bluuebunny/crossref_metadata_2025 huggingface dataset:
Quasi‐symmetry
Authors: Hatzinger Reinhold
Published: July 2005

In its simplest form, quasi‐symmetry is a model for square contingency tables that states that certain odds ratios are symmetric around the main diagonal. A special case is complete symmetry, by which probabilities themselves are symmetric around the main diagonal. Another special case is quasi‐independence, by which the odds ratios equal 1. This is equivalent to quasi‐symmetry for 3 × 3 tables. This article discusses its multiplicative formula, gives an example, discusses generalizations to higher‐dimensional tables, discusses tests related to complete symmetry and marginal symmetry, and mentions related models.
Scores
Authors: Podgor Marvin J. , Gastwirth Joseph L.
Published: July 2005

This article discusses the choice of scores when conducting analyses with ordered categorical variables. Types of scores discussed include rank scores and scores that assume an underlying distribution. Examples are provided, as well as discussion of related topics and recent developments.
Trend Test for Counts and Proportions
Authors: Williams Paige L.
Published: July 2005

Trend tests evaluate whether counts or proportions Y1, …, Ykincrease (or decrease) with respect to some ordered covariate, x1, …, xk. This section reviews the general development of trend tests, and then describes specific applications for the case of binomial, Poisson, and multinomial random variables. The null and alternative hypotheses are provided, appropriate link functions are discussed, and both asymptotic and exact tests are described. An illustrative example is included for each type of random variable (binomial, Poisson, and multinomial). Choice of scores for trend tests are discussed, as are various extensions, such as incorporation of historical control data, tests for goodness of fit, isotonic regression, stratified data, multiple outcomes, and clustered data.
Psychometrics, Overview
Authors: Sireci Stephen G. , Wainer Howard , Braun Henry
Published: July 2005

Psychometrics includes educational testing, measurement of attitudes and perceptions, personality testing, opinion surveys, health inventories, and other forms of psychological measurement. Psychometric techniques created for measuring the human mind have also proven useful in biometrics and other fields that deal with measuring the intangible. This article is a comprehensive summary of the techniques for psychometric analysis including reliability, generalizability, and validity assessments. The article also presents the evolution of psychometric theory, popular measurement models, and a snapshot of new directions in the field.
Reduced Rank Regression
Authors: Jackson J. Edward
Published: July 2005

Problems can arise inmultiple regressionwhen the predictor variables are highly correlated. These include inflation of standard errors of the regression coefficients and ill‐conditioned covariance matrices of the predictors. Historically,stepwiseregression andridgeregression have been employed in these situations. With the advent of more powerful computers, it has become a common practice to obtain the principal components of the predictors variables and use them as predictors. However, there is no guarantee that the largest components will be good predictors.Latent root regressionis a possible alternative in which the response variable is included in the principal components solution. Two newer techniques arePartial Least Squares RegressionandMaximum Redundancy, both of which use the relationships between the predictors and the responses in obtaining a solution and both can handle the case of multiple responses.
Multivariate Techniques, Robustness
Authors: Krzanowski W. J.
Published: July 2005

In this article, three broad areas of robustness in the multivariate case are considered: inference, ordination and classification, and latent variable models. Also, the authors give an overview of the robustness of the main techniques encountered under each of them. In summary, across all the different areas the broad concerns are the effects that are departures from normality and/or equal dispersion matrices. Departures from normality are relatively unimportant, but lack of equality of dispersion matrices can have much more serious consequences.
Aalen's Additive Regression Model
Authors: Aalen Odd O. , Scheike Thomas H.
Published: July 2005

In survival analysis, the well‐known Cox regression is based on modeling the hazard rate. This method assumes a proportional hazard structure. Aalen's additive model, however, assumes a linear parametric structure on the hazard rate. This approach may be justified within counting process theory, and produces martingale type estimates. These are used for plotting purposes and shows time‐varying effects of covariates on survival. The plots are easy to compute and have a well‐developed statistical theory. There exist several software packages for carrying out the analysis.
Additive Hazard Models
Authors: Preston Dale L.
Published: July 2005

A review of additive hazard models is given with special emphasis on parametric models and applications to studies of radiation effects. Some hybrid models including both additive and multiplicative hazards are discussed. Inference using Poisson regression is studied.
Additive–Multiplicative Intensity Models
Authors: Scheike Thomas H.
Published: July 2005

The two standard regression models in survival analysis are the multiplicative Cox model and the additive Aalen model. These models postulate that covariate effects are either multiplicative or additive. This contribution reviews some recent work that combines multiplicative and additive effects into flexible models.
Compliance and Survival Analysis
Authors: Goetghebeur Els
Published: July 2005

Partial compliance with assigned treatment can act as both cause and effect of survival experiences in randomized clinical trials. Structural failure time models have been developed for causal inference on the effect of resulting exposure patterns on survival time. In this setting, we review randomization based estimation methods that protect the standard α level. We also point to alternative methods relying on the assumption of no unmeasured time‐dependent confounders.
Discrete Survival‐Time Models
Authors: Fahrmeir Ludwig
Published: July 2005

Survival times are rarely intrinsically discrete (an important counterexample being time to pregnancy, which is naturally measured in menstrual cycles). If times are recorded in intervals, grouped survival times methods may be relevant, but it is alternatively sometimes preferable to consider time as discrete. This leads to various forms of generalized linear models, notably repeated logistic regressions or discrete proportional hazards (complementary log–log link functions). Smoothing methods are considered.

Generalizations to competing risks and to frailty models have also been studied.
Excess Mortality
Authors: Væth Michael
Published: July 2005

The term excess mortality is used to describe the additional hazard rate in a study population relative to that seen in a reference population with the same sex and age distributions. A formal model is introduced and a nonparametric estimate of the integrated excess mortality rate is given. As a by‐product, an estimate of the expected survival function is also obtained. For the special case with a constant excess mortality rate, simple explicit moment estimate is also available. The methodology can be generalized to both parametric and nonparametric regression models.
Explained Variation Measures in Survival Analysis
Authors: Graf Erika
Published: July 2005

In linear regression, the proportion of explained variation,
R
2
, is a familiar measure used to quantify the predictive value of a model including covariates compared to the null model of no covariate effect. Similar measures have been proposed for survival analysis where the situation is typically complicated by the presence of censored observations. In addition, squared error may not be felt to be an adequate loss function in this context. We introduce the concept of explained variation in the survival setting and illustrate it by selected measures.
Frailty
Authors: Hougaard Philip
Published: July 2005

A frailty model is a random effects model for survival data or other time‐to‐event data. Conceptually, frailty models are similar to the normal distribution mixed models, but they are dedicated to the survival data case, where observations may be censored and where updating is natural as the events happen over time. The frailty is a common random term and is assumed to enter multiplicatively in the hazard function. The gamma distribution is the most commonly used distribution for the frailty, but other distributions may also be useful. Over the last 20 years, theory as well as applications have developed so that by now the frailty models have almost become standard models.
Kaplan–Meier Estimator
Authors: Borgan Ørnulf
Published: July 2005

It is described how the Kaplan–Meier estimator can be used to provide a nonparametric estimate of the survival distribution from right‐ censored data. Problems with estimating the survival distribution with left‐truncated data are considered, and it is pointed out that for such data, it may be more useful to estimate conditional survival distributions. Estimation of the median and mean survival times is discussed, and it is described how the Kaplan–Meier estimator may be derived by the redistribute‐to‐the‐right algorithm and how it is the unique self‐consistent estimator for right‐censored survival data. It is shown how the Kaplan–Meier estimator can be given as the product‐integral of the Nelson–Aalen estimator for the cumulative hazard rate, and it is indicated how the product‐integral formulation is helpful in the study the of statistical properties of the estimator. Finally, the construction of confidence bands for the survival distribution is considered.
Median Survival Time
Authors: Brookmeyer Ron
Published: July 2005

The median is an important summary measure for survival data. A confidence interval for the median estimated from censored survival data is easily obtained from the pointwise confidence interval around the Kaplan–Meier estimator. A test for equality ofkmedians is also given.
Risk Set
Authors: Andersen Per Kragh
Published: July 2005

The notion of the risk set in survival analysis is introduced and its role in non‐ and semiparametric inference from censored survival data is explained. A generalized risk set concept for multistate models is also considered.
Smoothing Hazard Rates
Authors: Wang Jane‐Ling
Published: July 2005

The nonparametric approach to estimate hazard rates for lifetime data is flexible, model‐free, and data‐driven. No shape assumption is imposed other than that the hazard function is a smooth function. Such an approach typically involves smoothing of an initial hazard estimate, with arbitrary choice of smoother. We describe methods for grouped lifetime data observed at certain time intervals and for continuously observed lifetime data. There are some intrinsic differences between the smoothing approaches for these two types of data. More specifically, smoothing an initial hazard estimate based on the life table is adopted for grouped lifetime data; while for continuous data, smoothing is employed to increments of the Nelson–Aalen cumulative hazard estimate aiming at the derivative of the cumulative hazard function. A few nonparametric hazard regression methods are also discussed.
Staggered Entry
Authors: Slud E. V.
Published: July 2005

Clinical trials and epidemiologic studies, which measure duration until some endpoint of interest, often follow subjects who may arrive at random or “staggered” times and may remain under observation until occurrence either of the endpoint or censoring/withdrawal. An important theme of statistical methodological development over the past three decades has been to understand how to handle such data. Building on earlier ideas (Lexis Diagrams, life tables), the treatment of staggered data has required sophisticated mathematical treatment (counting processes, martingales, repeated significance tests), but has practical impact in the interim analysis and early stopping of survival experiments.
Bayesian Model Selection in Survival Analysis
Authors: Ibrahim Joseph G. , Chen Ming‐Hui
Published: July 2005

Bayesian model selection in survival analysis has become common practice with recent advances in computational technology and software. In this article, we review some Bayesian model selection techniques for the analysis of survival data.
Baseline Adjustment in Longitudinal Studies
Authors: Senn Stephen
Published: July 2005

A baseline variable may be a demographic characteristic such as sex, a pretreatment observation on the variable to be used for measuring outcome, or a pretreatment observation on a variable correlated with the outcome. Adjustment by the analysis of covariance is preferable to the analysis of changes. Topics discussed include testing for baseline balance, adjustment for baselines taken after start of treatment, cut‐off designs, cohort studies, and uncontrolled studies.
Correspondence Analysis of Longitudinal Data
Authors: Van Der Heijden Peter G. M.
Published: July 2005

Correspondence analysis is an exploratory tool for the analysis of associations between categorical variables, the results of which may be displayed graphically. For longitudinal data with two time points, an analysis of the transition matrix (showing the relative frequencies for pairs of categories) provides insight into the structure of departures from independence in the transitions. Transitions between more than two time points can be studied simultaneously. Examples are provided.
Generalized Estimating Equations
Authors: Pickles Andrew
Published: July 2005

Longitudinal data occurs frequently in medical studies, particularly in clinical trials. Often, the response variable is nonnormal, for example, it may be a binary variable, that is, “improved” or “not improved”. The approach to the analysis of such data discussed in this entry is the use of generalized estimating equations, which, while making weaker distributional assumptions than those required for a fully likelihood‐based model, maintains the properties of consistency and asymptotic normality of parameter estimates.
Lagged Dependent Variables
Authors: Jones B. , Wang J.
Published: July 2005

A regression model for longitudinal data may include the dependent variable, lagged by one or more time units, as an explanatory variable or variables, as in serial correlation, and may also involve covariates measured at one or more previous times. There may also be random subject effects. Methods of estimation are described.
Longitudinal Data Analysis, Overview
Authors: Dunn Graham , Pickles Andrew
Published: July 2005

The article reviews several approaches to the analysis of data containing repeated or serial measurements, including data arising from both observational studies and experiments. Examples include fitting growth curves and other forms of trend analysis. Methods for the analysis of quantitative responses include adaptations of analysis of variance (ANOVA), multivariate analysis of variance (MANOVA), and analysis of covariance (ANCOVA) models allowing for autocorrelation. Structural equation models involving latent variables are also explained briefly, as are random effects models. Methods for counts and categorical responses include loglinear models, latent class models, random effects models, and marginal models involving generalized estimating equations.
Non‐Fourier Waveforms
Authors: McCoy E. J.
Published: July 2005

The discrete Fourier transform (DFT), which gives us information about the “frequency” domain characteristics of data, is probably the most ubiquitous orthogonal data transform, and uses basis functions made up of sines and cosines at different frequencies. Often, we have to analyze signals whose frequency behavior changes over time. In such cases, we can utilize transforms with non‐Fourier basis functions, which can better represent nonstationary behavior.
Nonlinear Growth Curve
Authors: Crowder M. J.
Published: July 2005

Growth curves are important in several areas of medicine, for example, the monitoring of height or weight of children over several years. In this article, growth curves in which the parameters defining the curve occur nonlinearly are considered.
Nonlinear Time Series Analysis
Authors: Tong Howell
Published: July 2005

This article considers both parametric and nonparametric approaches to the analysis of nonlinear time series. Such series are important in several areas of medicine including epidemiology and physiology.
Summary Measures Analysis of Longitudinal Data
Authors: Matthews John N. S.
Published: July 2005

There are many methods for the analysis of longitudinal data and the method of summary measures is one of the simplest and most effective. The observations on each unit are combined into a summary measure, which measures a relevant aspect of the response over time. In this way, the data from each unit is reduced to a single value and these can, within limits, be analyzed as the response of the unit. For example, two groups of individuals might be compared by performing at‐test on the summary measures. Summary measures can be whatever adequately captures the essential features of the response over time and more than one summary can be used if more than one aspect of the response is of interest. Typical summaries are the area under the curve, the slope of a regression line or the time to the maximum response. Some difficulties arise with missing data and irregular observations but these problems are generally not more severe for this approach than for other methods that analyze longitudinal data.
Axes in Multivariate Analysis
Authors: Krzanowski W.J.
Published: July 2005

Visualization of data is valuable in statistical analysis to reveal features that are difficult to detect in numerical summaries. When
p
variables are measured on
n
subjects it can be difficult to graphically depict patterns due to limits of dimensionality. Multivariate graphical analysis identified directions that can be projected into subspaces to highlight sample features. Projecting the original data into a subspace generated from several orthogonal linear combinations allows for low‐dimensional approximation to the full data representation.
Communality
Authors: D'agostino, Sr Ralph B. , Russell Heidy K.
Published: July 2005

In factor analysis, a mathematical model is used to explain the interrelationships of a set of
p
manifest variables by a smaller number of
m
underlying latent factors that cannot be observed or measured directly. If
X
1
,
X
2
, …,
X

p

represent the standardized measurement of the
p
manifest variables, then each standardized measurement can be represented as a linear combination of common factors and a unique factor. Communality is defined as the proportion of a variable's total variance that is accounted for by the common factors. The squared multiple correlation is the most commonly used estimate for the communality. It generally approaches the true communality as the number of variables increases.
Factor Analysis, Confirmatory
Authors: D'agostino Ralph B. , Russell Heidy K.
Published: July 2005

In factor analysis, a common‐factor analysis model involves representing each observed variable as a weighted sum of
m
common‐factor scores and one unique‐factor score. The collection of these equations for all
p
variables is called a factor pattern. Sometimes, an investigator may have a hypothesized factor pattern or there may exist a factor matrix from a previous study. An investigator might wish to estimate and test the fit of the data to the hypothesized factor pattern or to the model obtained from a previous study. Performing an analysis to find out how the estimated factor matrix from the data matches the hypothesized factor matrix is called confirmatory factor analysis. Software packages available for this analysis are highlighted.
Eigenvalue
Authors: Searle Shayle R.
Published: July 2005

An eigenvalue is a number that originates from a determinant involving a square matrix. If the matrix hasnrows andncolumns it hasneigenvalues. They characterize many attributes of the matrix, and are very useful in many applications.
G
ramian Matrix
Authors: Sullivan Lisa M.
Published: July 2005

A real symmetric matrix A is said to be Gramian if there exists a matrix of real numbers B such that BB' = A. Gramian matrices and Gramian determinants arise in principal components analysis and factor analysis applications.
Independence of a Set of Variables, Tests of
Authors: Krzanowski W. J.
Published: July 2005

If
p
variables have been divided a priori into distinct sets, then within each set, individual values for each variable may be highly correlated; however, variables in different sets could be treated as independent. Testing independence of such sets of variables is considered through development of a likelihood ratio test. Derivation from this likelihood ratio test to canonical correlation and a union–intersection test is also shown.
Multidimensional Scaling
Authors: De Soete Geert , Carroll J. Douglas
Published: July 2005

Multidimensional scaling includes a set of models and associated methods for constructing geometrical representations of proximity or dominance relationships between elements in one or more sets of entities. The most common techniques of multidimensional scaling are described and illustrated.
Multivariate Distributions, Overview
Authors: Jensen D. R.
Published: July 2005

Multivariate distributions arise throughout statistics and applied probability and they are defined on finite‐dimensional spaces. They serve as probabilistic models for dependent outcomes of random experiments. Biometric data typically comprises observations on multiple characteristics for each experimental subject, and joint distributions are central to the modeling and analyses of such data. From multivariate distributions, the distributions of various sample statistics of note in statistical inference can be derived. Multivariate distributions also characterize the behavior of stochastic processes through properties of their finite‐dimensional projections.
Multivariate Multiple Regression
Authors: Krzanowski W. J.
Published: July 2005

Multivariate multiple regression analysis is used to model the relationship between
q
(>1) dependent variables and a set of explanatory variables. Maximum likelihood estimates of the parameter error matrix,
H
, and the corresponding test statistics are specified. The test of independence of dependent variables on a subset of explanatory variables, which is the basis for stepwise selection, is also described.
Multivariate Normality, Tests of
Authors: Thode Henry C.
Published: July 2005

Testing for multivariate normality in a sample of observations presents greater difficulty than testing for univariate normality. Tests for multivariate normality are mostly based on either the unique properties of the multivariate normality, generalizations of tests for univariate normality, or generalizations of univariate tests for goodness‐of‐fit. Several methods of assessing multivariate normality are presented here, with a focus on those tests that are easily applied and/or have good power properties. Some plots that can be used for subjectively determining the normality of a sample of observations are also described.
MultivariatetDistribution
Authors: Jensen D. R.
Published: July 2005

Multivariate versions of the Student'stdistribution derive through Studentization from multivariate normal models. Multivariatetdistributions are useful to model errors of a random experiment offering greater flexibility and heavier tails than multivariate normal models. There are two types of multivariatetdistributions. Type I distributions are derived by scaling each component of a normal vector by a single random scalar. Type II distributions involve separate scalings by elements of a further random vector, itself having a joint multivariate distribution.
Oblique Rotation
Authors: Jackson J. Edward
Published: July 2005

Any rotation of vectors associated with
principal components analysis
or
factor analysis
designed to produce new set approximating
simple structure
is said to be
oblique
if the resultant set of vectors are not at right angles to each other. Oblique rotations may obtain a better approximation to simple structure but with the loss of orthogonality. Some of these rotations are obtained by a general procedure called
Oblimin rotation
but some of the more widely used rotations such as
Orthoblique, Promax
and
Optres
do not use this form but instead are two‐ and three‐stage procedures, usually using orthogonal rotations as the first stage.
Pattern Recognition
Authors: Lange Nicholas
Published: July 2005

Pattern recognition refers to a technology that recognizes and analyzes patterns automatically by machine. There are some differences between pattern recognition and its relatives. Classical pattern recognition proceeds in two stages: feature detection and extraction followed by classification. One important property of feature detectors and classifiers for pattern recognition is their invariance with respect to minor pattern distortions or deformations. Pattern recognition systems built according to well‐established criteria from statistical decision and information theory are known to outperform systems built using ad hoc methods. The construction of pattern recognition systems and the evaluation of their performance usually involve some form of prediction. Artificial neural networks embody many key pattern recognition ideas and principles.
Projection Pursuit
Authors: Sun Jiayang
Published: July 2005

This article introduces projection pursuit and its history. It describes two basic elements of projection pursuit and ideas for choosing the elements in practice. How to judge the interestingness of a projection pursuit solution is discussed via p‐values of projection pursuit. The other variants or applications of projection pursuit, such as projection pursuit regression and projection pursuit density estimation, are also provided. The article concludes with a discussion on interactive projection pursuit.
Confidentiality and Computers
Authors: Maindonald John H. , Stott Helen P.
Published: July 2005

Computer systems that hold data of a private or confidential nature should be designed so that data are “disclosed only to authorized persons, entities and processes at authorized times and in the authorized manner” 24. Rights of privacy and confidentiality may be enshrined in legislation. Such rights are typically not absolute; they may, especially where data is required for research purposes, be weighed against a wider public interest. Demands for privacy and confidentiality impose responsibilities on designers and managers of computer systems that store sensitive data. Advances in computer technology, and especially the use of the internet, have created huge security challenges.
Database Systems
Authors: Walker N. , Lawrence Catherine
Published: July 2005

The word “database” is used here in two overlapping ways to refer to a collection of related data and its storage method, or to programming system software used to organize and integrate a collection of related data. It is management, access, and control (including efficiency and recoverability), and not necessarily size, which distinguishes a database from a data set. This article concentrates on database software: choosing a database management system (DBMS), designing databases, object‐oriented approaches, and closes with an overview of resources that might be helpful.
Dynamic Programming
Authors: Booth Hilary S.
Published: July 2005

Dynamic programming is a method of selecting one of a group of multistage strategies, to maximize a specified function. A decision is made at each stage, of the optimal strategy for that stage alone, following Bellman's principle of optimality. Applications to molecular biology are described, providing an efficient method of finding the optimal alignment of two molecular sequences.
Graphical Displays
Authors: Martin Michael A. , Welsh A. H.
Published: July 2005

Graphical methods are fundamental tools of modern data analysis. Graphics are useful for communicating complex information quickly and easily, and are thus excellent tools for describing data and for providing direction in data analysis. They are essential tools for building statistical models from data, and for assessing model adequacy. This article surveys the use of statistical graphics in describing data and for statistical modeling, including models for relating variables, for spatiotemporal dependence, and for survival data. Graphics are also critical tools for visualizing and analyzing high‐dimensional data. The article illustrates the use of graphics in statistical modeling through numerous examples drawn from the life sciences.
Internet
Authors: Bowman Adrian W. , Crawford Ewan , Currall James , Young Stuart G.
Published: July 2005

The internet (lower case i) is the world's biggest computer network connecting millions of computers worldwide. The Internet (upper case I) is a term describing what can be done over the internet. It opens up new vistas for academics, the business community, and the general public, providing easier access to information and faster means of communication.
M
onte
C
arlo Methods
Authors: O'Neill T. J. , Barry S. C. , Puza B.
Published: July 2005

Monte Carlo techniques are used in situations in which the analytic solution of the problem is either intractable or time consuming. Instead of calculating exact quantities, simulation is used to produce stochastic approximations to the solution.
Neural Network
Authors: Hastie Trevor
Published: July 2005

A neural network has a set of input units, which are related to one or more sets of “hidden” units, which are in turn related to a set of output units. Models are typically overparameterized and can be made very flexible by varying the structure and the types of relationship. Approaches to model fitting and validation are outlined.
Pseudo‐Random Number Generator
Authors: Kemp C. D.
Published: July 2005

One of the most frequently called functions on a scientific computer is the random number generator. Random numbers are required for many purposes. A major use is in simulation. Large quantities of random numbers are needed to generate the random samples from distributions (theoretical and/or empirical) that are the basis of any stochastic simulation. Another use for random numbers is the Monte Carlo evaluation of multivariate integrals. Ideally, what is required in a simulation is a stream of independently and uniformly distributed random variables taking values between 0 and 1 [i.e. a random sample from the uniform distribution on (0,1)]. At best this ideal can only be realized approximately.
R
Authors: Venables W. N. , Maindonald John H.
Published: July 2005

The R system is an independent, open source, freely available implementation of a computer language that is well adapted to the development of scientific calculations, data analysis, and graphics. It has gained particular importance for developing and promulgating new statistical, data analytical, or graphical techniques. Versions of R are available for Microsoft Windows, Unix, Linux, and MacOS 8 or later.
Spline Smoothing
Authors: Turlach Berwin A.
Published: July 2005

Over the past few decades, several methods for fitting smooth curves to data have been proposed and intensively studied. Among these methods are kernel smoothers, orthogonal series estimators, wavelet smoothers, and spline smoothers. This entry provides a brief description of the commonly used spline smoothers – regression splines, smoothing splines, and penalized splines.
S
‐
PLUS
and
S
Authors: Venables W. N.
Published: July 2005

S‐PLUS is a commercial software environment for data analysis and graphics based on the programming language S, designed and initially implemented by John M. Chambers at Bell Laboratories. In addition to a large suite of software for standard tasks, the system aims to provide users with the means to implement new techniques or combine standard methodology in possibly new ways. The language is interpreted rather than compiled, and has some object‐oriented features. Versions of S‐PLUS are current for Unix, Linux, and Windows. The system has a command‐line interface only, but the Windows implementation offers a graphical user interface as well.
Ancillary Statistics
Authors: Reid N.
Published: July 2005

An ancillary statistic is a function of the data whose distribution is free of the parameters in the model. Ancillary statistics are complementary in a certain sense to sufficient statistics. In the theory of statistics, it is often argued that inference should be conditional on ancillary statistics, when this is possible. Fisher showed that in a location model, the conditional distribution of the maximum likelihood estimator could be obtained by renormalizing the likelihood function, or, in other words, treating the likelihood function as a posterior distribution for the location parameter. This result has been generalized to provide an approximate conditional distribution to the maximum likelihood estimator, given an approximately ancillary statistic, a result known as thep* approximation.
Axioms of Probability
Authors: Rosalsky Andrew , Scheaffer Richard
Published: July 2005

One often thinks of probability as a limiting relative frequency; if a balanced coin is tossed often enough, the fraction of heads will be close to one half. The simple notion belies the nature and depth of mathematics that is required to make it rigorous, namely, the Kolmogorov axioms of probability and the Borel strong law of large numbers. Once the notion of long‐run relative frequency is made precise through measure theory, it can be used to build practical probability models for real‐world systems. Measure theory gives probability its rigor; repeated experimentation gives it its application and inspiration.
Bartlett's Test
Authors: Boos Dennis D.
Published: July 2005

Bartlett's test for the equality of variances ofknormal populations is based on a modification of the asymptotic form of the likelihood ratio statistic. The null distribution of the statistic is extremely sensitive to deviations from the assumption of normal populations.
Critical Region
Authors: D'Agostino Ralph B.
Published: July 2005

In a test of a null hypothesis, the critical region defines the set of possible outcomes that would lead to rejection of the hypothesis, these being chosen to have predetermined probability called the size of the test. In a simple example, the distinction is made between one‐ and two‐tailed tests.
Delta Method
Authors: Cox C.
Published: July 2005

The Delta Method, also known as the Method of Propagation of Errors, refers to applications of the result that a smooth function of an asymptotically normal estimator also has an asymptotic normal distribution. The delta method has two principal groups of applications, first to the computation of a variance stabilizing transformation, and second to the computation of the asymptotic variance and confidence intervals for a nonlinear function of a set of previously estimated parameters, typically by the method of maximum likelihood. This article discusses the proof of both the univariate and multivariate versions of the theorem and gives numerous examples.
E
dgeworth Expansion
Authors: Hall Peter
Published: July 2005

The Edgeworth expansion expresses one probability density in terms of another by the use of cumulants, and is particularly important when the reference distribution is the standard normal. There are applications to the distributions of statistics including the sample mean, sample variance, variance ratio and correlation coefficient, and to the coverage of confidence intervals.
Efficiency and Efficient Estimators
Authors: Rotnitzky Andrea
Published: July 2005

The efficiency of a point estimator of a parameter reflects its tendency to be distributed closely around the parameter value. Asymptotic efficiency refers to the limiting distribution in large samples, which is often asymptotically normal, the minimum variance being given by the Cramér–Rao lower bound in terms of the information matrix. Concepts discussed here include asymptotic relative efficiency, semiparametric efficiency, partial likelihood, and quasi‐likelihood.
F

Distributions
Authors: Selvin S.
Published: July 2005

The
F
‐distribution is fundamental to much of statistical analysis, particularly the analysis of variance. The article entitled “
F
‐Distribution” is a brief but complete description of this probability distribution. Its properties are first discussed, followed by a description of the relationship between the
F
‐distribution and other common statistical distributions (e.g. beta, t‐, chi‐square, and binomial distributions). The article concludes with a typical application of this distribution to a small set of sampled observations.
Fiducial Probability
Authors: Seidenfeld Teddy
Published: July 2005

Fiducial probability was introduced by R. A. Fisher in 1930, as a method by which probability statements could (apparently) be made about an unknown parameter value without recourse to the Bayesian tools of prior and posterior probability. In a simple univariate one‐parameter case, this involves inversion of a probability statement from the distribution function. In other cases, it may involve a pivotal quantity whose distribution is independent of the parameter. The resulting statements are often similar to Bayesian posteriors with uniform priors, and fiducial intervals are often numerically equivalent to confidence intervals, although the logical structure is quite different. In some cases, notably the ratio of two normal means (the Creasy–Fieller problem) and the difference between two normal means with different variances (the Behrens–Fisher problem) fiducial and confidence intervals differ. The relationship between fiducial and Bayesian inference is discussed here in some detail.
Gamma Distribution
Authors: Bowman K.O. , Shenton L.R.
Published: July 2005

The density function of this positively skew, unimodal distribution is a gamma function. The distribution is a Pearson type III and includes the important χ2distribution. There are three parameters, defining the location, scale, and shape. Properties and examples are described.
Hypothesis Testing
Authors: Salsburg David
Published: July 2005

A significance test of a null hypothesis involves the choice of a test statistic whose null distribution is known, and a judgment as to whether, or how well, the observed value of the statistic accords with the known distribution. The article reviews the history of this approach, with emphasis on the Neyman–Pearson formulation, taking account of the behavior of the test statistic under alternative hypotheses.
Inverse Gaussian Distribution
Authors: Folks J. Leroy
Published: July 2005

The inverse Gaussian distribution arises as the distribution of the first passage time to a boundary in a Wiener process. It is positively skewed and is a member of the exponential family. Methods of parameter estimation are briefly described.
K
ullback–
L
eibler Information
Authors: Dykstra Richard
Published: July 2005

The Kullback–Leibler information number,
I
(
P
‖
Q
), determined for two probability measures defined on the same measurable space often serves as a type of “distance” between the two measures.

Projections (
I
‐projections) defined in terms of minimizing the Kullback–Leibler information number over appropriate sets of probability measures are often useful for finding desirable estimators.

Kullback–Leibler information numbers also play a useful role in hypothesis testing problems.
Lehmann Alternatives
Authors: Razzaghi Mehdi
Published: July 2005

In a nonparametric comparison of two groups, the test statistic may not be distribution‐free under alternatives to the null hypothesis. Lehmann proposed a family of alternatives under which the distribution function of the response variable in one group is a specified function of that in the other. A power relationship is usually considered. The distribution of the rank vector under the alternative then depends only on the specified function and rank statistics are distribution‐free. An extension covers the situation where a proportion of the population is unaffected by treatment.
Likelihood Principle
Authors: Lindsey J. K.
Published: July 2005

The likelihood function is central to all approaches to statistical inference. Principles have been proposed to justify the equivalence of inferences based on data with likelihood functions identical apart from a proportionality factor independent of parameters. The weak principle refers to replications with the same design; the more contentious strong principle refers to results from different designs. The latter, implied by Bayesian theory, needs to be supported by strong confidence in the validity of the model.
Maximum Likelihood
Authors: Tosteson Tor D. , Demidenko Eugene
Published: July 2005

Maximum likelihood is a general method of parameter estimation of great practical and theoretical significance in biostatistics. The method is defined and the optimal properties of maximum likelihood estimates are explained along with the underlying assumptions. An example is given and computational methods are discussed.
Mean Square Error
Authors: Schluchter Mark D.
Published: July 2005

If

is an estimator of a parameter θ, then the mean‐square(d) error (MSE) of the estimator is defined as the expectation of the squared deviation of

from θ:

. The MSE is a measure of closeness of the estimator to the true value. It can be shown that

That is, MSE reflects both the bias (accuracy) of the estimator, that is, how much its expected value deviates systematically from the true value, as well as the variance (precision) of the estimator, which indicates how much it varies about its expected value due to sampling variability.
Noncentral

t

Distribution
Authors: Rust Philip F.
Published: July 2005

This is the distribution of the ratio of a normally distributed variable with unit variance to the square root of a ratio of a chi‐square variable to its degrees of freedom. When the noncentrality parameter (the mean of the normal distribution) is zero, the distribution is Student's
t
distribution. The noncentral distribution provides the power function for a
t
test, and other applications concern tolerance intervals and confidence intervals. Details are given here.
Outliers
Authors: Hawkins Douglas M.
Published: July 2005

Outliers are observations highly discordant from the rest of the data, either in a univariate distribution or in relation to a model with explanatory variables. They may arise through contamination, errors in data gathering, or misspecification of the model. Approaches to detection include “high‐breakdown” methods involving partitions of the data into acceptable and discordant sets; and sequential identification of individual outliers by forward selection or, preferably, backward elimination. Likelihood ratios or multiple comparison methods may be used. Special cases are considered and the topic is illustrated by an example using data on body weight and brain weight of animals and dinosaurs.
Power
Authors: Muller K. E.
Published: July 2005

In the Neyman–Pearson approach to the theory of hypothesis testing, power is the probability of rejecting the null hypothesis when an alternative hypothesis is true.

It therefore depends on the specific alternative, and may lead to the selection of a test statistic that provides satisfactory power. In particular, one might choose a uniformly most powerful unbiased test (unbiasedness here indicating minimum power at the null hypothesis). Power considerations are often used in the determination of sample size.
Profile Likelihood
Authors: Aitkin M.
Published: July 2005

This is the likelihood for a set of parameters maximized over nuisance parameters. It formalizes the use of likelihood ratio tests to construct confidence regions for a parameter in the presence of nuisance parameters. It may result in overprecision by ignoring uncertainty in the nuisance parameters, and various adjustments have been proposed. The Bayesian viewpoint is discussed.
Pseudo‐likelihood
Authors: Kalbfleisch John D.
Published: July 2005

A pseudo‐likelihood is a function of the data and parameters, which has properties similar to those of a likelihood function. Its primary use is to define an estimating equation by setting a pseudo‐score function to zero. Examples are given.
Restricted Maximum Likelihood
Authors: Cook Nancy R.
Published: July 2005

Restricted maximum likelihood estimation (REML) aims to maximize likelihood over a restricted parameter space. In a general linear model with multivariate normal error distribution, REML leads to unbiased estimators. Transformation of the data enables the log‐ likelihood to be split so that variances are estimated from error contrasts. There is a connection with Bayesian methods. The method has been applied to a random‐coefficients model for longitudinal data, and to situations where the parameters satisfy order restrictions.
Sampling Distributions
Authors: Begg Melissa D.
Published: July 2005

In frequentist inference, the effect of random variability on a calculated statistic is assessed by assuming that the data were drawn randomly from a hypothetical population defined by parameters, and statistical theory shows the extent to which the statistic would vary in repeated random samples from such a distribution. The sampling distribution of the sample mean tends to be nearly normally distributed in large samples, and so significance tests and confidence intervals for the mean make much use of the normal distribution, the standard deviation of which (the “standard error of the mean”) is the population standard deviation divided by the square root of the sample size.
Studentized Range
Authors: Hochberg Yosef
Published: July 2005

In a comparison of the means of several samples of size
n
, the studentized range is the range of the sample means divided by the usual standard error of a single mean. It may be used in successive stages of a multiple comparison procedure. Two approaches are the Newman–Keuls method and Tukey's
T
‐method. In a single sample of size
n
, the internally studentized range is the ratio of the sample range to the standard deviation, and may be used in a test of normality. Various applications are discussed.
Sufficient Statistic
Authors: Strawderman William E.
Published: July 2005

A sufficient statistic is such that the conditional distribution of the data given the sufficient statistic does not depend on any unknown parameters. In this sense, it contains all the information in the original data. A necessary and sufficient condition is the factorization theorem. A minimal sufficient statistic is a function of every other sufficient statistic, and is conveniently found for distributions in the exponential family. Topics described here include complete sufficient statistics, minimum variance estimation, uniformly most powerful unbiased tests, Bayesian methods, and invariance.
Utility
Authors: Parmigiani Giovanni
Published: July 2005

Utility measures the desirability of an outcome of a decision. A standard procedure in decision theory is to choose the action that maximizes the expected utility. Savage used utility theory as a basis for Bayesian statistical inference.
Wilcoxon‐Type Scale Tests
Authors: Chakraborti Subha
Published: July 2005

Wilcoxon‐type tests are surveyed for a comparison of scale parameters of two continuous distributions. Considerations are given to situations where the locations are equal and known, equal but unknown, and unequal and unknown. The tests are distribution‐free and include some that are linear rank tests and some that are not.
Saddlepoint Approximation
Authors: Kolassa John E.
Published: July 2005

Saddlepoint approximations are asymptotic approximations to the density or tail‐area probability of a statistic. The basic result, using the cumulant generating function for a univariate random variable, may be generalized to apply to random vectors and developed to deal with tail probabilities, conditional distributions, and discrete random variables.