bluuebunny/crossref_metadata_2025_split huggingface dataset:
Procrustes Rotation
Authors: Jackson J. Edward 
Published: July 2005

Abstract

Most rotation procedures in principal component analysis and factor analysis attempt to rotate the original principal component or factor vectors into a new set intended to approximate
simple structure
used either to simplify the interpretation of the principal components or factors or to cluster the original variables. In
Procrustes rotation
, the situation is reversed; there are two sets of vectors and it is desired to find the rotation that will best approximate one from the other. Solutions may be either
orthogonal
or
oblique
. These techniques may be used for any two sets of vectors, not just the ones obtained from principal component analysis or factor analysis. A numerical example is included.
P
romax Rotation
Authors: Jackson J. Edward 
Published: July 2005

Abstract

Orthoblique rotation
is a two‐stage procedure for obtaining
oblique rotations
used to transform vectors associated with principal component analysis or factor analysis to
simple structure
. The first stage will usually be an orthogonal rotation such as
Varimax
. The nature of the second stage will depend to some extent on the degree of obliqueness desired. A numerical example is included with two choices for the second stage.
Simplex Models
Authors: Browne Michael W. 
Published: July 2005

Abstract
Simplex models are used for the analysis of relationships among variables that can be arranged according to a logical ordering. Most methods for fitting the simplex require a prior knowledge of the ordering of the variables, although there are two‐stage procedures where the ordering is estimated in the first stage and procedures that do not require a prior specification. There are two types, a perfect simplex where no allowance is made for error of measurement and a quasi‐simplex where measurement error is taken into account.
Clustering, Single Linkage
Authors: Massaro Joseph M. 
Published: July 2005

Abstract

Single linkage clustering is used in hierarchical clustering 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.
Structural Equation Models
Authors: Bollen Kenneth A. 
Published: July 2005

Abstract
Structural equation models refer to general statistical procedures for multiequation systems that include continuous latent variables, multiple indicators of concepts, errors of measurement, errors in equations, and observed variables. An analysis that uses structural equation models has several components. These include (a) model specification, (b) the implied moment matrix, (c) identification, (d) estimation, (e) model fit, and (f) respecification. Historical origins of structural equation models are also described.
Varimax Rotation
Authors: Jackson J. Edward 
Published: July 2005

Abstract

Varimax rotation
is the most common form of
orthogonal rotation
used to transform vectors associated with
principal component analysis
or
factor analysis
to
simple structure
. It is a special case of
orthomax rotation
, which maximizes the sums of squares of the coefficients within each of the resultant vectors as opposed to
Quartimax
, which maximizes the sums of squares of the coefficients across the vectors for each of the original variables. A numerical example is included.
Artificial Intelligence
Authors: Hand David J. 
Published: July 2005

AbstractArtificial intelligence (AI) has been defined as the study of artificial systems that exhibit intelligent behavior. Research in AI is aimed either towards yielding greater understanding of how naturally intelligent systems, like human brains, function (i.e. a subdomain of cognitive psychology) or at building systems like computer programs or robots, say, that behave in apparently intelligent ways (i.e. a subdomain of engineering). In this overview, some important facets of AI research are examined.
Computer Architecture and Organization
Authors: Nielsen Ole M. 
Published: July 2005

Abstract
Computer technology has made amazing progress since the first general‐purpose electronic computer was created half way through the last century. This rapid rate of improvement is mainly driven by advances in the technology used to build computers but is also (and increasingly so) due to innovations in computer design needed to harness the tremendous capabilities of integrated circuits. Examples are memory hierarchies, pipelining, and parallelism.
This article gives an overview of modern computer architectures and outlines programming practices that will lead to efficient use of the hardware we are likely to encounter in biostatistics. The article concludes with qualified guesses as to the nature of computing in the future.
Computer Languages and Programs
Authors: Lawrence Catherine , Maindonald John H. 
Published: July 2005

Abstract

Computer languages are artificial languages that enable humans to give instructions to computer systems. Texts written in these languages are called
computer programs
or
code
. Influences that explain differences between languages include differences of historical development, of conceptual framework, and of intended use. The use of computer languages for writing large and complex programs raises major issues of program design. The use of computer languages for statistical analysis raises further issues that have been the subject of extensive discussion and research.
EMAlgorithm
Authors: Laird Nan M. 
Published: July 2005

AbstractIn the practice of biostatistics, one is often faced with the problem of estimation with incomplete or missing data. Some common examples include grouped, censored or truncated data, and multivariate data with some individuals having missing responses. Many simple techniques have been developed for this problem, which are based on a “filling in” or “successive substitution” algorithm. The idea is based on the intuitive notion that (i) if we had the values of the missing observations we could estimate the parameters in the “standard” way, often without iteration, and (ii) if we knew the parameters, we could “fill in” the missing observations by setting them equal to their expected values under the model. This suggests estimation can proceed iteratively, alternating between computing expectations for the incomplete data and estimating parameters with “complete data”.
Markov ChainMonteCarlo
Authors: Gilks W. R. 
Published: July 2005

AbstractMarkov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Successive random selections form a Markov chain, the stationary distribution of which is the target distribution. It is particularly useful for the evaluation of posterior distributions in complex Bayesian models. In the Metropolis–Hastings algorithm, items are selected from an arbitrary “proposal” distribution and are retained or not according to an acceptance rule. The Gibbs sampler is a special case in which the proposal distributions are conditional distributions of single components of a vector parameter. Various special cases and applications are considered.
Numerical Analysis
Authors: Smyth Gordon K. 
Published: July 2005

Abstract
Numerical analysis is concerned with the accurate and efficient evaluation of mathematical expressions, especially on computers with floating point arithmetic. This article discusses the fundamental ideas of numerical accuracy including rounding error, approximation error, conditioning, and stability. The problem of computing a sample variance is considered as a simple illustrative example. The article goes on to describe briefly a number of key topics in numerical analysis, which are treated in more depth in separate articles, including linear equations and matrix computations, optimization and nonlinear equations, interpolation and approximation, and numerical integration. Brief pointers are given to available software.
Numerical Integration
Authors: Smyth Gordon K. 
Published: July 2005

AbstractThis article focuses on the process of approximating a definite integral from values of the integrand when exact mathematical integration is not available. This problem arises in statistics when marginal density functions or expected values of random variables are required. The article describes classical univariate quadrature methods, including the trapezoidal rule, Simpson's rule, Newton–Cotes formulas, Clenshaw–Curtis integration, and Gaussian quadrature. Refinements including adaptive methods, treatment of singularities, and progressive rules of the Gaussian type are also mentioned. A survey is given of the possibilities and limitations of multiple integration methods, including product rules, globally adaptive methods, rules of polynomial degree, lattice methods, and Monte Carlo integration. Detailed pointers are given to available software.
Polynomial Approximation
Authors: Smyth Gordon K. 
Published: July 2005

Abstract
Polynomials are very simple mathematical functions, which have the flexibility to represent very general nonlinear relationships. Approximation of more complicated functions by polynomials is a basic building block for many numerical techniques. This article considers two distinct but related applications. The first is polynomial regression in which polynomials are used to model a nonlinear relationship between a response variable and an explanatory variable. The advantages of using orthogonal polynomials as predictor variables are illustrated using a data set on the height and age of preadult girls. The second problem is that of approximating a difficult to evaluate function, such as a density or a distribution function, with the aim of fast evaluation on a computer. The use of Chebyshev polynomials is illustrated for the purpose of obtaining a uniformly accurate approximation to a function over a finite interval.
Software Reliability
Authors: Dalal S. R. , Lyu M. R. , Mallows C. L. 
Published: July 2005

AbstractThe demand for complex software systems has increased more rapidly than the ability to design, implement, test, and maintain them, and the reliability of software systems has become a major concern for our modern society. In the last decade of the twentieth century, many reported system outages or machine crashes were traced down to computer software failures. Consequently, recent literature is replete with horror stories regarding software problems. This entry focuses on software reliability models and measurements.
Spreadsheet
Authors: Hunt N. 
Published: July 2005

Abstract

The term
spreadsheet
is derived from the sheet of paper employed by an accountant to set out financial calculations, often so large that it had to be spread out on a table. The first electronic spreadsheet, VisiCalc, was developed in 1979 by Bob Frankston and Dan Bricklin at Harvard. Originally designed as a business problem‐solving tool, the remarkable versatility of the spreadsheet package has made it a popular working environment among computer users in many professions. The most commonly used spreadsheet packages include Excel, Lotus 1‐2‐3, Quattro Pro, and SuperCalc.
Markov ChainMonteCarlo, Recent Developments
Authors: Mengersen K. 
Published: July 2005

AbstractThere has recently been an explosion of interest in Markov chain Monte Carlo (MCMC) for biostatistical modeling and analysis, particularly, but not exclusively in a Bayesian framework. The aim of this update is to describe some of the new developments in MCMC methodology and their application in biostatistics. MCMC algorithms, including variations on Metropolis and Gibbs, reversible jump MCMC, slice sampling perfect simulation, delayed rejection, and sequential Monte Carlo are briefly discussed. The important issue of MCMC convergence is also addressed. Applications of MCMC in areas of interest in biostatistics are illustrated through references to CART, latent variable models, hidden Markov models, time series models, factorial experiments, correlated data, meta‐analysis, clinical trials, and bioinformatics. The expanding array of software available for MCMC is acknowledged, followed by references to recent biostatistics‐related books on the topic.
A
spin–
W
elch Test
Authors: Algina James 
Published: July 2005

Abstract
The Aspin–Welch test can be used to test the hypothesis that two means, from independent samples, are equal without making the assumption of equal variances. The article presents the formulas for the test statistic and for two critical values, cites the literature for another critical value, identifies problems in the test, cites literature for solutions to these problems, and cites literature for extensions to the Aspin–Welch test for comparing means in multiple independent samples in one‐way and multi‐way layouts, and for comparing mean vectors in multivariate designs.
BANEstimates
Authors: Bemis Kerry G. 
Published: July 2005

AbstractBAN (Best Asymptotically Normal) estimates and their corresponding chi‐squared test statistics form the core of classical categorical data analysis. Historically, these estimates were developed for the multinomial distribution. This article describes the optimal properties of these classical estimators and the form of their chi‐squared test statistics. It is also shown that these are special cases of BAN estimators and chi‐squared test statistics from a more general family of discrete multivariate distributions (Sum Symmetric Power Series Distributions), which include multinomial, negative multinomial, and multivariate Poisson as special cases. These results are in turn generalized to the multivariate exponential family. BAN estimators and their test statistics take on familiar forms and properties as the classes of distributions are generalized. Estimation and hypothesis testing are described from the point of view of constraint equations and the more familiar freedom equations.
B
ayes' Theorem
Authors: Lindley Dennis V. 
Published: July 2005

Abstract
Bayes' Theorem, which plays a central role in Bayesian inference, relates the probability of an uncertain event after additional knowledge has been obtained, to the probability before the acquisition of that knowledge, by multiplying by the likelihood (i.e. the probability of the additional information given the occurrence of the event). An alternative formulation is that the posterior odds on the event is obtained from the prior odds by multiplying by the likelihood ratio.
Bonferroni Inequalities and Intervals
Authors: Costigan T. 
Published: July 2005

AbstractThe first‐order Bonferroni inequality relates the probability that at least one of a series of events occurs to the sum of the probabilities of the individual events. In statistical applications, it provides an upper bound to the probability of falsely rejecting at least one of a series of true null hypotheses, and therefore, a possible correction to a significance level to account for multiplicity of hypotheses tested. Further developments include sequential rejection procedures and the use of higher‐order Bonferroni inequalities.
Cauchy Distribution
Authors: Glynn Robert J. 
Published: July 2005

AbstractThe Cauchy distribution, discovered by S.D. Poisson in 1824, arises as the ratio of two independent standard normal variables. It does not have finite moments. The sample mean has the same distribution as a single observation, and so is not a consistent estimator of the median. It is a useful model for situations where error distributions have very long tails.
Cramér–Rao Inequality
Authors: Bhapkar V. P. 
Published: July 2005

AbstractThis inequality provides a lower bound for the variance of any unbiased estimator of a one‐dimensional parameter or more generally of a function of the parameter. It involves the Fisher information (the expectation of the squared derivative of the log likelihood). If equality is attained, the statistic is a minimum variance unbiased estimator, and the density function belongs to a one‐dimensional exponential family. The inequality can be generalized for a multidimensional parameter.
Decision Theory
Authors: Lewis Roger J. , Berry Donald A. 
Published: July 2005

AbstractStatistical decision theory is a set of techniques for making optimal decisions in the setting of uncertainty. Both frequentist and Bayesian approaches have been developed. The definition of a Bayesian decision problem requires the specification of an unknown parameter and its prior probability density function (pdf), observed data (if any), a likelihood function, a set of possible decisions or actions, and a loss or utility function whose expectation is to be optimized. The expected loss can be defined in a number of ways, yielding a frequentist risk, the conditional Bayes risk, and the Bayes risk. The goal is generally to select a decision rule that maps the possible observed data to a set of actions so that the Bayes risk is minimized. Alternatively, a frequentist may seek to minimize the maximum frequentist risk that might be experienced over all possible fixed values of the unknown parameter, yielding the optimal “minimax” decision rule. Examples in the chapter include a problem of selecting drug therapy when the patient diagnosis is unclear and the decision to order a test to determine the disease present, choosing a sample size when the goal is to estimate an unknown mean of a normally distributed variable, sequential stopping rules determined through back induction, and sequential allocation or assignment (bandits).
Degrees of Freedom
Authors: Potthoff Richard F. 
Published: July 2005

Abstract

This short article deals mainly with (i) defining of the term
degrees of freedom
; (ii) the relation of degrees of freedom to the χ
2
,
t
, and
F
distributions and associated hypothesis testing; (iii) relations among these three distributions, and between them and other distributions (normal, exponential, and Cauchy), when degrees‐of‐freedom parameters assume certain values; (iv) noninteger degrees of freedom; and (v) issues pertaining to reporting of results for publication.
Density Estimation
Authors: Scott D. W. 
Published: July 2005

Abstract
The “density estimation” article highlights the dual roles of the graphical exploration of data as well as the nonparametric estimation of probabilities. Estimators described in detail include the histogram, frequency polygon, averaged shifted histogram, and kernel method. The basic theory of estimation is outlined. Data‐based calibration by cross‐validation is described. Several concrete examples are given.
Estimating Functions
Authors: Desmond A. F. , Godambe V. P. 
Published: July 2005

AbstractThe aim of this article is twofold. First, we demonstrate that the methodology of estimating functions produces a unification and extensions of approaches to statistical modeling and inference, both parametric and semiparametric. Secondly, we illustrate via selected examples, how this methodology applies in biostatistics. We have limited our discussion, here, to case–control studies, prospective and retrospective sampling, and overdispersion.
Exchangeability
Authors: Greenland Sander , Draper David 
Published: July 2005

Abstract

In probability theory, the
random variables
Y
1
, …,
Y

N

are said to be
exchangeable
(or
permutable
or
symmetric
) if their joint distribution
F
(
y
1
, …,
y

N

) is symmetric; that is, if
F
is invariant under permutation of its arguments, so that

whenever
z
1
, …,
z

N

is a permutation of
y
1
, …,
y

N

. There is a related epidemiologic usage, which is described in the article on confounding. In many ways, sequences of exchangeable random variables play a role in subjective Bayesian theory analogous to that played by independent identically distributed (iid) sequences in classical frequentist theory. In particular, the assumption that a sequence of random variables is exchangeable allows the development of inductive statistical procedures for inference from observed to unobserved members of the sequence [1, 2, 3, 5, 6, 9].
Exponential Family
Authors: Pierce Donald A. 
Published: July 2005

AbstractThe overview focuses on aspects of exponential families important in biostatistics, particularly by emphasizing not identically distributed observations involving covariables. This is closely related to generalized linear models. The distinction is made between regular and curved exponential families. In the former, the maximum likelihood estimator is sufficient, the expected and observed information are the same, and there are exact frequency methods for some inferences. In the latter, there is information not contained in the maximum likelihood estimator, much of which is carried by the ratio of observed to expected information, and asymptotic methods must always be used. Of these, likelihood ratio and score methods are preferable to the Wald method. Modern higher‐order asymptotic methods, including the modified profile likelihood, are mentioned. The tractability of Bayesian inference in exponential families is indicated.
Foundations of Probability
Authors: Lindley Dennis V. 
Published: July 2005

AbstractProbability, as a mathematical subject, is essentially unique and its formal rules are described. There are many different interpretations, of which three are mentioned: classical, frequentist, and in terms of belief. The many and varied justifications for thinking that beliefs should conform to the rules of probability, the Bayesian view, are described. The connections with decision‐making, the concept of utility, and the role of subjectivity are discussed. It is pointed out that the frequentist and Bayesian interpretations lead to distinct, practical implications of probability, especially in connection with the likelihood principle, and illustrations are provided. The role of exchangeability, which links the two views, is developed. Alternatives to probability are briefly mentioned.
Inference
Authors: Zelen M. 
Published: July 2005

AbstractStatistical inference is the process of drawing conclusions from data governed by probability laws. Conclusions are uncertain, and a principal goal is to measure the uncertainty. In practical applications, the aim may be to draw conclusions about models that may have generated the data. A central concept, in almost all approaches, is the likelihood, regarded as a function of the unknown parameters. The most widely used approach is that of the frequentist school, based on error probabilities, that is, probabilities of particular observations given hypotheses about parameters. Key concepts here are significance tests, confidence intervals, and point estimation. The likelihood school uses likelihood ratios as measures of relative support. The Bayesian school uses inverse probability, that is, probabilities of hypotheses. Bayes' theorem relates posterior probabilities (postdata) to prior probabilities (predata) via the likelihood function. Details of these approaches are given in the article.
Inference, Foundations of
Authors: Fisher Lloyd D. 
Published: July 2005

AbstractThe foundations of statistical inference are summarized beginning with regularity as embodied in probability theory. Both objective external probability and subjective internal Bayesian probability are discussed. The law of large numbers is the theoretical underpinning of the usefulness of probability. Objective large probability methods have led to concepts such as confidence intervals, hypothesis testing, andp‐values. Subjective probability uses data to go from prior to posterior distributions. No known method of statistical inference can withstand all rational criticism. Important areas of inference arise from descriptive and graphical methods of summarizing data and are mentioned.