bluuebunny/crossref_metadata_2025 huggingface dataset:
Multilevel Models
Authors: Goldstein Harvey
Published: September 2014

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.
Nonparametric Regression ‐ An Introduction
Authors: Hazelton Martin L.
Published: September 2014

Nonparametric methods of regression seek to describe the relationship between an explanatory and response variable while making only the most general type of assumptions about the functional form of this relationship. Techniques for computing nonparametric regression functions include kernel estimation, local polynomial regression, and spline smoothing. In each case, there is a balance to be reached between the smoothness of the regression function and its goodness of fit to the data. Simple nonparametric regression models (with one explanatory variable) can be extended to deal with multiple covariates by employing so‐called generalized additive models.
Orthogonality
Authors: Chandler R. E.
Published: September 2014

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.
Transformation: Introduction
Authors: Farewell Vern T.
Published: September 2014

Transformation of variables and/or parameters to facilitate the use of standard statistical methodology is described. Particular reference to regression models is made.
Loglinear Model
Authors: Carriquiry Alicia L. , Fienberg Stephen E.
Published: September 2014

The general structure of loglinear models emerged in the 1960s as a response to the need for methods to assess departures from independence (and conditional independence) for multiple categorical variables. This entry describes both frequentist and Bayesian statistical methods for the analysis of multiway contingency tables using loglinear models. Special attention is given to the subclass of graphical models, defined in terms of the simultaneous occurrence of one or more conditional independence relationships. The entry also includes a brief guide to computer programs for categorical data analysis.
Proportional Odds Model: Theoretical Background
Authors: McCullagh Peter
Published: September 2014

The proportional odds model is one member of the family of cumulative logistic regression models, designed for studying the effect of covariates on an ordinal responses variable. Relationships with other members of this family are described, emphasizing the effect of response aggregation. The connection with latent variable models, dispersion models, continuation‐ratio models, and log‐linear models is also discussed.
Aalen's Additive Regression Model
Authors: Aalen Odd O. , Scheike Thomas H.
Published: September 2014

In survival analysis, the well‐known Cox regression is based on modeling the hazard rate. This method assumes a proportional hazard structure. Aalen's additive model, however, assumes a linear parametric structure on the hazard rate. This approach may be justified within counting process theory, and produces martingale type estimates. These are used for plotting purposes and shows time‐varying effects of covariates on survival. The plots are easy to compute and have a well‐developed statistical theory. There exist several software packages for carrying out the analysis.
Additive Hazard Models
Authors: Preston Dale L.
Published: September 2014

A review of additive hazard models is given with special emphasis on parametric models and applications to studies of radiation effects. Some hybrid models including both additive and multiplicative hazards are discussed. Inference using Poisson regression is studied.
Additive–Multiplicative Intensity Models
Authors: Scheike Thomas H.
Published: September 2014

The two standard regression models in survival analysis are the multiplicative Cox model and the additive Aalen model. These models postulate that covariate effects are either multiplicative or additive. This contribution reviews some recent work that combines multiplicative and additive effects into flexible models.
Bayesian Methods
Authors: Dempster A. P.
Published: September 2014

Bayesian methods are used in the analysis of statistical data to provide posterior (postdata) probabilities for hypotheses. These are calculated, by Bayes' theorem, from prior (predata) probabilities and the likelihood function obtained from the observed data. Probabilities may be “objective” or “subjective”, representing degrees of belief. The history of this approach is surveyed. It fell out of fashion during the first half of the twentieth century, under the influences of the different frequentist approaches of Fisher and Neyman, but interest was maintained by the work of Jeffreys and de Finetti and revived in the mid‐1950s through the work of Savage and others. The article discusses problems arising in contemporary work, such as the involvement of many parameters, aspects of modeling, and availability of computing methods, and summarizes arguments for and against Bayesian methods.
Bivariate Normal Distribution
Authors: Morrison Donald F.
Published: September 2014

The bivariate normal distribution is defined by its density function. The regression function of one variable on the other is given by the linear conditional mean function. The conditional variance is expressed in terms of the correlation coefficient of the variables. References to tables of the bivariate normal distribution are included.
Convergence in Distribution and in Probability
Authors: Taqqu M. S.
Published: September 2014

The notions of convergence in distribution and convergence in probability are introduced, discussed and illustrated with examples.
Extreme Values
Authors: Davison A. C.
Published: September 2014

Statistics of extremes concerns the largest or smallest few observations in a sample. These are important in environmental, financial, and engineering applications, as representing rare events such as floods, high tides, or stock market crashes, but the same statistical ideas are useful in many other situations. This article provides a short review of the topic, with many further references.
Hypothesis Testing, with an Historical Perspective
Authors: Salsburg David
Published: September 2014

A significance test of a null hypothesis involves the choice of a test statistic whose null distribution is known, and a judgment as to whether, or how well, the observed value of the statistic accords with the known distribution. The article reviews the history of this approach, with emphasis on the Neyman–Pearson formulation, taking account of the behavior of the test statistic under alternative hypotheses.
Inference
Authors: Zelen M.
Published: September 2014

Statistical inference is the process of drawing conclusions from data governed by probability laws. Conclusions are uncertain, and a principal goal is to measure the uncertainty. In practical applications, the aim may be to draw conclusions about models that may have generated the data. A central concept, in almost all approaches, is the likelihood, regarded as a function of the unknown parameters. The most widely used approach is that of the frequentist school, based on error probabilities, that is, probabilities of particular observations given hypotheses about parameters. Key concepts here are significance tests, confidence intervals, and point estimation. The likelihood school uses likelihood ratios as measures of relative support. The Bayesian school uses inverse probability, that is, probabilities of hypotheses. Bayes' theorem relates posterior probabilities (postdata) to prior probabilities (predata) via the likelihood function. Details of these approaches are given in the article.
Isotonic Inference ‐ Further Developments
Authors: Hirotsu C.
Published: September 2014

Various approaches to the order restricted inference other than the isotonic regression approach are developed. They include the testing procedures, the multiple comparisons approach, and the method of the confidence intervals. They have many applications in analyzing a dose–response curve, ordered categorical responses, including two‐ and higher‐way contingency tables and so on. The distribution theories of the concerned statistics are naturally complicated for the restricted parameters as compared with the unrestricted case but in some cases very efficient exact algorithms for calculating the distribution functions are obtained.
Kolmogorov‐Smirnov Test: Basic
Authors: Wilcox R.
Published: September 2014

This is a distribution‐free method for comparing two empirical distributions, based on the largest vertical distance between the two cumulative distribution functions. The Kolmogorov test is a special case where one distribution function is known, and hence is a test of goodness‐of‐fit. Various properties, including power, are discussed.
K
ullback
L
eibler Information
Authors: Dykstra Richard
Published: September 2014

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

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

Kullback–Leibler information numbers also play a useful role in hypothesis testing problems.
Likelihood
Authors: Hills Michael
Published: September 2014

Likelihood is the probability (or probability density) of the observed data, given a probability model, regarded as a function of the unknown parameters. It needs to be defined only up to a constant of proportionality, because ratios of likelihoods are used to compare different parameter values. It is used extensively as the basis for procedures of frequentist inference, such as maximum likelihood estimation, score tests and likelihood ratio tests, and is central to Bayesian inference. Some systems of inference are based on likelihood alone.
Lognormal Distribution ‐ Basic
Authors: Ratnaparkhi M.
Published: September 2014

This is the distribution of a positive‐valued random variable, the natural logarithm of which follows a normal distribution with mean μ and variance σ2. A generalization to the three‐parameter lognormal distribution arises if a location parameter is introduced, giving a nonzero lower bound for the variable. Properties, estimation procedures, and applications are described.
Method of Moments
Authors: Lindsay Bruce G.
Published: September 2014

This is a method for obtaining point estimators of parameters, by solving a set of equations that equate a number of sample moments to their expectations. The method is consistent but inefficient. Topics considered here include the extension to multivariate data, the generalized and least squares methods of moments, distribution theory, and various specific examples.
Most Powerful Test
Authors: Peña Edsel A. , Rohatgi Vijay K.
Published: September 2014

The Neyman–Pearson approach to testing of hypotheses is described. Neyman–Pearson fundamental lemma and its ramifications are discussed.
Normal Scores
Authors: Davison A. C.
Published: September 2014

Normal scores are expected values of order statistics of a sample from the normal distribution. They are used for assessing the fit of this distribution to a sample, and for transformations to normality.
Ordered Alternatives
Authors: Lee Y. J.
Published: September 2014

Ordered alternatives occur naturally in many experimental and observational studies. The dose‐response study is a typical one. The age‐related response is another one. Tendency to lie may increase with the age. The more the alcohol that is consumed during the pregnancy the higher is the risk of congenital malformation. There are numerous other examples of this nature. The independent variable that induces ordered response could be quantitative or ordinal (ordered categorical). The response variable observed for the ordered alternatives can be continuous counts, ordinal, dichotomous (success or failure), or even multinomial. Data may be collected in blocks or per dose group. Methods appropriate for dichotomous outcomes, Poisson counts, multinomial outcomes, and nonparametric data are reviewed. A reference for nonparametric methods appropriate for blocked data with missing observations is also given but not reviewed.
Partial Likelihood
Authors: Harrington David
Published: September 2014

In a model with many parameters, it may be possible to isolate a factor in the full likelihood representing a conditional distribution depending on one parameter or a subset of the parameter set. An example arises in the Cox proportional hazards model for survival data. Many of the standard asymptotic properties of maximum likelihood estimates hold also for partial likelihood.
Power, Sensitivity and Alignment
Authors: Muller K. E.
Published: September 2014

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

It therefore depends on the specific alternative, and may lead to the selection of a test statistic that provides satisfactory power. In particular, one might choose a uniformly most powerful unbiased test (unbiasedness here indicating minimum power at the null hypothesis). Power considerations are often used in the determination of sample size.
Rank Correlation
Authors: Kraemer Helena Chmura
Published: September 2014

When observations on two variables are made on each of a number of units, the association between the variables is often measured by the Pearson product‐moment correlation coefficient. This is inappropriate if the variables are ordinal or when their distributions are far from normal. Rank correlation coefficients are based on the ranks of the observations. Spearman's coefficient is obtained by using the ranks in the usual formula for the product‐moment coefficient. Kendall's tau coefficient is expressed through comparisons between all pairs of units, noting, for each pair, whether the two variables order the units in the same or the opposite direction. Methods of inference are outlined.
Restricted Maximum Likelihood: Introduction
Authors: Cook Nancy R.
Published: September 2014

Restricted maximum likelihood estimation (REML) aims to maximize likelihood over a restricted parameter space. In a general linear model with multivariate normal error distribution, REML leads to unbiased estimators. Transformation of the data enables the log‐ likelihood to be split so that variances are estimated from error contrasts. There is a connection with Bayesian methods. The method has been applied to a random‐coefficients model for longitudinal data, and to situations where the parameters satisfy order restrictions.
Scedasticity
Authors: Morrison Donald F.
Published: September 2014

Homoscedasticity is the condition under which a series of observations have the same degree of variation. Heteroscedasticity is its negation. Homoscedasticity is assumed in the standard models for least squares linear regression, the two‐sample
t
test and the analysis of variance, and the effects of heteroscedasticity in these situations are discussed here.
Simultaneous Inference
Authors: Geller Nancy L. , O'Brien Peter C.
Published: September 2014

This covers statistical procedures to assess either more than one parameter or one parameter repeatedly during the course of a study. It includes multiple comparisons, multiple endpoints, sequential methods, and longitudinal data analysis. A common approach is to control the per‐family error rate (expected number of false positive conclusions per study) or the experimentwise error rate (probability of at least one false positive). Examples are given, with particular reference to problems of data monitoring of clinical trials.
Uniform Distribution: Definitions and Properties
Authors: Deng Lih‐yuan
Published: September 2014

A discrete uniform distribution has constant probability for each integer within a stated range. A continuous uniform distribution has constant probability density within a stated range. The continuous uniform distribution in the range (0, 1) has connections with the probability integral transformation, and with the exponential, logistic, and beta distributions. Some characterizations of functions of two independent random variables are given.
Uniform Random Numbers
Authors: Deng Lih‐Yuan
Published: September 2014

Pseudorandom numbers, uniformly distributed on the interval (0, 1), are widely used in the development of statistical methods when analytic methods become intractable. The following methods of generation are described: linear congruential, shift register, multiple recursive, matrix congruential, and combination generator. The merits of each are discussed.
Unimodality
Authors: Krieger Abba M.
Published: September 2014

This entry considers the restriction that the density is unimodal on two related problems. The first problem is to obtain refined bounds on the probability in the tail through extensions of Chebyshev's inequality. The second problem is to refine interpolation of distribution functions by obtaining tighter bounds.
U

‐Statistics
Authors: Janssen Paul
Published: September 2014

A U‐statistic, calculated from a random sample of size
n
, is an average of a symmetric function of
m
sample values, calculated for all
m
‐tuples in the sample. Examples are the sample variance, the Cramér‐von Mises statistic of goodness of fit, and the Kaplan–Meier and Nelson–Aalen estimators in survival analysis. Asymptotic properties are described.
W
ald's Identity
Authors: Lee Mei‐Ling Ting
Published: September 2014

This identity, due to A. Wald, arises in the study of boundary problems in a random walk. It provides expressions for the mean and variance of the stopping time in sequential analysis.
Youden Squares and Row–Column Designs
Authors: Gupta Sudhir
Published: September 2014

Often several other factors potentially contribute to the variability of the observed response in addition to the treatments or factors under investigation. For example, gender, age, and center may also affect the blood pressure of patients in a placebo‐controlled hypertensive clinical trial. The technique of blocking is used to control the experimental error variation due to these types of extraneous sources. Two sources can be controlled using a Youden square or a row–column design, the rows and columns comprising two different systems of blocking. This article provides the details for the least squares analysis of these designs. Designs having variance balance, supplemented balance, row‐orthogonality, adjusted‐orthogonality, designs for factorial experiments, and computer algorithms for constructing designs are also briefly discussed.
A
alen–
J
ohansen Estimator
Authors: Borgan Ørnulf
Published: September 2014

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.
Compliance and Survival Analysis
Authors: Goetghebeur Els
Published: September 2014

Partial compliance with assigned treatment can act as both cause and effect of survival experiences in randomized clinical trials. Structural failure time models have been developed for causal inference on the effect of resulting exposure patterns on survival time. In this setting, we review randomization based estimation methods that protect the standard α level. We also point to alternative methods relying on the assumption of no unmeasured time‐dependent confounders.
Event History Analysis
Authors: Andersen Per Kragh , Keiding Niels
Published: September 2014

An introduction to event history analysis via multistate models is given. Examples include the two‐state model for survival analysis, the competing risks and illness–death models, and models for bone marrow transplantation. Statistical model specification via transition intensities and likelihood inference is introduced. Consequences of observational patterns are discussed.
Ghosts
Authors: Keiding Niels
Published: September 2014

Turnbull coined the termghoststo intuitively explain inverse‐probability‐weighted estimation survival analysis, similar to Horvitz–Thompson estimation.
Hazard Models Based on First‐Passage Time Distributions
Authors: Aalen Odd O.
Published: September 2014

When considering the occurrence of events in survival analysis, it is natural to think of them as being due to some underlying process crossing a limit. Myocardial infarction, for instance, occurs when occlusion of blood vessels in the heart reaches a certain level. The idea of viewing the event as due to the passage of a process beyond a limit may be formalized within standard models like Markov chains or Wiener processes. Although these models are highly idealized, they may still yield useful insights. The concept of quasi‐stationarity is of use in this context.
Interval Censoring: Overview
Authors: Sun Jianguo
Published: September 2014

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.
Kaplan–Meier Estimator
Authors: Borgan Ørnulf
Published: September 2014

It is described how the Kaplan–Meier estimator can be used to provide a nonparametric estimate of the survival distribution from right‐ censored data. Problems with estimating the survival distribution with left‐truncated data are considered, and it is pointed out that for such data, it may be more useful to estimate conditional survival distributions. Estimation of the median and mean survival times is discussed, and it is described how the Kaplan–Meier estimator may be derived by the redistribute‐to‐the‐right algorithm and how it is the unique self‐consistent estimator for right‐censored survival data. It is shown how the Kaplan–Meier estimator can be given as the product‐integral of the Nelson–Aalen estimator for the cumulative hazard rate, and it is indicated how the product‐integral formulation is helpful in the study the of statistical properties of the estimator. Finally, the construction of confidence bands for the survival distribution is considered.
Parametric Models in Survival Analysis
Authors: Lawless J. F.
Published: September 2014

Survival analysis deals with the analysis of times to events, or lifetimes. Parametric models are used to represent the distributions of lifetimes, and their relationship to explanatory variables, or covariates. This entry reviews common parametric models such as the exponential, Weibull, and lognormal distributions, and associated methods of inference. Parametric regression analysis using linear models for log lifetimes (accelerated failure time models) is described and illustrated. Parametric proportional hazards models and models for competing risks or multiple lifetimes are discussed briefly.
Phase‐Type Distributions in Survival Analysis
Authors: Aalen Odd O.
Published: September 2014

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.
Product‐integration
Authors: Gill Richard D.
Published: September 2014

Product integration is a mathematical theory doing for products, what integration does for sums. It lies behind the relationship between hazard rates and densities, unifying both the discrete time and the continuous time cases. It comes into its own when studying Markov processes and event history analysis, since when multiplying matrices, the time order of the matrices makes a difference to the result. The theory becomes a powerful tool, which takes full account of these potential problems.
Quality of Life and Survival Analysis
Authors: Olschewski Manfred
Published: September 2014

In clinical research, where time to a specific endpoint like, for example, death is of interest, methods of survival analysis are usually applied. In recent years, such research is more and more accompanied by the inclusion of the patients subjective assessment of treatment circumscribed by the term
health‐related quality of life
(HQoL). In the analysis of HQoL, methods of longitudinal analysis of variance are adequate. In this chapter, we put emphasis on statistical analysis methods that allow a joint analysis of both quantity and quality of survival. The major approach here is defined by quality‐adjusted survival (QAS) times. Conceptually, this means that the survival time of a patient is weighted according to his/her subjective assessment of that time. Advantages and limitations of such a joint analysis are discussed in relation to the separate analyses of the components.
Rank Regression
Authors: Cuzick Jack
Published: September 2014

In rank regression, the ranks of the dependent variable are regressed on a set of covariates. A formal specification iswheregis any strictly increasing transformation of the dependent variablet,zis a vector of covariates,βare the corresponding regression coefficients and ϵ is the error term, usually taken to have mean zero. If ϵ has the extreme‐value distributionthe well‐known proportional hazards model results. Other choices of ϵ lead to other important models, including frailty models.

A different use of the term “rank regression” refers to models based on ranks of residuals in censored regression models, an early example being the Buckley–James approach.
Staggered Entry
Authors: Slud E. V.
Published: September 2014

Clinical trials and epidemiologic studies, which measure duration until some endpoint of interest, often follow subjects who may arrive at random or “staggered” times and may remain under observation until occurrence either of the endpoint or censoring/withdrawal. An important theme of statistical methodological development over the past three decades has been to understand how to handle such data. Building on earlier ideas (Lexis Diagrams, life tables), the treatment of staggered data has required sophisticated mathematical treatment (counting processes, martingales, repeated significance tests), but has practical impact in the interim analysis and early stopping of survival experiments.
Structural Nested Failure Time Models
Authors: Robins James M.
Published: September 2014

Structural nested failure time models are causal models for the effect of a time‐dependent treatment or exposure on a survival time outcome in the presence of time‐dependent confounders. A time‐dependent confounder is a repeatedly measured covariate, which acts as a confounder for future exposure measurements but is on the causal path between earlier exposure measurements and ultimate response (i.e. here it acts as intermediate variable). Standard epidemiologic methodology fails if time‐dependent confounders are present.

Under the essential condition ofno unmeasured confounders(the mathematical definition of which is a central issue) these models allow for unbiased causal inference, generalizing Robins'sg‐computation algorithm. The models use as building blocks time‐dependent accelerated failure time models.

Important applications are intricate endogenous (feedback) selection effects in occupational epidemiology (healthy worker effect) and clinical epidemiology (AIDS treatment).
Software for Survival Analysis
Authors: Klein John P. , Zhang Mei‐Jie
Published: September 2014

Statistical software for survival analysis is discussed. We examine the abilities of five major software packages, BMDP, SAS, SPSS, S‐PLUS, and STATA to perform basic calculations for censored data. Packages are compared based on their performance in computing summary statistics, in hypothesis testing, in performing Cox semiparametric regression modeling, and in performing parametric regression modeling.
Turnbull Estimator
Authors: Frydman H.
Published: September 2014

The Turnbull estimator is a nonparametric maximum likelihood estimator of a one‐dimensional distribution based onNindependent, arbitrarily censored and/or truncated observations (including interval censoring). Turnbull's algorithm is based on Efron's idea of self‐consistency. Asymptotic properties of the estimator depend essentially on the particular patterns of censoring and truncation.
Bayesian Model Selection in Survival Analysis
Authors: Ibrahim Joseph G. , Chen Ming‐Hui
Published: September 2014

Bayesian model selection in survival analysis has become common practice with recent advances in computational technology and software. In this article, we review some Bayesian model selection techniques for the analysis of survival data.
Data Archives
Authors: Smith George Davey
Published: September 2014

Data archives that allow secondary analysis are valuable in allowing replication of analyses, individual data meta‐analyses, and possibilities to detect fraud.
Data Quality in Vital and Health Statistics
Authors: Swerdlow A. J.
Published: September 2014

In this chapter, the importance of data quality in the design and interpretation of analyses of routinely collected vital and health data is discussed. Issues considered include: the quality of information in underlying data sources; completeness of data collection; duplication; late registrations and alterations; validity and precision of data collection; coding, bridge‐coding, and assignment of underlying cause; data processing and editing; quality of denominators; and numerator/denominator discrepancies. Separate consideration is given to particular issues for analysis of clusters and other rare events, and record linkage. While not intrinsically interesting, data quality is worth considerable attention if analyses of routinely collected data are to be correctly interpreted.
Gestational Age
Authors: Macfarlane Alison
Published: September 2014

The World Health Organization defines the duration of a pregnancy as the time since the first day of the last normal menstrual period. This can be problematic if women conceive soon after stopping oral contraception or have irregular periods; so ultrasound estimates are increasingly used in countries where it is in common use. The World Health Organization also subdivides gestational ages into preterm, term and postterm.
Social Classifications
Authors: Kunst Anton E.
Published: September 2014

This article provides recommendations on the measurement of the three key indicators of socioeconomic status: educational level, income level, and occupational class.
Ageing Models
Authors: Müller Hans‐Georg
Published: September 2014

Models for aging have come under intensive scrutiny in recent years due to increasing interest in aging, not least due to the ever‐increasing human lifespan. The methods used to model aging are largely similar to those in survival analysis, with unique features such as the use of lifetables and less prominence of incomplete data. An increasingly prominent role is played by event history analysis. Nonparametric, frailty, event history, and extreme value approaches are increasingly prominent due to the uncertainties regarding the shape of population hazard functions, which display remarkable plasticity across sexes, species, and environmental conditions. These newer approaches complement classical likelihood methods associated with the Gompertz and Gompertz–Makeham parametric models and are therefore the methods of choice in the emerging field of biodemography. Recent problems where aging models are relevant concern the influence of genetic variants and signaling pathways on mortality, the modeling of plateaus in the right tail of hazard functions, and the complex interplay between reproduction, diet, and mortality.
Bayesian Methods for Categorical Data
Authors: Gutiérrez‐Peña Eduardo
Published: September 2014

The Bayesian approach to statistical inference is based on a subjective interpretation of probability. Given a prior distribution describing our prior beliefs concerning the value of an unknown quantity, Bayes' theorem allows us to update those beliefs in the light of observed data. The resulting posterior distribution then summarizes all the available information about the quantity of interest, conditional on the posited model. Bayesian methods for the analysis of categorical data use the same classes of models as the classical approach. However, Bayesian analyses can be more informative – the posterior distribution of the parameter contains more information than mere point estimates or test statistics – and may provide more natural solutions in certain situations such as those involving sparse or missing data, or unidentifiable parameters.
Number of Matches and Magnitude of Correlation
Authors: Rovine Michael J.
Published: September 2014

A definition of the correlation as the proportion of matches between two continuous variables is described. This definition is compared to other coefficients; in particular, coefficients that are used to describe the association in a 2 × 2 contingency. These include Yule's Q, Peirce's coefficient, the φ‐coefficient, and the Binomial Effect Size Display (BESD).
Power Analysis for Categorical Methods
Authors: Erdfelder Edgar , Faul Franz , Buchner Axel
Published: September 2014

This article shows how to assess and control the power of chi‐square tests for categorical data. It discusses both the case of simple null hypotheses (i.e., the category or cell probabilities are completely specified
a priori
) and the case of composite null hypotheses for parameterized multinomial models requiring parameter estimation (e.g., log‐linear models). Power calculations are illustrated for chi‐square tests of association and chi‐square goodness‐of‐fit tests using standard examples from behavioral research.
Tetrachoric Correlation
Authors: Hershberger Scott L.
Published: September 2014

The tetrachoric correlation is used to estimate the Pearson product‐moment correlation between two continuous, bivariate‐normally distributed variables from dichotomized versions of those variables. Two common areas of application of the tetrachoric correlation are to the behavior genetic analysis of twin data and to the factor analysis of item data from cognitive ability tests.
Computational Models
Authors: Ployhart Robert E. , Harold Crystal M.
Published: September 2014

Computational models provide a powerful methodology useful for answering many statistical and substantive questions. We address the topic from two perspectives. The first considers computational modeling as a methodology for answering substantive questions. The second considers statistical ‘Monte Carlo simulations’ as a special case of computational models.
Jackknife
Authors: Rodgers Joseph Lee
Published: September 2014

The jackknife is a resampling method that can be used to do hypothesis testing, to estimate standard errors, to define confidence intervals, and to estimate bias. This entry treats the history of the development of the jackknife, organizes the jackknife procedure in relation to other resampling procedures, and gives several examples.
Randomization Based Tests
Authors: Ludbrook John
Published: September 2014

I argue that when an experiment in behavioral psychology is designed, it almost always involves taking a nonrandom sample and, by a process of randomization, dividing it into two or more ‘treatment’ groups. This places such experiments within the randomization model of inference, not the better‐known population model. The appropriate tests of significance under the randomization model are permutation (randomization) tests.
Confounding Variable
Authors: Onghena Patrick , Noortgate Wim
Published: September 2014

A confounding variable is a variable that is related to an explanatory variable and at the same time has an effect on a response variable that is entangled with a possible effect of the explanatory variable, possibly leading to wrong conclusions about the latter. Confounding may also occur in situations involving more than one explanatory or response variable, and more than one confounding variable may be present. Observational studies, quasi‐experiments, as well as randomized experiments might be affected by confounding. If confounding variables cannot be avoided or ruled out by design, a final option is to model them as covariates.
Multiple Baseline Designs
Authors: Ferron John , Scott Heather
Published: September 2014

Multiple‐baseline designs are a type of single‐case design used to study treatment effects across multiple participants, multiple behaviors, or multiple settings. Baselines are established by repeated observation. Interventions are then implemented at different times for the different participants, behaviors, or settings. Effects are demonstrated when changes are observed that coincide with the interventions.
Internal Consistency
Authors: Schmitt Neal
Published: September 2014

The internal consistency of a measure refers to the degree to which items in the measure assess the same underlying construct. Coefficient alpha is widely used as an index of internal consistency though it has limitations in that regard. Limitations and other possibilities are discussed.
Binomial Distribution: Estimating and Testing Parameters
Authors: Berger Vance W. , Zhou YanYan
Published: September 2014

We consider categorical variables with only two possible values. Such data are referred to as binary data, Bernoulli variables, dichotomous data, and sometimes as 0/1 data. In contrast with continuous data, which are generally summarized with sums and averages, binary data are generally summarized with counts and proportions. The binomial distribution is generally used to model the proportion of binary trials that result in a given outcome. When testing a binomial population to determine if the success probability appears to be a given specified value, the binomial test is used.
Sampling Distributions
Authors: Landau Sabine
Published: September 2014

A sampling distribution is the probability distribution of a statistic that is based on a random sample from some population. The sampling distribution is a key concept in statistics because it quantifies the precision of estimators in parameter estimation and is the distribution from which
P
values are derived in statistical tests.
Finite Mixture Distributions
Authors: Everitt Brian S.
Published: September 2014

Finite mixture distribution are a particular class of probability distributions that are particularly useful for modeling data thought to contain relatively distinct subgroups of clusters of observations. They can be used as the basis of a cluster analysis of the data via estimation of both, the number of component distributions in the mixture and the parameters of these distributions.
Average Deviation
Authors: Howell David C.
Published: September 2014

The average deviation is a measure of scale based on the average of the absolute deviations from the center of the distribution. For a normal distribution, the average deviation is somewhat less efficient than the standard deviation as a measure of scale, but this advantage quickly reverses for distributions with heavier tails.
Replicability of Results
Authors: Thompson Bruce
Published: September 2014

Because result replicability is important to researchers, and statistical significance does not evaluate replicability, both external and internal replicability analyses are important. Some researchers find true external replication studies impractical, and therefore turn to ‘internal’ methods. Three internal replicability analyses have been proposed: cross‐validation, the jackknife, and the bootstrap.
Hodges‐Lehman Estimator: Basic
Authors: Lunneborg Clifford E.
Published: September 2014

The Hodges–Lehmann one‐sample procedure provides an efficient alternative to the sample median as an estimator of the location of a population, while the two‐sample procedure estimates the location shift parameter linking two populations.
Asymptotic Relative Efficiency: Basics
Authors: Lunneborg Clifford E.
Published: September 2014

Two competing test statistics may differ in their sensitivity to deviations from the null hypothesis. A standard comparison is the asymptotic relative efficiency (ARE) – the limit, as sample sizes grow, of the ratio of the sample sizes needed for the two tests to attain the same power.
Laws of Large Numbers: Applications
Authors: Kay Jim W.
Published: September 2014

The laws of large numbers concern when the average of
n
random variables converges to a fixed value as
n
tends to infinity. The
n
random variables could arise in various ways. They could, for example, be mutually independent and identically distributed and represent a simple random sample of observations taken from some fixed population, or they could represent a sequence of
n
realizations of a stationary stochastic process. The laws are defined under the assumptions that the random variables are mutually independent and follow the same probability distribution; generalizations obtained under more general conditions are mentioned. Illustrations are provided, which demonstrate the usefulness of the laws in providing theoretical justification for: the frequentist concept of probability; the strong consistency of the sample mean and variance estimators; the consistent estimation of the
P
value associated with a Monte Carlo test.
Heteroscedasticity and Complex Variation
Authors: Goldstein Harvey
Published: September 2014

In the standard generalized linear model, the residual variance is assumed to be a constant or a well‐defined function of the linear predictor. In many applications, however, this assumption may not hold and interest will often focus on modeling the residual variation as a function of further explanatory variables, such as gender or age. The article shows how to formulate such models and provides extensions to the multilevel case and to the modeling of a full covariance matrix.
Index Plots
Authors: Lovie Pat
Published: September 2014

An index plot is a scatterplot of values on some measure plotted serially against the associated observation (case) number within the sample. It can help in highlighting extreme values in a sample and in indicating trends when the data are, for instance, in spatial or time order.
Star and Profile Plots
Authors: Everitt Brian S.
Published: September 2014

Star and profile plots are graphical displays for multivariate data. They consist of an array of small individual plots, each corresponding to a single multivariate observation. In any individual plot, every variable is represented by a line whose length is proportional to the value of the variable. In a star plot, the lines form spokes radiating from a central point, while in a profile plot, they are vertical spikes. In each case, the ends of the lines are connected across variables.

These displays can provide useful insight into the structure of the data, especially when groups of plots with similar shapes, or particular individuals that seem different from the rest, can be identified.
Symmetry Plot
Authors: Lovie Sandy
Published: September 2014

Like EQQ plots, symmetry plots are another instance of added value scatterplots, which, in this case, allows the analyst to perform a graphical test as to the equality of the shape of a sample about a central value.
Pie Chart
Authors: Wright Daniel B. , Williams Siân E.
Published: September 2014

Pie charts are used for displaying frequency or categorical data. The slices of a pie chart represent the proportion of values in a sample falling within given categories.
History of Analysis of Variance
Authors: Tweney Ryan D.
Published: September 2014

First used by behavioral scientists in the 1930s, use of Analysis of Variance (ANOVA) grew quickly after World War II, and exactly paralleled incorporation of statistical significance testing in general. Detailed consideration of this history suggests several reasons for the pattern of slow growth, followed by rapid institutionalization. In particular, ANOVA stood duty as a warrant of scientific legitimacy among behavioral scientists, a fact that may also be relevant to understanding recent critiques by psychologists of its overuse and misuse.
History of Psychometrics
Authors: Buchanan Roderick D. , Finch Susan J.
Published: September 2014

Psychometrics developed as a means for measuring psychological abilities and attributes, usually via a standardized psychological test. Its emergence as a specialized area of psychology combined entrenched social policy assumptions, existing educational practices, contemporary evolutionary thinking, and novel statistical techniques. Psychometrics came to offer a pragmatic, scientific approach that both fulfilled and encouraged the need to rank, classify, and select people.
History of Intelligence Measurement
Authors: Weidman Nadine
Published: September 2014

The history of intelligence measurement can be roughly divided into three chronological periods: the first (1870–1920s), a period of initial interest in defining intelligence and the establishment and use of intelligence tests; the second (1930s–1960s), a period of repudiation of the hereditarian assumptions underlying the tests; and the third (1969 to present), a period of resurgence of belief in the connection between intelligence and heredity.
Development of Statistical Theory in the 20th Century
Authors: Lee Peter M.
Published: September 2014

This article covers the three most important approaches to statistical inference in the twentieth century: Fisher's fiducial inference, Neyman and Egon Pearson's null and alternative hypothesis testing, and the application and extension of Bayes theorem by Savage and de Finetti.