bluuebunny/crossref_metadata_2025 huggingface dataset:
Total Time on Test
Authors: Andersen Per Kragh , Keiding Niels
Published: July 2005

In survival analysis, the exponential distribution plays an important role as a null hypothesis. One classical technique from reliability analysis for testing this hypothesis is based on the total time on test concept. It is explained, in the framework of counting processes, how a number of graphical and numerical test procedures may be based on this concept. These procedures are illustrated using data from a clinical trial in liver cirrhosis.
Causal Direction, Determination
Authors: Jones B. , Wang J.
Published: July 2005

Causation is defined here in terms of probability, taking into account potential confounding variables. Path diagrams and approaches to causality in longitudinal studies are discussed briefly.
Analysis of Variance for Longitudinal Data
Authors: Dunn Graham
Published: July 2005

This entry describes how traditional analysis of variance (ANOVA) techniques might be adapted for use in the analysis of serial measurements. Typically, one might recognize data coming from a longitudinal study as coming from a nested or split‐plot design with serial or repeated measurements (subplots) being nested within each of the subjects (plots). The difference between the results of a typical split‐plot experiment and that involving repeated measurements arises from the fact that the latter are likely to be serially correlated. Methods of adapting ANOVA methods to allow for serial correlation (i.e. the Greenhouse–Geisser correction) are described, together with suggestions for alternative data analytic approaches.
Antidependence Models
Authors: Kenward M. G.
Published: July 2005

Ante‐dependence models are flexible nonstationary covariance structures for repeated measurements that generalize stationary autoregressive models. The structures can only be used for data in which the times of measurement are common across units. They can be defined in the Gaussian setting by conditional independence relationships: an
r
th order structure holds if and only if two repeated measurements are conditionally independent given
r
intermediate measurements. Equivalently, an
r
th order ante‐dependence covariance matrix has a inverse of band form of order r + 1, with no other constraints on the matrix, other than that it is positive definite. It provides a compromise between the unrealistic stationary structures, such as the autoregressive, and the potentially inefficient unstructured form. In certain repeated measurements settings, it allows test statistics to be constructed with known small sample distributions, that can be regarded as generalizations of Wilks' lambda statistic.
Broadband Smoothing
Authors: Castiglioni Paolo
Published: July 2005

Broadband smoothing is a procedure to reduce the estimation variance of fast Fourier transform (FFT) power spectra, which results in estimation variance and frequency resolution not constant over the frequency axis, as in conventionally smoothed spectra.
Change‐point Problem
Authors: Pawitan Yudi
Published: July 2005

The change‐point problem is to determine the unknown location τ, based on an ordered sequence of observationsx1, …,xn, such that the two groups of observationsx1, …,xτandxτ+1, …,xnfollow distinct models. In most applications, each model is a regression model, so the overall model is a nonlinear regression model. A commonly used change‐point model is the piecewise linear model. Statistical inference on the change parameter τ is nonstandard, since the likelihood function is not differentiable with respect to τ. The article provides a review of methodology from a general quasi‐likelihood framework, and discusses an application of the change point in breast cancer incidence as a function of age.
Coherence Between Time Series
Authors: Pawitan Yudi
Published: July 2005

Association between time series is a complex phenomenon that may involve a leading or lagging relationship. This entry is concerned with coherence, which is an approach to measuring the strength of the relationship and is, in some sense, the time series analogue of the standard correlation coefficient.
Direct and Indirect Effects
Authors: Pickles Andrew
Published: July 2005

Methods are described for studying the causal mechanism by which a risk exposure or treatment is effective. Distinctions are made between variables having direct causal and intermediate effects.
D
urbin–
W
atson Test
Authors: Chatfield Chris
Published: July 2005

This text checks for independent errors when fitting a multiple regression model to time series data.
Fast
F
ourier Transform (
FFT
)
Authors: McCoy E. J.
Published: July 2005

Data can be described either in the “time domain” or in the “frequency domain”. The frequency domain representations present details about the same signal in a different, potentially more illuminating way. The fast Fourier transform (FFT) provides a fast computational algorithm to obtain the coefficients of the discrete‐frequency representation.
Forecasting
Authors: Chatfield Chris
Published: July 2005

Predicting future values of a time series poses special problems because successive observations are not usually independent but are correlated through time. Consequently, methods such as multiple regression are not generally suitable. The models that are used often have to cope with two sources of variation, trend and seasonality. In this entry, several approaches to forecasting in time series are described including the application of autoregressive models.
Generalized Linear Models for Longitudinal Data
Authors: Zeger Scott L. , Diggle Peter J. , Huang W.
Published: July 2005

The analysis of longitudinal data needs to take into account the correlations between the repeated measures of the response variable to draw valid and efficient inferences about parameters of scientific interest. There are a variety of approaches that can be taken depending on what is the target of the estimation process. This article outlines marginal models, random effects models and transition models.
Instrumental Variables
Authors: Dunn Graham
Published: July 2005

We describe the application of instrumental variable methods to cope with model identification problems arising from either hidden selection effects or measurement errors. The problem of hidden selection effects is illustrated by reference to adjustment to non‐ or partial compliance with allocated treatment in a randomized controlled trial. Measurement error problems are illustrated by reference to two potential areas of application: (1) the estimation of the effect of environmental exposure on risk of disease when exposure is measured with error (i.e. measurement error models in epidemiology), and (2) comparative calibration of two measurement methods when both are subject to measurement errors (i.e. method comparison studies).
Kalman Filter
Authors: Janacek G. J.
Published: July 2005

The Kalman filter is a recursive algorithm to provide estimates of parameters in state‐space models of time series. Observations are linearly related to unobserved state‐space variables following an autoregressive model. Recursive algorithms enable prediction and updating of estimates of state‐space parameters with likelihood estimation. A simple example is provided.
Mining Time Series Data
Authors: Keogh Eamonn J.
Published: July 2005

Some of the problems raised by massive time series databases are briefly considered. They include indexing, clustering, classification, prediction, association detection, summarization, anomaly detection, and segmentation, most of which make explicit or implicit use of a distance measure. A key feature is the approximate representation of time series.
Nonignorable Dropout in Longitudinal Studies
Authors: Fitzmaurice Garrett
Published: July 2005

Longitudinal studies are very common in medical research and many suffer from patients not completing the intended number of visits, that is, some patients drop out of the study prematurely. Such dropouts can be classified into different types depending on how their probability of dropping out relates to observed and missing values of the response. In this article, nonignorable dropouts for which the probability of dropping out depends on unrecorded values of the response are considered.
Nonlinear Mixed Effects Models for Longitudinal Data
Authors: Bates Douglas M.
Published: July 2005

The analysis of longitudinal data may require a mixed‐effects model, incorporating parameters for fixed effects associated with the whole population and also parameters describing distributions of random effects associated with individual subjects. These may enter the model nonlinearly, as in compartment models used in pharmacokinetics. Maximum likelihood estimation is carried out by numerical optimization.
Panel Study
Authors: Jones B. , Wang J.
Published: July 2005

In a panel study, a number of individuals are followed for a given period of time, measurements being taken at predetermined time points. Loss to follow‐up may be important. Variants include the rotating panel study, with parts of the panel being replaced at various times, and the split panel design, where new panels are recruited periodically but all panels are followed up. Models for analysis typically involve random subject and covariate effects and serial correlation. Methods of analysis are summarized.
Population Growth Models
Authors: Diamond Ian
Published: July 2005

Models for forecasting the growth of human populations have been of interest for at least two hundred years. In this article, several models are described including the logistic curve, and ARMA and ARIMA time series models.
Reverse Arrangement Test
Authors: Castiglioni Paolo
Published: July 2005

The reverse arrangement test is used for evaluating whether a sequence of ordered data is derived from independent observations of the same random variable by detecting whether a significant trend underlies the observations.
Reversibility
Authors: Lawrance A. J.
Published: July 2005

Reversibility is a concept applying to stochastic processes that are stationary in time but whose structure makes no recognition of the direction of time. However, the contrary is often the case, and interest is in features, which are not the same when time is considered in the reverse direction.
Seasonal Time Series
Authors: Pawitan Yudi
Published: July 2005

Time series containing seasonal cyclic variation occur frequently in medical studies, for example, monthly pulmonary deaths. In this article, both static and dynamic models for seasonality are discussed.
S
lutzky–
Y
ule Effect
Authors: Loynes R. M.
Published: July 2005

Smoothing a time series by forming a moving average is a commonly employed approach. In this article, some of the problems that arise are discussed, in particular, the introduction of correlations even when the observations in the original series were independent.
Stationarity
Authors: Lawrance A. J.
Published: July 2005

Stationarity is an important property of some time series, which in very general terms means that the series does not evolve over time–the behavior of the series over periods of sufficient length is statistically identical. In this article, a formal definition of the property is given.
Time Series Regression
Authors: Campbell Michael J.
Published: July 2005

In multiple regression, one of the key assumptions is that the outcome variable is a sequence of independent observations. When this assumption is implausible, for example, when the outcome variable is ordered in fine, then we must modify the model fitting and testing procedure. This article discusses models for continuous and count outcomes.
Wavelet Analysis
Authors: Brammer M.J.
Published: July 2005

Wavelet analysis is an approach to signal representation that overcomes some of the limitations of Fourier series representation by capturing transient behavior in signals.
Y
ule–
W
alker Equations
Authors: Cheng Bing
Published: July 2005

In a stationary autoregressive time series of order
p
, the Yule–Walker equations are a set of
p
linear equations for the regression coefficients in terms of the autocorrelation functions. By estimating the latter, one can obtain estimates of the former.
Z
Analysis
Authors: Castiglioni Paolo
Published: July 2005

Z
‐analysis is a method of measuring the association between two probabilistic events by a statistical link coefficient ranging from −1 for mutually exclusive events to +1 for independent events.
Bagging and Boosting
Authors: De Veaux Richard
Published: July 2005

Bagging and boosting are examples of ensemble learning methods from data mining. These methods combine the predictions of many different models (usually of the same type) into an ensemble prediction. For classification problems, the normal combining technique is voting, taking the modal prediction of all the models as the ensemble prediction. For regression problems, averaging or other summaries of the various model outputs can be used. In principle, bagging and boosting can be applied to any class of models, although the most widely studied class of models to which bagging and boosting have been applied are decision trees. The result of bagging is an estimate that has reduced variance when compared to the predictions of the individual models. Boosting, by contrast, appears to be able to reduce both the bias and the variance of the predictions. Both methods share the disadvantage, however, that the resulting model is not as interpretable as the constituent models because they average over a large collection of predictions.
Canonical Correlation
Authors: Das Shubhabrata , Sen Pranab K.
Published: July 2005

The article starts with the genesis of canonical correlation from the product moment correlation and defines canonical correlation from the various angles including algebraic and geometric approaches. The different issues related to interpreting canonical correlation analysis, for example, canonical coefficients, canonical loadings, and redundancy coefficients are highlighted. Relationship with other multivariate methods like principal component analysis, factor analysis, and MANOVA have also been outlined. Then the article dwells on the sampling distribution of the sample canonical correlation and the related inference problems, while touching upon the relevant resampling methods in this context. Various generalizations and extensions are taken up next, with emphasis on the treatment of dependence between more than two sets of variables as well as constrained dependence under problem specific restrictions on the coefficients. The article concludes with a narration of issues related to applications, specially in the field of Biostatistics and a reference to key articles in this domain.
Classification, Overview
Authors: Gower John C.
Published: July 2005

Classification is concerned with (i) constructing classes, and (ii) assigning objects to one of several predetermined classes. Classes may be constructed to balance measures of their hopefully large external differences and hopefully small internal differences, or so that subsequent assignment is optimal. Objectives for assigning to classes may be to minimize errors of misclassification and/or to minimize costs. Both (i) and (ii) exist in probabilistic and nonprobabilistic forms. The nature of the things being classified is crucial. Are they single items representative of many identical items, or representative of a population of similar items characterized by a probability distribution? Are they initially totally unstructured, or are they already explicitly, or implicitly, assigned to some previous classification? Class‐construction includes the multivariate mixture problem (probabilistic), partition problems and many methods for hierarchical classification (nonprobabilistic). Assignment problems include discriminant analysis (probabilistic), diagnostic keys and tables (nonprobabilistic), and expert systems for diagnosis (hybrid).
Cluster Analysis, Variables
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

Cluster analysis involves grouping objects, subjects or variables, with similar characteristics into groups. Similarity or dissimilarity of objects is measured by a particular index of association. The focus here is on clustering of variables instead of subjects. Types of methods that cluster variables based on correlation structure of variables or factorial structure of variables are considered, where factorial structure refers to a structure obtained from principal component analysis or factor analysis. The approaches described for cluster analysis uses various indices of association such as factor loadings, correlation matrix, and cosine matrix. Examples of these are shown and the advantages and disadvantages of each approach are discussed.
Cluster Score
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

Cluster analysis of variables produces groupings or clusters of variables with similar characteristics. One means of summarizing the clusters is with cluster scores, which are simply sums of the original measurements on the variables that comprise the cluster. Often, the variables have unequal standard deviations, in which case, the variables are standardized. Cluster scores then become the sum of these standardized measurements. If the variables of the cluster are equally reliable, then this is a good strategy; if they are not equally reliable, then the two other methods described can be used instead.
C
ronbach's Alpha
Authors: Dukes Kimberly A.
Published: July 2005

Cronbach's alpha is used to assess the internal consistency reliability of a multiple item instrument. A multiple item instrument is internally consistent if its items are highly correlated. Cronbach's alpha is computed as the ratio of the true variation to the total variation.
Dendrogram
Authors: CaliŃski Tadeusz
Published: July 2005

The term “dendrogram” is used in numerical taxonomy for any graphical drawing or diagram giving a treelike description of a taxonomic system. More generally, a dendrogram is a two‐dimensional diagram representing a tree of relationships, whatever their nature.

The representation of a taxonomic system by a dendrogram is particularly suitable in connection with a cluster analysis applied to investigate the structure of the corresponding operational taxonomic units, such as individual patients or case histories initially used to construct disease classifications. This becomes apparent when it is desirable to interpret the results of the analysis in terms of a natural nonoverlapping taxonomic hierarchy. The usual basis for a cluster analysis is a resemblance matrix, its rows and columns referring to the operational taxonomic units, and its entries being the estimates of the resemblances between the corresponding units.

Examples of dendrograms and methods of their presentation are described in many textbooks on numerical taxonomy and cluster analysis.

Among the various clustering methods, the single linkage cluster analysis is particularly relevant to a dendrogram representation.
Factor Analysis, Second‐Order
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

In second‐order factor analysis, we first obtain the second‐order correlation matrix
R
s
. From
R
s
, we obtain the reduced second‐order correlation matrix

. We then factor this

to obtain
F
s
, the second‐order initial factor matrix, and rotate
F
s
to obtain
V
s
, the second‐order rotated factor matrix. Second order factor analyses with two, three, four, five, and more first‐order factors are illustrated.
Factor Loading Matrix
Authors: D'agostino Ralph B. , Russell Heidy K.
Published: July 2005

A factor loading matrix is a matrix of coefficients (or weights) for a set of linear equations relating
p
observed variables to
m
factors. The rows of the matrix correspond to the observed variables and the columns correspond to the factors. The term “factor loading matrix” is also used to refer to the matrix that contains the correlations between the variables and factors.
G
uttman Scale
Authors: Sullivan Lisa M.
Published: July 2005

In a Guttman scale, a set of items are developed to measure a latent construct or attribute. The items are ordered hierarchically such that an affirmative response to one implies affirmative responses to each of the items preceding it. The sum of affirmative responses represents the subject's level of the attribute under study.
Hotelling's

T

2
Authors: Morrison Donald F.
Published: July 2005

Hotelling's T
2
is a generalization of the squared univariate
t
and used for testing hypotheses about a mean vector
μ
. Specifically, T
2
is a measure of the distance between a sample mean vector and a hypothesized population vector in the metric of the sample covariance matrix. T
2
can also be used for testing the equality of two independent multivariate normal mean vectors assuming that the populations have a common covariance matrix, for testing whether
p
means measured over time are equal, or the equality of response vectors measured over time. Rejection of a test based on T
2
does not indicate which of the
p
responses may have contributed to that decision.
Lawley–Hotelling Trace
Authors: Dasgupta Somesh
Published: July 2005

The Lawley–Hotelling trace is used to test the equality of mean vectors ofkp‐variate normal distributions with common but unknown covariance matrix. The explicit form of the null distribution of Tis theFdistribution. The asymptotic null distribution is the chi‐square distribution. The power function of the test is described and its power is compared with the likelihood ratio test.
L
ikert Scale
Authors: Dukes Kimberly A.
Published: July 2005

A Likert scale is computed by summing responses over several items hypothesized to measure the same latent variable or construct. Likert scales are often measured on Likert response formats, which often have three, four, or five response options and represent degrees of endorsement of items. When creating a Likert sale, usually each item is equally weighted, and assumed to have a linear relationship to the overall scale score.
LISREL
Authors: Bollen Kenneth A.
Published: July 2005

LISREL is derived from LInear Structural RELations and it refers to either structural equations models or Jöreskog & Sörbom's software program, which estimates such statistical models. The LISREL model consists of a latent variable model, which formulates the relation between latent variables, free of the confounding effects of measurement errors, and a measurement model, which provides the linkages between the latent and observed variables.
Multiple Endpoints, Multivariate Global Tests
Authors: D'agostino Ralph B. , Russell Heidy K.
Published: July 2005

In clinical trials, multiple endpoints are often used to investigate a treatment effect. The statistical procedures available to analyze multiple endpoints fall into two categories: multivariate global methods and endpoint‐specific orPlevel methods. Multivariate global methods are used to make an omnibus assessment of the efficacy of the treatments, providing an overallPvalue to reflect whether there are any differences among the treatments on several correlated endpoints. Within multivariate global methods there are two types of tests. The first type involves combining multiple endpoints into a single test statistic, which provides an overall statement to declare whether there are any significant differences among the treatments. The second method arises from Bonferroni‐type adjustments. They adjust the univariate observedPvalues of the individual tests to maintain the familywise type I error rate at a prescribed level, and they provide an overall probability for the trial.
Multiple Endpoints,PLevel Procedures
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

In clinical trials, multiple endpoints are often used to investigate a treatment effect. There has been substantial debate in determining the appropriate approach to analyze these multiple endpoints. Global procedures provide a single statistic and are used to compare treatments in an overall sense but do not look at how the treatments affect each endpoint individually.

Endpoint‐specific procedures are intended to be used to identify the significance of the individual endpoints. These procedures are also calledPlevel procedures because they usually involve making statistical decisions on the basis of thePlevel, resulting from examining the endpoints separately.Plevel procedures can be used in the primary analyses or as a second‐step analyses in conjunction with multivariate global tests. There are three categories of endpoint specific methods, Bonferroni‐type tests, normal‐based tests and resampling procedures.
MultivariateBartlett Test
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

The multivariate Bartlett test is used to test for homogeneity of covariance matrices. This is important, for example, prior to employing Hotelling'sT2test or multivariate analysis of variance to test the equality of two or more mean vectors as these procedures assume homogeneity of covariance matrices.
Multivariate Graphics
Authors: Carr Daniel B.
Published: July 2005

This extensive tour of multivariate graphics leads into methods that can be applied to biostatistics. Basic design issues and techniques, such as smoothing and density estimation, are considered. Issues of choice of color, depth, and structure are discussed. Multivariate representations basically fall into a few classes: glyph plots, linked plots, nested plots, conditioned plots, geometric section plots, series plots, and composite plots. Each of these classes is described in detail. Once the graphics are created, important considerations of viewing the graphics in terms of dimensionality, cognostics, and projection pursuit arise. Ways of handling these issues by starting from low‐dimensional plots and seeing what happens through cognostics or projection pursuit, and sectioning are pursued. The ongoing challenges of producing graphics from massive datasets is also considered.
Multivariate Normal Distribution
Authors: Jensen D. R.
Published: July 2005

Biometric data typically entail observations on multiple characteristics for each experimental subject. Multivariate normal distributions are often central to the modeling and analysis of such data. Empirical evidence often points towards the normality of multivariate data. Biometric measurements, especially, may emerge as the result of many small increments due to heredity and environment, so that the approximate multivariate normality of such data rests on multidimensional central limit theory. In addition, many normal‐theory procedures, both univariate and multivariate, remain exact for many nonnormal multivariate distributions exhibiting suitable symmetries.
O
blimin Rotation
Authors: Jackson J. Edward
Published: July 2005

Oblimin rotation
is a general form for obtaining oblique rotations used to transform vectors associated with principal component analysis or factor analysis to
simple structure
. Oblimin is similar to the
Orthomax rotation
procedures used in orthogonal rotation in that it, too, includes an arbitrary constant used to obtain different rotational properties. A numerical example is included. While most orthogonal rotations use some form of Orthomax rotation, this is no longer the case with Oblimin rotation for the oblique case.
O
ptres Rotation
Authors: Jackson J. Edward
Published: July 2005

Optres Rotation
is a three‐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 be an
orthogonal
rotation such as
Varimax
. The second will be an oblique rotation such as
Promax
. The third rotation requires, for each vector, a specification of which coefficients are to be maximized and which are to be minimized. A numerical example is included.
Orthogonal Rotation
Authors: Jackson J. Edward
Published: July 2005

Any rotation of vectors associated with principal components analysis or factor analysis designed to produce a new set of approximating
simple structures
is said to be
orthogonal
if the resultant set of vectors are still at right angles to each other. Most such rotations are obtained by a procedure called
Orthomax rotation
whose general expression is given. Part of the expression is multiplied by arbitrary constant changes which will produce new vectors with different properties. The two most common orthogonal rotations are
Varimax
and
Quartimax
. An alternate procedure,
Minimum Entropy
, is also presented.
Path Analysis
Authors: Bollen Kenneth A.
Published: July 2005

Assuming that a model consists of a system of equations and assumptions about the relationships among a set of variables, where variables can be observed, unobserved (latent), or disturbances (error), path analysis provides an algorithm to understand the direct, indirect, and total effect of one variable on another in a hypothesized model. It also allows a test of whether the hypothesized model is consistent with the covariances (correlations) of observed variables. It is not intended as a tool to discover causal relations. The primary components of path analysis: path diagram, estimation of path coefficients, and decomposition of effects are illustrated. The current form of path analysis is best shown through structural equation modeling.
Pillai's Trace Test
Authors: DasGupta Somesh
Published: July 2005

Pillai proposed the trace test for the following three tests: (a) equality of mean vectors oflp‐variate normal distributions with the common but unknown covariance matrix, (b) independence between two sets of variates distributed jointly as a normal distribution with unknown mean vector, and (c) equality of covariance matrices of twop‐variate normal distributions with unknown mean vectors.
Trees, Probabilistic Functional
Authors: Gama Joäo
Published: July 2005

Decision Trees are one of most used models for prediction tasks in machine learning, data mining, and statistics. The representation language of a standard decision tree uses splitting tests based on a single input variable and constants at tree leaves. The most recent advances in learning decision trees from data involve the use of splitting tests based on combinations of the input variables and functions at tree leaves. These types of models are denominatedfunctional trees. In this article, we present the issues and actual trends in learning functional trees from data.
Procrustes Rotation
Authors: Jackson J. Edward
Published: July 2005

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

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

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

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.
Sphericity Test
Authors: Das Gupta Somesh
Published: July 2005

An overview of the sphericity test is given for ap‐variate normal distribution. The sphericity test presented is a likelihood ratio test for the sample covariance matrix. Forp = 2, the asymptotic distribution is described. An application of the sphericity test to repeated measures analysis is outlined.
Structural Equation Models
Authors: Bollen Kenneth A.
Published: July 2005

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.
Tree‐Structured Statistical Methods
Authors: Zhang Heping , Crowley John , Sox Harold C. , Olshen Richard A.
Published: July 2005

In this article, we describe the binary tree‐structured methods and illustrate the methods with applications in biomedical research. The methods include decision trees, regression trees, survival trees, and trees for a vector response. We also provide information with regard to the software.
Varimax Rotation
Authors: Jackson J. Edward
Published: July 2005

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

Artificial 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.
Bioinformatics
Authors: Wilson Susan R.
Published: July 2005

A brief summary of this emerging field is given.
Computer Architecture and Organization
Authors: Nielsen Ole M.
Published: July 2005

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

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

In 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”.
Floating Point Arithmetic
Authors: Maindonald John H.
Published: July 2005

Floating point number representations are a response to the demand to store and manipulate in a computer, using a fixed number of digits, numbers that range widely in magnitude. The consequent finite precision of floating point arithmetic has large implications for practical computation, of a kind that this article explores.
Jackknife Method
Authors: Schucany William R.
Published: July 2005

The jackknife is a general methodology for approximate confidence intervals and tests in certain settings. Characteristics of the relevant sampling distribution are estimated using a process that systematically holds out subsets of the samples. Bias and standard error estimators are based upon elementary summary statistics.
Machine Learning
Authors: Wilson Susan R.
Published: July 2005

A brief overview is given of machine learning and its relationship with various commonly used statistical techniques.
Markov ChainMonteCarlo
Authors: Gilks W. R.
Published: July 2005

Markov 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.
Matrix Computations
Authors: Maindonald John H. , Smyth Gordon K.
Published: July 2005

This article gives an overview of modern matrix algorithms, and of widely used high quality matrix routines, with especial reference to statistical computations. The article begins with an account of linear least squares calculations, which are fundamental to many other statistical computations, and from there moves on to discuss other matrix computations.
Numerical Analysis
Authors: Smyth Gordon K.
Published: July 2005

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

This 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

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

The 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.
Software, Biostatistical
Authors: Goldstein Richard
Published: July 2005

This article is a brief overview of statistical software available for biostatisticians. Included are historical notes, contact information for some of the more major packages, and some guidelines on how to assess such software.
Spreadsheet
Authors: Hunt N.
Published: July 2005

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

There 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.
Alternative Hypothesis
Authors: Schork M. A.
Published: July 2005

In testing a null hypothesis, it is often useful to consider possible alternatives. A distinction is drawn between one‐sided and two‐sided alternatives.
A
spin–
W
elch Test
Authors: Algina James
Published: July 2005

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

BAN (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.
Bayes Factors
Authors: Pauler Donna K.
Published: July 2005

The Bayes factor is the ratio of posterior odds to prior odds, in a comparison of two competing hypotheses. Consideration is given to computing methods, and to the choice of the prior distribution.
B
ayes' Theorem
Authors: Lindley Dennis V.
Published: July 2005

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.
Bivariate Distributions
Authors: Kocherlakota Subrahmaniam
Published: July 2005

Bivariate distributions play an important role in biological studies. In this article we discuss continuous, discrete and mixed bivariate distributions. Examples of such distributions with references to pertinent data are given.