bluuebunny/crossref_metadata_2025 huggingface dataset:
Prior Distribution
Authors: Kass R. E.
Published: September 2014

In Bayesian inference, the unknown parameter is given a prior distribution, and Bayes' theorem combines this with the likelihood from the observed data, to give the posterior distribution. The prior may be informative, subjective, or have a standard, reference form (usually representing vague knowledge). In hierarchical models, the prior distribution may have its own parameters that themselves have priors.
Quasi‐likelihood
Authors: Zeger Scott L. , Liang Kung‐yee
Published: September 2014

Quasi‐likelihood is a method of estimation for regression analysis which requires the specification of only the first two moments of the response distribution, and the term refers also to the objective function analogous to the likelihood. In generalized linear models, the iterative weighted least squares algorithm for solving the score equations depends only on the mean and variance of the response. The solution of the estimating equations is consistent even when the variance expression is incorrect. Various extensions are described.
R
adon–
N
ikodym Theorem: Definitions and Applications
Authors: Alexander Kenneth S.
Published: September 2014

The Radon–Nikodym Theorem, in the context of probability measures, states that if all events having probability 0 under one probability measure also have probability 0 under a second measure, then the second measure is obtained by integrating some density function with respect to the first measure. Applications in probability and statistics arise in the contexts of likelihood functions, conditional expectation, Girsanov's formula, spectral measures and local time. The theorem and these applications are discussed.
Robustness
Authors: Prescott Philip
Published: September 2014

A short history of the development of ideas of robustness is provided, from Gauss' introduction of the normal distribution in 1821 to the detailed examination of the conflicting criteria for robustness as discussed by Huber, Hampel, Tukey, and others, which led to consideration of validity robust and efficiency robust procedures. Robust estimators of location using M‐estimators (based on maximum likelihood), L‐estimators (based on order statistics), and R‐estimators (based on ranking methods) are described and examples given. These methods produce a wide range of robust estimators including trimmed means, Winsorized means, and the median of all pairwise averages. Adaptive and resistant procedures are discussed briefly.
Shrinkage Estimation
Authors: Casella George
Published: September 2014

Shrinkage estimation developed from the work of Stein for the estimation of the mean of a multivariate normal distribution. The observed vector is, from a decision theory viewpoint, inadmissible. The James–Stein estimator shrinks the observations toward zero. Later work involves estimators shrunk toward the sample mean. Shrinkage estimators are, however, not dominant in problems with a finite sample space, such as the binomial distribution. There are connections with Bayesian and empirical Bayes solutions.
Skewness
Authors: Hall Peter
Published: September 2014

Several definitions of skewness are given, and their main features discussed. Connections to relationships among the mean, median, and mode, and to work of Haldane, Pearson, and van Zwet, are noted.
Studentized Range including Neuman‐Keuls and Tukey's T Methods
Authors: Hochberg Yosef
Published: September 2014

In a comparison of the means of several samples of size
n
, the studentized range is the range of the sample means divided by the usual standard error of a single mean. It may be used in successive stages of a multiple comparison procedure. Two approaches are the Newman–Keuls method and Tukey's
T
‐method. In a single sample of size
n
, the internally studentized range is the ratio of the sample range to the standard deviation, and may be used in a test of normality. Various applications are discussed.
Sufficient Statistic: Theoretical Background
Authors: Strawderman William E.
Published: September 2014

A sufficient statistic is such that the conditional distribution of the data given the sufficient statistic does not depend on any unknown parameters. In this sense, it contains all the information in the original data. A necessary and sufficient condition is the factorization theorem. A minimal sufficient statistic is a function of every other sufficient statistic, and is conveniently found for distributions in the exponential family. Topics described here include complete sufficient statistics, minimum variance estimation, uniformly most powerful unbiased tests, Bayesian methods, and invariance.
Tolerance Interval
Authors: Vangel Mark G.
Published: September 2014

Tolerance intervals cover at least a specified proportion of a population, either on average or else with a stated level of confidence. The theory is described in relation to simple random samples and also linear regression. Topics discussed include random effects tolerance limits, simultaneous tolerance intervals, and applications to calibration. Methods are illustrated by an example on serum glucose measurements.
Tolerance Region: Definition
Authors: Vangel Mark G.
Published: September 2014

In a sample from a multivariate distribution, a tolerance region is a region for which there is a specified high probability (theconfidence) that at least a specified proportion of the distribution (thecoverage) is included. It is thus a generalization of the univariate tolerance interval. The construction using statistically equivalent blocks is described briefly.
Unbiasedness
Authors: Link W. A.
Published: September 2014

Unbiasedness is defined and illustrated as a criterion in estimation and hypothesis testing. Limitations of the criterion are illustrated by examples: that unbiasedness is not transformation invariant, that unbiased estimators need not exist, nor be unique, and that even when a unique unbiased estimator exists, it might be clearly unreasonable.
U
‐Shaped Distribution
Authors: Everitt Brian S.
Published: September 2014

This is a probability distribution or frequency distribution shaped roughly like the letter U, with highest frequency at the two extremes, although not necessarily symmetrically. The variables are sometimes called
bounded scores
.
Utility
Authors: Parmigiani Giovanni
Published: September 2014

Utility measures the desirability of an outcome of a decision. A standard procedure in decision theory is to choose the action that maximizes the expected utility. Savage used utility theory as a basis for Bayesian statistical inference.
Saddlepoint Approximation: Introduction
Authors: Kolassa John E.
Published: September 2014

Saddlepoint approximations are asymptotic approximations to the density or tail‐area probability of a statistic. The basic result, using the cumulant generating function for a univariate random variable, may be generalized to apply to random vectors and developed to deal with tail probabilities, conditional distributions, and discrete random variables.
Accelerated Failure‐time Models
Authors: James Ian
Published: September 2014

The accelerated failure‐time model assumes a survival function of the formS(t) = S0(θt), whereS0is an underlying survival function and θ may depend on a number of covariates. This is equivalent to a location‐shift model for the log failure time, and in particular to a loglinear regression model when θ is loglinear. Parametric and semiparametric approaches to analyses based on these models with possibly censored failure‐time responses are reviewed. The accelerated failure‐time assumption is compared and contrasted with that of proportional hazards.
Bayesian Survival Analysis
Authors: Ibrahim Joseph G. , Chen Ming‐Hui , Sinha Debajyoti
Published: September 2014

Great strides in the analysis of survival data using Bayesian methods have been made in the past ten years due to advances in Bayesian computation and the feasibility of such methods. In this chapter, we review Bayesian advances in survival analysis and discuss the various semiparametric modeling techniques that are now commonly used. We review parametric and semiparametric approaches to Bayesian survival analysis, with a focus on proportional hazards models. Reference to other types of models are also given.
Bootstrapping in Survival Analysis
Authors: Veraverbeke Noël
Published: September 2014

We survey the main developments in bootstrapping survival data since the paper of Efron [17] on resampling procedures for censored data. We consider bootstrap versions of the product‐limit estimator and also deal with resampling in semi‐ and nonparametric regression models.
Counting Process Methods in Survival Analysis
Authors: Chang I‐Shou , Hsiung Chao Agnes
Published: September 2014

A brief review of the counting process approach to generalized survival analysis is given. Examples are provided to illustrate that counting processes together with their filtration‐specific intensities are excellent modeling tools for survival analysis, where data are often incompletely observed; that the associated martingale theories are useful in proposing inference procedures, analyzing their distributional properties, and studying their efficiency; that the representation of the likelihood process in terms of its intensity offers extra convenience in the data analysis. Several examples are also given to indicate that counting process may provide good statistical model but offer limited help in the analysis.
Cure Models
Authors: Farewell Vern T.
Published: September 2014

Cure models are used for time to event data when a subset of subjects who will not experience the event of interest may exist. A formulation of such models is presented and special cases in the literature are highlighted. Estimation of such models is discussed. The issues of identifiability of parameters and nonstandard hypothesis testing are highlighted.
Delayed Entry
Authors: Keiding Niels
Published: September 2014

Methods for analyzing survival data with delayed entry are surveyed. Particular emphasis is put on the definition of random left truncation, the counting process approach and filtering, delayed entry and covariates, and independent and dependent delayed entry. As specific examples are mentioned the prevalent cohort study, confirmatory analysis of a chance finding at interim analysis, and left censoring, right truncation, time‐reversal, and the retro‐hazard.
Discrete Survival‐Time Models
Authors: Fahrmeir Ludwig
Published: September 2014

Survival times are rarely intrinsically discrete (an important counterexample being time to pregnancy, which is naturally measured in menstrual cycles). If times are recorded in intervals, grouped survival times methods may be relevant, but it is alternatively sometimes preferable to consider time as discrete. This leads to various forms of generalized linear models, notably repeated logistic regressions or discrete proportional hazards (complementary log–log link functions). Smoothing methods are considered.

Generalizations to competing risks and to frailty models have also been studied.
Duration Dependence
Authors: Oakes David
Published: September 2014

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.
Excess Mortality
Authors: Væth Michael
Published: September 2014

The term excess mortality is used to describe the additional hazard rate in a study population relative to that seen in a reference population with the same sex and age distributions. A formal model is introduced and a nonparametric estimate of the integrated excess mortality rate is given. As a by‐product, an estimate of the expected survival function is also obtained. For the special case with a constant excess mortality rate, simple explicit moment estimate is also available. The methodology can be generalized to both parametric and nonparametric regression models.
Expected Number of Deaths
Authors: Væth Michael
Published: September 2014

In epidemiological studies, the mortality in a study population is often related to that of a reference population by computing the number of deaths that is expected if the mortality rates of the reference population are applied to the follow‐up experience of the study population. This method is based on a multiplicative model for the mortality rate and is closely related to standardization techniques. Two approaches to the calculation of the expected number of deaths, the prospective model, and the person‐years method are described. In the simplest form the latter method is essentially an indirect standardization, but the model is easily generalized to include regression on covariables.
Expected Survival Curve
Authors: Keiding Niels
Published: September 2014

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.
Explained Variation Measures in Survival Analysis
Authors: Graf Erika
Published: September 2014

In linear regression, the proportion of explained variation,
R
2
, is a familiar measure used to quantify the predictive value of a model including covariates compared to the null model of no covariate effect. Similar measures have been proposed for survival analysis where the situation is typically complicated by the presence of censored observations. In addition, squared error may not be felt to be an adequate loss function in this context. We introduce the concept of explained variation in the survival setting and illustrate it by selected measures.
Exponential Distribution as a Survival Model
Authors: Klein John P. , Andersen Per Kragh , Keiding Niels
Published: September 2014

Characterizations of the exponential distribution and the lack of aging properties of this distribution are discussed. Maximum likelihood estimation for the exponential distribution is presented. The two parameter exponential distribution is discussed.
Grenander Estimators
Authors: Keiding Niels
Published: September 2014

Grenander's nonparametric maximum likelihood estimators of a decreasing density and an increasing hazard are reviewed with particular focus on current status data.
Historical Controls in Survival Analysis
Authors: Keiding Niels
Published: September 2014

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”.
Inverse Probability Weighting in Survival Analysis
Authors: Rotnitzky Andrea , Robins James M.
Published: September 2014

Survival studies usually incorporate high‐dimensional covariate data, but interest may focus on low‐dimensional characteristics of the survival distribution. Inverse probability weighted augmented estimation (AIPW) provides a robust approach, largely insensitive to model misspecification. The method was originally introduced to deal with coarsened (incompletely observed) data, and is discussed here with and without the assumption of random coarsening.
Length Bias
Authors: Wang Mei‐Cheng
Published: September 2014

Length bias occurs when the sampling scheme implies that the probability of observing a positive‐valued random variable is proportional to the value of the valuable. Length bias is well documented in renewal theory, but is also important in epidemiology. Examples include prevalent cohort studies (where left truncation is an important remedy), other sampling schemes originating from cross‐sectional samples, and screening for chronical diseases.

Comments are given on necessary modification of methods from nonparametric survival analysis.
Life Table: Overview
Authors: Hoem Jan M.
Published: September 2014

A life table is a traditional representation of estimates related to grouped survival times, sometimes called the “actuarial” method. Besides the survivor function, it usually contains estimates of residual life expectancy. There are also so‐called multiple‐decrement tables relevant for competing risks.

The history of the life table is long and well documented. Many special terms have been attached to the life table during its service in demography and actuarial mathematics.
Marker Processes
Authors: Kalbfleisch John D.
Published: September 2014

In longitudinal studies in survival analysis, repeated measures are often taken of more or less specific health measures. Suchmarker processesmay serve as (a) time‐dependent covariates for improving estimation of a survival distribution, (b) surrogate for survival in a comparative trial, (c) providing information on time of onset of disease, when this is not directly observable.
Median Survival Time
Authors: Brookmeyer Ron
Published: September 2014

The median is an important summary measure for survival data. A confidence interval for the median estimated from censored survival data is easily obtained from the pointwise confidence interval around the Kaplan–Meier estimator. A test for equality ofkmedians is also given.
N
elson–
A
alen Estimator
Authors: Borgan Ørnulf
Published: September 2014

It is described how the Nelson–Aalen estimator can be used to provide a nonparametric estimate of the cumulative hazard rate function based on right censored and/or left truncated survival data, and how the estimator can be used to estimate the integrated transition intensities in Markovian multistate models. It is also pointed out how the Nelson–Aalen estimator can be adopted to estimate other quantities, like the integrated relative mortality in a multiplicative model with external rates and the cumulative infection rate in a simple epidemic model. It is shown that counting processes provide a framework that allows for a unified treatment of all these diverse situations, and the most important small and large sample properties of the estimator are summarized. Finally, the construction of confidence bands is considered.
Risk Set
Authors: Andersen Per Kragh
Published: September 2014

The notion of the risk set in survival analysis is introduced and its role in non‐ and semiparametric inference from censored survival data is explained. A generalized risk set concept for multistate models is also considered.
Survival Analysis, Overview
Authors: Andersen Per Kragh , Keiding Niels
Published: September 2014

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: September 2014

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 to Pregnancy
Authors: Keiding Niels
Published: September 2014

Time to pregnancy, the duration that a couple waits from initiating attempts to conceive until conception occurs, is regarded as one of the direct measures of natural fecundity. Statistical tools for designing and analyzing time‐to‐pregnancy studies belong to the general area of survival analysis, but several special features have developed.
Time‐Dependent Covariate
Authors: Andersen Per Kragh
Published: September 2014

Models for survival data are often specified from the hazard function thereby allowing regression models to include covariates that change with time. This article reviews different types of time‐dependent covariates in survival analysis. Potential problems in connection with estimation of the survival function in models with time‐dependent covariates are discussed. Estimation in Cox's multiplicative regression model and other regression models with time‐dependent covariates, including additive models, are reviewed.
Total Time on Test
Authors: Andersen Per Kragh , Keiding Niels
Published: September 2014

In survival analysis, the exponential distribution plays an important role as a null hypothesis. One classical technique from reliability analysis for testing this hypothesis is based on the total time on test concept. It is explained, in the framework of counting processes, how a number of graphical and numerical test procedures may be based on this concept. These procedures are illustrated using data from a clinical trial in liver cirrhosis.
Type‐Specific Covariates in Survival Analysis
Authors: Andersen Per Kragh
Published: September 2014

In regression analysis of event history data using multistate models it is sometimes useful to consider models where some covariates have the same effect on several transition intensities. It is demonstrated how this may be achieved using type‐specific covariates.
Birthweight
Authors: Macfarlane Alison
Published: September 2014

The World Health Organization defines birthweight as the first weight of the fetus or baby after birth. It also defines low birthweight, very low birthweight and extremely low birthweight. The limited definition of “prematurity”, based on birthweight, was abandoned in the 1970s and replaced by a definition of preterm birth based on gestational age.
Case Fatality
Authors: Hakulinen T.
Published: September 2014

Case fatality indicates how serious a disease condition is in causing death to patients, usually within a defined period of time. Conceptually, case fatality may be seen as a complement to survival, and thus the methods of survival analysis can be employed in assessing case fatality.
Conception
Authors: Macfarlane Alison
Published: September 2014

Conception rates are difficult to compile because of the difficulty in collecting data about miscarriage and these difficulties may be compounded if gestational age is not recorded at birth registration. In England and Wales, estimated conception rates are derived from live and stillbirth registrations combined with notifications of legal abortions.
Demography
Authors: Timæus Ian M.
Published: September 2014

Demography is the study of human populations with respect to their size and structure and of the components of population change, in particular, fertility, mortality, and migration. This article outlines the more important sources of data, substantive issues, concepts, methods, and models in demography. The discipline's central concern is to measure, explain, and model the dynamics of human populations. Thus, one crucial application of demography is the forecasting of population change by means of cohort‐component projections. Classical demographic measures and techniques of analysis were developed largely for the study of vital statistics with census‐based denominators. They include the calculation of age‐specific rates, standardization, and life‐table analysis. More recently, the growing importance of individual‐level analysis of survey data has led to convergence between demographic methods and other forms of statistical analysis.
Geographic Patterns of Disease
Authors: Elliott Paul , Best Nicola
Published: September 2014

The study of geographic variations in disease is a classic epidemiological technique in which location is used as a surrogate for the mix of lifestyle, environmental, and possibly genetic factors that may underlie variations in rates of disease across populations. This article reviews issues involved in disease mapping (the usual means of presenting descriptive geographic data on disease occurrence) and geographic correlation studies (where the aim is to explore geographic variation in disease in terms of underlying spatially varying “risk factors”). The emphasis is on small‐area applications, where a number of recent advances in methodology have been made.
Occupational Mortality, Background on
Authors: Lynge Elsebeth
Published: September 2014

The study of occupational mortality involves the systematic tabulation of mortality by occupational or socio‐economic group. Three main methods are used to conduct these studies: cross‐sectional studies, death certificate studies, and follow‐up studies. Cross‐sectional studies were undertaken in England and Wales from 1851 to 1979–1983, and these studies have provided key data on social inequalities in health. Death certificate studies have been used for identification of occupational groups with high excess risks from specific diseases. Follow‐up studies require linkage of individual records, typically from censuses and death certificates. They have been undertaken in, for example, all the Nordic countries. The follow‐up studies have shown a high mortality for all marginal groups of the labor market, a social gradient in overall mortality, low mortality among farmers, and a widening gap between the mortality rates of blue and white collar workers.
Scan Statistics for Disease Surveillance
Authors: Wallenstein S.
Published: September 2014

Scan statistics provide the researcher with a measure of unusualness (statistical significance) associated with a clustering of events in time, as would occur when observing an unexpectedly large number of events in a short time period. The statistic can also be used to detect seasonal clustering, and as a method to detect geographical, time, or space–time clustering. The scan statistic can also be generalized and then applied to the analysis of DNA sequence data focusing on detecting a certain pattern.
Multivariate Analysis, Bayesian: Overview with Examples
Authors: Press Jim
Published: September 2014

This article discusses Bayesian multivariate analysis. It focuses on higher‐dimensional statistical inference based upon the posterior distribution, so inference involves multidimensional prior, posterior, and predictive probability distributions. The article describes credibility intervals and regions, and hypothesis testing. There is reference to both vague and natural conjugate prior distributions. Computer‐intensive methods of numerical evaluation such as MCMC and Metropolis–Hastings algorithms are considered. There is an illustrative example involving Bayesian factor analysis.
Optimal Design for Categorical Variables
Authors: F. Berger Martijn P.
Published: September 2014

Optimal design research is important for behavioral research because it enables efficient estimation of parameters and maximum power with minimum sample size and costs of performing a study. In this paper, an elementary introduction to the concepts of optimal design research is given and illustrated by means of four different cases, namely, linear and logistic models, with and without dummy coding. Although different models have different optimal designs, a rule‐of‐thumb may be that if the independent variables are treated as being quantitative, efficiency will increase if the variance of the independent variable increases, and if the design is as orthogonal as possible. For qualitative independent variables, a design that has equal sample sizes for all categories is often optimal.
Prediction Analysis of Cross‐Classifications
Authors: Szabat Kathryn A.
Published: September 2014

Prediction analysis is an approach to the analysis of cross‐classified, cross‐sectional or longitudinal data. It is a method for predicting events based on specification of predicted relations among qualitative variables and includes techniques for both stating and evaluating those event predictions. This article presents the basic methodology employed in prediction analysis and highlights approaches developed by Hildebrand, Laing, and Rosenthal (6), and by von Eye, Brandtstadter & Rovine, (12, 13). The Hildebrand et al. approach involves measuring prediction success of a set of predictions via a proportionate reduction in error (PRE) measure, known as
del
. The von Eye et al. approach models the prediction hypothesis via nonstandard log‐linear modeling and evaluates the extent to which the model adequately describes the data. Both approaches to prediction analysis provide a viable, valuable data analysis tool that researchers in the social and behavioral sciences can use in a wide range of research settings.
Goodness of Fit for Categorical Variables
Authors: Eye Alexander , Bogat G. Anne , Weber Stefan
Published: September 2014

The examination of frequency distributions of categorical variables often involves an evaluation of the appropriateness of probability models.Goodness‐of‐fitstatistics are suitable for this task. These statistics assess the discrepancy between the data and a probability model. Applications focus on (1) overall goodness‐of‐fit and (2) individual residuals. This entry describes the most frequently used goodness‐of‐fit statistics and presents an application in the context of logistic regression.
Incomplete Contigency Tables
Authors: Hershberger Scott L.
Published: September 2014

Incomplete contingency tables have cells with no observations. When empty cells are predicted on the basis of theory or characteristics of the population, the cells have structural zeros. When empty cells occur from deficiencies in sampling, the cells have sampling zeros. While the analytic consequences of structural zeros are usually benign, the consequences arising from sampling zeros are often not so. The primary problem is deterioration in the accuracy of theG2and χ2approximations for testing the fit of loglinear models. Under these conditions, various strategies can be used to improve model testing.
Incidence
Authors: Grietens Hans
Published: September 2014

Incidence is determined as the number of new cases of a disease in a population at a specified interval of time. Incidence rates provide information about the number of new cases or cases added to the sick population. Incidence studies are part of epidemiological research. They are conducted in medicine, psychiatry, and behavioral science. By means of an incidence study, one can truly identify individuals at risk for a disease, or risk factors. There is a dynamic relationship between prevalence and incidence, since prevalence rates are a function of both the incidence and the duration of a disease.
Yates' Correction
Authors: Stefanescu Catalina , Berger Vance W. , Hershberger Scott L.
Published: September 2014

Yates' correction for continuity adjusts the formula for Pearson's chi‐square test by subtracting 0.5 from each observed value in a 2 × 2 contingency table. This formula is used when at least one cell of the table has an expected frequency less than 5, although some sources say that this correction should be used when the expected frequency is less than 10.
Structural Equation Modeling: Categorical Variables
Authors: Skrondal Anders , Rabe‐Hesketh Sophia
Published: September 2014

In the behavioral sciences, response variables are often noncontinuous, common types being dichotomous, ordinal or nominal variables, counts and durations. Conventional structural equation models (SEMs) have thus been generalized to accommodate different kinds of responses.
Two by Two Contingency Tables
Authors: Liu Li , Berger Vance W.
Published: September 2014

Two by two contingency tables are generated by cross classifying subjects by two binary variables. In such tables, one may be interested in testing for an association between the two variables, and, if it is found, then also in estimating the strength of the association. The χ
2
test of independence and Fisher's exact test are illustrated to test independence. The difference in proportions, relative risk, and odds ratio are shown to measure the association. Sensitivity, specificity, the false positive rate, and the false negative rate are used to evaluate a test method. Simpson's paradox is introduced to illustrate the difference between marginal tables and partial tables.
Rater Agreement – Weighted Kappa
Authors: Mun Eun Young
Published: September 2014

One of the characteristics of Cohen's kappa (κ) is that any discrepancy between raters is equally weighted as zero. On the other hand, any agreement means
absolute agreement
between raters, and is equally weighted as one. Thus, the distinction between agreement and disagreement is categorical. For example, Table 1 from the Cohen's kappa section shows three agreement cells and six disagreement cells. In deriving Cohen's kappa, all six disagreement cells are treated equally. However, there could be situations in which the discrepancy between adjacent categories such as Categories 1 and 2 can be considered as partial agreement as opposed to complete disagreement. In those situations, Cohen's weighted kappa can be used to reflect the magnitude of agreement.
Trend Tests for Counts and Proportions
Authors: Liu Li , Berger Vance W. , Hershberger Scott L.
Published: September 2014

In a contingency table, one may be interested in whether there is a trend along the row variable or the column variable. In an R by 2 or 2 by C contingency table, where one variable is binary and the other is ordinal, the Cochran–Armitage test for trend can be used to test whether the binomial proportions have a linear trend across the ordinal variable. In an R by C contingency table, where one variable is ordinal and the other variable is nominal, the Jonckheere–Terpstra trend test can used to determine whether, within each level of the nominal variable, the proportions are distributed identically.
Chi‐Square Decomposition
Authors: Rindskopf David
Published: September 2014

When analyzing a contingency table, the most frequently used procedure is to test the association between variables using a chi‐square test of independence. If the variables are not independent, no further testing is typically done. However, the chi‐square can be decomposed, or partitioned, into components to test hypotheses concerning the nature of the differences among categories of the variables. One can determine which categories of one variable are (and are not) the same in their relation to the other variable.
Marginal Independence
Authors: Berger Vance W. , Zhang Jialu
Published: September 2014

Two variables, say
X
and
Y
, are marginally independent if their joint distribution is the product of the two marginal distributions. In the discrete case, inspecting only the marginal table of
X
‐by‐
Y
may lead to false conclusions if the association between
X
and
Y
is not the same across different levels of another covariate, say
Z
. To prevent such misleading conclusions, partial tables of
X
and
Y
at each level of the confounding covariate
Z
should be displayed.
Sampling Issues in Categorical Data
Authors: Hershberger Scott L.
Published: September 2014

The sampling method used to collect response on categorical variables determines both the probability of the specific configuration of a contingency table, and, more importantly, determines the interpretation of the data. Sampling methods leading to three probability distributions; the hypergeometric, the binomial, and the negative binomial, are discussed. The three sampling methods indicate the same probability of observing an example contingency table, but lead to different interpretations of the data.
Models for Matched Pairs
Authors: Berger Vance W. , Zhang Jialu
Published: September 2014

When randomization is impractical or impossible, matched studies are often used to enhance the comparability of comparison groups, so as to facilitate the assessment of the association between an exposure and an event while minimizing confounding. There are several approaches to the analysis of matched data, including the matched pairs
t
Test, stratified analyses, the McNemar test, and conditional logistic regression. Each approach may be best in a given situation.
M
onte
C
arlo Simulation
Authors: Gentle James E.
Published: September 2014

Monte Carlo simulation is one of the most important tools in all fields of science. Monte Carlo methods use samples of numbers that appear to be random, although the numbers are generated by deterministic computer algorithms. These “pseudorandom” samples can be used in solving a wide range of problems, including both deterministic problems such as evaluating a definite integral and problems in which there is a random component such as studying and comparing the performance of statistical procedures under various assumptions about the underlying distribution.
Bootstrap Inference
Authors: Canty A. J. , Davison Anthony C.
Published: September 2014

Bootstrap methods are powerful and widely useful simulation procedures, whose applications include assessing the uncertainty of a statistical estimator, constructing a confidence interval, and computing the significance probability associated with a hypothesis test.
Permutation Based Inference
Authors: Lunneborg Clifford E.
Published: September 2014

Permutation tests were developed to free analysts of unwarranted assumptions about the mathematical form of response distributions and are nearly synonymous with nonparametric or distribution‐free tests.
Software for Statistical Analyses
Authors: Silver N. Clayton
Published: September 2014

This article describes general purpose statistical software packages and special purpose packages for hierarchical linear modeling, meta‐analysis, power analyses, qualitative data, structural equation modeling, and confidence intervals for effect size indexes. The packages are compared and problems identified. Additional sources of statistical software are listed and future improvements in statistical software are addressed.
Internal Validity
Authors: Howell David C.
Published: September 2014

Internal validity refers to the degree to which any effect that is found for an experiment can be attributed to the action of the manipulated independent variable. This entry discusses 14 threats to internal validity and, in some cases, techniques for dealing with those threats.
Regression Discontinuity Design
Authors: Shadish William R. , Luellen Jason K.
Published: September 2014

The regression discontinuity design is a quasi‐experimental design in which the experimenter assigns participants to treatment conditions on the basis of a cutoff score on an assignment variable – usually a measure of need or merit. This design yields unbiased estimates of treatment effects when certain assumptions are met. Just as in a randomized experiment, the selection process is both completely known and perfectly measured in the regression discontinuity design. Numerous variations of the design are possible, including combinations with randomized experiments.
Causal‐Comparative Study
Authors: Busk Patricia L.
Published: September 2014

A nonexperimental design that is employed to explore cause‐and‐effect relationships is called
ex post facto
or causal‐comparative. The title
ex post facto
is applied to studies in which data are gathered retrospectively, whereas causal‐comparative is used when data are gathered from groups and the independent variable is not manipulated as in an experiment.
Matching‐Models and Examples
Authors: Berger Vance W. , Liu Li , Thach Chau
Published: September 2014

Evaluation is generally inherently comparative. For example, a treatment is generally neither ‘good’ nor ‘bad’ in absolute terms, but rather better than or worse than another treatment. The studies that allow for such comparative evaluations must then also be comparative. To isolate the effects of the treatments under study, avoid confounding treatment effects with unit effects, and possibly increase the power of the study, the units under study must be carefully matched across treatment groups. There are many matching techniques in practice, including crossover designs and case‐control studies. While randomization is often considered to be a competitor to matching, it actually may be considered to be a type of matching, except that here the matching is group‐to‐group.
Counterbalancing
Authors: DePuy Venita , Berger Vance W.
Published: September 2014

When subjects are consecutively given more than one treatment in an experiment, including for crossover design, earlier treatments may affect the results observed during later treatments. To counteract this, the treatment order may be varied between different subjects, or counterbalanced. While complete counterbalancing often requires large numbers of participants, partial counterbalancing, often through the use of a Latin squares design, can provide a reasonable alternative.
L
atin Squares Designs
Authors: Cotton John W.
Published: September 2014

Latin squares designs have an equal number of values for three variables. The designs provide information about the variables with fewer observations than three‐variable factorial designs, but confound interactions with main effects. Models for Latin squares are displayed and computational examples described, together with information about replicated Latin squares.
External Validity
Authors: Howell David C.
Published: September 2014

External validity refers to the degree to which any relationship that is found between an independent and a dependent variable can be generalized across people, settings, and times.
Quartiles
Authors: Clark‐Carter David
Published: September 2014

Quartiles are the result of partitioning an ordered set of numbers into quarters. They can contribute to measures of dispersion such as the interquartile range and are used in box plots.
Standard Error
Authors: Clark‐Carter David
Published: September 2014

The standard error is the standard deviation of the distribution of a sample statistic. The standard error of the mean (SEM) plays an important role in inferential statistics such as
z
tests and
t
Tests and in confidence intervals.
Median Absolute Deviation
Authors: Howell David C.
Published: September 2014

The median deviation is a measure of scale based on the median of the absolute deviations from the median of the distribution. For a normal distribution, the median deviation is somewhat less efficient than the standard deviation or the average deviation as a measure of scale, but this advantage quickly reverses for distributions with heavier tails.
Counternull Value of an Effect Size
Authors: Rosenthal Robert
Published: September 2014

The counternull value of an effect size is the nonnull magnitude of effect size that is supported by exactly the same amount of evidence as supports the null value of the effect size. That is, if the counternull value were taken as the null hypothesis, the resulting
P
value would be the same as the obtained
P
value for the actual null hypothesis. Reporting the counternull, in addition to the
P
value, virtually eliminates the common error of equating failure to reject the null with the estimation of the effect size as equal to zero.
Direct Maximum Likelihood Estimation
Authors: Enders Craig K.
Published: September 2014

Direct maximum likelihood (ML) is used to estimate models with missing data. Direct ML has theoretical advantages, and is unbiased and efficient when data are missing at random. The EM algorithm is one widely‐used method for obtaining ML parameter estimates with missing data.
Dot Chart
Authors: Everitt Brian S.
Published: September 2014

A dot chart is a graphical representation of data values associated with each class of a categorical variable. The value for each category is indicated by the position of a dot along a horizontal line. It has some similarities to a bar chart but can be used in a wider variety of situations.
Leverage Plot
Authors: Lovie Pat
Published: September 2014

A leverage plot (or residual versus leverage plot) is a scatterplot of studentized residuals versus hat values. It is one of the many tools available for checking the adequacy of data when fitting regression models and is especially valuable for gauging the combined effect of high leverage and regression outliers.
Bar Chart
Authors: Everitt Brian S.
Published: September 2014

A bar chart is graphical display of the observed frequencies associated with each category of a categorical variable. The height of a rectangular bar drawn over each category is proportional to the frequency.
Interaction Plot
Authors: Lovie Pat
Published: September 2014

An interaction plot is a simple line graph for examining interactions between variables in factorial designs. For a two‐factor design, the cell means on the response variable for each level of one factor are plotted over all the levels of the second. The resulting profiles are parallel when there is no interaction and nonparallel when interaction is present.
Empirical Quantile–Quantile Plots
Authors: Lovie Sandy
Published: September 2014

An empirical quantile–quantile plot is an instance of added value scatterplots, which allows the analyst to perform a simultaneous graphical comparison between the measures of location and spread of two equal‐sized samples.
Bubble Plot
Authors: Everitt Brian S.
Published: September 2014

A bubble plot is a way of representing the relationship between three variables on a scatterplot. Observations on two variables are plotted in the usual way using circles as symbols; the radii of the circles are made proportional to the associated values for the third variable.
S
hepard Diagram
Authors: Leeuw Jan
Published: September 2014

The Shepard diagram is a version of the residual plot familiar from regression analysis that is useful in nonmetric scaling. We discuss its properties and illustrate it with an example.
History of Discrimination and Clustering
Authors: Hershberger Scott L.
Published: September 2014

The history of discrimination and clustering is primarily defined by the development of two quantitative procedures, discriminant analysis, and cluster analysis. Although they have different goals, the procedures address complimentary problems in classification, and thus are frequently used together.
M
axwell,
A
lbert
E
rnest
Authors: Bartholomew David J.
Published: September 2014

A.E. Maxwell (1916–1996) spent most of his career at the Institute of Psychiatry, University of London, latterly as head of the Biometrics Unit. He is best known for his co‐authorship (with D.N. Lawley) of the book
Factor Analysis as a Statistical Method
. This established factor analysis as a branch of statistical methodology. His other publications include several books on psychological statistics and he also engaged in extensive consulting and advisory work.
F
echner,
G
ustav
T
Authors: Ross Helen
Published: September 2014

Fechner was a physicist who turned to psychophysics and philosophy. He was famous for Fechner's law ‐ that sensation increases with the logarithm of the stimulus intensity. His main contribution to statistics lay in the development of probabilistic procedures for measuring thresholds.
History of Mathematical Learning Theory
Authors: Lovie Sandy
Published: September 2014

The article outlines the story of modern mathematical learning theory from 1950. In particular, the account provides a brief comparison of the earlier work by Hull with the newer approaches by Estes, Bush and Mosteller, and Luce. It is noted that the latter were thoroughgoing probabilists in that they argued that all learning could only be modeled by stochastic mechanisms.