bluuebunny/crossref_metadata_2025 huggingface dataset:
SIREpidemic Models
Authors: Lefèvre Claude
Published: July 2005

These models describe the spread of infectious diseases in which an individual may move from Susceptible to Infected to Removed. Simple models, including the Reed–Frost chain binomial and the general epidemic, are Markovian with the implication of an exponentially distributed infectious period. The generalized epidemic removes this restriction. More complex models allow for individual or structural heterogeneity in susceptibility, infectivity, and/or mixing behavior.
Tumor Growth
Authors: White R. Allen
Published: July 2005

The measurement of tumor growth rates has applications in such areas as tumor detection, prognosis, and control. The proliferation of tumors can be measured at different levels including gross measures of size, population progression through the cell cycle, and at the molecular level with biomarkers. Each of these methods has specific requirements and problems, particularly, including the heterogeneity both within and between tumors. Since tumors typically consist of a very heterogeneous mixture of normal and tumor cells, problems of interpretation of results must be made with care. Moreover, since tumors typically are measured only at detection with variable unknown histories and, after detection, at only a single or, at most, a few time points, problems in analysis require special methods.
Viral Population Growth Models
Authors: Tuckwell Henry C.
Published: July 2005

The steps involved in viral reproduction are described briefly. Mathematical models for the growth of viral populations without random effects take the form of systems of differential equations and these are reviewed, including the standard HIV equations, with correction terms. The validity of the “mass action” assumption is questioned for low virion levels. Stochastic differential equation models for HIV are described, including those with both discontinuous and diffusion processes. Simulation results are given for early HIV population growth, including the distribution of the time to reach a detection level, for which an analytical theory is also available. The long‐term model of Tan and Wu is discussed briefly as well as mutation and its effect on drug treatments. Despite some useful analysis of patient data with respect to simple differential equations, it is concluded that modeling, either deterministic or stochastic, of viral populations requires much further development to give an accurate and useful representation of the underlying biological processes.
Allometry
Authors: Sprent P.
Published: July 2005

Allometry is a study of shape changes associated with size. Most living organisms change shape as they grow in order to overcome biological or physical constraints. Simply, allometry occurs when two size measurements exhibit an approximately linear relationship between their logarithms. Suggestions made for multivariate extensions range from applications of principal component analysis to studies of size variables and shape vectors. The limitations imposed by Euclidean geometry and other disadvantages with these approaches have been removed by considering curvature and tangential directions at a series of points called landmarks and by coordinate transformations.
General Practice
Authors: Peters T. J.
Published: July 2005

Research in general practice is principally focused on common, chronic medical conditions, covers organizational as well as clinical issues and is a multidisciplinary endeavor. At the same time, conditions seen frequently in particular specialities are relatively uncommon, and this has implications for the statistical performance of diagnostic and screening tests. Generally, biostatistical methods in general practice are characterized by sources of variation operating at various levels, including the primary care center, practitioner, and individual. Both study design and data analysis must make reasonable allowance for this, in particular, in experimental studies where the unit of randomization is a group of individuals. Practical, ethical, and financial constraints mean that a variety of study designs will remain valuable in general practice research, but the central role of the pragmatic randomized trial will continue.
Midwifery, Obstetrics, and Neonatology
Authors: Macfarlane Alison , Elbourne Diana
Published: July 2005

Midwifery is the profession responsible for care in normal pregnancy and labor, obstetrics is the profession that deals with pregnancies with clinical complications, and neonatology is concerned with care for sick newborn babies. Members of these professions and others concerned with evaluating the care they give in the context of clinical, environmental, and genetic factors have measured the outcome in terms of maternal and infant mortality and morbidity and used a range of descriptive, analytical, and experimental approaches to do so. From the mid‐nineteenth century onwards, routine vital statistics have been used to describe trends and variations. In latter half of the twentieth century, these were analyzed by correlation, regression, and later multivariate analysis, with techniques being developed for record linkage. Clinicians in these profession and associated researchers were among the first to use randomized trials to evaluate care and meta‐analyses to synthesis the results of trials.
Psychiatry
Authors: Everitt Brian S.
Published: July 2005

Psychiatry is the medical speciality that deals with mental disorders. In early medieval times, beating, incarceration, and restraint were the “treatments” endured by the majority of the mentally ill, and a more enlightened perspective has evolved only slowly over the intervening 600 years. In the twentieth century, psychiatry struggled to attain scientific respectability by making quantitative observations on mentally ill patients, and subjecting these observations to a variety of statistical methodology ranging from simple descriptive techniques to complex multivariate analyses.
Radiation
Authors: Darby Sarah C. , Gilbert Ethel S.
Published: July 2005

Biostatistical applications in the area of ionizing radiation are primarily directed towards obtaining quantitative risk estimates for various cancers following radiation exposure. The relevant data arise mostly from observational studies on humans. The major populations studied include the survivors of the atomic bombings of Hiroshima and Nagasaki, patients given radiotherapy for a variety of malignant and other conditions, workers in the nuclear industry, persons exposed in countries of the former Soviet Union, uranium miners, and populations living in areas where residential radon concentrations are high. Experimental animal data are also available. The large quantity of information and the availability of quantitative dose estimates in many studies have enabled the development of a variety of complex risk models. Recent work has included bringing together data from several different studies, usually with the objective of producing more precise estimates of risk, and the development and application of methods that account for uncertainties in the estimates of radiation dose.
Reproduction
Authors: Farley Timothy M. M.
Published: July 2005

Assessing objectively the efficacy and effectiveness of contraceptive methods is challenging because of the large number of different factors affecting the intrinsic fertility of a couple, the timing of intercourse, and the way the method is used. While effectiveness, or the typical use pregnancy rate, is easier to observe, it is difficult to generalize to other populations since it includes behavioral factors, such as how or whether the method was used. By contrast, efficacy, or the perfect use pregnancy rate, can be indirectly estimated from patterns of intercourse and method use, and is more easily generalized to other women or couples. For methods that are not coitus‐dependent, typical and perfect use pregnancy rates are similar, but can be very different for methods requiring a high degree of user compliance.
Transfusion Medicine
Authors: Casbard Angela
Published: July 2005

Transfusion medicine is the specialty of medicine devoted to all aspects of the use of blood products and components, from donor selection and care, through testing and component preparation, to storage, distribution, and use. It also includes pregnancy testing for blood group incompatibilities. This article provides a brief overview, including a history of blood transfusion, current practice, usage of statistics, and recent studies.
Dummy Variables
Authors: Darlington G. A.
Published: July 2005

Categorical explanatory variables can be included in models via the use of dummy variables. In a regression model with an intercept,
K
 − 1 linearly independent dummy variables must be defined to represent a
K
level categorical variable. Indicator variables are commonly used to define a set of dummy variables. In a regression model with an intercept, indicators are derived for
K
 − 1 of the
K
possible levels and the remaining level is the referent or baseline category. Overall tests regarding the categorical variable will not depend on the choice of dummy variable definitions. However, the interpretation of the coefficient for each dummy variable will depend on the coding used.
Errors in the Measurement of Covariates
Authors: Yi Grace Y. , Cook Richard J.
Published: July 2005

Measurement or misclassification errors in covariates lead to biased estimates of regression parameters. Corrections typically, require addition date such as replication of errors observations, validation through direct observation of correct values, or the introduction of further “instrument” variables. Methods described include regression calibration, simulation extrapolation, and the use of estimating equations. An example is based on a case–control study of cervical cancer.
Errors in Variables
Authors: Raboud J. M.
Published: July 2005

When independent variables in regression models are measured with error, the resulting coefficient estimates are biased. Formulae are given to adjust estimated coefficients obtained from linear regression models and proportional hazards models. References are given to estimate the degree of bias of estimated coefficients in logistic regression and other types of general linear models.
Explanatory Variables
Authors: Darlington G. A.
Published: July 2005

A model may be developed to investigate the relationship between an outcome and one or more explanatory variables. An explanatory variable can be quantitative, requiring consideration of how to include it in the model or an explanatory variable can be qualitative whereby indicator variables (or dummy variables) must be defined.
Fixed Effects
Authors: Farewell Vern T.
Published: July 2005

The concept of fixed effects in a regression model is explained and contrasted with that of random effects.
Goodness of Fit
Authors: Atkinson A. C.
Published: July 2005

Pearson's classical chi‐square test compares observed and expected frequencies in various classes. It tests for unspecified departures from the model, is based on aggregate rather than partial data, and uses an asymptotic distribution. Goodness of fit in multiple regression may be explored by including further explanatory variables in the model, or by comparing residual variation against external predictions. Tests based on residuals are described, as are extensions to generalized linear models.
Hierarchical Models
Authors: Farewell Vern T.
Published: July 2005

Usage of the term “hierarchical models” is described. In particular, a hierarchy of models is differentiated from multilevel models and Bayesian models.
Least Squares
Authors: Sprott D. A.
Published: July 2005

Gauss's development of Least Squares is outlined. It is based on assumptions that are probably not generally appropriate to biostatistics. But it has computational applications in obtaining initial values required to start iterative procedures of estimation.
Mallows'CpStatistic
Authors: Stone M.
Published: July 2005

This entry covers the seminal model choice criterion of H.L. Jones, attributed to and developed by C.L. Mallows.
Multilevel Models
Authors: Goldstein Harvey
Published: July 2005

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

The article describes models for all these kinds of data structures, increasing in complexity as they move from simple hierarchies with continuously distributed responses, to cross‐classifications, multiple unit membership and multivariate data, and to discrete responses. Various extensions and special cases will also be considered. The emphasis is on model specification with a brief section on estimation.
Multiple Linear Regression
Authors: Matthews David E.
Published: July 2005

Multiple linear regression involves finding the best‐fitting surface of a suitable functional form that relates the values of explanatory variables,X1, …,Xk, and the mean value of a response variable,Y, given values ofX1, …,Xk. The objectives of regression modeling are to determine whetherYand one or more of the explanatory variables are associated in some systematic way, and to estimate or predict the value ofY, or its mean, corresponding to known values of a selected subset ofX1, …,Xk. We describe methods of estimation, variable selection, statistical inference, and diagnostic checking of the assumed model and any associated unknown parameters, giving due importance to their statistical and scientific interpretations. An example concerning the relationship between oxygen uptake and age, weight, sex, the time required to run 1.5 miles, and various pulse rates for participants in a physical fitness workshop illustrates these concepts concretely. Finally, we outline the close connection between ordinary and weighted linear regression.
Nonlinear Regression
Authors: Kolkiewicz Adam W.
Published: July 2005

The defining feature of nonlinear regression models is that the expectation function depends on some parameters in a nonlinear way. In such cases, the classical results for linear models, including explicit estimation, normality of estimators, exact tests, and analysis of variance, break down. In nonlinear regression, estimates are usually found by iteration and for the distribution of estimators and test statistics, we have to rely on approximations, which are often based on linear models. Several measures of nonlinearity have been developed to provide information on the adequacy of such approximations and to indicate situations where, by changing the parameters, a nonlinear model behaves more like a linear one. In the article, we provide several examples of nonlinear models used in practice. We also give a brief description of some measures of nonlinearity and outline basic nonlinear optimization techniques used in estimation procedures.
Orthogonality
Authors: Chandler R. E.
Published: July 2005

The concept of orthogonality is encountered in many areas of mathematics and statistics, although in biostatistics it is probably used most often in the context of experimental design. This article provides a summary of the underlying mathematics in general terms, and discusses applications in design, analysis of variance, regression, likelihood theory, time series analysis, and probability theory.
Regression
Authors: Farewell Vern T.
Published: July 2005

Statistical usage of the term regression is discussed.
Response Variable
Authors: Farewell Vern T.
Published: July 2005

The concept of a response variable is introduced and contrasted with that of an explanatory variable. Univariate, multivariate, and intermediate response variables are discussed.
Association, Measures of
Authors: Khamis Harry J.
Published: July 2005

This contribution provides an overview of the commonly used measures of association for categorical variables. The discussion of these measures is broken down by (1) the size of the contingency table, (2) the measurement scale of the variables involved, and (3) the interpretability of the measure.
B
reslow–
D
ay Test
Authors: Liu I‐Ming
Published: July 2005

The Breslow–Day test is a test for assessing the homogeneity of the odds ratio in several 2 × 2 contingency tables. For
K
tables, it has approximately a chi‐squared null distribution with df = 
K
‐1.
Capture–Recapture
Authors: Chao Anne
Published: July 2005

Capture–recapture methods are used to estimate population sizes and related parameters such as survival rates, birth rates, and migration rates. A typical capture–recapture experiment places traps in the study area and samples the population several times. At each trapping sample, one records and attaches a unique tag to every unmarked animal, records the capture of any animal that has been previously tagged, and returns all animals to the population. This article describes the history of the method, discusses applications to epidemiology, discusses potential models, and discusses the challenges inherent in estimating the population size well.
Categorical Data Analysis
Authors: Imrey Peter B. , Koch Gary G.
Published: July 2005

Categorical data consist of observations counted and subclassified into one among a set of disjoint, exhaustive groups. The groups may be defined on the basis of a single or a combination of qualitative characteristics and/or quantitative measurements. When such categorizations are to be modeled as probabilistic “responses” or “outcomes”, statistical analysis must be based upon discrete distributions and requires extensive generalization of the classical notions of regression and analysis of variance commonly used for continuous observations. This overview article reviews statistical inference for categorical data, from the basic issues in analyses of 2 × 2 contingency tables through generalized linear mixed models, with emphasis on approaches that have increasingly unified our concepts of categorical and continuous data modeling.
Chi‐Square Tests
Authors: Koehler Kenneth
Published: July 2005

A chi‐square test investigates whether observed data are consistent with a proposed probability model. Best‐known chi‐square statistics are the Pearson statistic and the likelihood‐ratio statistic. This article also mentions various other statistics that have limiting chi‐square distributions, all of which are special cases of the power divergence statistic. The article begins by introducing simple goodness‐of‐fit tests, then discusses a variety of chi‐square statistics. Chi‐square tests are then presented that apply with fixed intervals or with random intervals. Applications are then discussed for categorical data, such as testing independence. Extensions are discussed for small samples and sparse tables, and for correlated responses and complex surveys.
Chi‐Square, Partition of
Authors: Gilula Z. , Haberman Shelby J.
Published: July 2005

A chi‐square statistic suitable for testing a primary hypothesis can be partitioned into components such that each component gives a test for a corresponding secondary hypothesis. Some partitionings are exact and some are approximate. The theory is based on the Fisher–Cochran theorem about decomposing quadratic functions of normal variables. The history of this technique is surveyed. Applications are described for contingency tables, with the main focus on the two‐way table and likelihood‐ratio statistics. Brief mention is also made of partitioning into non‐chi‐square components, such as a decomposition that forms the basis of correspondence analysis.
Contagious Distributions
Authors: Kemp Adrienne W. , Kemp C. D.
Published: July 2005

Polya proposed a model for the spread of an infection, whereby the probability of catching a disease depended on the existing number of cases. This leads to a distribution that has a limiting negative binomial form. Neyman proposed an alternative approach that arises from a Poisson distribution of clumps, with the number of cells for each clump being Poisson. This article surveys probability distributions and corresponding generating functions that result from such constructions.
Continuity Correction
Authors: Coull Brent A.
Published: July 2005

The term “continuity correction” has traditionally referred to an adjustment made when using a continuous distribution to approximate a discrete distribution. Corrections of this nature are common in both univariate settings, such as for inference on a single binomial proportion, and multivariate situations, such as contingency table analyses. In this entry, we briefly outline the historical development of the traditional continuity corrections and review the performance of such methods in the broader context of small‐sample strategies for categorical data. In particular, we consider the role of continuity corrections for inference on a single binomial proportion, a single 2 × 2 contingency table, and stratified 2 × 2 tables. In such settings, statistics based on the traditional continuity correction closely approximate “exact” results, yielding conservative inferences in small samples because of the discreteness of the exact distribution. We also review other more recent adjustments for discreteness. In many categorical data settings, if the nominal type I error rate of a test (or analogously the nominal confidence level for a confidence interval) need not be strictly maintained at every element of the parameter space, one can reasonably base inference on either an uncorrected score statistic or the mid‐Pvalue.
G
oodman–
K
ruskal Measures of Association
Authors: Upton G.
Published: July 2005

The measures of association proposed by Goodman and Kruskal between 1954 and 1972 can all be described as proportional reduction in error (PRE) measures. They describe the association between two categorical variables in terms of the increase in the probability of correctly predicting the category of one variable when information is supplied about the category of the other variable. The measures λ and τ are for use with any categorical variables, whereas the measure γ is for use when both variables are ordinal.
Logistic Regression
Authors: Lemeshow Stanley , Hosmer David W.
Published: July 2005

This article deals with the use of logistic regression to model data, where the response variable is binary and there are potentially numerous explanatory or predictor variables. Methods of estimating the coefficients in the model and procedures for hypothesis testing are considered. The correspondence of the coefficients of the logistic regression model to the odds ratio is noted along with methods for computing point estimates and confidence intervals for the odds ratio. Interpretation of coefficients for dichotomous, polytomous, and continuous independent variables is carefully developed. Confounding and effect modification are easily dealt with through the use of this modeling technique. All these concepts are illustrated through the analysis of low birth weight data.
McNemar Test
Authors: Lachenbruch Peter A.
Published: July 2005

The McNemar test is used for hypothesis testing with paired dichotomous data. The test involves only the off‐diagonal elements and results in a 1 degree of freedom χ
2
test. The dependence on only the discordant means that sample size estimation must rely on assumptions about the proportion of discordant observations. It can be shown that this test is equivalent to a paired t‐test under the assumption of equal means.
Misclassification Models
Authors: Albert Paul S.
Published: July 2005

Classification errors in categorical data may distort the results of statistical analyses. Misclassification models have been developed in various areas of biostatistical application to account for this distortion. This article reviews selected methods that have been proposed for (i) modeling the natural history of a disease or growth process, (ii) evaluating diagnostic tests, and (iii) identifying epidemiological risk factors. We consider methods that incorporate known estimates of misclassification rates, methods that estimate misclassification rates from a subsample in which the presumably “true” classification is known, and methods in which both the true process and the misclassification rates are simultaneously modeled from misclassified data. The various methodological approaches are discussed along with their limitations, and we note an important issue for additional research.
Multinomial Distribution
Authors: Zelterman Daniel
Published: July 2005

The multinomial distribution is the generalization of the binomial to more than two dichotomous categorical outcomes. This multivariate discrete distribution describes the joint behavior of frequencies of each of
k
possible outcomes following
n
independent trials. The
k
individual frequencies follow a binomial distribution and are mutually negatively correlated. The probabilities of each of the
k
outcomes are usually estimated using loglinear models or Bayesian methods. Goodness‐of‐fit and exact tests assess the adequacy of the model to explain the empirical data. These topics are illustrated with an example from genetics.
Nominal Data
Authors: Wasserman Stanley
Published: July 2005

Nominal data arise from a specific type of
measurement
. As defined by the psychologist S.S. Stevens, measurement is the assignment of numbers to entities according to some rule. Close examination of such rules allows us to classify the various types of measurements into a small set of possibilities. Measurements are, of course, taken for many different reasons to answer many different questions, usually (but not always) on experimental units or subjects or on observational units. Measurements produce data, which take on values along some measurement scale. The measurement type(s) of data to be analyzed dictates the analyses that yield justifiable and meaningful interpretations.
Poisson Regression
Authors: Seeber Gilg U. H.
Published: July 2005

Poisson regression models are regression models that assume a Poisson distribution for a count response variable. Usually they have loglinear form, modeling the log of the mean in terms of the explanatory variables. Such models are special cases of generalized linear models. A common application is in multiplicative modeling of rates, for which there are connections with Poisson process models and proportional hazards models. This article introduces these types of models, shows how to fit them using maximum likelihood, and discusses checking of model fit.
Polytomous Data
Authors: Tutz Gerhard
Published: July 2005

The article describes regression models for categorical responses, distinguishing between nominal, ordinal, and nested response categories. Basic concepts of estimation, testing, and goodness‐of‐fit statistics are given.
Proportions, Inferences, and Comparisons
Authors: Connor Jason T. , Imrey Peter B.
Published: July 2005

This article selectively reviews asymptotic, exact, and Bayesian methods for inference about one proportion, or for comparing two proportions from separate samples. For a single proportion, interval estimation is emphasized, and coverage properties of Wald, Jeffreys, Wilson, Agresti–Coull, and Clopper–Pearson (exact) confidence intervals are compared. Hypothesis testing is also discussed, primarily in the context of inverting interval estimates. For comparing proportions between two samples, we consider tests including the power‐divergence family (which incorporates all asymptotic tests commonly used); both conditional and unconditional exact tests; Bayesian tests; and several interval estimates. Some general aspects of power computation and sample size determination are discussed briefly in both contexts.
Quantal Response Models
Authors: Morgan Byron J.T.
Published: July 2005

In a quantal response experiment, a particular organism is selected and exposed in groups to different concentrations of a drug for a fixed time, and a discrete response is observed (such as death or survival). In the binary response case, a tolerance or threshold assumption yields a model for the probability of response at a particular dose that has the form of the cumulative distribution function of the tolerances. The median effective dose (ED50) is the dose level that results in an expected 50% response under certain model assumptions. The Spearman–Karber estimate is a nonparametric estimate of the ED50. This article also surveys parametric estimates, which can have improved precision. Parametric models are discussed, including extensions for polytomous responses and multivariate responses. Also discussed are the effect of overdispersion, and design issues such as sequential methods for finding the point at which a certain probability of response occurs.
Rasch Models
Authors: Carriquiry Alicia L. , Fienberg Stephen E.
Published: July 2005

A fundamental component of model item response theory in psychometrics is the family of Rasch models introduced by the Danish statistician Georg Rasch. The basic Rasch model represents the log odds of the probability that an individual responds positively to theith binary test item as the sum of an individual parameter representing ability and an item parameter representing the difficulty of the item. This entry relates the Rasch model to traditional loglinear models and, in particular, models of multidimensional quasi‐symmetry. The implications of this relationship are illustrated in the context of an epidemiological example, and applications in the context of capture–recapture methods are discussed briefly.
Simpson's Paradox
Authors: Dong Jianping
Published: July 2005

Simpson's paradox occurs when the direction of a measure of association between two variables is reversed after pooling over a covariate. For example, a treatment can be effective for both males and females, but ineffective when the data for males and females are combined. Since Simpson's original example in his 1951 paper, numerous real‐life examples of Simpson's paradox have been reported in many areas and the paradox has been generalized to many forms of association reversals. Necessary and sufficient conditions for Simpson's paradox as well as generalized forms of association reversals are given so we know, for example, when a high‐dimensional contingency table can be safely collapsed, avoiding the paradox. It is also of interest to know what the appropriate conclusion is when Simpson's paradox does occur.
Spatial Models for Categorical Data
Authors: Waller Lance A.
Published: July 2005

After reviewing characteristics of spatially correlated counts, rates, and proportions that compromise traditional methods of inference, we briefly survey general approaches for the analysis of such spatially referenced categorical data. The survey focuses on auto‐models, generalized linear models, and generalized linear mixed models, emphasizing structural and conceptual aspects rather than computational details.
Y
ates's Continuity Correction
Authors: Hilton Joan F.
Published: July 2005

This correction refers to the Pearson chi‐squared statistic for testing independence in a 2 × 2 contingency table. With this correction, the
P
‐value obtained from the chi‐squared distribution with df = 1 approximates the
P
‐value obtained with the hypergeometric distribution in Fisher's exact test.
R
‐ and
Q
‐Analysis
Authors: Jackson J. Edward
Published: July 2005

The starting point for principal component analysis, factor analysis, or multidimensional scaling is an
nxp
data matrix representing
n
observations on
p
variables. In
R‐Analysis
, a
pxp
matrix, such as a covariance or correlation matrix, is formed from which components or factors are obtained in terms of the variables. In
Q‐Analysis
, the situation is reversed. The matrix is
nxn
and the components or factors are in terms of the observations.
N‐Analysis
, also referred to as
Singular Value Decomposition
, operates on the data matrix directly and obtains results for both the variables and observations. In
M‐Analysis
, the data matrix has been corrected for both its row and column means.
Rotation of Axes
Authors: Jackson J. Edward
Published: July 2005

In both
principal component analysis
and
factor analysis
, it is sometimes desirable to perform a linear transformation of the original components or factors. This operation is performed to arrive at
simple structure
where the coefficients of the vectors related to these new transformed variables are either as large as possible or close to zero. These new coefficients may be used either to aid in the interpretation of the original components or factors, or to cluster the original variables. This linear transformation amounts to a rotation of the axes related to the components or factors. The rotation may be
orthogonal
in which case the resultant new set of vectors are still at right angles to each other or
oblique
in which they are not. Loss of orthogonality may result in better interpretation of the original vector sets or clustering of the original variables. Several numerical examples are given to illustrate these procedures including one where rotation is not appropriate.
Roy's Maximum Root Criteria
Authors: Dasgupta Somesh
Published: July 2005

To illustrate his union–intersection principle, Roy proposed the maximum characteristic root eigenvalue test statistic.
Scree Test
Authors: D'agostino Ralph B. , Russell Heidy K.
Published: July 2005

The scree test is a test for determining the number of factors to retain in a factor analysis or principal components analysis. The scree test involves plotting the eigenvalues in descending order of their magnitude against their factor numbers and determining where they level off. The break between the steep slope and a leveling off indicates the number of meaningful factors, different from random error. The technique is illustrated and compared with alternative techniques for determining the number of factors to retain.
Similarity, Dissimilarity, and Distance Measure
Authors: Gower John C.
Published: July 2005

Many coefficients exist that give measures of resemblance between a pair of cases, or samples, as opposed to association between pairs of variables; others measure resemblance between a pair of populations. Some are based on quantitative variables, some on categorical variables that may be dichotomous or ordered, and others on a mixture of both types.
Simple Structure
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

In factor analysis, a simple structure is the ideal structure in which each factor is defined by a subset of the original variables with as little overlap as possible. This article defines the criteria for simple structure and gives some guidelines for application.
Beta‐Binomial Distribution
Authors: Rosner Bernard
Published: July 2005

The beta‐binomial distribution results from a beta distribution mixture of binomials. It is useful for clustered binary data, for which the ordinary binomial distribution is usually inappropriate because different observations are correlated rather than independent. Models using this distribution permit the mean to depend on explanatory variables. An implicit assumption of the distribution is that subunits are exchangeable within a cluster. Generalizations of it allow for data structures with multiple levels of nesting.
A
alen–
J
ohansen Estimator
Authors: Borgan Ørnulf
Published: July 2005

The Aalen–Johansen estimator is a matrix version of the Kaplan–Meier estimator, which can be used to estimate the transition probability matrix of a Markov process with a finite number of states. The estimator is first presented for the competing risks model and the Markov illness–death model for a chronic disease. For these two simple Markov processes, the elements of the Aalen–Johansen estimator take an explicit form. Then a general finite state Markov process, modeling the life histories of individuals from a homogeneous population, is considered. It is described how the Aalen–Johansen estimator may be obtained as the product‐integral of the matrix of Nelson–Aalen estimators for the cumulative transition intensities. Finally, it is briefly indicated how the product‐integral formulation of the Aalen–Johansen estimator is useful for the study of its statistical properties.
Censored Data
Authors: Andersen Per Kragh
Published: July 2005

The concepts of independent right censoring and noninformative right censoring in survival analysis are defined. Dependent censoring is a kind of selection bias and adds to biased inference for the survival time distribution. Informative censoring typically leads to inefficient inference. Examples are presented and relations between right censoring and competing risks are discussed. Other kinds of censoring, including interval censoring are touched upon.
Coarsening at Random
Authors: Nielsen Søren Feodor
Published: July 2005

Statistical analysis is often complicated by the presence of incomplete observations. Unfortunately, incomplete observations are unavoidable in many studies. However, if the incompleteness is ignorable, some simplifications to the statistical analysis occur. The concept of coarsening at random is a formalization of what is meant by ignorability of incomplete data.
Cox Regression Model
Authors: Sasieni Peter
Published: July 2005

In 1972, D.R. Cox proposed the semiparametric modelfor the hazard at timetfor an individual with covariatesz. Here, the hazard is the instantaneous conditional probability of failure attgiven that failure has not happened beforet.

The Cox model has had enormous influence in the theory and practice of biostatistics, and this article surveys its definition, history, basic use, estimators and algorithms, asymptotic properties, use of time‐dependent covariates, model checking, and alternatives and extensions.
Duration Dependence
Authors: Oakes David
Published: July 2005

A brief survey is given of some of the many possible extensions of classical survival analysis models to problems involving multiple failures and, therefore, multiple possible measures of time.
Expected Survival Curve
Authors: Keiding Niels
Published: July 2005

An expected survival curve aims at describing how a concrete group of patients would have survived under “standard” (historical) conditions. Several approaches to handling this are surveyed and compared, and some further topics and pitfalls regarding the use of Cox regression models in this area are mentioned.
Ghosts
Authors: Keiding Niels
Published: July 2005

Turnbull coined the termghoststo intuitively explain inverse‐probability‐weighted estimation survival analysis, similar to Horvitz–Thompson estimation.
Grenander Estimators
Authors: Keiding Niels
Published: July 2005

Grenander's nonparametric maximum likelihood estimators of a decreasing density and an increasing hazard are reviewed with particular focus on current status data.
Grouped Survival Times
Authors: Zhang Mei‐Jie
Published: July 2005

The grouped survival data consists of occurrence and exposure data over given time intervals and possible covariate strata. Grouping data from such large sample sizes into tabular presentations (life tables) often provides a convenient format for presenting and summarizing life information. Grouped survival data can involve censored data (right censoring, left censoring, or double censoring) and/or truncated data. Moreover, the exact censoring or truncation times may be unknown for grouped data. In this article, we provide the literature on analyzing grouped survival data, and discuss the estimating procedure for the hazard function and survival function under nonparametric or parametric models, test statistics for comparing the survival probabilities among different population groups, and large sample properties for these estimators and test statistics.
Hazard Plotting
Authors: Keiding Niels
Published: July 2005

A brief introduction is given to the contrast between model‐based hazard plotting in the counting process approach and the marginal approach pioneered by W. Nelson and carefully studied by J. Lawless and colleagues.
Hazard Ratio Estimator
Authors: Andersen Per Kragh
Published: July 2005

The two‐sample proportional hazards model is studied and a series of semiparametric estimators for the (constant) hazard ratio are presented. These include both Cox's maximum partial likelihood estimator and explicit rank‐type estimators.
Historical Controls in Survival Analysis
Authors: Keiding Niels
Published: July 2005

Special technical features of the use of historical controls in survival analysis are outlined. They are more fully covered in the articles, “Expected number of deaths” and “Expected survival curve”.
Influence Function in Survival Analysis
Authors: Reid N.
Published: July 2005

The influence function of a statistic is a type of derivative that indicates how perturbations in the data affect the statistic. If the influence function is bounded, then the effect of outliers on the statistic is limited. Influence functions can be used in deriving the limiting distribution of a statistic, and can also be used to develop diagnostics for regression models. This latter application has been useful in proportional hazards regression.
Interim Analysis of Censored Data
Authors: Tsiatis A. A.
Published: July 2005

Many clinical trials of chronic diseases are monitored periodically and interim analyses are performed, both for reasons of “safety”, that is, to monitor possible side effects, and to avoid continuing a trial from which no more convincing evidence can be expected. The adequate handling of this monitoring requires close collaboration between the Data and Safety Monitoring Committee and statistical expertise.

Statistical methods have been developed for respecting basic rules of hypothesis testing when multiple views are involved. Usually some form ofgrouped sequentialapproach fits best to the practical situation. A complication is that often the entry of study subjects isstaggeredover time, so that time on trial does not correspond to calendar time.
Interval Censoring
Authors: Sun Jianguo
Published: July 2005

In interval censoring, a random variable is known only to lie in an interval. In survival or failure‐time data, this may arise, for instance, when a critical change has occurred sometime between a patient's successive visits to a clinic. The survival distribution may be estimated by nonparametric maximum likelihood. Attention is given to the estimation of parameters in regression models and to the comparison of survival functions.
L
exis Diagram
Authors: Keiding Niels
Published: July 2005

A Lexis diagram is a (time, age) coordinate system, representing individual lives by line segments of unit slope, joining (time, age) of birth and death. The diagram is an important descriptive tool in epidemiology and demography and it also has several applications in survival analysis and analytical epidemiology as a tool for several classes of statistical models.
Logrank Test
Authors: Keiding Niels
Published: July 2005

The nonparametric linear rank test statistic for survival data called the logrank test is presented.
Measurement Error in Survival Analysis
Authors: Prentice Ross L.
Published: July 2005

This entry describes various approaches to estimating the regression coefficient in the Cox failure time regression model when some of the regression variables are poorly measured. Regression parameter estimation may be considered if there is a validation subsample within which the “true” regression variable is available, or if there is a reliability subsample within which multiple estimates of the regression variable having independent measurement errors are available. The former context is closely connected to corresponding missing data estimation problems. Simplead hocestimators are reviewed, along with effective but inconsistent regression calibration estimators, and more complex nonparametric maximum likelihood and augmented inverse probability weighted estimators. With only a reliability sample available, estimation procedures include regression calibration and an extended risk set regression calibration approach, along with corrected score estimation procedures. Some comments on procedures for measurement error accommodation in an additive hazards model are also provided. This rapidly developing measurement error research area has implications for the analysis of follow‐up studies in a number of important application areas.
Phase‐Type Distributions in Survival Analysis
Authors: Aalen Odd O.
Published: July 2005

Consider a finite‐state, time‐continuous, homogeneous Markov chain with an absorbing state. The time to absorption in such a state is said to have a phase‐type distribution. Markov models are useful because they may incorporate an understanding of how a phenomenon, for example, a disease, moves through different stages. Densities and hazard rates of phase‐type distributions can be calculated from standard Markov chain theory. Hazard rates are usually asymptotically constant due to quasi‐stationarity; that is, conditional on nonabsorption, the distribution on the transient space converges to a limiting distribution. Quasi‐stationarity can also give an explanation of the various shapes of hazard rates that may occur.
Sample Size Determination in Survival Analysis
Authors: Collett D.
Published: July 2005

This article outlines how the sample size for a study can be determined when the response variable is a survival time. This is a two‐stage process in which the number of events required to meet prespecified power requirements is first determined. The occurrence of censoring generally means that not everyone recruited to a study will experience the end point, and so the next step is to calculate the number of individuals who need to be recruited to give the required number of events. This number will depend on parameters such as the anticipated recruitment rate and the follow‐up period. The procedure is illustrated using a numerical example.
Survival Analysis, Overview
Authors: Andersen Per Kragh , Keiding Niels
Published: July 2005

Survival analysis is the generic name for analysis of time‐to‐event data. The distinguishing feature of such data is the presence of right‐censored observations. Thus, for some subjects, instead of observing the real time to the event, only a lower limit for this time, the right‐censoring time, is known. This kind of incomplete data calls for special statistical techniques for the analysis. This article reviews such methods with cross‐references to other articles dealing with special techniques.
Survival Distributions and Their Characteristics
Authors: Klein John P.
Published: July 2005

The four basic functions that characterize the probability distribution of the time to an event are presented. These functions, the survival function, the density or mass function, the hazard rate, and the mean residual life function are presented, and the relationship between the functions is discussed. Classes of distributions that reflect aging properties are presented. These classes include the increasing hazard rate (IHR) class, the increasing hazard rate on the average class (IHRA), the decreasing mean residual life (DMRL) class, the new better than used (NBU) class, the new better than used in expectation class (NBUE) and the harmonic new better than used in expectation class (HNBUE).
Time Origin, Choice of
Authors: Keiding Niels
Published: July 2005

Subject‐matter issues as well as technical issues regarding choice of suitable time origin in survival and event history analysis are discussed. In rare cases, alternative time
scales
may be relevant.