bluuebunny/crossref_metadata_2025 huggingface dataset:
G
ramian Matrix
Authors: Sullivan Lisa M.
Published: September 2014

A real symmetric matrix A is said to be Gramian if there exists a matrix of real numbers B such that BB' = A. Gramian matrices and Gramian determinants arise in principal components analysis and factor analysis applications.
G
ram–
S
chmidt Process
Authors: Dukes Kimberly A.
Published: September 2014

The Gram–Schmidt process is a technique for constructing an orthogonal basis from a basis spanning the same subspace. The technique is used in a number of multivariate applications including battery reduction analyses and multiple regression applications. In the former, the goal is to reduce the number of variables at the same time explaining as much variance in the original variables as possible. In the latter, the goal is to reduce the
n
‐dimensional space (observations) to a
p
‐dimensional space (independent variables).
Matrix Algebra
Authors: Searle Shayle R.
Published: September 2014

Matrix algebra is the algebra of rectangular (including square) arrays of numbers (and/or symbols) set out neatly in rows and columns. This structure leads to a wide variety of operations different from the traditional algebra of numbers and letters. Those operations have application in many different uses of mathematics and statistics, especially when large amounts of data are involved.
Multiple Endpoints,PLevel Procedures
Authors: B. D'Agostino Ralph , Russell Heidy K.
Published: September 2014

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

Endpoint‐specific procedures are intended to be used to identify the significance of the individual endpoints. These procedures are also calledPlevel procedures because they usually involve making statistical decisions on the basis of thePlevel, resulting from examining the endpoints separately.Plevel procedures can be used in the primary analyses or as a second‐step analyses in conjunction with multivariate global tests. There are three categories of endpoint specific methods, Bonferroni‐type tests, normal‐based tests and resampling procedures.
Multivariate Classification Rules: Calibration and Discrimination
Authors: Schmid Christopher H. , Griffith John L.
Published: September 2014

This article presents a number of measures of discrimination and calibration, along with graphical representations of calibration and discrimination assessment. It emphasizes multivariate classification rules for models, where the classification is into one of two possible states, and also discusses extensions to multistate classifications. The c‐index and the Hosmer–Lemeshow χ2statistic are the most widely used measures of discrimination and calibration.
Multivariate Graphics
Authors: Carr Daniel B.
Published: September 2014

This extensive tour of multivariate graphics leads into methods that can be applied to biostatistics. Basic design issues and techniques, such as smoothing and density estimation, are considered. Issues of choice of color, depth, and structure are discussed. Multivariate representations basically fall into a few classes: glyph plots, linked plots, nested plots, conditioned plots, geometric section plots, series plots, and composite plots. Each of these classes is described in detail. Once the graphics are created, important considerations of viewing the graphics in terms of dimensionality, cognostics, and projection pursuit arise. Ways of handling these issues by starting from low‐dimensional plots and seeing what happens through cognostics or projection pursuit, and sectioning are pursued. The ongoing challenges of producing graphics from massive datasets is also considered.
Multivariate Normal Distribution: Overview
Authors: Jensen D. R.
Published: September 2014

Biometric data typically entail observations on multiple characteristics for each experimental subject. Multivariate normal distributions are often central to the modeling and analysis of such data. Empirical evidence often points towards the normality of multivariate data. Biometric measurements, especially, may emerge as the result of many small increments due to heredity and environment, so that the approximate multivariate normality of such data rests on multidimensional central limit theory. In addition, many normal‐theory procedures, both univariate and multivariate, remain exact for many nonnormal multivariate distributions exhibiting suitable symmetries.
Multivariate Weibull Distribution: Theory
Authors: Jensen D. R.
Published: September 2014

The univariate Weibull distribution is frequently used in accelerated life testing and failure time models. It reduces to the exponential distribution when the shape parameter is equal to one. Multivariate Weibull distributions encompass distributions exhibiting Weibull characteristics and relate to multivariate exponential distributions of various types. Applications include reliability analysis, survival analysis, and repeated failure time analysis.
Oblique Rotation
Authors: Jackson J. Edward
Published: September 2014

Any rotation of vectors associated with
principal components analysis
or
factor analysis
designed to produce new set approximating
simple structure
is said to be
oblique
if the resultant set of vectors are not at right angles to each other. Oblique rotations may obtain a better approximation to simple structure but with the loss of orthogonality. Some of these rotations are obtained by a general procedure called
Oblimin rotation
but some of the more widely used rotations such as
Orthoblique, Promax
and
Optres
do not use this form but instead are two‐ and three‐stage procedures, usually using orthogonal rotations as the first stage.
Orthogonal Rotation
Authors: Jackson J. Edward
Published: September 2014

Any rotation of vectors associated with principal components analysis or factor analysis designed to produce a new set of approximating
simple structures
is said to be
orthogonal
if the resultant set of vectors are still at right angles to each other. Most such rotations are obtained by a procedure called
Orthomax rotation
whose general expression is given. Part of the expression is multiplied by arbitrary constant changes which will produce new vectors with different properties. The two most common orthogonal rotations are
Varimax
and
Quartimax
. An alternate procedure,
Minimum Entropy
, is also presented.
Primary Factors
Authors: B. D'Agostino Ralph , Russell Heidy K.
Published: September 2014

One of the aims of factor analysis is to display the configuration of variables in as simple a manner as possible. There are several techniques for achieving this. Primary axes are the factor axes that are rotated to a good fit to the configurations of the variables. They are the unit vectors that lie in the intersection of the primary factors, which represent the ideal variables, each of which measures only one common factor with no unique factors. Reference vectors are defined as the axes that are perpendicular to the primary factors. Various techniques are illustrated for obtaining projections of variable vectors onto the different sets of axes. These are extremely important for interpretation of results.
Principal Coordinates Analysis
Authors: Gower John C.
Published: September 2014

Principal coordinates analysis, also known as Classical scaling, is a metric multidimensional scaling method based on projection, which uses spectral decomposition to approximate a matrix of distances/dissimilarities by the distances between a set of points in few dimensions. The points may be used in visualizations.
P
romax Rotation
Authors: Jackson J. Edward
Published: September 2014

Orthoblique rotation
is a two‐stage procedure for obtaining
oblique rotations
used to transform vectors associated with principal component analysis or factor analysis to
simple structure
. The first stage will usually be an orthogonal rotation such as
Varimax
. The nature of the second stage will depend to some extent on the degree of obliqueness desired. A numerical example is included with two choices for the second stage.
Structural Equation Models including Historical Origins
Authors: Bollen Kenneth A.
Published: September 2014

Structural equation models refer to general statistical procedures for multiequation systems that include continuous latent variables, multiple indicators of concepts, errors of measurement, errors in equations, and observed variables. An analysis that uses structural equation models has several components. These include (a) model specification, (b) the implied moment matrix, (c) identification, (d) estimation, (e) model fit, and (f) respecification. Historical origins of structural equation models are also described.
Adaptive Sampling
Authors: F. Seber George A. , Salehi M. Mohammad
Published: September 2014

Animal populations are often highly clustered. For example, fish can form large, widely scattered schools with few fish in between. Even rare species of animals may form small groups that are hard to find. Applying standard sampling methods such as simple random sampling of plots to such a population could yield little information, with most of the plots being empty. Adaptive sampling is based on the simple idea that when some animals are located on a sample plot, the neighboring plots (and possibly their neighbors as well) are added to the sample; in this way, the whole cluster can be sampled. The scheme lends itself to a variety of modifications, for example, sampling with and without replacement, sequential and inverse sampling, stratified sampling, two‐stage sampling, and so‐called adaptive allocation, as well as extensions to multivariate data and incomplete detectability.
Composite Estimators
Authors: Schaible Wes
Published: September 2014

A composite estimator is a weighted combination of two (or more) component estimators that, with appropriate weights, will have a mean square error smaller than that of either component estimator. Early applications of composite estimators occurred in conjunction with continuing surveys with rotational designs. These designs allow the production of two distinct component estimators for the current time period—one using the overlap data and the estimate for the previous time period, and the second using only the data from the current time period. A second type of application occurs when the sample size in a particular domain is too small to produce a standard, direct estimate of sufficient reliability. In such cases, a second indirect estimate can be made and a weighted combination of the direct and indirect estimates used to make a composite estimate.
Interpenetrating Samples
Authors: Ghosh Subir
Published: September 2014

This article presents an overview of interpenetrating sampling (IPS), also known as interpenetrating subsampling and replicated sampling. IPS was introduced in the pioneering contribution of P.C. Mahalanobis. It was originally proposed in assessing the nonsampling errors as the so‐called “interviewer errors”. IPS provides a quick, simple, and effective way of estimating the variance of an estimator even in a complex survey. In fact, IPS is the foundation of modern resampling methods like Jackknife, bootstrap, and replication methods. In IPS, three basic principles of experimental designs, namely, randomization, replication, and local control, are used. IPS is used extensively not only in agriculture, but also in social sciences, demography, epidemiology, public health, and many other fields.
Sample Surveys in the Health Sciences
Authors: Brock Dwight B. , Beckett Laurel A. , Bienias Julia L.
Published: September 2014

The use of sample surveys in biostatistics arises from the need to answer questions about health issues in the community. The earliest uses of survey methodology in health research in the United States were motivated more by policy issues than by epidemiologic questions, and paralleled the use of surveys by the Census Bureau and other Federal agencies to address social and economic concerns. More recently, it has been recognized that research into the causes and outcomes of disease can benefit from studies in the entire community, rather than just in the formal health care system, particularly in countries like the United States, where access to health care is not uniform. A number of practical concerns about conducting studies in the community then lead to the use of sample surveys, rather than complete enumeration. In this article, we review the history of sample surveys in the health sciences, present an overview of the major elements and features in the design of sample surveys and analysis of data from such studies, and discuss major methodological advances especially in the last decade. Finally, we provide a large set of references that might be of use to the reader wishing to delve more deeply into specific areas.
Sampling With and Without Replacement
Authors: Hsia Jason
Published: September 2014

Sampling with replacement is a class of sampling procedures in which the selected elements are replaced in the selection pool following each “draw” and may be reselected on subsequent draws. In contrast, sampling without replacement is a class of procedures in which the already selected elements are not replaced in the pool and cannot be selected again.
Snowball Sampling: Introduction
Authors: Johnson Timothy P.
Published: September 2014

Snowball sampling is a well‐known, nonprobability method of survey sample selection that is commonly used to locate hidden populations. This method relies on referrals from initially sampled respondents to other persons believed to have the characteristic of interest. Limitations of this approach include nonrandom selection procedures, correlations between network size and selection probabilities, reliance on the subjective judgments of informants, and confidentiality concerns. Advantages include cost and efficiency.
Superpopulation Models in Survey Sampling
Authors: Dorfman Alan H. , Valliant Richard
Published: September 2014

In sampling from a finite population, we often find it reasonable to posit a probability model (“superpopulation model”) that characterizes relations among variables that pertain to the units of the population. Such a model enables us to make inferences about population characteristics based on sample measurements and “auxiliary information”. In effect, we use the model to predict values for those units of the population that have not been sampled. In such prediction‐based (or model‐based) sampling and inference, it is important to guard against problems arising from model failure. Selecting a balanced sample or weighted balanced sample provides bias protection in the important case of estimating means or totals. Robust variance estimators guard against misapprehension of the variance structure.
Fan Plot
Authors: Atkinson A. C. , Riani Marco
Published: September 2014

This is a graphical procedure for determining the effect of one or more observations on the transformation parameter in the Box–Cox family of power transformations of the response variable in regression. It is based on a “Forward Search” through the data to fit subsets of increasing numbers of observations, with outliers being included toward the end of the search.
Bayesian Measures of Goodness of Fit
Authors: Rice Kenneth
Published: September 2014

We consider methods for critical assessment of the appropriateness of fitted Bayesian models. This is done for situations where no particular alternative model is specified, and where the discrepancy between model and data is measurable in a user‐defined way. We review methods for testing goodness of fit in nonhierarchical models before discussing hierarchical models.
Bayesian Methods for Model Comparison
Authors: Rice Kenneth
Published: September 2014

We consider Bayesian methods for comparing competing models. The Bayes factor is the simplest and best‐known tool for this purpose, but other criterion‐based methods are also considered, together with model averaging.
Diagnostics
Authors: Atkinson A. C.
Published: September 2014

Diagnostics are methods for identifying discrepancies between the data and a fitted model. Concepts described here include outliers, decision residuals, leverage, influence, Cook's distance, multiple deletion, and very robust regression methods as illustrated from a study of the relationship between mandible length and gestational age.
Errors in Variables
Authors: Raboud J. M.
Published: September 2014

When independent variables in regression models are measured with error, the resulting coefficient estimates are biased. Formulae are given to adjust estimated coefficients obtained from linear regression models and proportional hazards models. References are given to estimate the degree of bias of estimated coefficients in logistic regression and other types of general linear models.
Explanatory Variables
Authors: Darlington G. A.
Published: September 2014

A model may be developed to investigate the relationship between an outcome and one or more explanatory variables. An explanatory variable can be quantitative, requiring consideration of how to include it in the model or an explanatory variable can be qualitative whereby indicator variables (or dummy variables) must be defined.
Fixed Effects
Authors: Farewell Vern T.
Published: September 2014

The concept of fixed effects in a regression model is explained and contrasted with that of random effects.
Goodness of fit ‐ Examples
Authors: Atkinson A. C.
Published: September 2014

Pearson's classical chi‐square test compares observed and expected frequencies in various classes. It tests for unspecified departures from the model, is based on aggregate rather than partial data, and uses an asymptotic distribution. Goodness of fit in multiple regression may be explored by including further explanatory variables in the model, or by comparing residual variation against external predictions. Tests based on residuals are described, as are extensions to generalized linear models.
Linear Regression, Simple
Authors: Matthews David E.
Published: September 2014

Simple linear regression involves finding the best‐fitting curve of a suitable functional form that relates the value of an explanatory variable,X, and the mean value of a response variable,Y, givenX. The goals of regression modeling are to determine whetherYandXare associated in some systematic way, and to estimate or predict the value ofY, or its mean, corresponding to a known value ofX. We describe estimation, statistical inference, and diagnostic checking of the assumed model and any associated unknown parameters, giving due importance to their statistical and scientific interpretations. An example concerning the relationship between systolic blood pressure and age in men illustrates these concepts concretely.
Mallows' Cp statistic II
Authors: Stone M.
Published: September 2014

This entry covers the seminal model choice criterion of H.L. Jones, attributed to and developed by C.L. Mallows.
Parsimony
Authors: Farewell Vern T.
Published: September 2014

Parsimony is introduced as a principle in the development of statistical models.
Prediction
Authors: Farewell Vern T.
Published: September 2014

Statistical usage of the term prediction is discussed. The sources of error in prediction are identified.
Rank Transformation
Authors: Farewell Vern T.
Published: September 2014

Ranking of observations is described. Inference based on ranks of a response variable is discussed, as are rank transformations of explanatory variables in a regression model.
Residuals
Authors: Atkinson A. C.
Published: September 2014

Residuals are the differences between observed responses and those predicted by a model. Any pattern in the residuals is evidence of an inadequate model or of irregularities in the data, such as outliers, and suggests how the model may be improved. Various graphical methods are described, for both the normal regression and the generalized linear models, and are illustrated for the analysis of various data sets.
Response Variable
Authors: Farewell Vern T.
Published: September 2014

The concept of a response variable is introduced and contrasted with that of an explanatory variable. Univariate, multivariate, and intermediate response variables are discussed.
Standardized Coefficients
Authors: Farewell Vern T.
Published: September 2014

The use of standardized regression coefficients to facilitate the comparison of coefficients is described.
Two‐stage Least Squares Regression
Authors: Sprent P.
Published: September 2014

In systems of simultaneous equations, if some explanatory variables are correlated with errors, ordinary least squares regression gives inconsistent estimators. Two‐stage least squares overcomes this difficulty. The approach is closely related to procedures that use instrumental variables in models that allow for errors in explanatory variables.
Marginal Models
Authors: Lindsey J. K.
Published: September 2014

Repeated observations on the same subject are likely to be correlated. This situation may arise in longitudinal studies or in other cases where observations are clustered in some way. A multivariate model will refer both to the marginal distributions over the whole data and to the conditional distributions for individual subjects or clusters. A marginal model puts primary emphasis on marginal aspects. Relationships between responses and covariates may differ between marginal and conditional models. The issues are discussed with particular reference to paired binary data.
Misclassification Models
Authors: Albert Paul S.
Published: September 2014

Classification errors in categorical data may distort the results of statistical analyses. Misclassification models have been developed in various areas of biostatistical application to account for this distortion. This article reviews selected methods that have been proposed for (i) modeling the natural history of a disease or growth process, (ii) evaluating diagnostic tests, and (iii) identifying epidemiological risk factors. We consider methods that incorporate known estimates of misclassification rates, methods that estimate misclassification rates from a subsample in which the presumably “true” classification is known, and methods in which both the true process and the misclassification rates are simultaneously modeled from misclassified data. The various methodological approaches are discussed along with their limitations, and we note an important issue for additional research.
Ante‐dependence models
Authors: Kenward M. G.
Published: September 2014

Ante‐dependence models are flexible nonstationary covariance structures for repeated measurements that generalize stationary autoregressive models. The structures can only be used for data in which the times of measurement are common across units. They can be defined in the Gaussian setting by conditional independence relationships: an
r
th order structure holds if and only if two repeated measurements are conditionally independent given
r
intermediate measurements. Equivalently, an
r
th order ante‐dependence covariance matrix has a inverse of band form of order r + 1, with no other constraints on the matrix, other than that it is positive definite. It provides a compromise between the unrealistic stationary structures, such as the autoregressive, and the potentially inefficient unstructured form. In certain repeated measurements settings, it allows test statistics to be constructed with known small sample distributions, that can be regarded as generalizations of Wilks' lambda statistic.
Nonlinear Mixed Effects Models for Longitudinal Data
Authors: Bates Douglas M.
Published: September 2014

The analysis of longitudinal data may require a mixed‐effects model, incorporating parameters for fixed effects associated with the whole population and also parameters describing distributions of random effects associated with individual subjects. These may enter the model nonlinearly, as in compartment models used in pharmacokinetics. Maximum likelihood estimation is carried out by numerical optimization.
Random Coefficient Repeated Measures Model
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 that 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 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 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.
J
‐Shaped Distribution
Authors: Ghosh Pankaj
Published: September 2014

This is a probability distribution or frequency distribution shaped roughly like a letter J or its mirror image, with highest frequency at one end of the range. Formal characterizations are given, relating to the characteristic function.
Alternative Hypothesis
Authors: Schork M. A.
Published: September 2014

In testing a null hypothesis, it is often useful to consider possible alternatives. A distinction is drawn between one‐sided and two‐sided alternatives.
Ancillary Statistics
Authors: Reid N.
Published: September 2014

An ancillary statistic is a function of the data whose distribution is free of the parameters in the model. Ancillary statistics are complementary in a certain sense to sufficient statistics. In the theory of statistics, it is often argued that inference should be conditional on ancillary statistics, when this is possible. Fisher showed that in a location model, the conditional distribution of the maximum likelihood estimator could be obtained by renormalizing the likelihood function, or, in other words, treating the likelihood function as a posterior distribution for the location parameter. This result has been generalized to provide an approximate conditional distribution to the maximum likelihood estimator, given an approximately ancillary statistic, a result known as thep* approximation.
BANEstimates
Authors: Bemis Kerry G.
Published: September 2014

BAN (Best Asymptotically Normal) estimates and their corresponding chi‐squared test statistics form the core of classical categorical data analysis. Historically, these estimates were developed for the multinomial distribution. This article describes the optimal properties of these classical estimators and the form of their chi‐squared test statistics. It is also shown that these are special cases of BAN estimators and chi‐squared test statistics from a more general family of discrete multivariate distributions (Sum Symmetric Power Series Distributions), which include multinomial, negative multinomial, and multivariate Poisson as special cases. These results are in turn generalized to the multivariate exponential family. BAN estimators and their test statistics take on familiar forms and properties as the classes of distributions are generalized. Estimation and hypothesis testing are described from the point of view of constraint equations and the more familiar freedom equations.
Bartlett's Test
Authors: Boos Dennis D.
Published: September 2014

Bartlett's test for the equality of variances ofknormal populations is based on a modification of the asymptotic form of the likelihood ratio statistic. The null distribution of the statistic is extremely sensitive to deviations from the assumption of normal populations.
Bonferroni Inequlities and Intervals: Practical Aspects
Authors: Costigan T.
Published: September 2014

The first‐order Bonferroni inequality relates the probability that at least one of a series of events occurs to the sum of the probabilities of the individual events. In statistical applications, it provides an upper bound to the probability of falsely rejecting at least one of a series of true null hypotheses, and therefore, a possible correction to a significance level to account for multiplicity of hypotheses tested. Further developments include sequential rejection procedures and the use of higher‐order Bonferroni inequalities.
Cauchy Distribution
Authors: Glynn Robert J.
Published: September 2014

The Cauchy distribution, discovered by S.D. Poisson in 1824, arises as the ratio of two independent standard normal variables. It does not have finite moments. The sample mean has the same distribution as a single observation, and so is not a consistent estimator of the median. It is a useful model for situations where error distributions have very long tails.
Confidence Intervals and Sets
Authors: Cook Nancy R.
Published: September 2014

A confidence interval for a fixed parameter, or a confidence set for a multidimensional parameter, provides a plausible range of values for the parameter(s) consistent with the observed data. It is so constructed that there is a predetermined probability (the confidence level or coefficient, or the coverage probability) that the random interval or set contains the fixed parameter.
Consistent Estimator
Authors: Liang Kung‐Yee , Rathouz P. J.
Published: September 2014

An estimator of a statistical parameter is consistent if it converges in probability to the parameter as the sample size increases. Two methods of obtaining consistent estimators are described by minimizing a specific function of the data and the parameter (e.g. maximum likelihood,M‐estimation, weighted least squares), or by solving an equation involving an estimating function (e.g. quasi‐likelihood).
Cramer‐Rao Inequality
Authors: Bhapkar V. P.
Published: September 2014

This inequality provides a lower bound for the variance of any unbiased estimator of a one‐dimensional parameter or more generally of a function of the parameter. It involves the Fisher information (the expectation of the squared derivative of the log likelihood). If equality is attained, the statistic is a minimum variance unbiased estimator, and the density function belongs to a one‐dimensional exponential family. The inequality can be generalized for a multidimensional parameter.
Critical Region
Authors: B. D'Agostino Ralph
Published: September 2014

In a test of a null hypothesis, the critical region defines the set of possible outcomes that would lead to rejection of the hypothesis, these being chosen to have predetermined probability called the size of the test. In a simple example, the distinction is made between one‐ and two‐tailed tests.
Decision Theory: Biostatistical Aspects
Authors: Lewis Roger J. , Berry Donald A.
Published: September 2014

Statistical decision theory is a set of techniques for making optimal decisions in the setting of uncertainty. Both frequentist and Bayesian approaches have been developed. The definition of a Bayesian decision problem requires the specification of an unknown parameter and its prior probability density function (pdf), observed data (if any), a likelihood function, a set of possible decisions or actions, and a loss or utility function whose expectation is to be optimized. The expected loss can be defined in a number of ways, yielding a frequentist risk, the conditional Bayes risk, and the Bayes risk. The goal is generally to select a decision rule that maps the possible observed data to a set of actions so that the Bayes risk is minimized. Alternatively, a frequentist may seek to minimize the maximum frequentist risk that might be experienced over all possible fixed values of the unknown parameter, yielding the optimal “minimax” decision rule. Examples in the chapter include a problem of selecting drug therapy when the patient diagnosis is unclear and the decision to order a test to determine the disease present, choosing a sample size when the goal is to estimate an unknown mean of a normally distributed variable, sequential stopping rules determined through back induction, and sequential allocation or assignment (bandits).
E
dgeworth Expansion
Authors: Hall Peter
Published: September 2014

The Edgeworth expansion expresses one probability density in terms of another by the use of cumulants, and is particularly important when the reference distribution is the standard normal. There are applications to the distributions of statistics including the sample mean, sample variance, variance ratio and correlation coefficient, and to the coverage of confidence intervals.
Efficiency and Efficient Estimators
Authors: Rotnitzky Andrea
Published: September 2014

The efficiency of a point estimator of a parameter reflects its tendency to be distributed closely around the parameter value. Asymptotic efficiency refers to the limiting distribution in large samples, which is often asymptotically normal, the minimum variance being given by the Cramér–Rao lower bound in terms of the information matrix. Concepts discussed here include asymptotic relative efficiency, semiparametric efficiency, partial likelihood, and quasi‐likelihood.
Estimation, Interval
Authors: Kupper Lawrence L.
Published: September 2014

This article discusses methods for constructing confidence intervals. The basic principles underlying interval estimation are discussed, followed by simple examples of exact confidence intervals based on normality assumptions. Maximum likelihood‐based approximate confidence interval methods are then described, including interval estimation methods based on the use of likelihood ratio tests.
Experimentwise Error Rate in Practice
Authors: Iglewicz Boris
Published: September 2014

Multiple comparisons are frequently performed in medical experiments. These occur when dealing with multiple treatments, multiple endpoints, multiple subgroups, or multiple looks. Use of the nominal significance level for each comparison results in an inflated overall type I error rate, called the
experiment‐wise error rate
. Approaches for controlling this error rate are discussed and a number of references are provided.
Exponential Family: Overview
Authors: Pierce Donald A.
Published: September 2014

The overview focuses on aspects of exponential families important in biostatistics, particularly by emphasizing not identically distributed observations involving covariables. This is closely related to generalized linear models. The distinction is made between regular and curved exponential families. In the former, the maximum likelihood estimator is sufficient, the expected and observed information are the same, and there are exact frequency methods for some inferences. In the latter, there is information not contained in the maximum likelihood estimator, much of which is carried by the ratio of observed to expected information, and asymptotic methods must always be used. Of these, likelihood ratio and score methods are preferable to the Wald method. Modern higher‐order asymptotic methods, including the modified profile likelihood, are mentioned. The tractability of Bayesian inference in exponential families is indicated.
F

Distributions
Authors: Selvin S.
Published: September 2014

The
F
‐distribution is fundamental to much of statistical analysis, particularly the analysis of variance. The article entitled “
F
‐Distribution” is a brief but complete description of this probability distribution. Its properties are first discussed, followed by a description of the relationship between the
F
‐distribution and other common statistical distributions (e.g. beta, t‐, chi‐square, and binomial distributions). The article concludes with a typical application of this distribution to a small set of sampled observations.
F
ieller's Theorem
Authors: Buonaccorsi John P.
Published: September 2014

This provides confidence limits for the ratio of means of two variables following a bivariate normal distribution, where estimates of the elements of the covariance matrix are available as known multiples of a chi‐square variable. Approximate intervals are available when the distributional assumptions are not strictly satisfied.
Frequency Distribution
Authors: Smith W.
Published: September 2014

This is a method of summarizing a set of numerical data in tabular or graphical form, by forming class intervals of the variable and displaying the number of observations in each class. Results are usually displayed graphically by a histogram.
Generalized Maximum Likelihood
Authors: Crowder M. J.
Published: September 2014

Maximum likelihood estimation plays a central role in statistical methodology but can fail in certain situations. Various modified versions have been proposed, some of which are described here.
Half‐Normal Distribution
Authors: Byers R. H.
Published: September 2014

The half‐normal distribution occurs when sampling from a normally distributed population where the signs of the negative observations are lost or are not relevant. This article includes the formulae for the probability density, mean, variance, skewness and kurtosis, and the maximum likelihood estimator of the distribution's parameter.
Lehmann Alternatives with Emphasis on Biostatistics
Authors: Razzaghi Mehdi
Published: September 2014

In a nonparametric comparison of two groups, the test statistic may not be distribution‐free under alternatives to the null hypothesis. Lehmann proposed a family of alternatives under which the distribution function of the response variable in one group is a specified function of that in the other. A power relationship is usually considered. The distribution of the rank vector under the alternative then depends only on the specified function and rank statistics are distribution‐free. An extension covers the situation where a proportion of the population is unaffected by treatment.
Likelihood Principle
Authors: Lindsey J. K.
Published: September 2014

The likelihood function is central to all approaches to statistical inference. Principles have been proposed to justify the equivalence of inferences based on data with likelihood functions identical apart from a proportionality factor independent of parameters. The weak principle refers to replications with the same design; the more contentious strong principle refers to results from different designs. The latter, implied by Bayesian theory, needs to be supported by strong confidence in the validity of the model.
Location–Scale Family
Authors: Datta Gauri Sankar
Published: September 2014

A set of random variables has a location‐scale family distribution if their joint distribution function can be expressed as a function of the standardized observations (xi ‐μ)/s, where μ and s are, respectively, the location and scale parameters. Important examples are the uniform, normal, and Cauchy distributions. Least squares estimation and minimum risk equivariant estimation are described.
Logistic Distribution
Authors: George E. Olusegun
Published: September 2014

Properties of the logistic distribution, including characterizations, generalizations, and applications in generalized linear models, are summarized. Some definitions of multivariate logistic distribution are also highlighted.
Combining P‐values
Authors: Westfall Peter H.
Published: September 2014

When a null hypothesis is tested on different sets of data, the individual
P
values may usefully be combined to provide a more powerful test than any individual one. If the null hypothesis is true, and the statistical model is correct, each
P
value follows a uniform distribution. Various methods of combination have been suggested for situations where the data sets are independent, and some comparisons of power against alternative hypotheses are presented for a particular case. Most of these methods are invalid for dependent data.
Graeco–Latin Square Designs
Authors: Martin Richard J. , Nadarajah Saraleesan
Published: September 2014

A Graeco‐Latin square is a pair of orthogonal Latin squares (each letter combination occurs exactly once). Such squares exist for all positive integers except for 1, 2, and 6. They, and the extension to sets of mutually orthogonal Latin squares, are useful for constructing experimental designs in several situations in which there are four or more blocking or treatment structures. However, they are limited because essentially the numbers of treatments, blocks, or factor levels must all be the same.
Information
Authors: Zucker D. M.
Published: September 2014

This entry discusses concepts of information in statistics, including Fisher and Kullback information and the notion of precision as a measure of information. Application of these concepts is illustrated in two common biostatistical contexts: weighted combination of independent estimates (stratified analysis and meta analysis) and sequential monitoring of clinical trials (concept of information time).
Statistics, an Overview
Authors: Cox D. R.
Published: September 2014

The current topic may be called “statistical science”, encompassing statistical theory and its applications in science and technology. One pillar is probability theory, underpinning statistical inference and forming the basis for special models. Various types of models are distinguished. A second pillar is the design, analysis and interpretation of investigations. The discussion here touches on measurement issues, methods of analysis, detailed techniques, causation, and decision analysis. Formal theories of inference are discussed. Issues arising in the provision and use of official statistics are briefly discussed.
Marginal Likelihood
Authors: Kalbfleisch John D.
Published: September 2014

In a multivariate situation, where inference is required about a subset of the parameters, the others being nuisance parameters, it may be possible to factorize the full likelihood so that the parameters of interest are associated with one aspect of the data. This gives the marginal likelihood. Examples include the partial likelihood used in the Cox model for survival data.
Marginal Probability
Authors: Wypij David
Published: September 2014

In a multivariate probability distribution of many (perhaps correlated) random variables, a marginal distribution describes the univariate distribution of a single variable integrated over all the others. Marginal distributions do not uniquely determine the joint multivariate distribution.
Median
Authors: Hettmansperger T.
Published: September 2014

In a set of ordered data, the median is the middle observation, or the average of the two middle ones for an even sample size. It is similarly defined for a probability distribution, and the sample median is a robust estimate of the population median. Various properties of the sample median are summarized, including the fact that it minimizes the mean deviation about a point.
Moment Generating Function
Authors: Koehler Kenneth
Published: September 2014

Moment generating functions are used to derive moments of distributions, establish the distribution of sums and differences of independent random variables, and derive limiting distributions of sequences of random variables. It is an integral transform of the density function and the terms of its Taylor expansion about the origin give moments about zero. Central moments are easily obtained. Properties discussed briefly include the derivation of the MGF for a sum as the product of the separate MGFs, the multivariate case and the derivation of the factorial moment generating function.
Moments
Authors: Koehler Kenneth
Published: September 2014

The
r
th moment of a random variable
X
is the expectation of the
r
th power of
X
. Moments may also be taken about a constant
c
by taking powers of
X
 − 
c
, and central moments take
c
as the mean of the distribution. Related concepts are absolute moments, factorial moments, and cumulants. The method of moments is a method of estimation by equating estimated moments to theoretical values expressed in terms of parameters.
Noncentral

t

Distribution: Introduction
Authors: Rust Philip F.
Published: September 2014

This is the distribution of the ratio of a normally distributed variable with unit variance to the square root of a ratio of a chi‐square variable to its degrees of freedom. When the noncentrality parameter (the mean of the normal distribution) is zero, the distribution is Student's
t
distribution. The noncentral distribution provides the power function for a
t
test, and other applications concern tolerance intervals and confidence intervals. Details are given here.
Normality Tests: Overview
Authors: B. D'Agostino Ralph
Published: September 2014

The normal (Gaussian) distribution is of central importance in statistical theory, and numerous techniques for judging departures from normality have been developed. The article describes moment tests, normal probability plots, the chi‐square test, tests based on the empirical distribution function, transformation tests and correlation techniques. Power characteristics of different tests are summarized.
Paired Comparisons
Authors: David Herbert A.
Published: September 2014

This article focuses on the method of paired comparisons in which several “treatments” are subjectively compared in pairs, the actual measurement being irrelevant or not feasible.
Pairwise Independence
Authors: Wermuth Nanny
Published: September 2014

If two explanatory variables are independent, with additive effects on a third, response, variable, then the overall effect of one explanatory variable is qualitatively the same as the effect conditional on the other explanatory variable. If the explanatory variables are dependent, reversal can occur. In multivariate analysis, pairwise independence of all variables does not imply mutual independence.
Pearson Distributions
Authors: Armitage Peter
Published: September 2014

This is a family of probability distributions, often called Pearson
curves
, developed by Karl Pearson in the period between 1894 and 1895 to provide flexible descriptions of the nonnormal distributions encountered in his biometric research. It includes many well‐known distributions. The types can be distinguished by the standard indices of skewness and kurtosis.
Penalized Maximum Likelihood
Authors: Simonoff Jeffrey S.
Published: September 2014

Nonparametric density estimation methods are designed to estimate the density functionfthat generates a sample of observations, without assuming a parametric form forf. Rather,fis only assumed to be smooth, where smoothness is defined as the condition that at least a specified number of derivatives are square integrable. Penalized maximum likelihood estimators accomplish this by maximizing a penalized form of the log‐likelihood, where the penalty discourages roughness in the final estimate. The estimator has a Bayesian interpretation, with the penalty function corresponding to the logged prior for the density. The resultant estimator often takes the form of a polynomial spline with knots at the order statistics, and asymptotically is approximately a local‐bandwidth kernel estimator. The method can be adapted to large sparse contingency tables, resulting in cell probability estimators that are consistent under asymptotics that approximate such tables.
Power Transformations
Authors: Atkinson A. C.
Published: September 2014

Simple regression analysis assumes normally distributed errors of constant variance. These assumptions may be more nearly satisfied by transformation of either the response variable, the explanatory variable(s), or both. In using the power transformation of Box and Cox, the power parameter may be estimated by maximum likelihood. This family of transformations includes the square root, logarithm, and reciprocal.