bluuebunny/crossref_metadata_2025 huggingface dataset:
Utility in Health Studies
Authors: Torrance George W.
Published: July 2005

Utility is a quantitative expression of strength of preference. The more something is preferred, the greater is its utility. Formal utility theory for decision making under uncertainty was defined by von Neumann & Morgenstern. Utilities in their theory are measured using the standard gamble. Alternatively, time trade‐off and visual analog scales are used to measure preferences, and these also are sometimes called utilities. Utilities are an integrative measure of health‐related quality of life. Utilities, representing quality of life can be combined with quantity of life to form quality‐adjusted life years. These, in turn, are used in cost–utility analyses. For most clinical studies, the simplest and the preferred way to measure utilities is to use one of the multiattribute health status classification systems that include a utility scoring formula; for example, EuroQol EQ‐5D, Health Utilities Index, or Quality of Well‐Being Scale.
Anticipation
Authors: Dawson Deborah V.
Published: July 2005

Anticipation denotes a phenomenon characterized by increasing severity and/or earlier onset of a disease in successive generations. It could be induced by ascertainment bias and this cast serious doubts among many geneticists about its reality as a biological, phenomenon. However, it is now recognised that it may occur as result of trinucleotide repeat expansion. The number of repeats has been found to correlate with severity and onset age, and this is often associated with a parent‐of‐origin effect. Anticipation has also been reported for a growing number of disorders not found to be associated with trinucleotide repeat expansion.
Disease‐Marker Association
Authors: Schaid D.
Published: July 2005

Disease–marker association can be due to both genetic and nongenetic causes. The former are either that the marker plays a functional role in the disease process or that it is in close proximity to a disease locus (in linkage disequilibrium with it). Association can also be due to population stratification or admixture. Usual contingency table chi‐square methods assume independence of alleles and/or individuals. In the absence of this independence, it is possible to allow for population heterogeneity by using genomic control (i.e. markers that are unlinked to the disease) or family‐member controls, which provide various degrees of matching on ethnicity. Association studies can also use extended pedigree data provided that allowance is made for the statistical dependence of the members in the same pedigree.
E
lston–
S
tewart Algorithm
Authors: Elston Robert C.
Published: July 2005

The Elston–Stewart algorithm is a recursive algorithm that allows one to calculate the pedigree likelihood of an oligogenic model as a function of founder genotypic probabilities, penetrance functions, and genetic transition probabilities. The amount of time required for the computation increases linearly with the size of the pedigree, but exponentially with the number of loci in the genotype. In contrast, the Lander & Green algorithm has computational time that increases exponentially with the size of the pedigree, but linearly with the number of loci. An analogous algorithm allows the likelihood to be calculated for a pedigree of
n
individuals under polygenic inheritance without needing the inversion of an
n
 × 
n
symmetric matrix.
Pedigrees, Sequential Sampling
Authors: Vieland Veronica
Published: July 2005

Families are frequently selected for genetic epidemiologic study based upon (i) initial ascertainment of (affected) individuals, and (ii) subsequent intrafamilial sampling of (subsets of) relatives of those individuals. Such procedures must be taken into consideration when writing genetic likelihoods, estimating genetic parameters, or performing statistical tests based on ascertained pedigree data.
Privacy in Genetic Studies
Authors: Pelias Mary Kay
Published: July 2005

The evolution of the concept of privacy has included protections of property, personal space, personal activities, personal autonomy, and personal information. These ideas expanded over the twentieth century to include the individual's interests in medical treatment and participation in biomedical research. In response to the human genome project, strong moral and legal principles are now in place for protecting the confidentiality of genetic information and the rights of individuals whose genotypes are elucidated in medical practice and in genetics research.
Segregation Ratios
Authors: Elston Robert C.
Published: July 2005

Whereas genetic transition probabilities refer to the distribution of offspring genotype classes that arise from specific mating types, segregation ratios are the expected ratios of offspring phenotype classes. Typical Mendelian segregation ratios are 1 : 2 : 1 and (in the case of dominance) 3 : 1.
Minimum Therapeutically Effective Dose
Authors: Liu Jen‐pei , Chow Shein‐chung
Published: July 2005

The concept and definitions of minimum therapeutic effective dose (MTED) and therapeutic window of a drug are presented. Design and analysis for estimating MTED are overviewed. Design issues such as inclusion of a placebo concurrent control, selection of dose levels, and sample size determination are addressed. Statistical analyses such as the application of analysis of variance and model‐based approach to estimating MTED are summarized. Discussion on internal validity vs. external evidence is also provided.
F
ix–
N
eyman Process
Authors: Chiang Chin Long
Published: July 2005

Introduced in 1951 in their study of survival of cancer patients, Fix–Neyman process is a continuous time stochastic process with two transient states
S
1
and
S
2
and two absorbing states
R
1
and
R
2
, where
S
1
is the wellness state,
S
2
the illness state,
R
1
the death state from cancer and
R
2
the death state from other causes.

Let
p

ij

(0,
t
) be the transition probability from state si at
t
 = 0 to state
s

j

at time
t
. Two simultaneous differential equations are established and solutions derived for
p

ij

(0,
t
), for
i
,
j
 = 1, 2. Formulas for the death transition probabilities from
s

i

to
R

d

,
Q

id

(0,
t
), are also given.

Another useful concept is the expected duration of stay in each of the four states during (0,
t
).

The model is useful in areas of research where states
S
1
and
S
2
can be defined.
Forward Search
Authors: Atkinson A. C. , Riani Marco
Published: July 2005

This is a powerful robust statistical method for exploring the relationship between data and fitted models. It aids discovery of clusters of observations and previously unidentified important subsets of the data as well as revealing groups of outliers. Examples are given with regard to the use of the forward search for regression and generalized linear models, and for multivariate data. The method is to search through the data, fitting subsets of increasing numbers of observations, with outliers being included towards the end.
Stochastic Processes
Authors: Chiang Chin Long
Published: July 2005

Several articles in the Encyclopedia of Biostatistics describe various topics in stochastic process, while the present paper covers the main elements of the field, such as random walk, gambler's ruin, Markov chains, a general birth process, an equality in stochastic processes, death process and survival distribution, staging process, birth–death processes, and finite Markov processes. These topics together constitute an adequate representation of the field.

The main feature of this article is the explicit solution for every process presented, especially the K‐M epidemic model, the higher order transition probabilities in Markov chains, and the explicit solution for the Kolmogorov differential equations.

These solutions make it easy for the application of stochastic processes in research in biostatistics, epidemiology, health care, and medical sciences.
Rheumatology
Authors: Farewell Vern T. , Gladman D. D.
Published: July 2005

Rheumatology deals with clinical disorders, which involve the musculoskeletal system. A survey of the main rheumatic diseases is given along with a historical overview. Current treatments are discussed. Methodological issues arising in rheumatological research are highlighted, including the choice of outcome measures, multiplicity and longitudinal data analysis.
Interaction
Authors: Farewell Vern T.
Published: July 2005

A basic introduction to interaction terms in regression models is given.
Linear Regression, Simple
Authors: Matthews David E.
Published: July 2005

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.
Parsimony
Authors: Farewell Vern T.
Published: July 2005

Parsimony is introduced as a principle in the development of statistical models.
Rank Transformation
Authors: Farewell Vern T.
Published: July 2005

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.
Univariate Response
Authors: Farewell Vern T.
Published: July 2005

A univariate response variable is contrasted with a multivariate response variable. The terms univariate analysis and multivariate analysis are also discussed.
Generating Functions
Authors: Kemp Adrienne W.
Published: July 2005

A generating function is a way of documenting a sequence of items, such as probabilities or moments, using generators of formulas for the individual terms in a sequence. Common generating functions have the form of power series or of exponential series. For discrete distributions, they are particularly valuable for combining sequences of probabilities in various ways. This article discusses moment generating functions and gives examples. It then discusses the cumulant generating function, which is sometimes simpler to handle, and gives examples. Probability generating functions and factorial moment generating function are useful tools for studying discrete distributions.
Iterative Proportional Fitting
Authors: Speed Terry P.
Published: July 2005

Iterative proportional fitting, also known as iterative proportional scaling, is an algorithm for constructing tables of numbers satisfying certain constraints. In its simplest form, the algorithm enables one to construct two‐way tables with specified row and column margins and prescribed association, measured by a family of cross‐ratios. The algorithm can be used to obtain maximum likelihood estimates of table entries under hierarchical loglinear models, and it can be used to adjust counts in survey sampling, where it is known as raking. Natural extensions exist to three and higher‐way tables, and to more general exponential families.
Negative Binomial Distribution
Authors: Biggeri Annibale
Published: July 2005

The negative binomial distribution is an alternative to the Poisson distribution for modeling counts. It is useful when counts show more variation than the Poisson allows (overdispersion). This article shows two ways of deriving it, one way as a waiting time distribution for a certain number of successes with binary data, and the other way as a gamma mixture of Poisson distributions. It then discusses ways of estimating parameters, the use of the negative binomial distribution in regression models, and tests for overdispersion.
Quasi‐Independence
Authors: Mcdonald John W.
Published: July 2005

The quasi‐independence model is a generalization of the model of independence for a two‐dimensional contingency table. It states that a particular subset of cells satisfies the independence structure whereby the cell expected count is a product of a row effect and a column effect. It is often used for square tables (number of rows equal number of columns), the relevant subset being all cells not falling on the main diagonal. This article also discusses connections with loglinear models, extension to multidimensional tables, inference and testing of model fit, and applications and relationships with other models.
S
tat
X
act
Authors: Mehta Cyrus R. , Patel Nitin R. , Senchaudhuri Pralay , Corcoran Christopher D.
Published: July 2005

StatXact is a software package providing exact
P
values and confidence limits for categorical and nonparametric data, including contingency tables. Computations are based on permutation distributions of appropriate statistics. An associated package, LogXact, deals with binary data in the presence of covariates using a logistic regression model.
Two‐By‐Two Table
Authors: Haber Michael
Published: July 2005

This article summarizes the most commonly used statistical techniques for analyzing two by two frequency tables. It presents methods for the estimation of measures of association (risk ratios, risk differences, odds ratios) and agreement (kappa). Tests of hypotheses are described for large samples (based on asymptotic theory) and for small samples (exact tests). Both conditional (Fisher's exact test) and unconditional exact tests are discussed, and the controversy regarding the use of conditional versus unconditional tests is summarized. The article advocates the use of the mid
p
‐value exact test for small sample sizes. Methods for the determination of sample sizes for tests on two by two tables are also described.
Balanced Incomplete Block Designs
Authors: Freels Sally
Published: July 2005

Random assignment of treatments within homogeneous blocks is an effective way to eliminate confounding due to differences between blocks when analyzing treatment differences. A complete block design includes observations within every combination of blocking factors. Balanced incomplete block designs provide a more efficient alternative in which all combinations are not represented, but every pair of blocks appears equally often.
Delayed Entry
Authors: Keiding Niels
Published: July 2005

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.
Expected Number of Deaths
Authors: Væth Michael
Published: July 2005

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.
Exponential Distribution
Authors: Klein John P. , Andersen Per Kragh , Keiding Niels
Published: July 2005

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.
Joint Modeling of Longitudinal and Event Time Data
Authors: Henderson Robin
Published: July 2005

The relationship between the development over time of a quantitative response variable and a parallel process recording events in time can be investigated through a joint model for the two outcome types. Interest in this topic arose from two separate backgrounds: the possibility and consequences of informative dropout in longitudinal trials, and measurement error problems for survival studies with time‐varying covariates. Modeling strategies include combining a marginal distribution for one response type with a conditional distribution for the other, or adopting a correlated random effects model with otherwise conditional independence. Naive methods can lead to significant bias, yet more sophisticated modeling approaches are often sensitive to assumptions, some of which cannot be checked from observable data. When simple summaries such as treatment effect estimators are of interest, sensitivity analyses can bring some reassurance as to whether important conclusions are maintained as assumptions are allowed to change. Overall, the topic is relatively new and research is continuing in many directions.
N
elson–
A
alen Estimator
Authors: Borgan Ørnulf
Published: July 2005

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.
Proportional Hazards, Overview
Authors: Andersen Per Kragh
Published: July 2005

The concept of proportional hazards in survival analysis is introduced and estimators for the (constant) hazard ratio are discussed. Relations to standardization in epidemiology are outlined.
Quality of Life and Survival Analysis
Authors: Olschewski Manfred
Published: July 2005

In clinical research, where time to a specific endpoint like, for example, death is of interest, methods of survival analysis are usually applied. In recent years, such research is more and more accompanied by the inclusion of the patients subjective assessment of treatment circumscribed by the term
health‐related quality of life
(HQoL). In the analysis of HQoL, methods of longitudinal analysis of variance are adequate. In this chapter, we put emphasis on statistical analysis methods that allow a joint analysis of both quantity and quality of survival. The major approach here is defined by quality‐adjusted survival (QAS) times. Conceptually, this means that the survival time of a patient is weighted according to his/her subjective assessment of that time. Advantages and limitations of such a joint analysis are discussed in relation to the separate analyses of the components.
Time to Pregnancy
Authors: Keiding Niels
Published: July 2005

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.
ARMAandARIMAModels
Authors: Helfenstein Ulrich
Published: July 2005

Notifications of diseases, entries in a hospital, and so on are frequently collected in regular intervals. Consecutive observations of such “time series data” are likely to be dependent. In environmental medicine, where series such as daily concentrations of pollutants are analyzed, it is evident that dependence of consecutive measurements may be important. A flexible class of models allowing representing of the stochastic dependence of consecutive data are ARIMA models (autoregressive integrated moving average models). They may be particularly useful for forecasting. Forecasts of epidemiological time series are needed since it is of interest to know what frequencies of diseases might be expected in the future, in order to better plan the distribution of resources. It may also be of interest to assess relations between two ARIMA time series, a “response” series such as “daily number of patients coming to a clinic” and “explanatory” series such as daily concentrations of a pollutant, daily mean temperature, and so on. Such situations may be adequately represented by the so‐called “transfer function model”. Analogous questions arise when studying time series data recorded in an individual subject. This article reviews basic concepts of the topic. The methods are illustrated by medical applications.
B
arahona–
P
oon Test
Authors: Castiglioni Paolo
Published: July 2005

The Barahona–Poon test is used to detect nonlinear dynamics in time series. The method compares linear and nonlinear models fitted to the data, and rejects the null hypothesis of a linear model if at least one nonlinear model is significantly more predictive than all the linear models considered.
D
iggle–
K
enward Model for Dropouts
Authors: Diggle Peter J.
Published: July 2005

Dropouts in longitudinal data analysis occur when some subjects withdraw prematurely from the study. When the processes that govern dropout are related to the measurement process, which is the primary focus of the study, interpretation of the data can be difficult. The Diggle–Kenward model gives a description of the association between the dropout and measurement processes in which completely random, random, or informative dropout models can be obtained by imposing simple and explicit restrictions on the model parameters. Typically, informative dropout parameters are poorly identified, and the associated inferences are therefore sensitive to distributional assumptions.
Distribution‐Free Methods for Longitudinal Data
Authors: Davis Charles S.
Published: July 2005

Many of the available methods for analyzing repeated measurements require distributional assumptions on either the joint distribution of a subject's repeated measurements or on the marginal distributions at each occasion. Although such methods are often adequate and appropriate, nonparametric approaches to the analysis of repeated measurements may also be indicated in certain situations. One approach is to use nonparametric summary statistics in order to reduce the vector of responses from each subject to a single measurement. Multivariate nonparametric approaches include generalizations of univariate distribution‐free methods such as the Wilcoxon–Mann–Whitney and Kruskal–Wallis tests, randomization model methods based on the use of Cochran–‐Mantel–Haenszel statistics, asymptotically distribution‐free analogs of parametric procedures for normally distributed outcomes, distribution‐free methods for incomplete repeated measurements based onUstatistics, the rank transform method, and nonparametric regression methods such as kernel estimation, weighted local least squares estimation, and smoothing splines.
Graphical Presentation of Longitudinal Data
Authors: Zeger Scott L. , Katz Joanne
Published: July 2005

Displaying longitudinal data graphically can be an aid in developing appropriate models and in interpreting more formal analyses of the data. In this entry, a variety of techniques are described and illustrated.
L
evinson–
D
urbin Algorithm
Authors: Castiglioni Paolo
Published: July 2005

The Levinson–Durbin algorithm is a method for finding the solution to a system of linear equations defined by a Toeplitz matrix (with constant diagonal elements).
Moving Average
Authors: Janacek G. J.
Published: July 2005

Given a series of observations made over time, a time series, it is often useful to examine the raw data sequence. One common approach is to apply a moving average. This is, as the name suggests, a series of averages (possibly weighted) of the data sequence. While there are many kinds of moving average, the most common are used to eliminate seasonal or other cyclic effects in time series. While the method is very simple, it can be very effective.

The name, often abbreviated to MA, is used to describe both the output and the computational process. The term Moving Average is also used to describe a time series model in the ARIMA family.
Multivariate Adaptive Splines for Analyzing Longitudinal Data
Authors: Zhang Heping
Published: July 2005

In a longitudinal study, the outcome variable is measured repeatedly over time, together with covariates that may or may not vary over time. This article describes MASAL — a nonparametric method for analyzing and exploring longitudinal and growth curve data. It is based on a high‐dimensional smoothing technique, accommodates time‐varying covariates, and allows unrestricted interactions among covariates and between time and covariates.
Spline Function
Authors: Subba Rao T.
Published: July 2005

Spline functions are an approach to the smoothing of bivariate data (a spline is a term for a flexible strip of metal or rubber used by a draftsman to draw curves). Splines are polynomials within intervals of the
x
‐variable that are connected across different values of
x
. In this article, the theory behind such functions is described.
Vector Field Plot
Authors: Boker Steven M. , McArdle John J.
Published: July 2005

Longitudinal data are often analyzed using a series of dynamic simultaneous equations. A
slope field
plots the expected change in one variable with respect to another variable for ranges of each of these variables. A
vector field
plots the expected changes in variables with respect to each other and with respect to time. A
statistical slope field
or a
statistical vector field
offers a similar display, but are exploratory graphs generated from raw data and include information about the variations due to sampling.
Time Series Similarity Measures
Authors: Gunopulos Dimitrios
Published: July 2005

A distance or similarity function measures the extent to which different time series display different behavior, and can be used to define clusters of time series or to classify a new time series.
Time‐By‐Time Analysis of Longitudinal Data
Authors: Matthews John N. S.
Published: July 2005

Groups of subjects measured longitudinally over time are often analyzed by comparing the groups separately at each time the outcome was observed. Graphs of profiles of group means, with attendant error bars (of various kinds) are often displayed. This approach is flawed because it ignores the structure of the data, seldom answers a question of practical interest, and is difficult to interpret.
Transfer Function Models
Authors: Lawrence Clive J.
Published: July 2005

Time series data occur frequently in medicine and there are a number of approaches to their analysis. For assessing the relationship between two such series, for example, the daily number of patients attending a clinic for respiratory disease and the daily pollen count, transfer function models have proved very useful, and are described in this entry.
W
igner–
V
ille Distribution
Authors: Castiglioni Paolo
Published: July 2005

The Wigner–Ville distribution is a time–frequency distribution developed for the analysis of time‐varying spectra. The interpretation of classic Fourier analysis may become problematic when the frequency content changes over time. The power spectrum cannot indicate when specific spectral components occur or how they change in intensity and frequency. Thus, when the signal is characterized by a time‐varying spectrum, it is preferable to decompose the signal power by a joint function of time and frequency. The Wigner–Ville distribution quantifies the fraction of the power in a certain frequency band during a certain time range.
Battery Reduction
Authors: Massaro Joseph M.
Published: July 2005

Battery reduction is used to select a subset of
m
variables from an original set of
n
variables (
m
 < 
n
) that reproduce a large proportion of the variance in the original set of
n
variables. There are a number of procedures for performing battery reduction analysis. A popular method involves first performing a principal components analysis to select
m
components, which account for the salient variance in the original data. Gram–Schmidt orthogonal rotations are then performed to determine the
m
variables that account for the largest proportion of variance. The procedure is illustrated and reference made to a SAS macro for performing the analysis.
Centroid Method
Authors: D'agostino, Sr Ralph B. , Russell Heidy K.
Published: July 2005

The centroid method is a factoring procedure based on the idea that if the original variables are represented as a set of vectors, then the common factor can be thought of as the vector that passes through the centroid of the terminal points for this set of vectors. The centroid method was originally derived as an approximation to the more complex principal‐axes procedure. An outline of the procedure is provided along with references for further details.
Correspondence Analysis
Authors: Weller Susan C.
Published: July 2005

Correspondence analysis is a procedure for exploring the relationships among two or more sets of variables. A key feature of the analysis is the joint scaling of both row and column variables to provide information on the interrelationships among variables within a set (either row or column variables) and in interrelationships between row and column variables. Correspondence analysis can be used on qualitative or quantitative data. A final step in the analysis involves rescaling of characteristic vectors into optimal scores. Normalizing these optimal scores allows for assessment of relative importance of factors. Correspondence analysis can also be used to find the optimal ordering of variables for a given set of characteristics. If the technique is extended to three or more sets of variables, it is called multiple correspondence analysis.
Cosine of Angle between Two Vectors
Authors: Searle Shayle R.
Published: July 2005

A vector in 2‐space is easily represented by two coordinates. The angle between two such vectors can, by using right angle triangle trigonometry, be expressed as an algebraic function of the two pairs of coordinates, one pair on one vector, the other on the second vector. And this can be extended generally to two vectors in
n
‐space. The resulting expression has useful invariance properties, and can also be thought of as a product–moment correlation.
Eigenvector
Authors: Searle Shayle R.
Published: July 2005

An eigenvector is a vector associated with an eigenvalue of a square matrix. Premultiplying an eigenvector by that square matrix results in the eigenvector being multiplied by its associated eigenvalue.
Factor Analysis, Overview
Authors: Jolliffe Ian T.
Published: July 2005

Factor analysis is a latent variable technique. It reduces the dimension of a data set by assuming that underlying the measured variables is a smaller number of unobservable factors. The measured variables are assumed to be linearly related to the factors, apart from an error term. There are many ways of estimating the loadings that relate the variables to the factors, and predicting the scores of the factors themselves. Solutions are typically nonunique, and are made as “simple” as possible by means of rotation of loadings. Much of factor analysis has been developed within psychology, but it is useful more widely. The technique is often used in an exploratory or descriptive manner, but confirmatory versions are also available.
Mahalanobis Distance
Authors: Dasgupta Somesh
Published: July 2005

Mahalanobis distance is a measure of divergence between two populations based on
p
characteristics or variates. Mahalanobis distance is used in a variety of multivariate applications, including testing the equality of mean vectors of two
p
‐variate normal distributions, discriminant analysis, and classification analysis.
Multivariate Classification Rules: Calibration and Discrimination
Authors: Schmid Christopher H. , Griffith John L.
Published: July 2005

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.
Orthoblique Rotation
Authors: Jackson J. Edward
Published: July 2005

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 second‐stage options. This technique is sometimes referred to as the
Harris–Kaiser rotation
.
Primary Factors
Authors: D'Agostino Ralph B. , Russell Heidy K.
Published: July 2005

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.
Algorithm
Authors: Maindonald John H.
Published: July 2005

The term algorithm is generally used to describe a series of logical steps that, when implemented within a computer program, will perform a desired computational task. The task must be defined precisely. Issues that are generally important in finding or choosing such an algorithm include: computational cost, computer memory or storage requirements, accuracy and precision, simplicity of description, and the tuning of the algorithm to computer architecture and organization.
Bootstrap Method
Authors: De Angelis D. , Young G. A.
Published: July 2005

Bootstrap methods are procedures for the empirical estimation or approximation of sampling distributions and their characteristics. Their primary use lies in the estimation of accuracy measures, such as bias and variance, for parameter estimators, and in construction of confidence sets or hypothesis tests for population parameters. They are applied in circumstances in which the form of the population from which the observed data have been drawn is unknown. They prove particularly useful where very limited sample data are available and traditional parametric modeling and analysis are difficult or unreliable.
Computer‐Assisted Interviewing
Authors: Brinkley Eden , Finlay Elizabeth , Wensing Fred
Published: July 2005

Computer‐assisted interviewing (CAI) refers to the use of a computer to aid the interview process in survey data collection. The computer generally presents the question text on the screen, along with allowable response categories, and the interviewer or respondent records the answers directly into the computer. The computer can also be used to undertake various forms of processing at the time of the interview, and can facilitate electronic data transfer. The advantages and disadvantages of CAI relative to paper and pencil interviewing (PAPI) are discussed briefly.
Cross‐Validation
Authors: Marron J. S.
Published: July 2005

Cross‐validation is one of several methods for error assessment of a statistical method. Alternatives include the jackknife method, the bootstrap method, Akaike's AIC, and others.
Chi‐Square Distribution
Authors: Lancaster H. O. , Seneta E.
Published: July 2005

A chi‐square random variable is defined as the sum of squares of independently distributed standard normal random variables, which explains the additive property of independent chi‐square random variables. Its probability distribution is described by a gamma probability density. The chi‐square goodness‐of‐fit statistic, when sample size is large, is approximately a chi‐square random variable. Tests of hypotheses relating to contingency tables are also based on a statistic with approximate chi‐square distribution.
Estimation
Authors: Doksum Kjell A.
Published: July 2005

One of the principal objects of a statistical analysis is to estimate the value of a parameter from observed data. This article is concerned mainly with point (as distinct from interval) estimation. Methods described include the frequency plug‐in method, the method of moments, weighted and generalized least squares, maximum likelihood, and Bayesian estimation. Attention is given to the properties of unbiasedness and efficiency.
Identifiability
Authors: Greenland Sander
Published: July 2005

Consider a vector
Y
of random variables having a distribution
F
(
y;θ
) that depends on an unknown parameter vector
θ
.
θ
is
identifiable
by observation of
Y
if distinct values
θ
1
and
θ
2
for
θ
yield distinct distributions for
Y
, that is, if
θ
1
 ≠ 
θ
2
implies
F
(
y
;
θ
2
) for some
y
. A function
g
(
θ
) is
identifiable
by observation of
Y
if
g
(
θ
1
) ≠ 
g
(
θ
2
) implies
F
(
y
;
θ
1
) ≠ 
F
(
y
;
θ
2
) for some
y
. Note that
θ
is identifiable if and only if all functions of
θ
are identifiable.
Information Matrix
Authors: Zucker D. M.
Published: July 2005

Mathematical definitions are given of the observed and expected Fisher information matrix. A description is given of the role of Fisher information as a lower bound on the variance of unbiased parameter estimates (Cramer–Rao inequality) and as the asymptotic variance of the maximum likelihood estimate.
Likelihood
Authors: Hills Michael
Published: July 2005

Likelihood is the probability (or probability density) of the observed data, given a probability model, regarded as a function of the unknown parameters. It needs to be defined only up to a constant of proportionality, because ratios of likelihoods are used to compare different parameter values. It is used extensively as the basis for procedures of frequentist inference, such as maximum likelihood estimation, score tests and likelihood ratio tests, and is central to Bayesian inference. Some systems of inference are based on likelihood alone.
Most Powerful Test
Authors: Peña Edsel A. , Rohatgi Vijay K.
Published: July 2005

The Neyman–Pearson approach to testing of hypotheses is described. Neyman–Pearson fundamental lemma and its ramifications are discussed.
Paired Comparisons
Authors: David Herbert A.
Published: July 2005

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.
Prior Distribution
Authors: Kass R. E.
Published: July 2005

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.
R
adon–
N
ikodym Theorem
Authors: Alexander Kenneth S.
Published: July 2005

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.
Rank Correlation
Authors: Kraemer Helena Chmura
Published: July 2005

When observations on two variables are made on each of a number of units, the association between the variables is often measured by the Pearson product‐moment correlation coefficient. This is inappropriate if the variables are ordinal or when their distributions are far from normal. Rank correlation coefficients are based on the ranks of the observations. Spearman's coefficient is obtained by using the ranks in the usual formula for the product‐moment coefficient. Kendall's tau coefficient is expressed through comparisons between all pairs of units, noting, for each pair, whether the two variables order the units in the same or the opposite direction. Methods of inference are outlined.
Robustness
Authors: Prescott Philip
Published: July 2005

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.
Sign Tests
Authors: Dietz E. Jacquelin
Published: July 2005

The sign test is a simple distribution‐free test that can be applied easily in a variety of situations. Observations, which may be difference scores, are replaced by their positive or negative signs and analyzed using a binomial test. The specific applications discussed here are a location test for the median, the Cox–Stuart test for tend, and the McNemar test for correlated proportions.
Signed‐rank Statistics
Authors: Woolson R. F.
Published: July 2005

In a single sample, or with differences between paired observations in a matched‐pair study, the observations may be ranked in order of their absolute values. The sign of the rank is then restored, and an appropriate statistic is calculated from the signed ranks. The most usual example is the Wilcoxon signed‐rank test.
Standard Error
Authors: Gauvreau K.
Published: July 2005

This is a term given to the standard deviation of a statistic in its theoretical sampling distribution, and is used to assess the variability of the statistic and to facilitate hypothesis tests and the calculation of confidence intervals. The standard error may involve unknown parameter values that may have to be estimated from the observed data.
Subjective Probability
Authors: Dickey James M.
Published: July 2005

Subjective probability measures a person's degree of belief in a proposition. It may be used a basis for Bayesian inference. The approaches of B.De Finetti and L.J. Savage are described. Important concepts include coherence, conditional probability, and exchangeability.
Unimodality
Authors: Krieger Abba M.
Published: July 2005

This entry considers the restriction that the density is unimodal on two related problems. The first problem is to obtain refined bounds on the probability in the tail through extensions of Chebyshev's inequality. The second problem is to refine interpolation of distribution functions by obtaining tighter bounds.
Variance
Authors: Gauvreau K.
Published: July 2005

The variance of a random variable is the average value of the squared deviation from the mean. Its square root is the standard deviation. In a sample of size
n
, the corresponding sample measure usually has a divisor
n
 − 1 rather than
n
.
W
ald's Identity
Authors: Lee Mei‐Ling Ting
Published: July 2005

This identity, due to A. Wald, arises in the study of boundary problems in a random walk. It provides expressions for the mean and variance of the stopping time in sequential analysis.
Weighted Distributions
Authors: Patil G. P.
Published: July 2005

Traditional environmetric theory and practice have been occupied largely with randomization and replication. But in environmental and ecological work, observations also fall in the nonexperimental, nonreplicated, and nonrandom categories. The problems of model specification and data interpretation then acquire special importance and great concern. The theory of weighted distributions provides a unifying approach for these problems. Weighted distributions take into account the method of ascertainment, by adjusting the probabilities of actual occurrence of events, to arrive at a specification of the probabilities of those events as observed and recorded. Failure to make such adjustments can lead to incorrect conclusions.
Magic Square Designs
Authors: Martin Richard J.
Published: July 2005

An
n
by
n
magic square with entries 1 to n
∧
2 has all row and column sums equal. If
n
has
k
factors, then a factorial design with n
∧
2 runs and up to 2
k
factors can be obtained from this for which the main effects and at least some of the two‐factor interactions are linear‐trend‐free (their estimates are unaffected if there is a straight line trend over time). An unconnected experimental design called a
magic Latin square
is a Latin square with additional blocking obtained from structured rows and columns.
Optimal Design
Authors: Street D. J.
Published: July 2005

This article provides an overview of the various types of optimality that are used and illustrates the form of optimal designs for particular situations.
A
merican Public Health Association
Authors: Feinleib Manning
Published: July 2005

The
American Public Health Association
(APHA) is the oldest and largest organization of public health professionals in the world, with more than 50 000 members in over 50 occupations of public health in 2004. APHA covers issues affecting personal and environmental health, including funding for health programs, pollution control, programs and policies related to chronic and infectious diseases, a smoke‐free society, and professional education in public health. APHA's activities include scientific and practice programs, annual meetings, and an awards program; it also publishes the
American Journal of Public Health
and
The Nation's Health
.
B
enjamin,
B
ernard
Authors: Haberman Steven
Published: July 2005

Bernard Benjamin (1910–2002) was a distinguished British actuary, demographer, and health statistician. He held several leading posts, in university teaching and public administration, and wrote many influential books on various topics including vital and health statistics, mortality analysis, and medical records.
B
ernard,
C
laude
Authors: Fagot‐Largeault A.
Published: July 2005

A young man of a very modest extraction (his parents were vineyard workers), Claude Bernard received a pious and humanistic education, and started working as an assistant pharmacist. At 21 he ventured to Paris with the ambition of making a career in literature. He was advised rather to learn a job that would enable him to earn a living. He then turned to medicine, which he studied conscientiously, although clinical work did not much appeal to him. As a resident at the Hotel‐Dieu, under François Magendie, he realized what he really wanted to do: experimental work in physiology. For four years he assisted Magendie in his laboratory at the Collège de France, and became an expert at using animal vivisection to trace physiologic facts.
B
iometrika
Authors: Sowan B. J. , Titterington D. M.
Published: July 2005

A statistical journal,
Biometrika
places an emphasis on papers containing original theoretical contributions of direct or potential value in applications. Papers in bordering fields may also be published occasionally.