Data
Authors: Glynn Robert J. , Rosner Bernard
Published: July 2005

Longitudinal data in which the repeatedly measured response variable is binary often occur in medical investigations, particularly clinical trials. This article describes a number of suitable approaches to modeling such data including marginal models, mixed effects models, and transition models.
Noise and White Noise
Authors: Lawrence Clive J.
Published: July 2005

Noise and white noise arise in the description of time series data. Both are defined in this entry.
N
yquist Frequency
Authors: Janacek G. J.
Published: July 2005

If we sample a signal observed over time at intervals, perhaps by digitizing an analog signal, then we naturally expect some loss of information. If the observed signal has cyclic components, then the sampling may induce some distortion in the effects we see in the sampled results.

There are some cyclic effects that are not observable, at least directly, and these are ones with high frequencies (or short periods). The frequency beyond which cyclic effects are unobservable is called the
Nyquist or folding frequency
. This frequency is related to the sampling rate
Δt
and is in fact 1/2
Δt
cycles per unit time.
Random Coefficient Repeated Measures Model
Authors: Goldstein Harvey
Published: July 2005

Biostatistical data often have a hierarchical structure. Typically, these structures are naturally occurring ones: animal populations are characterized by individuals nested within parents, themselves often nested within groups or herds that may also be nested within spatial entities. In other cases, the structure may result from research designs, as in multicenter clinical trials. Other examples are repeated measure designs, where measurements are “nested” within individual subjects and multivariate response data, where measurements are “nested” within individuals. In addition to nesting relationships among data units, we may also have cross‐classifications. For example, an individual cow may be nested within a herd of cattle, but also be the offspring of parent stock where any parent may contribute to several herds: individual cows are thus cross‐classified by parents as well as nested within their herds. A further complexity is also often present whereby individual units at one level of a data hierarchy may be nested within more than one higher‐level unit as in spatial data.

The article describes models for all these kinds of data structures, increasing in complexity as they move from simple hierarchies with continuously distributed responses, to cross‐classifications and multivariate data and to discrete responses. Various extensions and special cases will also be considered. The emphasis is on model specification with a brief section on estimation.
Robust Methods in Time Series Analysis
Authors: Lawrence Clive J.
Published: July 2005

In this article, a number of techniques for analyzing time series data that are robust to the presence of outliers are discussed. The general approach is to remove or suppress the possible effects of contamination or spurious information by smoothing.
Spectral Analysis
Authors: Janacek G. J.
Published: July 2005

Identifying cyclic effects in time series data is often an extremely important part of the analysis of the series. In this article, a number of approaches are discussed, including harmonic analysis and linear filters.
Structural Time Series Models
Authors: Harvey A. , Koopman S. J.
Published: July 2005

Structural time series models are set up in terms of components, such as trends and cycles, which have a direct interpretation. Their statistical treatment is based on the state space form and the Kalman filter. Signal extraction is carried out by smoothing algorithms. Models may be constructed for multivariate series, and explanatory and intervention variables may be included. Nonlinear and non‐Gaussian models are handled using simulation techniques.
Time Series
Authors: Everitt Brian S.
Published: July 2005

Time series data consist of recordings of some quantity of interest made, usually at regular intervals, over a considerable period of time. Medical time series are common and include as examples, weekly admissions into a hospital and monthly mortality rates for a particular disease. In this article, an overview of the analysis of time series is given.
Variogram
Authors: Diggle Peter J.
Published: July 2005

When analyzing longitudinal data, it is often essential to learn something about the covariance structure of the data before deciding which models might be most appropriate. The variogram is a graphical display that is often helpful.
Window Estimate
Authors: Davies Neville
Published: July 2005

Window estimate is a term used both in frequency domain estimation in time series analysis and in kernel density estimation. In this article, both situations are considered.
Zero Padding
Authors: Castiglioni Paolo
Published: July 2005

Zero‐padding is a procedure by which the length of a time series is extended by the addition of zeros, enabling the power spectrum to be evaluated over a fuller range of frequencies.
Cluster Analysis of Subjects, Hierarchical Methods
Authors: Everitt Brian S.
Published: July 2005

Hierarchical cluster classification involves partitioning data into a series of groups. The first group consists ofnsingle‐member “clusters”, the last consists of a single group with allnindividuals. Hierarchical cluster classification could be represented by a diagram known as a dendrogram. Properties and problems of hierarchical clustering techniques are described. The two major types of algorithms that have been used to produce hierarchical classifications are agglomerative and divisive. Careful validation of solutions is a clear requirement in any clustering exercise. An example to illustrate hierarchical methods is presented.
Cluster Analysis of Subjects, Nonhierarchical Methods
Authors: Everitt Brian S.
Published: July 2005

Cluster analysis is concerned with investigating whether or not a given set of data consists of relatively distinct groups of observations. The cluster analysis methods produce a partition of individuals into a particular number of groups and a numerical index is assigned to the partition. This index indicates how successful the partition is in describing the data and therefore allows the comparison of competing partitions.

The numerical indices most commonly used arise from three matrices representing total dispersion, within‐group dispersion and between‐group dispersion. Several possible indices have been suggested which differ in the implicit assumptions made about the shape of any clusters present. Once the index has been selected, the partition that leads to its optimization is found. It is usually impractical to examine each individual partition, so algorithms designed to search for the optimum value of the clustering criterion have been developed.
Clustering, Complete Linkage
Authors: Massaro Joseph M.
Published: July 2005

Complete linkage clustering is used in hierarchical cluster analysis to partition a set of
n
observations into
g
groups or clusters on the basis of p variables measured on the
n
observations. The analysis is performed on an
n
 × 
n
matrix of pairwise distance measures between observations. The procedure starts by combining the two observations with the smallest distance between them. The procedure continues to combine clusters with the smallest distances until only one cluster, including the entire sample, remains. The number of groups,
g
, is determined to represent the final number of clusters to retain.
Covariance Matrix
Authors: Krzanowski W. J.
Published: July 2005

Inferential aspects of the sample covariance matrix are presented. A maximum likelihood estimator that is based on a frequentist approach as well as the other two estimators that are based on decision theoretic and Bayesian approaches are summarized. Inference is appropriate when sampling from a multivariate normal distribution. Elliptic distributions that encompass distributions other than normal distribution are described. The asymptotic distribution of the maximum likelihood statistic and its associated hypothesis tests are fully explained.
Discriminant Analysis, Linear
Authors: Hand David J.
Published: July 2005

Given a set of objects, each with a descriptive vector of measured variables, and each belonging to one of a finite set of classes, linear discriminant analysis is a tool for finding those linear combinations of the measured variables on which the classes are most separated. There are two main uses: using the linear combinations to predict the class membership of a new object for which only the vector of variables has been observed, and studying the vector of variables to characterize the nature of the differences between the classes. The method is based on the first and second moments of the multivariate distribution of the measured variables, and assumes that the covariance matrices of the different classes are the same. However, it is robust and can often be usefully applied in other situations. Many generalizations and extensions have been developed, and linear discriminant analysis continues to be widely used.
Factor
Authors: D'agostino Ralph B. , Russell Heidy K.
Published: July 2005

This article introduces the factors in factor analysis. A factor pattern is defined. Common factors and unique factor are explained. Once the number of factors and the final factor pattern are determined, the factor complexity is used to interpret the factors.
Factor Scores
Authors: D'AGOSTINO Ralph B. , Russell Heidy K.
Published: July 2005

After a set of interpretable factor loadings is determined, a next possible step is to compute factor scores. Exact factor scores can be obtained for a principal components analysis model. Methods that compute factor scores in the common factor analysis model are also presented. In general, there is limited value in the use of factor scores, for even though the factors are interpretable, the estimated factor scores are not well determined.
G
ram–
S
chmidt Process
Authors: Dukes Kimberly A.
Published: July 2005

The Gram–Schmidt process is a technique for constructing an orthogonal basis from a basis spanning the same subspace. The technique is used in a number of multivariate applications including battery reduction analyses and multiple regression applications. In the former, the goal is to reduce the number of variables at the same time explaining as much variance in the original variables as possible. In the latter, the goal is to reduce the
n
‐dimensional space (observations) to a
p
‐dimensional space (independent variables).
Lambda Criterion, Wilks'
Authors: Dasgupta Somesh
Published: July 2005

Wilk's Λ criterion is used to test the equality of the mean vectors ofkp‐variate normal distributions. The distribution of the test statistic can be expressed as a function of several independent beta distributions. The power function of this test is compared with the Lawley–Hotelling trace and Philbi's trace test for MANOVA.
Matrix Algebra
Authors: Searle Shayle R.
Published: July 2005

Matrix algebra is the algebra of rectangular (including square) arrays of numbers (and/or symbols) set out neatly in rows and columns. This structure leads to a wide variety of operations different from the traditional algebra of numbers and letters. Those operations have application in many different uses of mathematics and statistics, especially when large amounts of data are involved.
Multivariate Analysis of Variance
Authors: Morrison Donald F.
Published: July 2005

The multivariate analysis of variance (MANOVA) is an extension of the univariate analysis of variance to multidimensional, or vector‐valued, observations. The same experimental design or treatment layout applies to each of the observed response variables. The univariate assumption of a normal distribution is replaced by the multivariate normal distribution for the data vectors and the random error components in the mathematical model of the design.
Multivariate Analysis,Bayesian
Authors: Press S. James
Published: July 2005

Multivariate Bayesian Analysis uses Bayes' Theorem to make inferences about several generally correlated unknown quantities. This article explains the multivariate version of Bayes' Theorem. It explains the use of multidimensional credibility regions, hypothesis testing involving multidimensional unknown quantities, multivariate prior distributions: both multivariate vague and natural conjugate prior distributions, multivariate mixture prior distributions, exchangeability, multivariate numerical methods such as Markov Chain Monte Carlo methods, and computer‐assisted multivariate inference. There are references to some biostatistical multivariate Bayesian applications.
Multivariate Analysis, Overview
Authors: Morrison Donald F.
Published: July 2005

The article contains a survey of statistical methods based on the multivariate normal distribution. These include the estimators of the mean vector and covariance matrix and their distributions, hypothesis tests for the mean vector, the multivariate generalization of the analysis of variance and the general linear model, classification and discriminant functions, inferences about covariance matrices, principal components analysis, and factor analysis.
Multivariate Median and Rank Sum Tests
Authors: Sen Pranab K.
Published: July 2005

Univariate nonparametric procedures are extended to multivariate nonparametric procedures via the rank‐permutation principle. Univariate nonparametric procedures, such as the Wilcoxon signed‐rank test and ANCOVA, are reviewed. From this, expansion to the multivariate case is shown through emphasis on the rank‐permutation principle. This principle, based on the assets of a conditional procedure that renders conditionally distribution‐free properties, is applied to the multivariate case and allows for distribution‐freeness of a large class of rank statistics, such as the median and rank sum tests. This then carries into the development of multivariate multisample median tests and multivariate rank sum tests, such as the multivariate signed‐rank test. Furthermore, application of the principle is also used in the development of rank MANCOVA, the multivariate extension of ANCOVA.
Multivariate Outliers
Authors: Barnett Vic
Published: July 2005

Interest in outliers goes back to the origins of statistical enquiry. As observations in a univariate sample, they are not only extreme but are “extremely extreme”; they trigger concern for whether they are truly part of the population under investigation or reflect “contamination” from some other low incidence source.
MultivariateWeibull Distribution
Authors: Jensen D. R.
Published: July 2005

The univariate Weibull distribution is frequently used in accelerated life testing and failure time models. It reduces to the exponential distribution when the shape parameter is equal to one. Multivariate Weibull distributions encompass distributions exhibiting Weibull characteristics and relate to multivariate exponential distributions of various types. Applications include reliability analysis, survival analysis, and repeated failure time analysis.
Principal Components Analysis
Authors: D'agostino Ralph B.
Published: July 2005

Principal components analysis is a method for transforming a set ofncorrelated variables intom(<n) uncorrelated variables. This article provides a comprehensive summary of the technique, including its mathematical derivation and the various techniques that are employed for determining the number of factors to retain, for rotation and for development of component scores. The article illustrates the application of the technique with a comprehensive example from the Framingham Heart Study. The article also outlines the many applications of principal components analysis in a variety of fields.
Principal Coordinates Analysis
Authors: Gower John C.
Published: July 2005

Principal coordinates analysis, also known as Classical scaling, is a metric multidimensional scaling method based on projection, which uses spectral decomposition to approximate a matrix of distances/dissimilarities by the distances between a set of points in few dimensions. The points may be used in visualizations.
W
ishart Distribution
Authors: Lee Austin F.S.
Published: July 2005

Suppose
X
is a
p
‐variate multivariate normal random vector with mean μ and covariance matrix Σ. If

, the distribution of
V
is the Wishart distribution. The Wishart distribution is frequently used in likelihood ratio tests in multivariate tests.
Data Mining, Software Packages for
Authors: Haughton Dominique
Published: July 2005

The term
data mining
refers to the identification—within a typically large database—of new, valid, and interesting patterns. While data mining has become most popular in the context of, for example, database marketing, most of the methods under the data mining umbrella have been widely applied in biostatistics. We describe which main applications of data mining have arisen recently in biostatistics, and introduce the reader to some of the available data mining software packages with a reference to biostatistical needs.
Antithetic Variable
Authors: Morgan Byron J. T.
Published: July 2005

A brief overview of this concept and its application is given.
Computer Algebra
Authors: Kendall Wilfrid S.
Published: July 2005

This article begins by presenting basic common features, and by listing and giving references for current commercial computer algebra packages. There follow two simple introductory examples of the use of computer algebra in statistical contexts, and then a discussion of some typical constraints and features of computer algebra systems: it is important to be aware of these, since in this area, unrealistic expectations lead all too rapidly to frustration and disappointment. Finally, we list a few representative examples of research uses of computer algebra in statistical science, and conclude with suggestions for further reading.
Computer‐Intensive Methods
Authors: Ripley B. D.
Published: July 2005

One sense of “computer‐intensive” statistics is just statistical methodology that makes use of a large amount of computer time. (Examples include the bootstrap, jackknife, smoothing, image analysis, and many uses of the EM algorithm.) However, the term is usually used for methods that go beyond the minimum of calculations needed for an illuminating analysis, for example, by replacing analytic approximations by computational ones, or requiring numeric optimization or integration over high‐dimensional spaces. We introduce the subject with a very simple yet useful example, and then consider some of the areas in which computer‐intensive methods are used, to give a flavor of current research.
Optimization and Nonlinear Equations
Authors: Smyth Gordon K.
Published: July 2005

Optimization means finding that argument which minimizes or maximizes a given function such as a likelihood function or a sum of squares. Many optimization algorithms are derived from algorithms that solve the nonlinear equations defined by setting the derivative of the objective function equal to zero. This article discusses a number of methods for unconstrained optimization, including bisection and golden search in the univariate case and Newton's method and quasi‐Newton algorithms in the multivariate case. Applications to maximum likelihood estimation, Fisher's method of scoring, nonlinear regression, and generalized linear models are described. Restricted step modifications are discussed for preventing divergence of Newton‐type algorithms. A number of derivative‐free methods are discussed, including the Nelder–Mead simplex method, methods with numeric derivatives, conjugate methods, and the EM algorithm.
Simulation
Authors: Morgan Byron J.T.
Published: July 2005

Computer simulation is widely used to mimic the rules of complex systems. The resulting simulation models may then be used to study the effect of changes to those systems.
Statistical Map
Authors: Lawson A. B.
Published: July 2005

The concept of a map is defined, and the area of statistical map construction and related interpretational issues are discussed.
Support Vector Machines
Authors: Schölkopf Bernhard , Smola Alex
Published: July 2005

We describe the central ideas of Support Vector (SV) learning in a nutshell. This includes the notion of learning from data via risk minimization and the notion of kernels as an efficient means of encoding prior information. We give an overview over the optimization problems arising from SV classification and discuss various extensions such as kernel PCA and automatic margin adaptation.
Asymptotic Relative Efficiency (ARE)
Authors: Dasgupta Anirban
Published: July 2005

The efficiencies of two consistent estimators of a parameter may be compared by the ratio of their asymptotic variances. Alternative measures are the Pitman and Bahadur measures, which relate to the ratio of sample sizes needed to achieve equivalent asymptotic power. Topics examined are the relevance for finite samples, sensitivity to the underlying distribution, higher‐order efficiency (to compare different fully efficient estimators), and efficiency estimation in complex models.
Bayesian Methods
Authors: Dempster A. P.
Published: July 2005

Bayesian methods are used in the analysis of statistical data to provide posterior (postdata) probabilities for hypotheses. These are calculated, by Bayes' theorem, from prior (predata) probabilities and the likelihood function obtained from the observed data. Probabilities may be “objective” or “subjective”, representing degrees of belief. The history of this approach is surveyed. It fell out of fashion during the first half of the twentieth century, under the influences of the different frequentist approaches of Fisher and Neyman, but interest was maintained by the work of Jeffreys and de Finetti and revived in the mid‐1950s through the work of Savage and others. The article discusses problems arising in contemporary work, such as the involvement of many parameters, aspects of modeling, and availability of computing methods, and summarizes arguments for and against Bayesian methods.
Behrens–Fisher Problem
Authors: Cook Richard J. , Farewell Vern T.
Published: July 2005

The problem of testing for a difference between two normal means when no assumptions are made about the variances of the distributions, commonly known as the Behrens–Fisher problem, is presented. The properties of competing solutions to this problem are discussed. Inference when the variance ratio is fixed is outlined and the value of explicit consideration of the robustness of inferences as the variance ratio varies is explored.
Bivariate Normal Distribution
Authors: Morrison Donald F.
Published: July 2005

The bivariate normal distribution is defined by its density function. The regression function of one variable on the other is given by the linear conditional mean function. The conditional variance is expressed in terms of the correlation coefficient of the variables. References to tables of the bivariate normal distribution are included.
Central Limit Theory
Authors: Nolan Deborah
Published: July 2005

The sum of a large number of none too large independent random variables is approximately normally distributed. This important result enables the normal distribution to be used in a wide range of approximately valid procedures. The history of the central limit theorem is outlined, and various versions for different conditions on the random variables are described. Topics discussed include rates of convergence, a functional central limit theorem, results for dependent random variables, and stable laws for convergence to nonnormal distributions.
Characteristic Function
Authors: Koehler Kenneth
Published: July 2005

The characteristic function is a complex‐valued transform of the distribution function of a random variable, closely related to the moment generating function. It may be used to derive distributions of sums and other linear functions of independent random variables, to determine limiting distributions in large samples, and to derive moments. Its natural logarithm is the cumulant generating function. Various results are illustrated in terms of the binomial distribution.
Conditional Probability
Authors: Hill B.
Published: July 2005

Both the Bayesian and frequentist approaches to conditional probability are presented. The undesirable consequences of using any statistical decision procedure that is not directly based upon conditional probability are discussed. These include sure loss as in the coherency theory of de Finetti; and the uniform improvement in risk of any non‐Bayes statistical decision procedure by a computable Bayesian decision procedure. By means of the evaluation game, it is then shown how the latter translates into extremely poor performance for such a non‐Bayes procedure in repeated applications such as arise in the frequentist theory. A version of these results is then presented for prediction. In all such applications, any non‐Bayes decision procedure in common use can be replaced by a uniformly better Bayes procedure using simple linear programming methods. Finally, issues regarding finite versus infinite partitions and the countably additive approach of Kolmogorov are discussed.
Consistent Estimator
Authors: Liang Kung‐Yee , Rathouz P. J.
Published: July 2005

An estimator of a statistical parameter is consistent if it converges in probability to the parameter as the sample size increases. Two methods of obtaining consistent estimators are described by minimizing a specific function of the data and the parameter (e.g. maximum likelihood,M‐estimation, weighted least squares), or by solving an equation involving an estimating function (e.g. quasi‐likelihood).
EmpiricalBayes
Authors: Morris Carl N. , Christiansen Cindy L.
Published: July 2005

When certain estimation situations are encountered many times, the data from all these situations can be used to construct estimates for each individual case. This overview shows how empirical Bayes has evolved and expanded since its 1955 origins in order to useful practical procedures, which are good not only asymptotically but also in moderately sized samples. Various distributional models are described, as is an application to gene expression data.
Expectation
Authors: Proschan Michael A.
Published: July 2005

The expectation of a random variable is its mean, and may be thought of as the long‐run average of a random series of observations. Attention is paid here to the definition of conditional expectation, and to the properties of marginal and conditional expectations.
Experiment‐Wise Error Rate
Authors: Iglewicz Boris
Published: July 2005

Multiple comparisons are frequently performed in medical experiments. These occur when dealing with multiple treatments, multiple endpoints, multiple subgroups, or multiple looks. Use of the nominal significance level for each comparison results in an inflated overall type I error rate, called the
experiment‐wise error rate
. Approaches for controlling this error rate are discussed and a number of references are provided.
Exploratory Data Analysis
Authors: Hoaglin David C.
Published: July 2005

Exploratory data analysis describes the attitude, approach, and techniques for flexible probing of data associated particularly with John W. Tukey. Four main themes are resistance to misbehavior in data, the importance of examining residuals, re‐expression of data in a more useful scale, and graphical displays. An example illustrates these and other points.
Generalized Maximum Likelihood
Authors: Crowder M. J.
Published: July 2005

Maximum likelihood estimation plays a central role in statistical methodology but can fail in certain situations. Various modified versions have been proposed, some of which are described here.
Geometric Distribution
Authors: Ahsanullah M.
Published: July 2005

In a geometric distribution for a variable taking integral values, the probabilities form a decreasing geometric series. An example arises in coin tossing, for the number of tosses required to obtain the first head. Properties of the distribution, including generating functions, are given here.
K
olmogorov–
S
mirnov Test
Authors: Wilcox R.
Published: July 2005

This is a distribution‐free method for comparing two empirical distributions, based on the largest vertical distance between the two cumulative distribution functions. The Kolmogorov test is a special case where one distribution function is known, and hence is a test of goodness‐of‐fit. Various properties, including power, are discussed.
Likelihood Ratio
Authors: Birkes D.
Published: July 2005

In a statistical model for a set of data, the likelihood of a particular value of the parameter is the density function for that parameter evaluated at the observed data. For any two values of the parameter, the ratio of their likelihoods indicates how likely the first parameter value is relative to the second.
Likelihood Ratio Tests
Authors: Birkes D.
Published: July 2005

The likelihood ratio test is a widely used procedure for testing hypotheses. It rejects the null hypothesis when the maximum likelihood under the null hypothesis is significantly smaller than the maximum likelihood under the alternative hypothesis. In some situations, itsPvalue can be calculated exactly, but, in general, thePvalue must be approximated, usually by using a chi‐square table. Three examples are presented. Methods have been, and continue to be, developed for improving the approximation.
Locally Most Powerful Tests
Authors: Peña Edsel A. , Rohatgi Vijay K.
Published: July 2005

Locally most powerful tests, that is, tests that have maximum power against local alternatives are discussed.
Logistic Distribution
Authors: George E. Olusegun
Published: July 2005

Properties of the logistic distribution, including characterizations, generalizations, and applications in generalized linear models, are summarized. Some definitions of multivariate logistic distribution are also highlighted.
Marginal Likelihood
Authors: Kalbfleisch John D.
Published: July 2005

In a multivariate situation, where inference is required about a subset of the parameters, the others being nuisance parameters, it may be possible to factorize the full likelihood so that the parameters of interest are associated with one aspect of the data. This gives the marginal likelihood. Examples include the partial likelihood used in the Cox model for survival data.
Marginal Probability
Authors: Wypij David
Published: July 2005

In a multivariate probability distribution of many (perhaps correlated) random variables, a marginal distribution describes the univariate distribution of a single variable integrated over all the others. Marginal distributions do not uniquely determine the joint multivariate distribution.
Mode
Authors: Everitt Brian S.
Published: July 2005

The mode of a distribution is the value of the variable with the highest frequency (for discrete data) or density (for continuous data). An example is given of a bimodal (two‐mode) distribution.
Nonparametric Methods
Authors: Wolfe Douglas A.
Published: July 2005

The motivation behind distribution‐free hypothesis tests and confidence intervals based on signs and ranks is discussed and standard text references are provided. Distribution‐free test and confidence interval procedures for the one‐sample, two‐sample, one‐way analysis of variance, two‐way analysis of variance, and independence settings are presented. Point estimation techniques, including Hodges–Lehmann estimators and contrast estimation, and multiple comparisons procedures are described. Basic references for censored data approaches and bootstrapping are provided.
Normal Distribution
Authors: Tong Y. L.
Published: July 2005

This article provides a description of the univariate normal distribution and a brief discussion on the bivariate and multivariate normal distributions.
Order Statistics
Authors: Davison A. C. , Dorsaz F.
Published: July 2005

Order statistics are the ordered sample values. They are widely used for model checking through probability plots and as a basis for tests and estimators.
Ordered Alternatives
Authors: Lee Y. J.
Published: July 2005

Ordered alternatives occur naturally in many experimental and observational studies. The dose‐response study is a typical one. The age‐related response is another one. Tendency to lie may increase with the age. The more the alcohol that is consumed during the pregnancy the higher is the risk of congenital malformation. There are numerous other examples of this nature. The independent variable that induces ordered response could be quantitative or ordinal (ordered categorical). The response variable observed for the ordered alternatives can be continuous counts, ordinal, dichotomous (success or failure), or even multinomial. Data may be collected in blocks or per dose group. Methods appropriate for dichotomous outcomes, Poisson counts, multinomial outcomes, and nonparametric data are reviewed. A reference for nonparametric methods appropriate for blocked data with missing observations is also given but not reviewed.
P

Value
Authors: Goodman Steven N.
Published: July 2005

The
P
value is the probability, under a given simple hypothesis, of the observed or a more extreme value of a test statistic. Problems discussed include that of the choice of test statistic, the relevance of alternative hypotheses, ambiguity about the sample space, and the interpretation generally. The distinction between one‐sided and two‐sided tests is a source of confusion. Bayesian considerations are discussed.
Pairwise Independence
Authors: Wermuth Nanny
Published: July 2005

If two explanatory variables are independent, with additive effects on a third, response, variable, then the overall effect of one explanatory variable is qualitatively the same as the effect conditional on the other explanatory variable. If the explanatory variables are dependent, reversal can occur. In multivariate analysis, pairwise independence of all variables does not imply mutual independence.
Partial Likelihood
Authors: Harrington David
Published: July 2005

In a model with many parameters, it may be possible to isolate a factor in the full likelihood representing a conditional distribution depending on one parameter or a subset of the parameter set. An example arises in the Cox proportional hazards model for survival data. Many of the standard asymptotic properties of maximum likelihood estimates hold also for partial likelihood.
Penalized Maximum Likelihood
Authors: Simonoff Jeffrey S.
Published: July 2005

Nonparametric density estimation methods are designed to estimate the density functionfthat generates a sample of observations, without assuming a parametric form forf. Rather,fis only assumed to be smooth, where smoothness is defined as the condition that at least a specified number of derivatives are square integrable. Penalized maximum likelihood estimators accomplish this by maximizing a penalized form of the log‐likelihood, where the penalty discourages roughness in the final estimate. The estimator has a Bayesian interpretation, with the penalty function corresponding to the logged prior for the density. The resultant estimator often takes the form of a polynomial spline with knots at the order statistics, and asymptotically is approximately a local‐bandwidth kernel estimator. The method can be adapted to large sparse contingency tables, resulting in cell probability estimators that are consistent under asymptotics that approximate such tables.
Power Transformations
Authors: Atkinson A. C.
Published: July 2005

Simple regression analysis assumes normally distributed errors of constant variance. These assumptions may be more nearly satisfied by transformation of either the response variable, the explanatory variable(s), or both. In using the power transformation of Box and Cox, the power parameter may be estimated by maximum likelihood. This family of transformations includes the square root, logarithm, and reciprocal.
Quasi‐likelihood
Authors: Zeger Scott L. , Liang Kung‐yee
Published: July 2005

Quasi‐likelihood is a method of estimation for regression analysis which requires the specification of only the first two moments of the response distribution, and the term refers also to the objective function analogous to the likelihood. In generalized linear models, the iterative weighted least squares algorithm for solving the score equations depends only on the mean and variance of the response. The solution of the estimating equations is consistent even when the variance expression is incorrect. Various extensions are described.
Shrinkage Estimation
Authors: Casella George
Published: July 2005

Shrinkage estimation developed from the work of Stein for the estimation of the mean of a multivariate normal distribution. The observed vector is, from a decision theory viewpoint, inadmissible. The James–Stein estimator shrinks the observations toward zero. Later work involves estimators shrunk toward the sample mean. Shrinkage estimators are, however, not dominant in problems with a finite sample space, such as the binomial distribution. There are connections with Bayesian and empirical Bayes solutions.
Spearman Rank Correlation
Authors: Zar Jerrold H.
Published: July 2005

In a nonparametric (distribution‐free) approach to the correlation between two sets of measurements made on the same individuals, each set may be ranked in order of magnitude. The standard (Pearson) product‐moment correlation coefficient is calculated on the two sets of ranks. Simple formulae are available, and adjustments are made for tied ranks. Test and estimation procedures are described. The relationship with the Kendall rank correlation coefficient is discussed.
Tolerance Interval
Authors: Vangel Mark G.
Published: July 2005

Tolerance intervals cover at least a specified proportion of a population, either on average or else with a stated level of confidence. The theory is described in relation to simple random samples and also linear regression. Topics discussed include random effects tolerance limits, simultaneous tolerance intervals, and applications to calibration. Methods are illustrated by an example on serum glucose measurements.
Tolerance Region
Authors: Vangel Mark G.
Published: July 2005

In a sample from a multivariate distribution, a tolerance region is a region for which there is a specified high probability (theconfidence) that at least a specified proportion of the distribution (thecoverage) is included. It is thus a generalization of the univariate tolerance interval. The construction using statistically equivalent blocks is described briefly.
Wilcoxon Signed‐Rank Test
Authors: Woolson R. F.
Published: July 2005

This is a nonparametric test procedure for the analysis of matched‐pair data, based on differences, or for a single sample. The null hypothesis is that the differences, or individual observations in the single‐sample case, have a distribution centered about zero. The absolute values are ranked. The test statistic is the sum of the ranks for either the positive or the negative values. Examples and details, including large‐sample properties, are given.
Adaptive Sampling
Authors: Seber George A. F. , Mohammad Salehi M.
Published: July 2005

Animal populations are often highly clustered. For example, fish can form large, widely scattered schools with few fish in between. Even rare species of animals may form small groups that are hard to find. Applying standard sampling methods such as simple random sampling of plots to such a population could yield little information, with most of the plots being empty. Adaptive sampling is based on the simple idea that when some animals are located on a sample plot, the neighboring plots (and possibly their neighbors as well) are added to the sample; in this way, the whole cluster can be sampled. The scheme lends itself to a variety of modifications, for example, sampling with and without replacement, sequential and inverse sampling, stratified sampling, two‐stage sampling, and so‐called adaptive allocation, as well as extensions to multivariate data and incomplete detectability.
Bias from Stage Migration in Cancer Survival
Authors: Blesch Karen Smith
Published: July 2005

Stage migration due to different methods of cancer staging can artifactually inflate cancer survival rates by shifting patients with a marginal prognosis out of a better prognosis group into a worse prognosis group. This bias can be avoided by ensuring that comparable staging methods are used when comparing survival across cancer cohorts.
Response Surface Methodology
Authors: Raghavarao D. , Altan S.
Published: July 2005

Response Surface Methodology (RSM) is used to determine the optimal setting givenkquantitative experimental factors. A preliminary first‐order experiment based on a Plackett–Berman design identifies important factors with a first pass estimate of the optimal point by means of the method of steepest ascent. This is followed by a confirmatory second‐order experiment using a suitable fractional factorial, Central Composite or Box–Behnken design. RSM identifies the optimal setting by estimating the coefficients of a polynomial equation describing the responses from thenexperimental runs, each run corresponding to a prescribed choice of factor combinations. The relationship betweennandk, the points in the experimental design and their properties in relation to orthogonality, rotatability, and uniform precision constitute the major part of the RSM.
Sampling in Developing Countries
Authors: Krótki Karol P.
Published: July 2005

The survey process is complex and, if well implemented, becomes a very powerful tool for collecting vital data that are essential for developing policy. This article discusses many issues related to survey design and, in particular, how they operate in developing countries.
B
ayes,
T
homas
Authors: Lindley Dennis V.
Published: July 2005

Thomas Bayes (1701–1761) is remembered for a paper published in 1763 and often reprinted. In modern terminology, he showed how to pass from the observation of
r
successes in
n
binomial trials of unknown chance θ, to an inference about θ. To do this, he introduced and used what is nowadays always referred to as Bayes's theorem that
p
(θ/
r
) ∝ 
p
(
r
/θ)
p
(θ), a result that has wide applications throughout science.
B
iometrics
Authors: Davidian M. , McGilchrist C. A.
Published: July 2005

Biometrics
is a quarterly journal of the International Biometric Society (IBS), which is a highly regarded outlet for reports on statistical and mathematical research relevant to the biological sciences. We review the history, structure, and content of the journal.
B
ortkiewicz,
L
adislaus
V
on
Authors: Armitage Peter
Published: July 2005

Von Bortkiewicz contributed to statistics and economics, developing theories of overdispersion of binary data. His “law of small numbers”, illustrated by celebrated data on deaths from horse‐kicks in the Prussian army, showed that a near‐Poisson marginal distribution could hide considerable internal heterogeneity.
B
radley,
R
alph
A
.
Authors: Billard Lynne
Published: July 2005

Ralph Bradley's contributions to the world of statistics fall under two headings: his statistical research (especially the Bradley–Terry test used extensively in taste‐testing experiments) and his professional leadership role in statistical science, as evidenced by his development of statistical programs, by his presidency (1981) of the American Statistical Association, and by his editorial efforts.
B
rownlee,
J
ohn
Authors: Armitage Peter
Published: July 2005

Brownlee worked for the Medical Research Council in London and published papers on vital statistics, advocating the cohort method of displaying age‐specific death rates, and on the periodicity of epidemics.
Cooperative Cancer Trials
Authors: Rubinstein Lawrence V. , Ungerleider Richard S.
Published: July 2005

The clinical trials cooperative group program was established by the US National Cancer Institute in 1955. The number of groups increased to 18 by 1985 and then decreased to 11. This article describes the work of the groups since the mid‐1980s. It covers Phase I, II, and III studies, in a range of histologic groups, including breast, colorectal, lung, ovary, central nervous system, and prostate. Developments include intergroup studies, formalized interim monitoring systems, and increased emphasis on pediatric cancer and cancer prevention and control. Various methodologic developments are summarized.