bluuebunny/crossref_metadata_2025 huggingface dataset:
Bonferroni Inequalities and Intervals
Authors: Costigan T.
Published: July 2005

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: July 2005

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: July 2005

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.
Convergence in Distribution and in Probability
Authors: Taqqu M. S.
Published: July 2005

The notions of convergence in distribution and convergence in probability are introduced, discussed and illustrated with examples.
Cramér–Rao Inequality
Authors: Bhapkar V. P.
Published: July 2005

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.
Decision Theory
Authors: Lewis Roger J. , Berry Donald A.
Published: July 2005

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).
Degrees of Freedom
Authors: Potthoff Richard F.
Published: July 2005

This short article deals mainly with (i) defining of the term
degrees of freedom
; (ii) the relation of degrees of freedom to the χ
2
,
t
, and
F
distributions and associated hypothesis testing; (iii) relations among these three distributions, and between them and other distributions (normal, exponential, and Cauchy), when degrees‐of‐freedom parameters assume certain values; (iv) noninteger degrees of freedom; and (v) issues pertaining to reporting of results for publication.
Density Estimation
Authors: Scott D. W.
Published: July 2005

The “density estimation” article highlights the dual roles of the graphical exploration of data as well as the nonparametric estimation of probabilities. Estimators described in detail include the histogram, frequency polygon, averaged shifted histogram, and kernel method. The basic theory of estimation is outlined. Data‐based calibration by cross‐validation is described. Several concrete examples are given.
Estimating Functions
Authors: Desmond A. F. , Godambe V. P.
Published: July 2005

The aim of this article is twofold. First, we demonstrate that the methodology of estimating functions produces a unification and extensions of approaches to statistical modeling and inference, both parametric and semiparametric. Secondly, we illustrate via selected examples, how this methodology applies in biostatistics. We have limited our discussion, here, to case–control studies, prospective and retrospective sampling, and overdispersion.
Estimation, Interval
Authors: Kupper Lawrence L.
Published: July 2005

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.
Exchangeability
Authors: Greenland Sander , Draper David
Published: July 2005

In probability theory, the
random variables
Y
1
, …,
Y

N

are said to be
exchangeable
(or
permutable
or
symmetric
) if their joint distribution
F
(
y
1
, …,
y

N

) is symmetric; that is, if
F
is invariant under permutation of its arguments, so that

whenever
z
1
, …,
z

N

is a permutation of
y
1
, …,
y

N

. There is a related epidemiologic usage, which is described in the article on confounding. In many ways, sequences of exchangeable random variables play a role in subjective Bayesian theory analogous to that played by independent identically distributed (iid) sequences in classical frequentist theory. In particular, the assumption that a sequence of random variables is exchangeable allows the development of inductive statistical procedures for inference from observed to unobserved members of the sequence [1, 2, 3, 5, 6, 9].
Exponential Family
Authors: Pierce Donald A.
Published: July 2005

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.
Extreme Values
Authors: Davison A. C.
Published: July 2005

Statistics of extremes concerns the largest or smallest few observations in a sample. These are important in environmental, financial, and engineering applications, as representing rare events such as floods, high tides, or stock market crashes, but the same statistical ideas are useful in many other situations. This article provides a short review of the topic, with many further references.
F
ieller's Theorem
Authors: Buonaccorsi John P.
Published: July 2005

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.
Foundations of Probability
Authors: Lindley Dennis V.
Published: July 2005

Probability, as a mathematical subject, is essentially unique and its formal rules are described. There are many different interpretations, of which three are mentioned: classical, frequentist, and in terms of belief. The many and varied justifications for thinking that beliefs should conform to the rules of probability, the Bayesian view, are described. The connections with decision‐making, the concept of utility, and the role of subjectivity are discussed. It is pointed out that the frequentist and Bayesian interpretations lead to distinct, practical implications of probability, especially in connection with the likelihood principle, and illustrations are provided. The role of exchangeability, which links the two views, is developed. Alternatives to probability are briefly mentioned.
Frequency Distribution
Authors: Smith W.
Published: July 2005

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.
Half‐Normal Distribution
Authors: Byers R. H.
Published: July 2005

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.
Inference
Authors: Zelen M.
Published: July 2005

Statistical inference is the process of drawing conclusions from data governed by probability laws. Conclusions are uncertain, and a principal goal is to measure the uncertainty. In practical applications, the aim may be to draw conclusions about models that may have generated the data. A central concept, in almost all approaches, is the likelihood, regarded as a function of the unknown parameters. The most widely used approach is that of the frequentist school, based on error probabilities, that is, probabilities of particular observations given hypotheses about parameters. Key concepts here are significance tests, confidence intervals, and point estimation. The likelihood school uses likelihood ratios as measures of relative support. The Bayesian school uses inverse probability, that is, probabilities of hypotheses. Bayes' theorem relates posterior probabilities (postdata) to prior probabilities (predata) via the likelihood function. Details of these approaches are given in the article.
Inference, Foundations of
Authors: Fisher Lloyd D.
Published: July 2005

The foundations of statistical inference are summarized beginning with regularity as embodied in probability theory. Both objective external probability and subjective internal Bayesian probability are discussed. The law of large numbers is the theoretical underpinning of the usefulness of probability. Objective large probability methods have led to concepts such as confidence intervals, hypothesis testing, andp‐values. Subjective probability uses data to go from prior to posterior distributions. No known method of statistical inference can withstand all rational criticism. Important areas of inference arise from descriptive and graphical methods of summarizing data and are mentioned.
Information
Authors: Zucker D. M.
Published: July 2005

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).
James–Stein Estimator
Authors: Casella George
Published: July 2005

For a multivariate normal distribution with independent components, the observed vector is an inadmissible estimator of the mean vector. The James–Stein estimator defines a statistic with lower squared error loss than the observed vector. It has led to research into families of minimax estimators, some of which is summarized.
Kurtosis
Authors: D'Agostino Ralph B.
Published: July 2005

Kurtosis expresses the extent to which a probability distribution is more (or less) peaked and long‐tailed as compared to the normal distribution. More‐peaked distributions are called leptokurtic, less‐peaked are platykurtic. The usual index, β2, is the ratio of the fourth moment to the square of the second moment, and equals three for a normal distribution, with the associated index γ2(= β2) equal to zero. The indices may be estimated in terms of sample moments. Other measures have been proposed.
Law of Large Numbers
Authors: Janssen Paul
Published: July 2005

It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.
Limit Theorems
Authors: Zabell S. L.
Published: July 2005

These concern the asymptotic behavior of distributions of random variables in large samples. Classical results include the weak and strong laws of large numbers, the central limit theorem, and the law of the iterated logarithm. Other results discussed briefly are the ergodic theorem, the martingale convergence theorem, Skorokhod's theorem, and the use of coupling methods.
L
indley's Paradox
Authors: Shafer Glenn
Published: July 2005

A test of significance can disagree strongly with a Bayesian analysis when the Bayesian analysis assigns a substantial probability to the point hypothesis and distributes the remaining probability diffusely. The disagreement is sometimes called
Lindley's paradox
. If the diffuse distribution under the alternative hypothesis is based on evidence, the Bayesian analysis is convincing, but if the distribution is diffuse only because of ignorance, the test of significance may be more convincing.
Location–Scale Family
Authors: Datta Gauri Sankar
Published: July 2005

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.
Lognormal Distribution
Authors: Ratnaparkhi M.
Published: July 2005

This is the distribution of a positive‐valued random variable, the natural logarithm of which follows a normal distribution with mean μ and variance σ2. A generalization to the three‐parameter lognormal distribution arises if a location parameter is introduced, giving a nonzero lower bound for the variable. Properties, estimation procedures, and applications are described.
Loss Function
Authors: Tritchler D.
Published: July 2005

In decision theory, the loss function expresses the consequences of a specific decision when a particular state of nature holds. Negative loss is called
gain or utility
. The aim in Bayesian theory is to minimize the expected loss, integrating over the prior distribution. In frequentist theory, various restrictive conditions may be imposed, or a minimax solution may be sought. Statistical procedures such as hypothesis testing and estimation may be interpreted in terms of loss functions. Practical applications include the design of clinical trials and screening procedures.
Mean
Authors: Smith W.
Published: July 2005

The sample arithmetic mean is the average of a set of numerical measurements. For a random variable, the analog is the expectation. In a weighted mean, the individual observations may be given different weights before averaging, perhaps because they represent groups of different sizes. The geometric mean is obtained by averaging logarithms (of positive‐valued measurements) and then taking antilogarithms. The harmonic mean is obtained by taking reciprocals before averaging and then taking the reciprocal of the average. The relationship between the arithmetic, geometric, and harmonic means is described.
Mean Deviation
Authors: Hampel Frank
Published: July 2005

The mean deviation of a set of measurements is the mean of the absolute deviations from some location measure, which may be the sample mean or median, a population mean or some other constant. Its relation to the standard deviation is described. For normal samples, the standard deviation is a more efficient estimator of the population standard deviation, but the reverse is true for some forms of contaminated distributions.
Median
Authors: Hettmansperger T.
Published: July 2005

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.
Method of Moments
Authors: Lindsay Bruce G.
Published: July 2005

This is a method for obtaining point estimators of parameters, by solving a set of equations that equate a number of sample moments to their expectations. The method is consistent but inefficient. Topics considered here include the extension to multivariate data, the generalized and least squares methods of moments, distribution theory, and various specific examples.
Minimax Theory
Authors: Brown L. D.
Published: July 2005

In statistical decision theory and the theory of games, a minimax strategy or decision is one that minimizes (over decisions) the maximum risk (over possible distributions). The fundamental minimax theorem asserts that, under certain conditions, there is a minimax procedure, which is a Bayes procedure for a corresponding prior distribution.
Minimum Variance Unbiased (
MVU
) Estimator
Authors: Bethel J.
Published: July 2005

Determining the minimum variance of an unbiased estimator constitutes a fundamental topic in mathematical statistics. Two primary results are the Cramér–Rao inequality, which gives a lower bound on the variance of an unbiased estimator, and the Rao–Blackwell theorem, which supplies a method for improving the variance of unbiased estimators that are not functions of sufficient statistics. The strongest results are available for sufficient statistics from complete exponential families. However, any functions of sufficient statistics will attain the Cramér–Rao lower bound for sufficiently large sample sizes.
Moment Generating Function
Authors: Koehler Kenneth
Published: July 2005

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: July 2005

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.
N
eyman–
P
earson Lemma
Authors: Le Cam L. M.
Published: July 2005

J. Neyman and E.S. Pearson showed in 1933 that, in testing a simple null hypothesis against a simple alternative, the most powerful test is based on the likelihood ratio. Extensions to other situations, and the connection with statistical decision theory, are briefly described.
Normal Scores
Authors: Davison A. C.
Published: July 2005

Normal scores are expected values of order statistics of a sample from the normal distribution. They are used for assessing the fit of this distribution to a sample, and for transformations to normality.
Normality, Tests of
Authors: D'Agostino Ralph B.
Published: July 2005

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.
Nuisance Parameter
Authors: Schmid Christopher H.
Published: July 2005

Nuisance parameters are parameters for which inference is not desired, but which need to be taken account of in model estimation. Bayesian and non‐Bayesian approaches to handling nuisance parameters are discussed, including the idea of conditioning via sufficient statistics.
Paired

t

Test
Authors: Hsu Henry , Lachenbruch Peter A.
Published: July 2005

This is used to compare mean differences between treatments when the observations have been obtained in pairs. The difference between the paired values is assumed to be normally distributed, and the null hypothesis that the expectation is zero is tested by Student's
t
test. The robustness properties are discussed, as is the asymptotic relative efficiency of nonparametric alternatives.
P
areto Distribution
Authors: Childs A. , Balakrishnan N.
Published: July 2005

The classical Pareto distribution of the first kind has density function of the form 1–(
x
/σ)
−ν
. Pareto distributions of the second, third, and fourth kind involve extra parameters. The second kind arises as a gamma mixture of exponential distributions, and has been used in survival analysis. A multivariate generalization is described.
Pearson Distributions
Authors: Armitage Peter
Published: July 2005

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.
Pitman Efficiency
Authors: Zacks S.
Published: July 2005

The Pitman efficiency is an index for comparing test procedures or estimators. It is especially important for comparing procedures in large samples. If procedure P1 requires n1 observations to attain a certain power of a test, or a specified mean squared error, and procedure P2 requires n2 observations to achieve the same precision, the Pitman relative efficiency of P1 against P2 is n2/n1. Accordingly, if the mean squared error of an estimator E1 is v1/n + o(1/n), and if that of another estimator, E2, is v2/n + o(1/n), then the asymptotic (large sample) relative efficiency of E1 against E2 is v2/v1. Under smoothness regularity conditions, the variance of an unbiased estimator, in the one‐parameter case, is bounded below by the Cramer–Rao (CR) lower bound. Accordingly, the ratio of the CR‐lower bound to the variance of the estimator is defined as the efficiency index of the unbiased estimator. This is the Pitman relative efficiency of the estimator against the best possible one. Moreover, the CR‐lower bound contains the Fisher information function as a factor, and the inverse of the Fisher information function is the asymptotic variance of the maximum likelihood estimator (bf MLE). Therefore, the efficiency of an unbiased estimator is measured relative to the asymptotic variance of the maximum likelihood estimator.

The Pitman relative efficiency of consistent and asymptotically normal test statistics is defined as the ratio of their corresponding
efficacy ratios
. The asymptotic efficacy relates the rate of change of the mean function of the statistic to its asymptotic standard deviation. For further information, see the article.
Probability Theory
Authors: Steele J.M.
Published: July 2005

Probability theory is that part of mathematics relating to phenomena depending on chance or uncertainty. Probability is usually thought of either in terms of outcomes of repeated trials or as a measure of subjective belief. A.N. Kolmogorov provided an axiomatic basis for probability using measure theory. Fundamental notions include random variables and independence. Two basic theorems are the law of large numbers and the central limit theorem. Reference is made to theories for dependent data, involving Markov processes and martingales.
Quantiles
Authors: Milton Roy C.
Published: July 2005

Quantiles are the
N
 − 1 points that divide a distribution or sample data into
N
equal and ordered parts. Some sets of these
N
 − 1 points have well‐known names: median (
N
 = 2) and percentiles (
N
 = 100). Quantiles are commonly used in testing of hypotheses, summarizing sample data, and modeling risk relationships.
Random Variable
Authors: Basu Asit P.
Published: July 2005

Random variables are used to describe the uncertain outcomes of a study. Formally, a random variable is a function assigning a real value to each outcome of the sample space, which is the set of all possible outcomes. Related topics described here include the distinction between continuous and discrete random variables, the probability density function and cumulative distribution function, a bivariate distribution, conditional and marginal probability, and independence.
Randomization Tests
Authors: Green Sylvan B.
Published: July 2005

In an experiment with random treatment assignment, the distribution of any test statistic may be obtained by considering all the possible treatment assignments that might have been made, assuming the observed data. A test of the null hypothesis that the treatments have equal effects follows. Confidence intervals may similarly be obtained.
Randomness, Tests of
Authors: Whaley F. S.
Published: July 2005

Tests of randomness are used to assess whether data are truly random or have some sort of pattern. These include tests performed on data collected over a period of time or over geographical areas. Some tests are used to detect general nonrandomness, while other tests focus on detecting a particular pattern of nonrandomness, such as clustering or trends. Spatial tests of randomness can be used to assess the randomness of the location of data in a geographical region or assess the relationship between a characteristic of a population and its location. Tests of randomness can also be used to assess the randomness of a sequence of pseudo‐random numbers.
Range
Authors: Chen Hubert J.
Published: July 2005

The range of a set of observations is the difference between the largest and smallest values. In life testing or survival analysis with exponentially distributed observations, the range provides a useful test statistic for comparing location parameters, particularly for multiple comparisons. Details of its distribution are given. An application for multiple comparisons between normally distributed means is also given.
Ranks
Authors: Cabilio Paul
Published: July 2005

Ranks arise when objects are ordered according to a specific quality, or when numerical observations are arranged in order of magnitude. They form the basis of a wide range of nonparametric methods such as rank correlation. Allowance has usually to be made for the possible appearance of tied ranks.
Sample Size Determination
Authors: Lachin John M.
Published: July 2005

This concerns the evaluation of sample size required for a study before data are collected. This may be done by specifying either the required precision of an estimate, or the power of a test to detect an effect regarded as important. It may be necessary to consider the effects of missing data and of measurement unreliability. Many examples are considered in detail.
Scedasticity
Authors: Morrison Donald F.
Published: July 2005

Homoscedasticity is the condition under which a series of observations have the same degree of variation. Heteroscedasticity is its negation. Homoscedasticity is assumed in the standard models for least squares linear regression, the two‐sample
t
test and the analysis of variance, and the effects of heteroscedasticity in these situations are discussed here.
Sequential Analysis
Authors: Lai T. L.
Published: July 2005

This article gives an overview of sequential analysis and its applications to data and safety monitoring of clinical trials, determination of median effective dose and maximum tolerated dose, fixed‐width confidence intervals, and adaptive design of biomedical studies.
Shrinkage
Authors: Casella George
Published: July 2005

In the estimation of the means of more than two normal distributions, the set of sample means is not an admissible estimator. Under squared error loss, a better estimator shrinks the separate estimates toward a central point.
Simultaneous Confidence Intervals
Authors: Chen Tuhao , Hoppe Fred M.
Published: July 2005

These provide a confidence region for a multivariate parameter, comprising individual intervals for the separate components of the parameter. There is a specified confidence level of the simultaneous correctness of all the statements. The approach is useful for multiple comparisons of treatment means. The methods of Bonferroni, Tukey, Scheffé and Duncan are described, with examples.
Simultaneous Inference
Authors: Geller Nancy L. , O'Brien Peter C.
Published: July 2005

This covers statistical procedures to assess either more than one parameter or one parameter repeatedly during the course of a study. It includes multiple comparisons, multiple endpoints, sequential methods, and longitudinal data analysis. A common approach is to control the per‐family error rate (expected number of false positive conclusions per study) or the experimentwise error rate (probability of at least one false positive). Examples are given, with particular reference to problems of data monitoring of clinical trials.
Skewness
Authors: Hall Peter
Published: July 2005

Several definitions of skewness are given, and their main features discussed. Connections to relationships among the mean, median, and mode, and to work of Haldane, Pearson, and van Zwet, are noted.
Standard Deviation
Authors: Gauvreau K.
Published: July 2005

The variance of a random variable is the expectation of the squared deviation from the mean. The standard deviation is the square root of the variance and provides a measure of the spread of the distribution. The sample standard deviation may be calculated from a finite sample. The ratio of the standard deviation to the mean, expressed as a percentage, is called the coefficient of variation.
Statistical Dependence and Independence
Authors: Wermuth Nanny , Cox D. R.
Published: July 2005

Statistical dependence is a type of relation between different characteristics measured on the same units. At one extreme is deterministic dependence; at the other is statistical independence, where the distribution of one variable is the same for all levels of the other. With more than two variables, an important distinction is between marginal and conditional dependence. In many contexts, the degree of dependence may be summarized by a suitable measure of association, perhaps as part of a general model. Reference is made to graphical models.
Sufficiency
Authors: Strawderman William E.
Published: July 2005

According to the principle of sufficiency, introduced by R.A. Fisher in 1922, a “sufficient statistic” is equivalent, for all purposes of estimation, to the original data. Connections with the likelihood principle are noted, as is the case for using robust methods, which do not demand full confidence in the proposed statistical model.
Trimming and Winsorization
Authors: Wilcox R.
Published: July 2005

Distributions with very long tails or isolated outlying values can disturb the effectiveness of methods based on the normal distribution. Trimming involves the discarding of a proportion of sample observations at the two extremes before calculation of a mean. Winsorization involves the replacement of the discarded values by the most extreme retained values.
Unbiasedness
Authors: Link W. A.
Published: July 2005

Unbiasedness is defined and illustrated as a criterion in estimation and hypothesis testing. Limitations of the criterion are illustrated by examples: that unbiasedness is not transformation invariant, that unbiased estimators need not exist, nor be unique, and that even when a unique unbiased estimator exists, it might be clearly unreasonable.
Uniform Distribution
Authors: Deng Lih‐yuan
Published: July 2005

A discrete uniform distribution has constant probability for each integer within a stated range. A continuous uniform distribution has constant probability density within a stated range. The continuous uniform distribution in the range (0, 1) has connections with the probability integral transformation, and with the exponential, logistic, and beta distributions. Some characterizations of functions of two independent random variables are given.
Uniform Random Numbers
Authors: Deng Lih‐Yuan
Published: July 2005

Pseudorandom numbers, uniformly distributed on the interval (0, 1), are widely used in the development of statistical methods when analytic methods become intractable. The following methods of generation are described: linear congruential, shift register, multiple recursive, matrix congruential, and combination generator. The merits of each are discussed.
Union–Intersection Principle
Authors: Chinchilli V. M.
Published: July 2005

Multivariate analysis is the branch of statistics that involves inferential methods with two or more response (dependent) variables simultaneously. Therefore, multivariate analysis tends to be more complex than univariate analysis (one dependent variable) with respect to its mathematical development and its interpretation. The union–intersection principle is an approach that provides a basic approach for hypothesis testing in certain multivariate situations. We illustrate an application of the union–intersection principle to the situation in which there are pretreatment and posttreatment measurements onpdependent variables for a sample ofnexperimental units.
U
‐Shaped Distribution
Authors: Everitt Brian S.
Published: July 2005

This is a probability distribution or frequency distribution shaped roughly like the letter U, with highest frequency at the two extremes, although not necessarily symmetrically. The variables are sometimes called
bounded scores
.
U

‐Statistics
Authors: Janssen Paul
Published: July 2005

A U‐statistic, calculated from a random sample of size
n
, is an average of a symmetric function of
m
sample values, calculated for all
m
‐tuples in the sample. Examples are the sample variance, the Cramér‐von Mises statistic of goodness of fit, and the Kaplan–Meier and Nelson–Aalen estimators in survival analysis. Asymptotic properties are described.
Wilcoxon–Mann–Whitney Test
Authors: Moses Lincoln E.
Published: July 2005

This is a method of comparing the size of observations in two groups by ranking all the observations and comparing the average ranks of the two groups. The distribution theory is described. The test has high efficiency for the comparison of two normal distributions with the same variance, and is much better than thettest for some nonnormal distributions. Topics discussed include adjustment for tied data and the calculation of confidence limits for a difference in translation between the two distributions.
Combining
P
Values
Authors: Westfall Peter H.
Published: July 2005

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.
Chi‐Square Distribution; Properties
Authors: Klotz Jerome
Published: July 2005

This article complements the introductory article on the chi‐square distribution, with details of its derivation as the distribution of the sum of squares of standard normal deviates, the density being a gamma function, and the relationship between its distribution function and that of the Poisson distribution. When the component normal distributions have nonzero mean, the sum of squares has a noncentral chi‐square distribution, depending on a noncentrality parameter, the sum of squares of the normal means. This plays a role in determining the power of a chi‐square test.
Composite Estimators
Authors: Schaible Wes
Published: July 2005

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.
Denominator Difficulties
Authors: Kravitz Howard M.
Published: July 2005

This article describes the fallacy in calculating rates and proportions without having denominator data. Missing, wrong, and unknown denominators are discussed. Examples are presented to illustrate this fallacy.
Distance Sampling
Authors: Buckland Stephen T. , Anderson David R. , Burnham Kenneth P. , Laake Jeffrey L.
Published: July 2005

Distance sampling covers a range of methods in which distances of detected objects (usually animals) from a line or point are recorded, from which object density or abundance is estimated. The most widely used methods are line transect sampling and point transect sampling. Objects on the line or point are usually assumed to be detected with certainty, and a detection function, representing probability of detection as a function of distance from the line or point, is modeled. This allows estimation of the proportion of objects missed within surveyed strips or circles, extending a specified distance from the line or point. Appropriate survey design allows the resulting object density estimates to be extrapolated to the full survey region, to yield estimates of object abundance.
Graeco–Latin Square Designs
Authors: Martin Richard J. , Nadarajah Saraleesan
Published: July 2005

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.
Incomplete Follow‐up
Authors: Kravitz Howard M.
Published: July 2005

This article describes a fallacy associated with longitudinal (follow‐up) data analysis. Bias that can occur from not accounting for study duration and from inappropriate management of missing data are considered.
L
atin Square Designs
Authors: Wild Pascal , Grzebyk Michel
Published: July 2005

A Latin square design is a balanced incomplete experimental design for comparing
t
treatments while controlling for two factors with
t
possible values under the assumption of unit‐treatment additivity. The design construction as well as estimation and analysis of variance aspects are described. Some extensions are mentioned.
Lattice Designs
Authors: Martin Richard J.
Published: July 2005

Lattice designs are two types of experimental designs in which the number of treatments is the square of
k
. A square lattice design is a resolvable (0, 1) incomplete block design, with block size
k
. For
r
replicates, with
r
more than three, existence requires that there is a set of
r
mutually orthogonal Latin squares. If r = k + 1, the design is balanced. Extensions to cubic lattice designs and some rectangular lattice designs are possible. A lattice square design is a nested row–column design with
k
rows and
k
columns, with each block being a complete replicate.
Nonresponse
Authors: Kviz F. J.
Published: July 2005

Nonresponse is a potential source of error in survey estimates because it may cause important segments of a target population to be underrepresented and it reduces statistical power. This article reviews the causes of survey nonresponse, the nature of nonresponse error and its potential impact on survey estimates, and techniques for measuring and reporting nonresponse.
Partially Balanced Incomplete Block Design
Authors: Street D. J. , Street A. P.
Published: July 2005

This article provides a brief overview of the most common partially balanced incomplete block designs and discusses other designs to which they are related.
Probability Sampling
Authors: Hsia Jason
Published: July 2005

Probability sampling is a process used to select a sample from a defined population with the characteristic that every element of the population has a known, nonzero probability of being included in the sample. In probability sampling, we select the sample by a random mechanism under which each element of the population receives this known selection probability.
Randomized Complete Block Designs
Authors: Murray L.
Published: July 2005

This article discusses the general need for blocking when experimental units are not homogeneous. Complete, generalized, and incomplete block designs are discussed, together with issues of randomization, missing cells, and the usual assumption of “no interaction”. Two specific cases of the Randomized Complete Block Design are considered: the fixed case (treatments and blocks are both fixed) and the mixed case (treatments are fixed, blocks are random). For each case, the scalar linear models are defined in full‐rank and non‐full‐rank versions, together with necessary assumptions. Point estimates and standard errors are given for main‐effect means and cell means, and for variance components. The analysis of variance table is given, with expected mean squares, for testing hypotheses.
Resampling Procedures for Sample Surveys
Authors: Frankel Martin R.
Published: July 2005

This entry discusses the two basic resampling techniques that have found widespread application in survey sampling: Jackknife Repeated Replication (JRR) and Balanced Repeated Replication (BRR). To date, resampling methods based on the bootstrap have found only limited application in samples, which involve complex (stratified and clustered) sample design.
Sample Size Adequacy in Surveys
Authors: Penne Michael A. , Levy Paul S.
Published: July 2005

This entry presents concepts involved in determination of sample size adequacy in sample surveys. There is a broad discussion of issues such as stratification, sampling without replacement, finite population corrections, sampling of clusters, sampling in more than one stage, and unequal weighting of observations that have a profound effect on the distribution of estimates and on the sample sizes necessary to construct reliable estimates. We include formulas for determining sample size adequacy for some of the most widely used sample designs and some practical guidelines for statistical consulting on this issue. A set of references to texts and specific articles is included.
Sample Surveys in the Health Sciences
Authors: Brock Dwight B. , Beckett Laurel A. , Bienias Julia L.
Published: July 2005

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 Probability Proportional to Size
Authors: Czaja Ronald
Published: July 2005

Sampling with probability to proportionate to size (PPS), as the name implies, involves sampling elements that are typically arranged into natural groupings of different sizes. Sampling PPS is an efficient design when a list of the population is not available; the elements are in clusters or natural groups that differ greatly in size; it is important to control the sample size; costs and equalizing interviewer workloads are important; the use of natural divisions is possible and supplementary information is available by the natural divisions; and when the sample must be drawn in multiple stages. This entry presents a detailed example of a PPS design, a multistage area probability design, and discusses a random digit telephone sample design that uses PPS sampling.
Snowball Sampling
Authors: Johnson Timothy P.
Published: July 2005

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.