bluuebunny/crossref_metadata_2025_split huggingface dataset:
Analysis of Variance for Longitudinal Data
Authors: Dunn Graham 
Published: July 2005

Abstract
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

Abstract

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.
Change‐point Problem
Authors: Pawitan Yudi 
Published: July 2005

AbstractThe 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.
Forecasting
Authors: Chatfield Chris 
Published: July 2005

AbstractPredicting 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

AbstractThe 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

Abstract
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).
Nonignorable Dropout in Longitudinal Studies
Authors: Fitzmaurice Garrett 
Published: July 2005

Abstract
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

Abstract
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

Abstract
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.
Bagging and Boosting
Authors: De Veaux Richard 
Published: July 2005

Abstract
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

AbstractThe 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

Abstract
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

Abstract
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

Abstract
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.
Dendrogram
Authors: CaliŃski Tadeusz 
Published: July 2005

Abstract
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

Abstract

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

Abstract

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.
Hotelling's

T

2
Authors: Morrison Donald F. 
Published: July 2005

Abstract

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

AbstractThe 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

Abstract
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

Abstract
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

AbstractIn 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

AbstractIn 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

AbstractThe 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

Abstract
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

Abstract
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

Abstract

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

Abstract

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

Abstract

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

Abstract
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

AbstractPillai 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

AbstractDecision 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.